Stochastic Metafrontier Analysis
An R package for implementing various deterministic and stochastic metafrontier analyses for efficiency and performance benchmarking, assessing technical efficiencies (TE), metafrontier technical efficiencies (MTE), and computing metatechnology ratios (MTRs) for firms operating under different technologies.
The package is available on CRAN: 10.32614/CRAN.package.smfa
smfa provides routines for:
- Deterministic envelope metafrontier via linear programming (LP) and quadratic programming (QP), following Battese, Rao & O’Donnell (2004) and O’Donnell, Rao & Battese (2008).
- Stochastic two-stage metafrontier following Huang, Huang & Liu (2014).
In addition, the package implements:
- Latent class stochastic metafrontier analysis — when technology groups are unobserved, the latent class model (LCM) robustly identifies classes and routes them to the metafrontier for benchmarking following Greene and Hensher (2003), Orea and Kumbhakar (2004), Greene (2005), Parmeter and Kumbhakar (2014).
- Sample selection correction metafrontier models — corrects for sample selection bias following Heckman (1979), Greene (2010), Greene (2003).
Dependency:
smfadepends on thesfaRpackage by Dakpo, Desjeux & Latruffe (2023), which provides the underlying stochastic frontier estimation routines for all group-level models.
Installation
install.packages("smfa")
library("smfa")
# Install devtools if not already installed
#if (!require("devtools")) #install.packages("devtools")
# Install smfa from GitHub
#devtools::install_github("SulmanOlieko/smfa")Note You do not need to install
sfaRmanually,smfatakes care of that automatically.
Usage Examples
The following sections provide comprehensive examples covering all three group-level model types and all four metafrontier methods. Each section demonstrates the full workflow: data preparation → group frontier estimation → metafrontier → efficiency and MTR extraction.
Section 1: Standard SFA Group Frontier (groupType = "sfacross")
Let’s use the ricephil data from sfaR. In this data, group boundaries are observed (a farm-size variable). If we assume that the production technology varies by farm size, we can try to estimate three frontiers that correspond to three types of farm sizes, namely small, medium and large. We can create a group variable group that captures these groups. We can then estimate each group’s frontier separately using sfacross from the sfaR package. So, we will specify the option groupType = "sfacross" in the smfa().
Data Preparation
library(smfa)
data("ricephil")
# Create three technology groups based on farm area terciles
ricephil$group <- cut(ricephil$AREA,
breaks = quantile(ricephil$AREA, probs = c(0, 1/3, 2/3, 1), na.rm = TRUE),
labels = c("small", "medium", "large"),
include.lowest = TRUE
)
#This is the distrubition of the various farm types:
table(ricephil$group)
small medium large
125 104 115 1a. LP Metafrontier (groupType = "sfacross", metaMethod = "lp")
We can begin by estimating a deterministic linear programming envelope (Battese, Rao & O’Donnell, 2004) over the three group frontiers. The metafrontier parameter vector minimises the sum of absolute deviations while staying at or above all group frontier predictions. We will be using a Cobb-Douglas functional form with rice production PROD as the response variable, and AREA, LABOR and NPK as the inputs.
meta_sfacross_lp <- smfa(
formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
data = ricephil,
group = "group",
S = 1,
udist = "hnormal",
groupType = "sfacross",
metaMethod = "lp"
)
summary(meta_sfacross_lp)Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Linear Programming (LP) Metafrontier
Stochastic Production/Profit Frontier, e = v - u
Group approach : Stochastic Frontier Analysis
Group estimator : sfacross
Group optim solver : BFGS maximization
Groups ( 3 ): small, medium, large
Total observations : 344
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: small (N = 125) Log-likelihood: -50.98578
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 42
Log likelihood value: -50.98578
Log likelihood gradient norm: 9.40653e-06
Estimation based on: N = 125 and K = 6
Inf. Cr: AIC = 114.0 AIC/N = 0.912
BIC = 130.9 BIC/N = 1.048
HQIC = 120.9 HQIC/N = 0.967
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05318
Sigma(v) = 0.05318
Sigma-squared(u) = 0.23435
Sigma(u) = 0.23435
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.53622
Gamma = sigma(u)^2/sigma^2 = 0.81504
Lambda = sigma(u)/sigma(v) = 2.09921
Var[u]/{Var[u]+Var[v]} = 0.61558
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.38626
Average efficiency E[exp(-ui)] = 0.70643
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -54.80277
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 7.63398
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -3.57676
M3T: p.value = 0.00035
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.58745 0.51274 -3.0960 0.001962 **
log(AREA) 0.24014 0.11834 2.0292 0.042440 *
log(LABOR) 0.43464 0.12292 3.5361 0.000406 ***
log(NPK) 0.30516 0.05701 5.3523 8.682e-08 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.45093 0.29867 -4.858 1.186e-06 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.93406 0.35401 -8.288 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: medium (N = 104) Log-likelihood: -15.28164
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 41
Log likelihood value: -15.28164
Log likelihood gradient norm: 3.83566e-05
Estimation based on: N = 104 and K = 6
Inf. Cr: AIC = 42.6 AIC/N = 0.409
BIC = 58.4 BIC/N = 0.562
HQIC = 49.0 HQIC/N = 0.471
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.01058
Sigma(v) = 0.01058
Sigma-squared(u) = 0.22010
Sigma(u) = 0.22010
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.48030
Gamma = sigma(u)^2/sigma^2 = 0.95412
Lambda = sigma(u)/sigma(v) = 4.56034
Var[u]/{Var[u]+Var[v]} = 0.88314
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.37433
Average efficiency E[exp(-ui)] = 0.71330
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -21.11323
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 11.66318
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -2.91021
M3T: p.value = 0.00361
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -0.08182 0.50668 -0.1615 0.8717190
log(AREA) 0.47410 0.13984 3.3903 0.0006981 ***
log(LABOR) 0.17935 0.10201 1.7581 0.0787310 .
log(NPK) 0.20255 0.08130 2.4913 0.0127289 *
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.51367 0.23549 -6.4276 1.296e-10 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.54846 0.76429 -5.9512 2.661e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: large (N = 115) Log-likelihood: -8.02197
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 68
Log likelihood value: -8.02197
Log likelihood gradient norm: 4.01301e-05
Estimation based on: N = 115 and K = 6
Inf. Cr: AIC = 28.0 AIC/N = 0.244
BIC = 44.5 BIC/N = 0.387
HQIC = 34.7 HQIC/N = 0.302
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.01399
Sigma(v) = 0.01399
Sigma-squared(u) = 0.16751
Sigma(u) = 0.16751
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.42602
Gamma = sigma(u)^2/sigma^2 = 0.92293
Lambda = sigma(u)/sigma(v) = 3.46063
Var[u]/{Var[u]+Var[v]} = 0.81315
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.32656
Average efficiency E[exp(-ui)] = 0.74195
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -16.96836
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 17.89279
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -4.12175
M3T: p.value = 0.00004
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.31194 0.41859 -3.1342 0.0017234 **
log(AREA) 0.38278 0.14297 2.6772 0.0074236 **
log(LABOR) 0.42105 0.10992 3.8303 0.0001280 ***
log(NPK) 0.23143 0.06065 3.8160 0.0001356 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.78673 0.20176 -8.8555 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.26963 0.40584 -10.521 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (lp):
(LP: deterministic envelope - no estimated parameters)
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
small 125 125 0.71065 0.70090 0.64126 0.63244 0.89981
medium 104 104 0.71253 0.70965 0.68204 0.67929 0.95597
large 115 115 0.74772 0.74406 0.72186 0.71834 0.96521
MTR_JLMS
small 0.89981
medium 0.95597
large 0.96521
Overall:
TE_group_BC=0.7236 TE_group_JLMS=0.7182
TE_meta_BC=0.6817 TE_meta_JLMS=0.6767
MTR_BC=0.9403 MTR_JLMS=0.9403
------------------------------------------------------------
Total Log-likelihood: -74.28939
AIC: 184.5788 BIC: 253.7103 HQIC: 212.113
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12 Note: Since the metafrontier is estimated via linear programming, no estimated parameters are returned.
To harvest individual efficiency, metafrontier efficiency and MTR estimates:
efficiencies(meta_sfacross_lp)Toggle to see the output
id group u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal
1 1 medium 0.26971650 0.7635959 0.7673345 1.316036
2 2 large 0.35156422 0.7035867 0.7080897 1.430406
3 3 large 0.27745651 0.7577085 0.7623358 1.327899
4 4 medium 0.17104173 0.8427864 0.8461331 1.191355
5 5 large 0.21196294 0.8089947 0.8133556 1.242901
6 6 small 0.19874988 0.8197549 0.8275685 1.232467
7 7 medium 0.13165358 0.8766446 0.8794388 1.144463
8 8 medium 0.15361555 0.8576017 0.8607383 1.170418
9 9 large 0.30257423 0.7389136 0.7435320 1.361845
10 10 large 0.39007628 0.6770052 0.6813703 1.486626
11 11 small 0.92924589 0.3948514 0.4035011 2.588082
12 12 small 0.19404574 0.8236202 0.8312437 1.226330
13 13 small 0.28406465 0.7527180 0.7633015 1.348389
14 14 large 0.33207087 0.7174365 0.7219968 1.402741
15 15 small 0.37604956 0.6865683 0.6987013 1.483309
16 16 small 0.68309402 0.5050519 0.5160333 2.023157
17 17 large 0.48308967 0.6168745 0.6208681 1.631571
18 18 large 0.14598902 0.8641672 0.8676168 1.161971
19 19 large 0.15942841 0.8526310 0.8563290 1.178095
20 20 large 0.23870615 0.7876463 0.7921780 1.277019
21 21 large 0.53509305 0.5856148 0.5894067 1.718664
22 22 medium 0.13321561 0.8752763 0.8780978 1.146294
23 23 large 0.64904573 0.5225442 0.5259279 1.926106
24 24 large 0.30494980 0.7371604 0.7417757 1.365097
25 25 large 0.36231584 0.6960625 0.7005296 1.445889
26 26 small 0.35782089 0.6991983 0.7111323 1.455722
27 27 medium 0.68946140 0.5018463 0.5043865 2.002728
28 28 medium 0.14820764 0.8622521 0.8653125 1.163983
29 29 medium 0.43885427 0.6447747 0.6480379 1.558779
30 30 medium 0.78101223 0.4579422 0.4602602 2.194735
31 31 small 0.11027845 0.8955847 0.8993545 1.121690
32 32 small 0.73726525 0.4784205 0.4888663 2.135906
33 33 medium 0.28891406 0.7490766 0.7527904 1.341617
34 34 small 0.95871703 0.3833844 0.3917835 2.665492
35 35 large 0.13741225 0.8716108 0.8748846 1.151776
36 36 small 0.96866209 0.3795906 0.3879066 2.692133
37 37 small 0.19029853 0.8267123 0.8341817 1.221460
38 38 large 0.10253454 0.9025470 0.9049738 1.111089
39 39 small 0.63054578 0.5323012 0.5437851 1.919362
40 40 medium 0.66058287 0.5165502 0.5191648 1.945719
41 41 small 0.28900704 0.7490069 0.7597103 1.355372
42 42 medium 0.05156756 0.9497395 0.9506433 1.053962
43 43 medium 1.19309253 0.3032819 0.3048170 3.313952
44 44 medium 0.39809366 0.6715991 0.6749962 1.496516
45 45 large 0.21353124 0.8077269 0.8121007 1.244881
46 46 large 0.38104349 0.6831482 0.6875476 1.473249
47 47 medium 0.51706842 0.5962660 0.5992841 1.685593
48 48 large 0.28476023 0.7521946 0.7568241 1.337689
49 49 small 0.37632866 0.6863767 0.6985123 1.483735
50 50 medium 0.08768834 0.9160463 0.9178976 1.093943
51 51 medium 0.21551709 0.8061245 0.8097887 1.246212
52 52 large 0.16517228 0.8477476 0.8515417 1.185043
53 53 large 0.42324534 0.6549179 0.6591518 1.536784
54 54 small 0.95317568 0.3855148 0.3939604 2.650762
55 55 small 0.34332381 0.7094085 0.7211483 1.434089
56 56 small 0.51135451 0.5996828 0.6120410 1.702453
57 57 large 0.42818450 0.6516912 0.6559051 1.544395
58 58 small 0.39520496 0.6735420 0.6858329 1.512769
59 59 small 0.20997321 0.8106060 0.8188583 1.247219
60 60 large 1.06549989 0.3445556 0.3467867 2.921083
61 61 large 0.24996507 0.7788280 0.7834040 1.291622
62 62 large 0.08518681 0.9183407 0.9202802 1.091328
63 63 large 0.16346958 0.8491923 0.8529585 1.182980
64 64 large 0.71818311 0.4876374 0.4907951 2.063983
65 65 medium 0.35067251 0.7042143 0.7077662 1.427189
66 66 large 0.46547169 0.6278389 0.6319026 1.603076
67 67 large 0.37806815 0.6851838 0.6895943 1.468868
68 68 large 0.16780715 0.8455169 0.8493530 1.188241
69 69 small 0.52285816 0.5928237 0.6051259 1.722346
70 70 medium 0.32721034 0.7209321 0.7245549 1.394073
71 71 medium 0.08807987 0.9156877 0.9175488 1.094384
72 72 medium 0.74487271 0.4747947 0.4771980 2.116834
73 73 medium 0.54952972 0.5772212 0.5801429 1.741207
74 74 small 0.47200427 0.6237508 0.6362295 1.635970
75 75 small 0.29451014 0.7448964 0.7557281 1.363182
76 76 medium 0.30540107 0.7368278 0.7405077 1.363962
77 77 small 0.92216588 0.3976568 0.4063679 2.569822
78 78 medium 0.10315484 0.9019873 0.9042066 1.111494
79 79 small 0.51145818 0.5996206 0.6119783 1.702632
80 80 small 0.21306571 0.8081030 0.8164725 1.251310
81 81 large 0.23111098 0.7936514 0.7981445 1.267249
82 82 small 0.80877949 0.4454014 0.4551491 2.294323
83 83 medium 0.79471384 0.4517105 0.4539969 2.225013
84 84 small 0.16143515 0.8509217 0.8571342 1.184527
85 85 medium 0.10754566 0.8980355 0.9003525 1.116521
86 86 medium 1.25923700 0.2838705 0.2853074 3.540564
87 87 medium 0.23574646 0.7899809 0.7937044 1.271874
88 88 large 0.18389141 0.8320262 0.8360918 1.207917
89 89 large 0.40442074 0.6673633 0.6716723 1.508115
90 90 medium 0.04250673 0.9583840 0.9590537 1.044179
91 91 large 0.22202050 0.8008989 0.8053358 1.255640
92 92 small 0.28784656 0.7498766 0.7605523 1.353729
93 93 medium 0.13117685 0.8770627 0.8798484 1.143905
94 94 medium 0.52361332 0.5923762 0.5953747 1.696661
95 95 large 0.18380892 0.8320948 0.8361594 1.207815
96 96 large 0.15458454 0.8567711 0.8603834 1.172263
97 97 small 0.58297518 0.5582350 0.5701315 1.829844
98 98 small 0.34450415 0.7085716 0.7203285 1.435840
99 99 small 0.30123486 0.7399040 0.7508851 1.372776
100 100 large 0.26745807 0.7653224 0.7699395 1.314601
101 101 small 0.55191295 0.5758472 0.5879742 1.773543
102 102 small 0.32564277 0.7220631 0.7335205 1.408063
103 103 large 0.19828894 0.8201328 0.8243660 1.225756
104 104 large 0.17208990 0.8419035 0.8458052 1.193454
105 105 large 0.18406916 0.8318783 0.8359462 1.208136
106 106 large 0.15321665 0.8579438 0.8615312 1.170620
107 107 large 0.11095437 0.8949796 0.8976289 1.120796
108 108 medium 0.10455186 0.9007281 0.9029788 1.113091
109 109 large 0.25275205 0.7766604 0.7812451 1.295259
110 110 large 0.13767911 0.8713783 0.8746577 1.152092
111 111 large 0.17065865 0.8431093 0.8469895 1.191710
112 112 small 0.27205429 0.7618129 0.7720866 1.331542
113 113 medium 0.17764282 0.8372414 0.8406542 1.199366
114 114 medium 0.07735091 0.9255650 0.9271532 1.082351
115 115 medium 0.53654999 0.5847622 0.5877221 1.718753
116 116 small 0.42936131 0.6509247 0.6633797 1.566533
117 117 small 0.16499487 0.8478981 0.8542718 1.189026
118 118 small 0.18357605 0.8322886 0.8394761 1.212766
119 119 medium 0.33274854 0.7169505 0.7205572 1.401821
120 120 small 0.59015184 0.5542431 0.5660812 1.843089
121 121 medium 0.13365201 0.8748945 0.8777234 1.146805
122 122 small 0.40621404 0.6661676 0.6785271 1.529923
123 123 small 0.17263606 0.8414438 0.8481581 1.198737
124 124 large 0.14782818 0.8625793 0.8660649 1.164167
125 125 small 0.35093355 0.7040305 0.7158764 1.445411
126 126 medium 0.56752097 0.5669291 0.5697988 1.772817
127 127 small 0.16877679 0.8446974 0.8512407 1.193823
128 128 medium 0.13506819 0.8736563 0.8765095 1.148467
129 129 small 0.44504953 0.6407925 0.6532776 1.591773
130 130 medium 0.39273080 0.6752105 0.6786253 1.488511
131 131 large 0.09767654 0.9069422 0.9092361 1.105523
132 132 large 0.53056371 0.5882733 0.5920824 1.710897
133 133 medium 0.14522379 0.8648287 0.8678449 1.160444
134 134 large 0.17999850 0.8352715 0.8392858 1.203130
135 135 small 0.46197481 0.6300382 0.6425273 1.619400
136 136 medium 0.07823949 0.9247429 0.9263541 1.083343
137 137 medium 0.23091467 0.7938072 0.7975205 1.265702
138 138 large 0.34534978 0.7079727 0.7124952 1.421530
139 139 large 0.12894860 0.8790191 0.8821063 1.141788
140 140 medium 0.48103124 0.6181456 0.6212744 1.625930
141 141 small 0.11773649 0.8889303 0.8930643 1.130632
142 142 small 0.50645708 0.6026269 0.6150063 1.694048
143 143 medium 0.20748338 0.8126267 0.8162542 1.236147
144 144 small 0.30880101 0.7343269 0.7454667 1.383636
145 145 small 0.22358989 0.7996430 0.8083984 1.265320
146 146 large 0.07234960 0.9302056 0.9317673 1.076921
147 147 large 0.20371424 0.8156954 0.8199828 1.232535
148 148 large 0.91631481 0.3999904 0.4025805 2.516249
149 149 large 0.22053264 0.8020915 0.8065180 1.253749
150 150 large 0.68148964 0.5058629 0.5091385 1.989621
151 151 medium 0.07974173 0.9233548 0.9250048 1.085022
152 152 large 0.79474993 0.4516942 0.4546191 2.228223
153 153 large 0.24061641 0.7861431 0.7906834 1.279487
154 154 large 0.12853914 0.8793791 0.8824569 1.141307
155 155 small 0.49661330 0.6085883 0.6210053 1.677268
156 156 small 0.37695482 0.6859471 0.6980886 1.484691
157 157 small 0.29497587 0.7445496 0.7553918 1.363844
158 158 medium 0.11544390 0.8909706 0.8934553 1.125612
159 159 small 0.31511456 0.7297053 0.7409698 1.392752
160 160 small 0.42183943 0.6558393 0.6682703 1.554555
161 161 small 0.26677319 0.7658468 0.7759759 1.324190
162 162 small 0.29141595 0.7472048 0.7579650 1.358786
163 163 small 0.35786292 0.6991689 0.7111035 1.455786
164 164 medium 0.18990769 0.8270355 0.8305524 1.214370
165 165 medium 0.53645635 0.5848170 0.5877771 1.718592
166 166 small 0.21390204 0.8074275 0.8158283 1.252419
167 167 large 0.11796311 0.8887288 0.8915551 1.128933
168 168 small 0.23927515 0.7871983 0.7964921 1.286445
169 169 medium 0.50509722 0.6034469 0.6065014 1.665535
170 170 small 0.16742642 0.8458389 0.8523218 1.192108
171 171 medium 0.35984536 0.6977842 0.7013069 1.440346
172 172 medium 0.62839799 0.5334457 0.5361459 1.884094
173 173 medium 0.38557500 0.6800595 0.6834980 1.477896
174 174 large 0.09504929 0.9093281 0.9115489 1.102523
175 175 large 0.76321806 0.4661639 0.4691825 2.159059
176 176 medium 0.20974379 0.8107920 0.8144306 1.238971
177 177 large 0.40981596 0.6637724 0.6680600 1.516277
178 178 small 0.34752184 0.7064366 0.7182360 1.440326
179 179 medium 0.21664337 0.8052171 0.8088858 1.247628
180 180 medium 0.26293729 0.7687901 0.7725324 1.307112
181 181 large 0.26522456 0.7670337 0.7716472 1.311647
182 182 large 0.66574949 0.5138882 0.5172158 1.958549
183 183 medium 0.86695664 0.4202285 0.4223556 2.391703
184 184 small 0.44160911 0.6430009 0.6554816 1.586208
185 185 small 0.65924072 0.5172439 0.5284572 1.975381
186 186 medium 0.24201528 0.7850442 0.7887775 1.279921
187 187 small 0.33041455 0.7186258 0.7301644 1.415048
188 188 small 0.18863690 0.8280871 0.8354875 1.219306
189 189 large 0.05330220 0.9480935 0.9490925 1.055909
190 190 large 0.24348518 0.7838911 0.7884435 1.283200
191 191 large 0.20676877 0.8132077 0.8175235 1.236365
192 192 large 0.20225170 0.8168893 0.8211625 1.230705
193 193 large 0.32382733 0.7233751 0.7279557 1.391197
194 194 medium 0.68377484 0.5047082 0.5072629 1.991372
195 195 large 0.32854526 0.7199703 0.7245397 1.397793
196 196 large 0.93573295 0.3922982 0.3948385 2.565587
197 197 large 0.78871134 0.4544300 0.4573726 2.214808
198 198 small 0.22155462 0.8012722 0.8099545 1.262600
199 199 small 0.46176488 0.6301705 0.6426597 1.619055
200 200 small 0.25045891 0.7784435 0.7880938 1.301687
201 201 medium 0.73516393 0.4794269 0.4818536 2.096382
202 202 small 0.33014326 0.7188207 0.7303549 1.414651
203 203 small 0.18123530 0.8342390 0.8413268 1.209752
204 204 small 0.31271670 0.7314571 0.7426751 1.389284
205 205 medium 0.14931823 0.8612950 0.8643715 1.165302
206 206 small 0.73144769 0.4812118 0.4917152 2.123507
207 207 medium 0.50481428 0.6036177 0.6066730 1.665063
208 208 medium 0.52000480 0.5945177 0.5975270 1.690550
209 209 small 0.29340163 0.7457226 0.7565288 1.361605
210 210 large 0.52692648 0.5904168 0.5942398 1.704686
211 211 small 0.16286045 0.8497098 0.8559870 1.186327
212 212 medium 0.46566409 0.6277181 0.6308953 1.601135
213 213 small 0.34921399 0.7052422 0.7170649 1.442846
214 214 medium 0.52477163 0.5916905 0.5946854 1.698627
215 215 medium 0.25196513 0.7772718 0.7810137 1.292787
216 216 medium 0.75345388 0.4707379 0.4731206 2.135077
217 217 large 0.09400786 0.9102756 0.9124671 1.101336
218 218 large 0.15539558 0.8560765 0.8597034 1.173238
219 219 medium 0.34483786 0.7083352 0.7119053 1.418882
220 220 large 0.28710954 0.7504295 0.7550588 1.340852
221 221 small 0.24484526 0.7828257 0.7922999 1.294018
222 222 large 0.31498520 0.7297997 0.7343987 1.378914
223 223 medium 0.51375510 0.5982449 0.6012730 1.680017
224 224 large 0.45762302 0.6327860 0.6368812 1.590542
225 225 large 0.12203744 0.8851152 0.8880406 1.133686
226 226 medium 0.38224719 0.6823264 0.6857758 1.472985
227 227 small 0.14021798 0.8691688 0.8743907 1.158046
228 228 small 0.38393814 0.6811736 0.6933777 1.495382
229 229 medium 0.27137489 0.7623307 0.7660678 1.318228
230 230 small 0.29544272 0.7442020 0.7550549 1.364509
231 231 small 0.16769672 0.8456103 0.8521053 1.192451
232 232 large 0.08077514 0.9224011 0.9242118 1.086355
233 233 large 0.25664172 0.7736453 0.7782408 1.300350
234 234 large 0.12582971 0.8817650 0.8847802 1.138126
235 235 large 0.30866071 0.7344299 0.7390399 1.370191
236 236 large 0.08755236 0.9161709 0.9181787 1.094003
237 237 medium 0.58452852 0.5573686 0.5601898 1.803226
238 238 large 0.31852288 0.7272224 0.7318145 1.383816
239 239 large 0.36568287 0.6937228 0.6981782 1.450771
240 240 large 0.54483361 0.5799383 0.5836935 1.735487
241 241 small 0.23470048 0.7908077 0.7999491 1.280253
242 242 small 0.27735774 0.7577834 0.7681971 1.338960
243 243 small 0.30605962 0.7363427 0.7474262 1.379693
244 244 medium 1.08243900 0.3387683 0.3404830 2.966812
245 245 small 0.34953987 0.7050124 0.7168395 1.443332
246 246 small 0.16336613 0.8492802 0.8555803 1.186966
247 247 small 0.71841455 0.4875246 0.4981571 2.095986
248 248 small 0.24053589 0.7862064 0.7955416 1.288156
249 249 small 0.56709915 0.5671683 0.5791874 1.800862
250 250 medium 0.29364064 0.7455444 0.7492495 1.347987
251 251 medium 0.45068791 0.6371897 0.6404147 1.577335
252 252 small 0.29203957 0.7467390 0.7575137 1.359671
253 253 small 0.19569314 0.8222645 0.8299550 1.228476
254 254 small 0.20941780 0.8110563 0.8192874 1.246485
255 255 medium 0.45207002 0.6363096 0.6395302 1.579516
256 256 small 0.11309466 0.8930661 0.8969735 1.125057
257 257 medium 0.29737033 0.7427689 0.7464666 1.353034
258 258 medium 0.06194103 0.9399383 0.9411184 1.065291
259 259 medium 0.83579891 0.4335280 0.4357224 2.318332
260 260 large 0.27845218 0.7569545 0.7615823 1.329229
261 261 large 0.90210294 0.4057156 0.4083427 2.480741
262 262 medium 0.37714757 0.6858149 0.6892811 1.465491
263 263 large 0.30646524 0.7360441 0.7406573 1.367175
264 264 small 0.49552007 0.6092540 0.6216748 1.675414
265 265 large 0.34762954 0.7063605 0.7108760 1.424780
266 266 medium 0.65750169 0.5181442 0.5207669 1.939734
267 267 large 0.22715688 0.7967958 0.8012658 1.262188
268 268 large 0.40928937 0.6641220 0.6684117 1.515479
269 269 medium 1.04243306 0.3525958 0.3543805 2.850465
270 270 small 0.20423088 0.8152741 0.8233047 1.239653
271 271 small 0.35400226 0.7018734 0.7137594 1.449998
272 272 medium 0.14892700 0.8616320 0.8647029 1.164837
273 273 small 1.01742375 0.3615251 0.3694456 2.826660
274 274 small 0.14753714 0.8628304 0.8683992 1.167116
275 275 large 0.59087223 0.5538440 0.5574303 1.817254
276 276 large 0.31868474 0.7271047 0.7316965 1.384040
277 277 large 1.05323826 0.3488064 0.3510650 2.885484
278 278 large 0.36206496 0.6962371 0.7007051 1.445526
279 279 large 0.11851463 0.8882388 0.8910786 1.129576
280 280 medium 0.56049118 0.5709286 0.5738184 1.760398
281 281 large 0.90082987 0.4062324 0.4088629 2.477585
282 282 large 0.68894756 0.5021042 0.5053555 2.004515
283 283 large 0.40681477 0.6657675 0.6700670 1.511732
284 284 small 0.32825360 0.7201804 0.7316827 1.411882
285 285 small 0.58712316 0.5559243 0.5677872 1.837488
286 286 small 0.26499068 0.7672131 0.7772924 1.321715
287 287 medium 0.55761408 0.5725736 0.5754718 1.755341
288 288 small 0.53914604 0.5832461 0.5954556 1.750876
289 289 small 0.72014046 0.4866839 0.4972993 2.099610
290 290 small 0.44715734 0.6394433 0.6519304 1.595191
291 291 small 0.49780579 0.6078630 0.6202759 1.679293
292 292 small 0.66100224 0.5163336 0.5275299 1.978871
293 293 medium 0.14962157 0.8610338 0.8641147 1.165663
294 294 medium 0.69952325 0.4968221 0.4993369 2.022981
295 295 small 0.35783098 0.6991912 0.7111254 1.455738
296 296 small 0.26213416 0.7694078 0.7794058 1.317758
297 297 small 0.57148927 0.5646838 0.5766699 1.808833
298 298 medium 0.85849748 0.4237984 0.4259435 2.371556
299 299 small 0.26961386 0.7636743 0.7738819 1.328140
300 300 medium 0.48262620 0.6171605 0.6202843 1.628526
301 301 medium 0.43839191 0.6450729 0.6483376 1.558058
302 302 medium 0.19214074 0.8251907 0.8287242 1.217118
303 303 large 0.04633628 0.9547209 0.9555221 1.048342
304 304 large 0.16623744 0.8468451 0.8506563 1.186334
305 305 medium 0.35468269 0.7013960 0.7049352 1.432927
306 306 large 0.18339731 0.8324374 0.8364966 1.207308
307 307 small 0.40384508 0.6677476 0.6800937 1.526218
308 308 medium 0.09255992 0.9115946 0.9135655 1.099444
309 309 medium 0.18483443 0.8312419 0.8347186 1.208145
310 310 large 0.10609289 0.8993411 0.9018632 1.115183
311 311 large 0.14589422 0.8642491 0.8676969 1.161858
312 312 medium 0.41861446 0.6579578 0.6612871 1.527545
313 313 small 0.13859242 0.8705828 0.8757270 1.156041
314 314 small 0.27929584 0.7563161 0.7667798 1.341679
315 315 medium 0.22048841 0.8021269 0.8058099 1.252475
316 316 small 0.53495241 0.5856972 0.5979319 1.743489
317 317 small 0.16536791 0.8475818 0.8539723 1.189498
318 318 large 0.10368213 0.9015118 0.9039695 1.112408
319 319 large 0.09606154 0.9084081 0.9106571 1.103678
320 320 large 0.04783823 0.9532880 0.9541312 1.049968
321 321 large 0.13853364 0.8706340 0.8739315 1.153104
322 322 large 0.30251241 0.7389593 0.7435778 1.361761
323 323 medium 0.33992775 0.7118218 0.7154070 1.411928
324 324 large 0.32328901 0.7237646 0.7283465 1.390446
325 325 large 0.84248670 0.4306383 0.4334269 2.337171
326 326 large 0.07179505 0.9307216 0.9322668 1.076303
327 327 small 0.20451647 0.8150413 0.8230830 1.240028
328 328 medium 0.09623443 0.9082511 0.9103100 1.103610
329 329 small 0.21597733 0.8057536 0.8142318 1.255173
330 330 medium 0.22230257 0.8006731 0.8043621 1.254768
331 331 small 1.77202421 0.1699885 0.1737128 6.011634
332 332 small 0.08729466 0.9164070 0.9190677 1.094637
333 333 small 0.07767033 0.9252694 0.9274824 1.083545
334 334 small 0.16530631 0.8476340 0.8540218 1.189420
335 335 small 0.62939981 0.5329116 0.5444060 1.917157
336 336 medium 0.06048170 0.9413110 0.9424521 1.063689
337 337 medium 0.13407398 0.8745254 0.8773616 1.147300
338 338 small 0.27164171 0.7621273 0.7723899 1.330967
339 339 small 0.10454772 0.9007318 0.9042222 1.114872
340 340 small 0.30967136 0.7336880 0.7448454 1.384890
341 341 medium 0.26074881 0.7704744 0.7742174 1.304243
342 342 small 0.10813117 0.8975099 0.9011749 1.119130
343 343 medium 0.15387524 0.8573790 0.8605191 1.170728
344 344 medium 0.06691412 0.9352755 0.9365885 1.070767
uLB_g uUB_g m_g TE_group_mode teBCLB_g teBCUB_g
1 0.077581942 0.4657010 0.268585702 0.7644599 0.6276949 0.9253512
2 0.130356248 0.5739174 0.351182071 0.7038556 0.5633144 0.8777827
3 0.065447909 0.4980807 0.275016061 0.7595599 0.6076959 0.9366478
4 0.018022507 0.3583190 0.158856747 0.8531186 0.6988501 0.9821389
5 0.027125654 0.4268531 0.202315202 0.8168374 0.6525594 0.9732389
6 0.009050601 0.5251973 0.079980247 0.9231346 0.5914386 0.9909902
7 0.008752251 0.3079157 0.104055371 0.9011754 0.7349772 0.9912859
8 0.013162800 0.3369142 0.136034520 0.8728125 0.7139701 0.9869235
9 0.085680505 0.5241016 0.301219525 0.7399153 0.5920871 0.9178874
10 0.167828760 0.6126522 0.389950793 0.6770902 0.5419117 0.8454986
11 0.521202993 1.3372953 0.929241967 0.3948529 0.2625548 0.5938058
12 0.008642986 0.5171656 0.069500977 0.9328592 0.5962080 0.9913943
13 0.020090696 0.6533880 0.232905761 0.7922282 0.5202801 0.9801098
14 0.111995595 0.5541988 0.331426088 0.7178992 0.5745324 0.8940482
15 0.044863312 0.7679191 0.355902815 0.7005407 0.4639775 0.9561282
16 0.276449440 1.0908091 0.682709790 0.5052460 0.3359446 0.7584720
17 0.260417553 0.7057716 0.483084286 0.6168778 0.4937275 0.7707297
18 0.009432911 0.3441592 0.113132519 0.8930323 0.7088160 0.9906114
19 0.011791634 0.3624708 0.133648420 0.8748976 0.6959546 0.9882776
20 0.039974576 0.4567367 0.233063689 0.7921031 0.6333471 0.9608139
21 0.312407891 0.7577792 0.535092362 0.5856152 0.4687062 0.7316830
22 0.009013704 0.3100686 0.106481830 0.8989914 0.7333966 0.9910268
23 0.426358917 0.8717326 0.649045730 0.5225442 0.4182263 0.6528820
24 0.087712464 0.5265410 0.303671028 0.7381036 0.5906445 0.9160242
25 0.140695307 0.5847546 0.362032747 0.6962596 0.5572426 0.8687540
26 0.038458085 0.7464911 0.333459137 0.7164412 0.4740269 0.9622720
27 0.492507972 0.8864148 0.689461398 0.5018463 0.4121307 0.6110919
28 0.011918220 0.3300082 0.128544438 0.8793745 0.7189179 0.9881525
29 0.241908494 0.6358051 0.438851372 0.6447766 0.5295090 0.7851280
30 0.584058808 0.9779657 0.781012234 0.4579422 0.3760754 0.5576305
31 0.003482336 0.3468517 0.000000000 1.0000000 0.7069102 0.9965237
32 0.329746003 1.1451784 0.737107658 0.4784959 0.3181671 0.7191064
33 0.094649581 0.4852999 0.288257791 0.7495683 0.6155126 0.9096917
34 0.550669150 1.3667681 0.958714970 0.3833852 0.2549295 0.5765639
35 0.008164362 0.3319277 0.099075758 0.9056741 0.7175392 0.9918689
36 0.560613143 1.3767136 0.968660439 0.3795912 0.2524067 0.5708589
37 0.008329538 0.5106760 0.060930435 0.9408887 0.6000898 0.9917051
38 0.004438296 0.2763860 0.030178916 0.9702719 0.7585201 0.9955715
39 0.225873860 1.0378517 0.629687722 0.5327581 0.3542148 0.7978187
40 0.463629442 0.8575363 0.660582868 0.5165502 0.4242059 0.6289966
41 0.021006858 0.6600248 0.240283242 0.7864051 0.5168385 0.9792122
42 0.001609375 0.1633501 0.000000000 1.0000000 0.8492938 0.9983919
43 0.996139105 1.3900460 1.193092531 0.3032819 0.2490639 0.3693025
44 0.201186961 0.5950330 0.398077979 0.6716097 0.5515444 0.8177595
45 0.027773455 0.4286464 0.204174672 0.8153199 0.6513902 0.9726087
46 0.158946763 0.6035852 0.380878944 0.6832606 0.5468475 0.8530418
47 0.320115147 0.7140218 0.517068350 0.5962660 0.4896709 0.7260654
48 0.071072796 0.5056953 0.282695589 0.7537492 0.6030861 0.9313941
49 0.044968124 0.7682434 0.356240949 0.7003039 0.4638271 0.9560280
50 0.003689591 0.2391210 0.019494873 0.9806939 0.7873196 0.9963172
51 0.037849713 0.4087433 0.211015955 0.8097611 0.6644848 0.9628576
52 0.012957480 0.3700178 0.141945964 0.8676681 0.6907220 0.9871261
53 0.200699455 0.6458929 0.423201311 0.6549468 0.5241943 0.8181583
54 0.545128502 1.3612265 0.953173347 0.3855157 0.2563462 0.5797673
55 0.033952267 0.7290678 0.315051603 0.7297512 0.4823584 0.9666176
56 0.122155766 0.9157602 0.507042817 0.6022740 0.4002123 0.8850105
57 0.205615831 0.6508379 0.428147102 0.6517155 0.5216085 0.8141458
58 0.052552895 0.7899306 0.378765887 0.6847059 0.4538763 0.9488041
59 0.010090458 0.5438693 0.103833569 0.9013753 0.5804978 0.9899603
60 0.842813067 1.2881867 1.065499891 0.3445556 0.2757704 0.4304978
61 0.046559313 0.4689444 0.245506506 0.7823082 0.6256623 0.9545079
62 0.003206229 0.2440483 0.000000000 1.0000000 0.7834498 0.9967989
63 0.012601214 0.3677966 0.139512577 0.8697821 0.6922579 0.9874778
64 0.495496284 0.9408699 0.718183107 0.4876374 0.3902882 0.6092685
65 0.154032974 0.5475451 0.350581285 0.7042786 0.5783679 0.8572438
66 0.242814282 0.6881492 0.465461408 0.6278453 0.5025052 0.7844172
67 0.156030772 0.6005967 0.377888484 0.6853069 0.5484842 0.8555329
68 0.013527181 0.3734293 0.145670000 0.8644429 0.6883697 0.9865639
69 0.131069580 0.9277430 0.519125251 0.5950408 0.3954452 0.8771567
70 0.131000057 0.5239870 0.327009100 0.7210772 0.5921549 0.8772177
71 0.003719710 0.2398234 0.020452225 0.9797555 0.7867668 0.9962872
72 0.547919285 0.9418261 0.744872711 0.4747947 0.3899151 0.5781515
73 0.352576316 0.7464831 0.549529704 0.5772212 0.4740307 0.7028749
74 0.094201509 0.8742577 0.465063318 0.6280953 0.4171716 0.9100994
75 0.022072437 0.6673362 0.248366459 0.7800740 0.5130735 0.9781694
76 0.110023143 0.5020052 0.305000090 0.7371233 0.6053157 0.8958134
77 0.514124743 1.3302147 0.922161315 0.3976586 0.2644205 0.5980238
78 0.005048633 0.2654447 0.053800315 0.9476213 0.7668648 0.9949641
79 0.122234702 0.9158684 0.507152041 0.6022082 0.4001690 0.8849406
80 0.010394631 0.5488989 0.110145413 0.8957039 0.5775854 0.9896592
81 0.035928588 0.4483881 0.224519377 0.7989001 0.6386568 0.9647092
82 0.400860479 1.2167931 0.808735563 0.4454209 0.2961785 0.6697435
83 0.597760416 0.9916673 0.794713842 0.4517105 0.3709577 0.5500421
84 0.006214525 0.4576563 0.000000000 1.0000000 0.6327649 0.9938047
85 0.005505876 0.2724415 0.062448544 0.9394614 0.7615180 0.9945093
86 1.062283575 1.4561904 1.259237001 0.2838705 0.2331227 0.3456656
87 0.050854925 0.4303856 0.232991434 0.7921604 0.6502583 0.9504165
88 0.017533862 0.3936387 0.167430378 0.8458355 0.6745977 0.9826190
89 0.182005354 0.6270360 0.404340143 0.6674171 0.5341727 0.8335969
90 0.001249378 0.1399248 0.000000000 1.0000000 0.8694236 0.9987514
91 0.031503770 0.4382585 0.214107362 0.8072617 0.6451590 0.9689873
92 0.020788329 0.6584725 0.238561250 0.7877604 0.5176414 0.9794263
93 0.008673865 0.3072556 0.103309411 0.9018479 0.7354626 0.9913636
94 0.326660005 0.7205667 0.523613273 0.5923763 0.4864765 0.7213289
95 0.017510812 0.3935376 0.167322600 0.8459267 0.6746660 0.9826416
96 0.010884822 0.3559817 0.126443767 0.8812237 0.7004855 0.9891742
97 0.181962446 0.9895722 0.581285703 0.5591790 0.3717357 0.8336326
98 0.034300668 0.7305001 0.316570646 0.7286435 0.4816681 0.9662809
99 0.023442491 0.6761635 0.258066843 0.7725436 0.5085644 0.9768301
100 0.058140452 0.4875789 0.264405056 0.7676625 0.6141114 0.9435174
101 0.154855952 0.9577708 0.549346483 0.5773270 0.3837474 0.8565386
102 0.029101642 0.7072981 0.291820564 0.7469025 0.4929744 0.9713177
103 0.021998212 0.4109565 0.185733607 0.8304948 0.6630158 0.9782420
104 0.014502679 0.3789105 0.151620760 0.8593141 0.6846069 0.9856020
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340 0.025272467 0.6870793 0.269979005 0.7633955 0.5030432 0.9750442
341 0.070026998 0.4564718 0.259305895 0.7715870 0.6335149 0.9323686
342 0.003389504 0.3416122 0.000000000 1.0000000 0.7106237 0.9966162
343 0.013225467 0.3372424 0.136388779 0.8725034 0.7137358 0.9868616
344 0.002346241 0.1985332 0.000000000 1.0000000 0.8199326 0.9976565
u_meta TE_meta_JLMS TE_meta_BC MTR_JLMS MTR_BC
1 0.39444390 0.6740548 0.6773549 0.8827375 0.8827375
2 0.37798358 0.6852418 0.6896274 0.9739266 0.9739266
3 0.30495307 0.7371580 0.7416598 0.9728780 0.9728780
4 0.17104173 0.8427864 0.8461331 1.0000000 1.0000000
5 0.23792711 0.7882601 0.7925093 0.9743700 0.9743700
6 0.32952629 0.7192644 0.7261201 0.8774139 0.8774139
7 0.21037952 0.8102767 0.8128593 0.9242932 0.9242932
8 0.25024492 0.7786101 0.7814578 0.9078924 0.9078924
9 0.32025822 0.7259616 0.7304990 0.9824715 0.9824715
10 0.41975249 0.6572095 0.6614469 0.9707598 0.9707598
11 1.00109439 0.3674771 0.3755272 0.9306719 0.9306719
12 0.37878552 0.6846924 0.6910300 0.8313206 0.8313206
13 0.29326068 0.7458277 0.7563143 0.9908461 0.9908461
14 0.33207087 0.7174365 0.7219968 1.0000000 1.0000000
15 0.55461328 0.5742943 0.5844432 0.8364708 0.8364708
16 0.78062329 0.4581204 0.4680813 0.9070758 0.9070758
17 0.50185661 0.6054056 0.6093249 0.9814081 0.9814081
18 0.15756846 0.8542183 0.8576283 0.9884873 0.9884873
19 0.15942841 0.8526310 0.8563290 1.0000000 1.0000000
20 0.26903223 0.7641186 0.7685150 0.9701291 0.9701291
21 0.60171509 0.5478712 0.5514187 0.9355487 0.9355487
22 0.16248573 0.8500282 0.8527683 0.9711541 0.9711541
23 0.64904573 0.5225442 0.5259279 1.0000000 1.0000000
24 0.34018613 0.7116379 0.7160934 0.9653772 0.9653772
25 0.40073531 0.6698273 0.6741261 0.9623092 0.9623092
26 0.38544339 0.6801490 0.6917579 0.9727555 0.9727555
27 0.76140601 0.4670093 0.4693732 0.9305824 0.9305824
28 0.17464481 0.8397552 0.8427359 0.9739092 0.9739092
29 0.43885427 0.6447747 0.6480379 1.0000000 1.0000000
30 0.78101223 0.4579422 0.4602602 1.0000000 1.0000000
31 0.22498830 0.7985256 0.8018868 0.8916248 0.8916248
32 0.94035513 0.3904891 0.3990150 0.8162049 0.8162049
33 0.33687022 0.7140015 0.7175415 0.9531756 0.9531756
34 1.29227182 0.2746461 0.2806630 0.7163727 0.7163727
35 0.19878240 0.8197282 0.8228071 0.9404751 0.9404751
36 0.96866209 0.3795906 0.3879066 1.0000000 1.0000000
37 0.19029853 0.8267123 0.8341817 1.0000000 1.0000000
38 0.10253454 0.9025470 0.9049738 1.0000000 1.0000000
39 0.74618972 0.4741698 0.4843996 0.8907923 0.8907923
40 0.75640502 0.4693507 0.4717264 0.9086256 0.9086256
41 0.28900704 0.7490069 0.7597103 1.0000000 1.0000000
42 0.05156756 0.9497395 0.9506433 1.0000000 1.0000000
43 1.19948894 0.3013482 0.3028735 0.9936240 0.9936240
44 0.51974684 0.5946711 0.5976790 0.8854554 0.8854554
45 0.21353124 0.8077269 0.8121007 1.0000000 1.0000000
46 0.41483089 0.6604520 0.6647052 0.9667770 0.9667770
47 0.51706842 0.5962660 0.5992841 1.0000000 1.0000000
48 0.28700071 0.7505112 0.7551303 0.9977620 0.9977620
49 0.68406107 0.5045638 0.5134848 0.7351120 0.7351120
50 0.14671525 0.8635398 0.8652850 0.9426814 0.9426814
51 0.23045956 0.7941685 0.7977784 0.9851686 0.9851686
52 0.18426984 0.8317113 0.8354336 0.9810836 0.9810836
53 0.46648877 0.6272007 0.6312553 0.9576782 0.9576782
54 0.95317568 0.3855148 0.3939604 1.0000000 1.0000000
55 0.35110760 0.7039080 0.7155568 0.9922464 0.9922464
56 0.51378204 0.5982288 0.6105570 0.9975754 0.9975754
57 0.44957542 0.6378989 0.6420237 0.9788362 0.9788362
58 0.55729241 0.5727578 0.5832096 0.8503668 0.8503668
59 0.36184307 0.6963916 0.7034812 0.8591001 0.8591001
60 1.09626055 0.3341182 0.3362817 0.9697076 0.9697076
61 0.26733337 0.7654179 0.7699151 0.9827817 0.9827817
62 0.08518681 0.9183407 0.9202802 1.0000000 1.0000000
63 0.20566339 0.8141071 0.8177177 0.9586840 0.9586840
64 0.75354044 0.4706971 0.4737451 0.9652604 0.9652604
65 0.40535189 0.6667422 0.6701051 0.9467887 0.9467887
66 0.49074842 0.6121681 0.6161304 0.9750400 0.9750400
67 0.45408166 0.6350309 0.6391185 0.9268037 0.9268037
68 0.19207913 0.8252416 0.8289857 0.9760202 0.9760202
69 0.53469257 0.5858494 0.5980068 0.9882353 0.9882353
70 0.32721034 0.7209321 0.7245549 1.0000000 1.0000000
71 0.12482115 0.8826548 0.8844487 0.9639255 0.9639255
72 0.74487271 0.4747947 0.4771980 1.0000000 1.0000000
73 0.54952972 0.5772212 0.5801429 1.0000000 1.0000000
74 0.55018140 0.5768452 0.5883855 0.9248006 0.9248006
75 0.44268347 0.6423105 0.6516504 0.8622816 0.8622816
76 0.36224749 0.6961101 0.6995866 0.9447391 0.9447391
77 1.32624636 0.2654719 0.2712873 0.6675904 0.6675904
78 0.10315484 0.9019873 0.9042066 1.0000000 1.0000000
79 0.51145818 0.5996206 0.6119783 1.0000000 1.0000000
80 0.21306571 0.8081030 0.8164725 1.0000000 1.0000000
81 0.23111098 0.7936514 0.7981445 1.0000000 1.0000000
82 1.03354713 0.3557429 0.3635284 0.7987018 0.7987018
83 0.85676884 0.4245316 0.4266805 0.9398312 0.9398312
84 0.21235388 0.8086785 0.8145825 0.9503559 0.9503559
85 0.10754566 0.8980355 0.9003525 1.0000000 1.0000000
86 1.29488076 0.2739305 0.2753171 0.9649840 0.9649840
87 0.38204365 0.6824653 0.6856820 0.8639009 0.8639009
88 0.22588475 0.7978100 0.8017085 0.9588762 0.9588762
89 0.40917872 0.6641955 0.6684841 0.9952533 0.9952533
90 0.04250673 0.9583840 0.9590537 1.0000000 1.0000000
91 0.22235705 0.8006295 0.8050648 0.9996635 0.9996635
92 0.41336393 0.6614215 0.6708378 0.8820404 0.8820404
93 0.20797359 0.8122285 0.8148083 0.9260781 0.9260781
94 0.61397327 0.5411963 0.5439356 0.9136023 0.9136023
95 0.22690705 0.7969949 0.8008880 0.9578174 0.9578174
96 0.19103008 0.8261077 0.8295908 0.9642106 0.9642106
97 0.66429840 0.5146345 0.5256017 0.9218957 0.9218957
98 0.41168997 0.6625297 0.6735226 0.9350214 0.9350214
99 0.30123486 0.7399040 0.7508851 1.0000000 1.0000000
100 0.35609786 0.7004041 0.7046295 0.9151752 0.9151752
101 0.64856316 0.5227964 0.5338062 0.9078735 0.9078735
102 0.36928538 0.6912281 0.7021963 0.9572960 0.9572960
103 0.26179509 0.7696687 0.7736414 0.9384684 0.9384684
104 0.19762269 0.8206794 0.8244828 0.9747904 0.9747904
105 0.25098227 0.7780362 0.7818408 0.9352765 0.9352765
106 0.23206267 0.7928964 0.7962118 0.9241822 0.9241822
107 0.11327896 0.8929015 0.8955447 0.9976781 0.9976781
108 0.18478190 0.8312856 0.8333628 0.9229040 0.9229040
109 0.28030021 0.7555569 0.7600170 0.9728278 0.9728278
110 0.19014512 0.8268391 0.8299510 0.9488866 0.9488866
111 0.19820918 0.8201983 0.8239730 0.9728255 0.9728255
112 0.27205429 0.7618129 0.7720866 1.0000000 1.0000000
113 0.24163198 0.7853451 0.7885464 0.9380152 0.9380152
114 0.14711027 0.8631988 0.8646799 0.9326182 0.9326182
115 0.53654999 0.5847622 0.5877221 1.0000000 1.0000000
116 0.61966844 0.5381228 0.5484194 0.8267052 0.8267052
117 0.31210476 0.7319049 0.7374067 0.8631991 0.8631991
118 0.34483403 0.7083379 0.7144550 0.8510725 0.8510725
119 0.33274854 0.7169505 0.7205572 1.0000000 1.0000000
120 0.89734536 0.4076504 0.4163574 0.7355082 0.7355082
121 0.16987168 0.8437731 0.8465014 0.9644284 0.9644284
122 0.42172180 0.6559165 0.6680859 0.9846119 0.9846119
123 0.17263606 0.8414438 0.8481581 1.0000000 1.0000000
124 0.16506081 0.8478421 0.8512682 0.9829150 0.9829150
125 0.65052421 0.5217722 0.5305514 0.7411215 0.7411215
126 0.56752097 0.5669291 0.5697988 1.0000000 1.0000000
127 0.20671025 0.8132553 0.8195550 0.9627770 0.9627770
128 0.13506819 0.8736563 0.8765095 1.0000000 1.0000000
129 0.44504953 0.6407925 0.6532776 1.0000000 1.0000000
130 0.50696718 0.6023195 0.6053657 0.8920471 0.8920471
131 0.15303209 0.8581022 0.8602725 0.9461487 0.9461487
132 0.53559387 0.5853216 0.5891116 0.9949825 0.9949825
133 0.14522379 0.8648287 0.8678449 1.0000000 1.0000000
134 0.18808338 0.8285456 0.8325276 0.9919477 0.9919477
135 0.58473830 0.5572517 0.5682979 0.8844728 0.8844728
136 0.10458820 0.9006954 0.9022647 0.9739954 0.9739954
137 0.35136778 0.7037249 0.7070168 0.8865187 0.8865187
138 0.37004555 0.6907029 0.6951151 0.9756067 0.9756067
139 0.16565412 0.8473393 0.8503152 0.9639600 0.9639600
140 0.48103124 0.6181456 0.6212744 1.0000000 1.0000000
141 0.30515784 0.7370070 0.7404346 0.8290943 0.8290943
142 0.50645708 0.6026269 0.6150063 1.0000000 1.0000000
143 0.25674124 0.7735684 0.7770215 0.9519356 0.9519356
144 0.46053631 0.6309452 0.6405166 0.8592157 0.8592157
145 0.41590758 0.6597412 0.6669648 0.8250447 0.8250447
146 0.14138654 0.8681537 0.8696112 0.9332922 0.9332922
147 0.23064280 0.7940230 0.7981965 0.9734308 0.9734308
148 0.94788510 0.3875598 0.3900694 0.9689228 0.9689228
149 0.28493182 0.7520655 0.7562160 0.9376306 0.9376306
150 0.75089131 0.4719457 0.4750017 0.9329519 0.9329519
151 0.19303528 0.8244529 0.8259262 0.8928885 0.8928885
152 0.80936469 0.4451408 0.4480232 0.9854915 0.9854915
153 0.25957775 0.7713772 0.7758323 0.9812173 0.9812173
154 0.13980979 0.8695236 0.8725669 0.9887926 0.9887926
155 0.49661330 0.6085883 0.6210053 1.0000000 1.0000000
156 0.40403547 0.6676204 0.6794376 0.9732827 0.9732827
157 0.29497587 0.7445496 0.7553918 1.0000000 1.0000000
158 0.11544390 0.8909706 0.8934553 1.0000000 1.0000000
159 0.46717667 0.6267693 0.6364449 0.8589349 0.8589349
160 0.42183943 0.6558393 0.6682703 1.0000000 1.0000000
161 0.26677319 0.7658468 0.7759759 1.0000000 1.0000000
162 0.29267926 0.7462615 0.7570081 0.9987375 0.9987375
163 0.56948748 0.5658154 0.5754736 0.8092685 0.8092685
164 0.22972866 0.7947492 0.7981289 0.9609615 0.9609615
165 0.56689043 0.5672867 0.5701582 0.9700244 0.9700244
166 0.21390204 0.8074275 0.8158283 1.0000000 1.0000000
167 0.14281593 0.8669136 0.8696705 0.9754535 0.9754535
168 0.48502138 0.6156840 0.6229529 0.7821207 0.7821207
169 0.55684957 0.5730115 0.5759119 0.9495640 0.9495640
170 0.33178121 0.7176443 0.7231447 0.8484410 0.8484410
171 0.35984536 0.6977842 0.7013069 1.0000000 1.0000000
172 0.62839799 0.5334457 0.5361459 1.0000000 1.0000000
173 0.53349777 0.5865498 0.5895154 0.8624977 0.8624977
174 0.17043379 0.8432989 0.8453584 0.9273868 0.9273868
175 0.77177777 0.4621907 0.4651835 0.9914768 0.9914768
176 0.20974379 0.8107920 0.8144306 1.0000000 1.0000000
177 0.41741748 0.6587458 0.6630010 0.9924273 0.9924273
178 0.42537352 0.6535256 0.6644413 0.9251016 0.9251016
179 0.29964461 0.7410815 0.7444580 0.9203500 0.9203500
180 0.33735846 0.7136530 0.7171269 0.9282806 0.9282806
181 0.29409585 0.7452051 0.7496873 0.9715415 0.9715415
182 0.73236497 0.4807706 0.4838838 0.9355549 0.9355549
183 0.86695664 0.4202285 0.4223556 1.0000000 1.0000000
184 0.49688866 0.6084207 0.6202302 0.9462206 0.9462206
185 0.65924072 0.5172439 0.5284572 1.0000000 1.0000000
186 0.24201528 0.7850442 0.7887775 1.0000000 1.0000000
187 0.46468726 0.6283316 0.6384204 0.8743516 0.8743516
188 0.34979118 0.7048353 0.7111342 0.8511607 0.8511607
189 0.11450304 0.8918092 0.8927489 0.9406343 0.9406343
190 0.27187405 0.7619502 0.7663752 0.9720103 0.9720103
191 0.22754237 0.7964887 0.8007158 0.9794407 0.9794407
192 0.24883443 0.7797091 0.7837878 0.9544856 0.9544856
193 0.32382733 0.7233751 0.7279557 1.0000000 1.0000000
194 0.74047763 0.4768861 0.4792999 0.9448749 0.9448749
195 0.34738626 0.7065324 0.7110164 0.9813354 0.9813354
196 0.97742507 0.3762787 0.3787153 0.9591650 0.9591650
197 0.84088104 0.4313303 0.4341234 0.9491678 0.9491678
198 0.27949430 0.7561660 0.7643596 0.9437069 0.9437069
199 0.55212962 0.5757224 0.5871325 0.9135979 0.9135979
200 0.25045891 0.7784435 0.7880938 1.0000000 1.0000000
201 0.73516393 0.4794269 0.4818536 1.0000000 1.0000000
202 0.54417850 0.5803183 0.5896301 0.8073199 0.8073199
203 0.38074407 0.6833528 0.6891586 0.8191330 0.8191330
204 0.42240969 0.6554654 0.6655180 0.8961092 0.8961092
205 0.14931823 0.8612950 0.8643715 1.0000000 1.0000000
206 1.25788778 0.2842538 0.2904582 0.5907041 0.5907041
207 0.66032361 0.5166841 0.5192994 0.8559791 0.8559791
208 0.56523250 0.5682280 0.5711042 0.9557798 0.9557798
209 0.32860771 0.7199254 0.7303578 0.9654064 0.9654064
210 0.54322191 0.5808737 0.5846349 0.9838366 0.9838366
211 0.35256362 0.7028839 0.7080764 0.8272046 0.8272046
212 0.57387767 0.5633368 0.5661881 0.8974359 0.8974359
213 0.56347855 0.5692255 0.5787680 0.8071348 0.8071348
214 0.52477163 0.5916905 0.5946854 1.0000000 1.0000000
215 0.25196513 0.7772718 0.7810137 1.0000000 1.0000000
216 0.80930268 0.4451684 0.4474217 0.9456821 0.9456821
217 0.09400786 0.9102756 0.9124671 1.0000000 1.0000000
218 0.23240384 0.7926260 0.7959841 0.9258822 0.9258822
219 0.37989456 0.6839335 0.6873807 0.9655507 0.9655507
220 0.28710954 0.7504295 0.7550588 1.0000000 1.0000000
221 0.48353326 0.6166009 0.6240634 0.7876606 0.7876606
222 0.38966238 0.6772855 0.6815536 0.9280430 0.9280430
223 0.61504229 0.5406180 0.5433545 0.9036735 0.9036735
224 0.48396957 0.6163320 0.6203207 0.9739975 0.9739975
225 0.18935689 0.8274911 0.8302261 0.9348965 0.9348965
226 0.38224719 0.6823264 0.6857758 1.0000000 1.0000000
227 0.34668601 0.7070273 0.7112751 0.8134523 0.8134523
228 0.38393814 0.6811736 0.6933777 1.0000000 1.0000000
229 0.30120267 0.7399278 0.7435552 0.9706127 0.9706127
230 0.49297466 0.6108067 0.6197143 0.8207539 0.8207539
231 0.29989023 0.7408995 0.7465903 0.8761714 0.8761714
232 0.17144298 0.8424483 0.8441021 0.9133210 0.9133210
233 0.28319722 0.7533712 0.7578462 0.9737940 0.9737940
234 0.16347906 0.8491843 0.8520881 0.9630506 0.9630506
235 0.37484885 0.6873932 0.6917079 0.9359548 0.9359548
236 0.11399593 0.8922616 0.8942170 0.9739030 0.9739030
237 0.71101709 0.4911444 0.4936304 0.8811842 0.8811842
238 0.36383332 0.6950070 0.6993957 0.9557007 0.9557007
239 0.43242580 0.6489330 0.6531007 0.9354356 0.9354356
240 0.61284425 0.5418076 0.5453160 0.9342505 0.9342505
241 0.27373668 0.7605323 0.7693237 0.9617159 0.9617159
242 0.27735774 0.7577834 0.7681971 1.0000000 1.0000000
243 0.30605962 0.7363427 0.7474262 1.0000000 1.0000000
244 1.36845484 0.2544999 0.2557881 0.7512507 0.7512507
245 0.61437852 0.5409770 0.5500523 0.7673298 0.7673298
246 0.28933216 0.7487635 0.7543180 0.8816448 0.8816448
247 0.71841455 0.4875246 0.4981571 1.0000000 1.0000000
248 0.30376589 0.7380336 0.7467968 0.9387275 0.9387275
249 0.65487883 0.5195050 0.5305140 0.9159627 0.9159627
250 0.43478797 0.6474019 0.6506193 0.8683614 0.8683614
251 0.45068791 0.6371897 0.6404147 1.0000000 1.0000000
252 0.29203957 0.7467390 0.7575137 1.0000000 1.0000000
253 0.22442988 0.7989716 0.8064442 0.9716722 0.9716722
254 0.38054996 0.6834854 0.6904219 0.8427102 0.8427102
255 0.62145825 0.5371605 0.5398793 0.8441811 0.8441811
256 0.26436696 0.7676918 0.7710506 0.8596136 0.8596136
257 0.39266862 0.6752525 0.6786140 0.9091017 0.9091017
258 0.09230253 0.9118293 0.9129741 0.9700948 0.9700948
259 0.97068626 0.3788230 0.3807405 0.8738143 0.8738143
260 0.32065506 0.7256735 0.7301102 0.9586753 0.9586753
261 0.90210294 0.4057156 0.4083427 1.0000000 1.0000000
262 0.37714757 0.6858149 0.6892811 1.0000000 1.0000000
263 0.33517162 0.7152153 0.7196980 0.9717017 0.9717017
264 0.74097705 0.4766480 0.4863654 0.7823469 0.7823469
265 0.39539081 0.6734168 0.6777217 0.9533614 0.9533614
266 0.84351863 0.4301942 0.4323717 0.8302595 0.8302595
267 0.29463580 0.7448028 0.7489811 0.9347474 0.9347474
268 0.49047591 0.6123349 0.6162901 0.9220217 0.9220217
269 1.04243306 0.3525958 0.3543805 1.0000000 1.0000000
270 0.22190773 0.8009893 0.8088791 0.9824785 0.9824785
271 0.38972436 0.6772435 0.6887125 0.9649084 0.9649084
272 0.14892700 0.8616320 0.8647029 1.0000000 1.0000000
273 1.24497404 0.2879484 0.2942569 0.7964824 0.7964824
274 0.46429150 0.6285803 0.6326372 0.7285097 0.7285097
275 0.71488163 0.4892500 0.4924181 0.8833715 0.8833715
276 0.33281230 0.7169047 0.7214321 0.9859718 0.9859718
277 1.11104946 0.3292133 0.3313451 0.9438281 0.9438281
278 0.42956697 0.6507908 0.6549672 0.9347258 0.9347258
279 0.12783560 0.8799980 0.8828115 0.9907223 0.9907223
280 0.70191382 0.4956358 0.4981446 0.8681223 0.8681223
281 0.97090888 0.3787387 0.3811911 0.9323202 0.9323202
282 0.72107398 0.4862298 0.4893783 0.9683842 0.9683842
283 0.41579460 0.6598158 0.6640769 0.9910604 0.9910604
284 0.40451450 0.6673007 0.6779585 0.9265744 0.9265744
285 0.58712316 0.5559243 0.5677872 1.0000000 1.0000000
286 0.26499068 0.7672131 0.7772924 1.0000000 1.0000000
287 0.55761408 0.5725736 0.5754718 1.0000000 1.0000000
288 0.62211428 0.5368083 0.5480456 0.9203804 0.9203804
289 0.92922040 0.3948614 0.4034740 0.8113304 0.8113304
290 0.50650286 0.6025993 0.6143669 0.9423811 0.9423811
291 0.66748218 0.5129986 0.5234743 0.8439379 0.8439379
292 0.72377619 0.4849177 0.4954327 0.9391557 0.9391557
293 0.26719593 0.7655231 0.7682622 0.8890744 0.8890744
294 0.69952325 0.4968221 0.4993369 1.0000000 1.0000000
295 0.51492106 0.5975478 0.6077470 0.8546271 0.8546271
296 0.32231167 0.7244724 0.7338865 0.9415974 0.9415974
297 0.62822186 0.5335397 0.5448647 0.9448467 0.9448467
298 0.88865128 0.4112100 0.4132914 0.9702963 0.9702963
299 0.46876020 0.6257776 0.6341420 0.8194300 0.8194300
300 0.48262620 0.6171605 0.6202843 1.0000000 1.0000000
301 0.43839191 0.6450729 0.6483376 1.0000000 1.0000000
302 0.20232336 0.8168308 0.8203284 0.9898690 0.9898690
303 0.06686091 0.9353253 0.9361102 0.9796846 0.9796846
304 0.18397220 0.8319589 0.8357031 0.9824216 0.9824216
305 0.35468269 0.7013960 0.7049352 1.0000000 1.0000000
306 0.19634133 0.8217317 0.8257388 0.9871394 0.9871394
307 0.40384508 0.6677476 0.6800937 1.0000000 1.0000000
308 0.18479876 0.8312715 0.8330688 0.9118873 0.9118873
309 0.27721884 0.7578886 0.7610585 0.9117546 0.9117546
310 0.19762206 0.8206800 0.8229814 0.9125347 0.9125347
311 0.22681075 0.7970716 0.8002514 0.9222707 0.9222707
312 0.41861446 0.6579578 0.6612871 1.0000000 1.0000000
313 0.20403108 0.8154370 0.8202554 0.9366565 0.9366565
314 0.27929584 0.7563161 0.7667798 1.0000000 1.0000000
315 0.22048841 0.8021269 0.8058099 1.0000000 1.0000000
316 0.73065997 0.4815910 0.4916511 0.8222527 0.8222527
317 0.38787128 0.6784997 0.6836154 0.8005123 0.8005123
318 0.16469390 0.8481533 0.8504655 0.9408122 0.9408122
319 0.09939144 0.9053882 0.9076298 0.9966756 0.9966756
320 0.10658262 0.8989008 0.8996958 0.9429478 0.9429478
321 0.19742076 0.8208452 0.8239541 0.9428132 0.9428132
322 0.30911985 0.7340928 0.7386808 0.9934143 0.9934143
323 0.36046341 0.6973531 0.7008654 0.9796738 0.9796738
324 0.36264124 0.6958360 0.7002410 0.9614120 0.9614120
325 0.90853075 0.4031161 0.4057264 0.9360896 0.9360896
326 0.07179505 0.9307216 0.9322668 1.0000000 1.0000000
327 0.27135686 0.7623444 0.7698662 0.9353445 0.9353445
328 0.14034729 0.8690564 0.8710265 0.9568460 0.9568460
329 0.21597733 0.8057536 0.8142318 1.0000000 1.0000000
330 0.22230257 0.8006731 0.8043621 1.0000000 1.0000000
331 2.15923253 0.1154137 0.1179423 0.6789496 0.6789496
332 0.33902242 0.7124665 0.7145351 0.7774564 0.7774564
333 0.13322630 0.8752670 0.8773604 0.9459591 0.9459591
334 0.23995782 0.7866610 0.7925893 0.9280669 0.9280669
335 0.75712881 0.4690111 0.4791273 0.8800919 0.8800919
336 0.15594150 0.8556092 0.8566464 0.9089549 0.9089549
337 0.13796519 0.8711290 0.8739542 0.9961164 0.9961164
338 0.32666571 0.7213248 0.7310380 0.9464624 0.9464624
339 0.21804969 0.8040855 0.8072014 0.8927024 0.8927024
340 0.60349789 0.5468953 0.5552121 0.7454058 0.7454058
341 0.26074881 0.7704744 0.7742174 1.0000000 1.0000000
342 0.26164992 0.7697805 0.7729239 0.8576847 0.8576847
343 0.15387524 0.8573790 0.8605191 1.0000000 1.0000000
344 0.06691412 0.9352755 0.9365885 1.0000000 1.0000000
head(efficiencies(meta_sfacross_lp))Toggle to see the output
id group u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal
1 1 medium 0.2697165 0.7635959 0.7673345 1.316036
2 2 large 0.3515642 0.7035867 0.7080897 1.430406
3 3 large 0.2774565 0.7577085 0.7623358 1.327899
4 4 medium 0.1710417 0.8427864 0.8461331 1.191355
5 5 large 0.2119629 0.8089947 0.8133556 1.242901
6 6 small 0.1987499 0.8197549 0.8275685 1.232467
uLB_g uUB_g m_g TE_group_mode teBCLB_g teBCUB_g u_meta
1 0.077581942 0.4657010 0.26858570 0.7644599 0.6276949 0.9253512 0.3944439
2 0.130356248 0.5739174 0.35118207 0.7038556 0.5633144 0.8777827 0.3779836
3 0.065447909 0.4980807 0.27501606 0.7595599 0.6076959 0.9366478 0.3049531
4 0.018022507 0.3583190 0.15885675 0.8531186 0.6988501 0.9821389 0.1710417
5 0.027125654 0.4268531 0.20231520 0.8168374 0.6525594 0.9732389 0.2379271
6 0.009050601 0.5251973 0.07998025 0.9231346 0.5914386 0.9909902 0.3295263
TE_meta_JLMS TE_meta_BC MTR_JLMS MTR_BC
1 0.6740548 0.6773549 0.8827375 0.8827375
2 0.6852418 0.6896274 0.9739266 0.9739266
3 0.7371580 0.7416598 0.9728780 0.9728780
4 0.8427864 0.8461331 1.0000000 1.0000000
5 0.7882601 0.7925093 0.9743700 0.9743700
6 0.7192644 0.7261201 0.8774139 0.8774139
#To subset only for small farms
#head(efficiencies(meta_sfacross_lp)[efficiencies(meta_sfacross_lp)$group == "small", ])1b. QP Metafrontier (groupType = "sfacross", metaMethod = "qp")
We can also estimate a quadratic programming envelope that minimises the sum of squared deviations from group frontier predictions subject to the envelope constraint. We now switch to metaMethod = "qp".
meta_sfacross_qp <- smfa(
formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
data = ricephil,
group = "group",
S = 1,
udist = "hnormal",
groupType = "sfacross",
metaMethod = "qp"
)
summary(meta_sfacross_qp)Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Quadratic Programming (QP) Metafrontier
Stochastic Production/Profit Frontier, e = v - u
Group approach : Stochastic Frontier Analysis
Group estimator : sfacross
Group optim solver : BFGS maximization
Groups ( 3 ): small, medium, large
Total observations : 344
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: small (N = 125) Log-likelihood: -50.98578
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 42
Log likelihood value: -50.98578
Log likelihood gradient norm: 9.40653e-06
Estimation based on: N = 125 and K = 6
Inf. Cr: AIC = 114.0 AIC/N = 0.912
BIC = 130.9 BIC/N = 1.048
HQIC = 120.9 HQIC/N = 0.967
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05318
Sigma(v) = 0.05318
Sigma-squared(u) = 0.23435
Sigma(u) = 0.23435
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.53622
Gamma = sigma(u)^2/sigma^2 = 0.81504
Lambda = sigma(u)/sigma(v) = 2.09921
Var[u]/{Var[u]+Var[v]} = 0.61558
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.38626
Average efficiency E[exp(-ui)] = 0.70643
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -54.80277
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 7.63398
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -3.57676
M3T: p.value = 0.00035
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.58745 0.51274 -3.0960 0.001962 **
log(AREA) 0.24014 0.11834 2.0292 0.042440 *
log(LABOR) 0.43464 0.12292 3.5361 0.000406 ***
log(NPK) 0.30516 0.05701 5.3523 8.682e-08 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.45093 0.29867 -4.858 1.186e-06 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.93406 0.35401 -8.288 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: medium (N = 104) Log-likelihood: -15.28164
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 41
Log likelihood value: -15.28164
Log likelihood gradient norm: 3.83566e-05
Estimation based on: N = 104 and K = 6
Inf. Cr: AIC = 42.6 AIC/N = 0.409
BIC = 58.4 BIC/N = 0.562
HQIC = 49.0 HQIC/N = 0.471
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.01058
Sigma(v) = 0.01058
Sigma-squared(u) = 0.22010
Sigma(u) = 0.22010
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.48030
Gamma = sigma(u)^2/sigma^2 = 0.95412
Lambda = sigma(u)/sigma(v) = 4.56034
Var[u]/{Var[u]+Var[v]} = 0.88314
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.37433
Average efficiency E[exp(-ui)] = 0.71330
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -21.11323
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 11.66318
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -2.91021
M3T: p.value = 0.00361
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -0.08182 0.50668 -0.1615 0.8717190
log(AREA) 0.47410 0.13984 3.3903 0.0006981 ***
log(LABOR) 0.17935 0.10201 1.7581 0.0787310 .
log(NPK) 0.20255 0.08130 2.4913 0.0127289 *
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.51367 0.23549 -6.4276 1.296e-10 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.54846 0.76429 -5.9512 2.661e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: large (N = 115) Log-likelihood: -8.02197
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 68
Log likelihood value: -8.02197
Log likelihood gradient norm: 4.01301e-05
Estimation based on: N = 115 and K = 6
Inf. Cr: AIC = 28.0 AIC/N = 0.244
BIC = 44.5 BIC/N = 0.387
HQIC = 34.7 HQIC/N = 0.302
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.01399
Sigma(v) = 0.01399
Sigma-squared(u) = 0.16751
Sigma(u) = 0.16751
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.42602
Gamma = sigma(u)^2/sigma^2 = 0.92293
Lambda = sigma(u)/sigma(v) = 3.46063
Var[u]/{Var[u]+Var[v]} = 0.81315
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.32656
Average efficiency E[exp(-ui)] = 0.74195
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -16.96836
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 17.89279
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -4.12175
M3T: p.value = 0.00004
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.31194 0.41859 -3.1342 0.0017234 **
log(AREA) 0.38278 0.14297 2.6772 0.0074236 **
log(LABOR) 0.42105 0.10992 3.8303 0.0001280 ***
log(NPK) 0.23143 0.06065 3.8160 0.0001356 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.78673 0.20176 -8.8555 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.26963 0.40584 -10.521 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (qp):
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.6117795 0.0291793 -20.966 < 2.2e-16 ***
log(AREA) 0.3937843 0.0073209 53.789 < 2.2e-16 ***
log(LABOR) 0.2791273 0.0077215 36.150 < 2.2e-16 ***
log(NPK) 0.2409454 0.0046846 51.434 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
small 125 125 0.71065 0.70090 0.64037 0.63156 0.89972
medium 104 104 0.71253 0.70965 0.66998 0.66727 0.94053
large 115 115 0.74772 0.74406 0.72290 0.71937 0.96676
MTR_JLMS
small 0.89972
medium 0.94053
large 0.96676
Overall:
TE_group_BC=0.7236 TE_group_JLMS=0.7182
TE_meta_BC=0.6777 TE_meta_JLMS=0.6727
MTR_BC=0.9357 MTR_JLMS=0.9357
------------------------------------------------------------
Total Log-likelihood: -74.28939
AIC: 192.5788 BIC: 277.0729 HQIC: 226.2318
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12 As expected, the two approaches produce almost identical outputs.
Note: The estimation of the group frontiers use the same method,
sfacross, and the only difference is in how we compute the metafrontier. QP estimates a deterministic frontier, no stochastic variance parameters are returned.
1c. Two-stage SFA Metafrontier — Huang et al. (2014) (sfaApproach = "huang")
In this approach, the group-specific fitted frontier values are pooled together and serve as the dependent variable in a second-stage pooled SFA. The technology gap U and noise V are estimated stochastically.
meta_sfacross_huang <- smfa(
formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
data = ricephil,
group = "group",
S = 1,
udist = "hnormal",
groupType = "sfacross",
metaMethod = "sfa",
sfaApproach = "huang"
)
Warning: The residuals of the OLS are right-skewed. This may indicate the absence of inefficiency or
model misspecification or sample 'bad luck'
summary(meta_sfacross_huang)Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [Huang et al. (2014), two-stage]
Stochastic Production/Profit Frontier, e = v - u
SFA approach : huang
Group approach : Stochastic Frontier Analysis
Group estimator : sfacross
Group optim solver : BFGS maximization
Groups ( 3 ): small, medium, large
Total observations : 344
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: small (N = 125) Log-likelihood: -50.98578
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 42
Log likelihood value: -50.98578
Log likelihood gradient norm: 9.40653e-06
Estimation based on: N = 125 and K = 6
Inf. Cr: AIC = 114.0 AIC/N = 0.912
BIC = 130.9 BIC/N = 1.048
HQIC = 120.9 HQIC/N = 0.967
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05318
Sigma(v) = 0.05318
Sigma-squared(u) = 0.23435
Sigma(u) = 0.23435
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.53622
Gamma = sigma(u)^2/sigma^2 = 0.81504
Lambda = sigma(u)/sigma(v) = 2.09921
Var[u]/{Var[u]+Var[v]} = 0.61558
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.38626
Average efficiency E[exp(-ui)] = 0.70643
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -54.80277
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 7.63398
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -3.57676
M3T: p.value = 0.00035
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.58745 0.51274 -3.0960 0.001962 **
log(AREA) 0.24014 0.11834 2.0292 0.042440 *
log(LABOR) 0.43464 0.12292 3.5361 0.000406 ***
log(NPK) 0.30516 0.05701 5.3523 8.682e-08 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.45093 0.29867 -4.858 1.186e-06 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.93406 0.35401 -8.288 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: medium (N = 104) Log-likelihood: -15.28164
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 41
Log likelihood value: -15.28164
Log likelihood gradient norm: 3.83566e-05
Estimation based on: N = 104 and K = 6
Inf. Cr: AIC = 42.6 AIC/N = 0.409
BIC = 58.4 BIC/N = 0.562
HQIC = 49.0 HQIC/N = 0.471
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.01058
Sigma(v) = 0.01058
Sigma-squared(u) = 0.22010
Sigma(u) = 0.22010
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.48030
Gamma = sigma(u)^2/sigma^2 = 0.95412
Lambda = sigma(u)/sigma(v) = 4.56034
Var[u]/{Var[u]+Var[v]} = 0.88314
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.37433
Average efficiency E[exp(-ui)] = 0.71330
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -21.11323
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 11.66318
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -2.91021
M3T: p.value = 0.00361
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -0.08182 0.50668 -0.1615 0.8717190
log(AREA) 0.47410 0.13984 3.3903 0.0006981 ***
log(LABOR) 0.17935 0.10201 1.7581 0.0787310 .
log(NPK) 0.20255 0.08130 2.4913 0.0127289 *
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.51367 0.23549 -6.4276 1.296e-10 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.54846 0.76429 -5.9512 2.661e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: large (N = 115) Log-likelihood: -8.02197
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 68
Log likelihood value: -8.02197
Log likelihood gradient norm: 4.01301e-05
Estimation based on: N = 115 and K = 6
Inf. Cr: AIC = 28.0 AIC/N = 0.244
BIC = 44.5 BIC/N = 0.387
HQIC = 34.7 HQIC/N = 0.302
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.01399
Sigma(v) = 0.01399
Sigma-squared(u) = 0.16751
Sigma(u) = 0.16751
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.42602
Gamma = sigma(u)^2/sigma^2 = 0.92293
Lambda = sigma(u)/sigma(v) = 3.46063
Var[u]/{Var[u]+Var[v]} = 0.81315
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.32656
Average efficiency E[exp(-ui)] = 0.74195
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -16.96836
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 17.89279
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -4.12175
M3T: p.value = 0.00004
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.31194 0.41859 -3.1342 0.0017234 **
log(AREA) 0.38278 0.14297 2.6772 0.0074236 **
log(LABOR) 0.42105 0.10992 3.8303 0.0001280 ***
log(NPK) 0.23143 0.06065 3.8160 0.0001356 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.78673 0.20176 -8.8555 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.26963 0.40584 -10.521 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.0031443 0.0568874 -17.634 < 2.2e-16 ***
log(AREA) 0.3670206 0.0091533 40.097 < 2.2e-16 ***
log(LABOR) 0.3297853 0.0096542 34.160 < 2.2e-16 ***
log(NPK) 0.2648079 0.0058572 45.211 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: group_fitted_values
Log likelihood solver: BFGS maximization
Log likelihood iter: 582
Log likelihood value: 553.35240
Log likelihood gradient norm: 5.34979e-04
Estimation based on: N = 344 and K = 6
Inf. Cr: AIC = -1094.7 AIC/N = -3.182
BIC = -1071.7 BIC/N = -3.115
HQIC = -1085.5 HQIC/N = -3.156
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00235
Sigma(v) = 0.00235
Sigma-squared(u) = 0.00000
Sigma(u) = 0.00000
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.04844
Gamma = sigma(u)^2/sigma^2 = 0.00017
Lambda = sigma(u)/sigma(v) = 0.01294
Var[u]/{Var[u]+Var[v]} = 0.00006
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.00050
Average efficiency E[exp(-ui)] = 0.99950
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = 553.35242
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = -0.00003
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = 4.16139
M3T: p.value = 0.00003
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.00314 0.05689 -17.634 < 2.2e-16 ***
.X2 0.36702 0.00915 40.097 < 2.2e-16 ***
.X3 0.32979 0.00965 34.160 < 2.2e-16 ***
.X4 0.26481 0.00586 45.211 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -14.749 174.352 -0.0846 0.9326
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -6.05510 0.07698 -78.658 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
small 125 125 0.71065 0.70090 0.71030 0.70055 0.99950
medium 104 104 0.71253 0.70965 0.71217 0.70930 0.99950
large 115 115 0.74772 0.74406 0.74734 0.74369 0.99950
MTR_JLMS
small 0.99950
medium 0.99950
large 0.99950
Overall:
TE_group_BC=0.7236 TE_group_JLMS=0.7182
TE_meta_BC=0.7233 TE_meta_JLMS=0.7178
MTR_BC=0.9995 MTR_JLMS=0.9995
------------------------------------------------------------
Total Log-likelihood: 479.063
AIC: -910.126 BIC: -817.9506 HQIC: -873.4137
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12 1d. Two-stage SFA Metafrontier — O’Donnell et al. (2008) (sfaApproach = "ordonnell")
In this approach, the LP deterministic envelope is used as the response variable in the second stage and the SFA quantifies the stochastic variation around this envelope.
meta_sfacross_odonnell <- smfa(
formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK),
data = ricephil,
group = "group",
S = 1,
udist = "hnormal",
groupType = "sfacross",
metaMethod = "sfa",
sfaApproach = "ordonnell"
)
Warning: The residuals of the OLS are right-skewed. This may indicate the absence of inefficiency or
model misspecification or sample 'bad luck'
summary(meta_sfacross_odonnell)
Warning: 344 MTR value(s) > 1 detected in O'Donnell SFA approach. This
typically occurs when the second-stage SFA estimates near-zero inefficiency
(sigma_u -> 0), causing TE_meta ~= 1 and MTR = TE_meta/TE_group > 1. Consider
using metaMethod='lp' or sfaApproach='huang' instead.Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [O'Donnell et al. (2008), envelope]
Stochastic Production/Profit Frontier, e = v - u
SFA approach : ordonnell
Group approach : Stochastic Frontier Analysis
Group estimator : sfacross
Group optim solver : BFGS maximization
Groups ( 3 ): small, medium, large
Total observations : 344
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: small (N = 125) Log-likelihood: -50.98578
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 42
Log likelihood value: -50.98578
Log likelihood gradient norm: 9.40653e-06
Estimation based on: N = 125 and K = 6
Inf. Cr: AIC = 114.0 AIC/N = 0.912
BIC = 130.9 BIC/N = 1.048
HQIC = 120.9 HQIC/N = 0.967
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05318
Sigma(v) = 0.05318
Sigma-squared(u) = 0.23435
Sigma(u) = 0.23435
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.53622
Gamma = sigma(u)^2/sigma^2 = 0.81504
Lambda = sigma(u)/sigma(v) = 2.09921
Var[u]/{Var[u]+Var[v]} = 0.61558
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.38626
Average efficiency E[exp(-ui)] = 0.70643
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -54.80277
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 7.63398
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -3.57676
M3T: p.value = 0.00035
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.58745 0.51274 -3.0960 0.001962 **
log(AREA) 0.24014 0.11834 2.0292 0.042440 *
log(LABOR) 0.43464 0.12292 3.5361 0.000406 ***
log(NPK) 0.30516 0.05701 5.3523 8.682e-08 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.45093 0.29867 -4.858 1.186e-06 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.93406 0.35401 -8.288 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: medium (N = 104) Log-likelihood: -15.28164
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 41
Log likelihood value: -15.28164
Log likelihood gradient norm: 3.83566e-05
Estimation based on: N = 104 and K = 6
Inf. Cr: AIC = 42.6 AIC/N = 0.409
BIC = 58.4 BIC/N = 0.562
HQIC = 49.0 HQIC/N = 0.471
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.01058
Sigma(v) = 0.01058
Sigma-squared(u) = 0.22010
Sigma(u) = 0.22010
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.48030
Gamma = sigma(u)^2/sigma^2 = 0.95412
Lambda = sigma(u)/sigma(v) = 4.56034
Var[u]/{Var[u]+Var[v]} = 0.88314
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.37433
Average efficiency E[exp(-ui)] = 0.71330
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -21.11323
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 11.66318
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -2.91021
M3T: p.value = 0.00361
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -0.08182 0.50668 -0.1615 0.8717190
log(AREA) 0.47410 0.13984 3.3903 0.0006981 ***
log(LABOR) 0.17935 0.10201 1.7581 0.0787310 .
log(NPK) 0.20255 0.08130 2.4913 0.0127289 *
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.51367 0.23549 -6.4276 1.296e-10 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.54846 0.76429 -5.9512 2.661e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: large (N = 115) Log-likelihood: -8.02197
------------------------------------------------------------
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: log(PROD)
Log likelihood solver: BFGS maximization
Log likelihood iter: 68
Log likelihood value: -8.02197
Log likelihood gradient norm: 4.01301e-05
Estimation based on: N = 115 and K = 6
Inf. Cr: AIC = 28.0 AIC/N = 0.244
BIC = 44.5 BIC/N = 0.387
HQIC = 34.7 HQIC/N = 0.302
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.01399
Sigma(v) = 0.01399
Sigma-squared(u) = 0.16751
Sigma(u) = 0.16751
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.42602
Gamma = sigma(u)^2/sigma^2 = 0.92293
Lambda = sigma(u)/sigma(v) = 3.46063
Var[u]/{Var[u]+Var[v]} = 0.81315
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.32656
Average efficiency E[exp(-ui)] = 0.74195
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = -16.96836
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 17.89279
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -4.12175
M3T: p.value = 0.00004
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.31194 0.41859 -3.1342 0.0017234 **
log(AREA) 0.38278 0.14297 2.6772 0.0074236 **
log(LABOR) 0.42105 0.10992 3.8303 0.0001280 ***
log(NPK) 0.23143 0.06065 3.8160 0.0001356 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.78673 0.20176 -8.8555 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.26963 0.40584 -10.521 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.6114342 0.0414990 -14.734 < 2.2e-16 ***
log(AREA) 0.3937848 0.0072782 54.105 < 2.2e-16 ***
log(LABOR) 0.2791270 0.0076764 36.361 < 2.2e-16 ***
log(NPK) 0.2409454 0.0046573 51.735 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: lp_envelope
Log likelihood solver: BFGS maximization
Log likelihood iter: 436
Log likelihood value: 632.20951
Log likelihood gradient norm: 5.34711e-02
Estimation based on: N = 344 and K = 6
Inf. Cr: AIC = -1252.4 AIC/N = -3.641
BIC = -1229.4 BIC/N = -3.574
HQIC = -1243.2 HQIC/N = -3.614
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00148
Sigma(v) = 0.00148
Sigma-squared(u) = 0.00000
Sigma(u) = 0.00000
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.03851
Gamma = sigma(u)^2/sigma^2 = 0.00013
Lambda = sigma(u)/sigma(v) = 0.01121
Var[u]/{Var[u]+Var[v]} = 0.00005
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.00034
Average efficiency E[exp(-ui)] = 0.99966
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = 632.20952
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = -0.00003
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = 6.24028
M3T: p.value = 0.00000
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -0.61143 0.04150 -14.734 < 2.2e-16 ***
.X2 0.39378 0.00728 54.105 < 2.2e-16 ***
.X3 0.27913 0.00768 36.361 < 2.2e-16 ***
.X4 0.24095 0.00466 51.735 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -15.496 172.306 -0.0899 0.9283
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -6.51356 0.07665 -84.978 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
small 125 125 0.71065 0.70090 0.99966 0.99966 1.49276
medium 104 104 0.71253 0.70965 0.99966 0.99966 1.50575
large 115 115 0.74772 0.74406 0.99966 0.99966 1.41180
MTR_JLMS
small 1.51673
medium 1.51248
large 1.41943
Overall:
TE_group_BC=0.7236 TE_group_JLMS=0.7182
TE_meta_BC=0.9997 TE_meta_JLMS=0.9997
MTR_BC=1.4701 MTR_JLMS=1.4829
------------------------------------------------------------
Total Log-likelihood: 557.9201
AIC: -1067.84 BIC: -975.6648 HQIC: -1031.128
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12 O’Donnell et al. (2008) approach involves taking the deterministic envelope (the maximum) of all group frontier values at each data point (yMeta = apply(groupFrontierMat, 1, max)). Because this new yMeta surface is the maximum of several hyperplanes, its shape is purely convex and completely lacks statistical noise. When the second-stage sfacross tries to fit a single straight line (hyperplane) through this convex envelope, the residuals predominantly curve upward away from the line. In production frontiers, this results in right-skewed OLS residuals. SFA models naturally interpret right-skewed residuals as having near-zero inefficiency (sigma_u -> 0). Because the meta-inefficiency is estimated as near zero (u_meta -> 0), the meta-efficiency approaches 1 (TE_meta ~= 1). Because TE_meta = TE_group * MTR, we get MTR = TE_meta / TE_group. Since TE_meta is ~1 and TE_group < 1, this mathematically forces MTR > 1. Why does this happen? Because fitting a stochastic frontier via Maximum Likelihood onto a purely mathematical LP envelope does not strictly enforce the bounding constraint MTR <= 1 at every individual data point. The SFA line will inevitably cut through the LP envelope rather than sitting strictly above it for all points. This is a well-known theoretical and computational limitation of the producing MTR value(s) > 1, which is partly why Huang et al. (2014) proposed their alternative method.
How to solve it? This is exactly why the warning message was added to the code. If you require MTR <= 1 bounds: (1) Use metaMethod = “lp” or “qp” (which explicitly enforce the mathematical envelope constraints). (2) Use sfaApproach = “huang”, which avoids this “wrong skewness” problem by using the actual observations’ own-group fitted values (yhat_group) as the SFA dependent variable, and directly estimating the technology gap U_i, inherently bounding MTR = exp(-U_i) <= 1.
Section 2: Latent Class SFA Group Frontier (groupType = "sfalcmcross")
When technology groups are unobserved, a pooled latent class model (sfalcmcross) is fitted on all data. The resulting class assignments (by maximum posterior probability) serve as the technology groups for the metafrontier.
Data Preparation
data("utility", package = "sfaR")
# No group variable needed for pooled LCM (groupType = "sfalcmcross" with no group argument)2a. LCM + LP Metafrontier
meta_lcm_lp <- smfa(
formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
data = utility,
S = -1,
groupType = "sfalcmcross",
lcmClasses = 2,
metaMethod = "lp"
)Toggle to see the output
Initialization: SFA + halfnormal - normal distributions...
LCM 2 Classes Estimation...Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Linear Programming (LP) Metafrontier
Stochastic Cost Frontier, e = v + u
Group approach : Latent Class Stochastic Frontier Analysis
Group estimator : sfalcmcross
Group optim solver : BFGS maximization
(Pooled LCM - latent classes used as groups)
Groups ( 2 ): Class_1, Class_2
Total observations : 791
Distribution : hnormal
============================================================
------------------------------------------------------------
Pooled LCM (2 classes) on all data (N = 791) Log-likelihood: 61.35325
------------------------------------------------------------
-- Latent Class 1 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.4472e+00 3.9123e-05 -36992 < 2.2e-16 ***
log(y) 8.4541e-01 2.3364e-06 361846 < 2.2e-16 ***
log(wl/wf) 3.5408e-01 4.4754e-06 79118 < 2.2e-16 ***
log(wk/wf) 4.2883e-01 1.3682e-05 31343 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029863 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -3.8759e+01 3.2680e-13 -1.186e+14 < 2.2e-16 ***
Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=107911523.6712
-- Latent Class 2 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -2.0490e+00 2.5608e-05 -80011.07 < 2.2e-16 ***
log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
log(wl/wf) -2.5916e-02 6.3375e-05 -408.93 < 2.2e-16 ***
log(wk/wf) 8.8450e-01 6.9315e-05 12760.74 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653956 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219998 < 2.2e-16 ***
Sigma_u=0.2218 Sigma_v=0.0835 Sigma=0.2370 Gamma=0.8759 Lambda=2.6573
-- Class Membership (logit) --
Coefficient Std. Error z value Pr(>|z|)
Cl1_(Intercept) -6.0163e-01 5.0453e-07 -1192458 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Log likelihood status: successful convergence
------------------------------------------------------------
Metafrontier Coefficients (lp):
(LP: deterministic envelope - no estimated parameters)
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
Class_1 202 202 0.68291 0.68291 0.68291 0.68291 1.00000
Class_2 589 589 0.85142 0.84951 0.85142 0.84951 1.00000
MTR_JLMS
Class_1 1.00000
Class_2 1.00000
Overall:
TE_group_BC=0.7672 TE_group_JLMS=0.7662
TE_meta_BC=0.7672 TE_meta_JLMS=0.7662
MTR_BC=1.0000 MTR_JLMS=1.0000
------------------------------------------------------------
Posterior Class Membership (pooled LCM):
------------------------------------------------------------
% assigned Mean post. prob.
Class 1 25.5 0.354
Class 2 74.5 0.646
------------------------------------------------------------
Total Log-likelihood: 61.35325
AIC: -96.70649 BIC: -35.95362 HQIC: -73.35552
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12 Retrieve efficiencies including per-class posterior probabilities
# Retrieve efficiencies including per-class posterior probabilities
head(efficiencies(meta_lcm_lp))Toggle to see the output
id Group_c u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal
1 1 2 0.15842759 0.8534848 0.8557686 1.174838
2 2 2 0.11418774 0.8920905 0.8939942 1.123406
3 3 2 0.08540291 0.9181423 0.9196136 1.090950
4 4 2 0.08020641 0.9229258 0.9243036 1.085175
5 5 2 0.05774132 0.9438941 0.9448245 1.060519
6 6 2 0.08181796 0.9214397 0.9228469 1.086964
PosteriorProb_c PosteriorProb_c1 PriorProb_c1 u_c1 teBC_c1
1 0.7249992 0.2750008 0.3539715 0.19428008 0.8234272
2 0.7334016 0.2665984 0.3539715 0.16086081 0.8514106
3 0.7039780 0.2960220 0.3539715 0.10301513 0.9021133
4 0.6909119 0.3090881 0.3539715 0.07745685 0.9254670
5 0.5808752 0.4191248 0.3539715 0.05410185 0.9473356
6 0.6927818 0.3072182 0.3539715 0.05781993 0.9438199
teBC_reciprocal_c1 PosteriorProb_c2 PriorProb_c2 u_c2 teBC_c2
1 1.214436 0.7249992 0.6460285 0.15842759 0.8557686
2 1.174521 0.7334016 0.6460285 0.11418774 0.8939942
3 1.108508 0.7039780 0.6460285 0.08540291 0.9196136
4 1.080536 0.6909119 0.6460285 0.08020641 0.9243036
5 1.055592 0.5808752 0.6460285 0.05774132 0.9448245
6 1.059524 0.6927818 0.6460285 0.08181796 0.9228469
teBC_reciprocal_c2 ineff_c1 ineff_c2 effBC_c1 effBC_c2 ReffBC_c1 ReffBC_c2
1 1.174838 NA 0.15842759 NA 0.8557686 NA 1.174838
2 1.123406 NA 0.11418774 NA 0.8939942 NA 1.123406
3 1.090950 NA 0.08540291 NA 0.9196136 NA 1.090950
4 1.085175 NA 0.08020641 NA 0.9243036 NA 1.085175
5 1.060519 NA 0.05774132 NA 0.9448245 NA 1.060519
6 1.086964 NA 0.08181796 NA 0.9228469 NA 1.086964
u_meta TE_meta_JLMS TE_meta_BC MTR_JLMS MTR_BC
1 0.15842759 0.8534848 0.8557686 1 1
2 0.11418774 0.8920905 0.8939942 1 1
3 0.08540291 0.9181423 0.9196136 1 1
4 0.08020641 0.9229258 0.9243036 1 1
5 0.05774132 0.9438941 0.9448245 1 1
6 0.08181796 0.9214397 0.9228469 1 1
# Columns include: Group_c, u_g, TE_group_JLMS, TE_group_BC, TE_group_BC_reciprocal,
# PosteriorProb_c, PosteriorProb_c1, PosteriorProb_c2,
# PriorProb_c1, PriorProb_c2, u_c1, u_c2,
# teBC_c1, teBC_c2, u_meta, TE_meta_JLMS, TE_meta_BC, MTR_JLMS, MTR_BC2b. LCM + QP Metafrontier
meta_lcm_qp <- smfa(
formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
data = utility,
S = -1,
groupType = "sfalcmcross",
lcmClasses = 2,
metaMethod = "qp"
)Toggle to see the output
Initialization: SFA + halfnormal - normal distributions...
LCM 2 Classes Estimation...Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Quadratic Programming (QP) Metafrontier
Stochastic Cost Frontier, e = v + u
Group approach : Latent Class Stochastic Frontier Analysis
Group estimator : sfalcmcross
Group optim solver : BFGS maximization
(Pooled LCM - latent classes used as groups)
Groups ( 2 ): Class_1, Class_2
Total observations : 791
Distribution : hnormal
============================================================
------------------------------------------------------------
Pooled LCM (2 classes) on all data (N = 791) Log-likelihood: 61.35325
------------------------------------------------------------
-- Latent Class 1 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.4472e+00 3.9123e-05 -36992 < 2.2e-16 ***
log(y) 8.4541e-01 2.3364e-06 361846 < 2.2e-16 ***
log(wl/wf) 3.5408e-01 4.4754e-06 79118 < 2.2e-16 ***
log(wk/wf) 4.2883e-01 1.3682e-05 31343 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029863 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -3.8759e+01 3.2680e-13 -1.186e+14 < 2.2e-16 ***
Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=107911523.6712
-- Latent Class 2 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -2.0490e+00 2.5608e-05 -80011.07 < 2.2e-16 ***
log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
log(wl/wf) -2.5916e-02 6.3375e-05 -408.93 < 2.2e-16 ***
log(wk/wf) 8.8450e-01 6.9315e-05 12760.74 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653956 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219998 < 2.2e-16 ***
Sigma_u=0.2218 Sigma_v=0.0835 Sigma=0.2370 Gamma=0.8759 Lambda=2.6573
-- Class Membership (logit) --
Coefficient Std. Error z value Pr(>|z|)
Cl1_(Intercept) -6.0163e-01 5.0453e-07 -1192458 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Log likelihood status: successful convergence
------------------------------------------------------------
Metafrontier Coefficients (qp):
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.3923607 0.0310122 -44.897 < 2.2e-16 ***
log(y) 0.8649986 0.0012419 696.519 < 2.2e-16 ***
log(wl/wf) 0.2909728 0.0047965 60.664 < 2.2e-16 ***
log(wk/wf) 0.5028978 0.0069500 72.359 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
Class_1 202 202 0.68291 0.68291 0.67786 0.67786 0.99326
Class_2 589 589 0.85142 0.84951 0.84548 0.84359 0.99285
MTR_JLMS
Class_1 0.99326
Class_2 0.99285
Overall:
TE_group_BC=0.7672 TE_group_JLMS=0.7662
TE_meta_BC=0.7617 TE_meta_JLMS=0.7607
MTR_BC=0.9931 MTR_JLMS=0.9931
------------------------------------------------------------
Posterior Class Membership (pooled LCM):
------------------------------------------------------------
% assigned Mean post. prob.
Class 1 25.5 0.354
Class 2 74.5 0.646
------------------------------------------------------------
Total Log-likelihood: 61.35325
AIC: -88.70649 BIC: -9.26042 HQIC: -58.17061
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12 2c. LCM + Two-stage SFA Metafrontier — Huang et al. (2014)
meta_lcm_huang <- smfa(
formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
data = utility,
S = -1,
groupType = "sfalcmcross",
lcmClasses = 2,
metaMethod = "sfa",
sfaApproach = "huang"
)Toggle to see the output
Initialization: SFA + halfnormal - normal distributions...
LCM 2 Classes Estimation...Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [Huang et al. (2014), two-stage]
Stochastic Cost Frontier, e = v + u
SFA approach : huang
Group approach : Latent Class Stochastic Frontier Analysis
Group estimator : sfalcmcross
Group optim solver : BFGS maximization
(Pooled LCM - latent classes used as groups)
Groups ( 2 ): Class_1, Class_2
Total observations : 791
Distribution : hnormal
============================================================
------------------------------------------------------------
Pooled LCM (2 classes) on all data (N = 791) Log-likelihood: 61.35325
------------------------------------------------------------
-- Latent Class 1 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.4472e+00 3.9123e-05 -36992 < 2.2e-16 ***
log(y) 8.4541e-01 2.3364e-06 361846 < 2.2e-16 ***
log(wl/wf) 3.5408e-01 4.4754e-06 79118 < 2.2e-16 ***
log(wk/wf) 4.2883e-01 1.3682e-05 31343 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029863 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -3.8759e+01 3.2680e-13 -1.186e+14 < 2.2e-16 ***
Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=107911523.6712
-- Latent Class 2 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -2.0490e+00 2.5608e-05 -80011.07 < 2.2e-16 ***
log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
log(wl/wf) -2.5916e-02 6.3375e-05 -408.93 < 2.2e-16 ***
log(wk/wf) 8.8450e-01 6.9315e-05 12760.74 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653956 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219998 < 2.2e-16 ***
Sigma_u=0.2218 Sigma_v=0.0835 Sigma=0.2370 Gamma=0.8759 Lambda=2.6573
-- Class Membership (logit) --
Coefficient Std. Error z value Pr(>|z|)
Cl1_(Intercept) -6.0163e-01 5.0453e-07 -1192458 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Log likelihood status: successful convergence
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) -2.2495079 0.0686629 -32.7616 < 2.2e-16 ***
log(y) 0.9909918 0.0024687 401.4291 < 2.2e-16 ***
log(wl/wf) 0.0399586 0.0106166 3.7638 0.0001674 ***
log(wk/wf) 0.7890986 0.0143801 54.8745 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: group_fitted_values
Log likelihood solver: BFGS maximization
Log likelihood iter: 58
Log likelihood value: 759.62892
Log likelihood gradient norm: 2.06059e-03
Estimation based on: N = 791 and K = 6
Inf. Cr: AIC = -1507.3 AIC/N = -1.906
BIC = -1479.2 BIC/N = -1.870
HQIC = -1496.5 HQIC/N = -1.892
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00068
Sigma(v) = 0.00068
Sigma-squared(u) = 0.02713
Sigma(u) = 0.02713
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.16676
Gamma = sigma(u)^2/sigma^2 = 0.97554
Lambda = sigma(u)/sigma(v) = 6.31586
Var[u]/{Var[u]+Var[v]} = 0.93546
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.13141
Average efficiency E[exp(-ui)] = 0.88105
--------------------------------------------------------------------------------
Stochastic Cost Frontier, e = v + u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = 588.58962
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 342.07861
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = 10.23625
M3T: p.value = 0.00000
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -2.24951 0.06866 -32.7616 < 2.2e-16 ***
.X2 0.99099 0.00247 401.4291 < 2.2e-16 ***
.X3 0.03996 0.01062 3.7638 0.0001674 ***
.X4 0.78910 0.01438 54.8745 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -3.60721 0.05642 -63.932 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -7.29334 0.12966 -56.25 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
Class_1 202 202 0.68291 0.68291 0.51861 0.51845 0.76693
Class_2 589 589 0.85142 0.84951 0.80511 0.80308 0.94559
MTR_JLMS
Class_1 0.76669
Class_2 0.94532
Overall:
TE_group_BC=0.7672 TE_group_JLMS=0.7662
TE_meta_BC=0.6619 TE_meta_JLMS=0.6608
MTR_BC=0.8563 MTR_JLMS=0.8560
------------------------------------------------------------
Posterior Class Membership (pooled LCM):
------------------------------------------------------------
% assigned Mean post. prob.
Class 1 25.5 0.354
Class 2 74.5 0.646
------------------------------------------------------------
Total Log-likelihood: 820.9822
AIC: -1603.964 BIC: -1515.172 HQIC: -1569.836
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12 2d. LCM + O’Donnell et al. (2008) Stochastic Metafrontier
meta_lcm_odonnell <- smfa(
formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
data = utility,
S = -1,
groupType = "sfalcmcross",
lcmClasses = 2,
metaMethod = "sfa",
sfaApproach = "ordonnell"
)Toggle to see the output
Initialization: SFA + halfnormal - normal distributions...
LCM 2 Classes Estimation...Warning: hessian is singular for 'qr.solve' switching to 'ginv'
Warning: hessian is singular for 'qr.solve' switching to 'ginv'
summary(meta_lcm_odonnell)
Warning: 761 MTR value(s) > 1 detected in O'Donnell SFA approach. This
typically occurs when the second-stage SFA estimates near-zero inefficiency
(sigma_u -> 0), causing TE_meta ~= 1 and MTR = TE_meta/TE_group > 1. Consider
using metaMethod='lp' or sfaApproach='huang' instead.Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [O'Donnell et al. (2008), envelope]
Stochastic Cost Frontier, e = v + u
SFA approach : ordonnell
Group approach : Latent Class Stochastic Frontier Analysis
Group estimator : sfalcmcross
Group optim solver : BFGS maximization
(Pooled LCM - latent classes used as groups)
Groups ( 2 ): Class_1, Class_2
Total observations : 791
Distribution : hnormal
============================================================
------------------------------------------------------------
Pooled LCM (2 classes) on all data (N = 791) Log-likelihood: 61.35325
------------------------------------------------------------
-- Latent Class 1 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.4472e+00 3.9123e-05 -36992 < 2.2e-16 ***
log(y) 8.4541e-01 2.3364e-06 361846 < 2.2e-16 ***
log(wl/wf) 3.5408e-01 4.4754e-06 79118 < 2.2e-16 ***
log(wk/wf) 4.2883e-01 1.3682e-05 31343 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029863 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -3.8759e+01 3.2680e-13 -1.186e+14 < 2.2e-16 ***
Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=107911523.6712
-- Latent Class 2 --
Frontier:
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -2.0490e+00 2.5608e-05 -80011.07 < 2.2e-16 ***
log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
log(wl/wf) -2.5916e-02 6.3375e-05 -408.93 < 2.2e-16 ***
log(wk/wf) 8.8450e-01 6.9315e-05 12760.74 < 2.2e-16 ***
Var(u):
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653956 < 2.2e-16 ***
Var(v):
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219998 < 2.2e-16 ***
Sigma_u=0.2218 Sigma_v=0.0835 Sigma=0.2370 Gamma=0.8759 Lambda=2.6573
-- Class Membership (logit) --
Coefficient Std. Error z value Pr(>|z|)
Cl1_(Intercept) -6.0163e-01 5.0453e-07 -1192458 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Log likelihood status: successful convergence
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.4655e+00 1.4682e-05 -99822 < 2.2e-16 ***
log(y) 8.5052e-01 2.0413e-06 416655 < 2.2e-16 ***
log(wl/wf) 3.4196e-01 8.2433e-06 41483 < 2.2e-16 ***
log(wk/wf) 4.4325e-01 1.1455e-05 38693 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: lp_envelope
Log likelihood solver: BFGS maximization
Log likelihood iter: 1139
Log likelihood value: 1949.16740
Log likelihood gradient norm: 3.76958e+02
Estimation based on: N = 791 and K = 6
Inf. Cr: AIC = -3886.3 AIC/N = -4.913
BIC = -3858.3 BIC/N = -4.878
HQIC = -3875.6 HQIC/N = -4.900
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00000
Sigma(v) = 0.00000
Sigma-squared(u) = 0.00170
Sigma(u) = 0.00170
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.04117
Gamma = sigma(u)^2/sigma^2 = 1.00000
Lambda = sigma(u)/sigma(v) = 5005571.02335
Var[u]/{Var[u]+Var[v]} = 1.00000
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.03285
Average efficiency E[exp(-ui)] = 0.96798
--------------------------------------------------------------------------------
Stochastic Cost Frontier, e = v + u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = 1595.35974
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 707.61532
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = 30.29500
M3T: p.value = 0.00000
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) -1.46554 0.00001 -99822 < 2.2e-16 ***
.X2 0.85052 0.00000 416655 < 2.2e-16 ***
.X3 0.34196 0.00001 41483 < 2.2e-16 ***
.X4 0.44325 0.00001 38693 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -6.3799 0.0000 -1.8575e+13 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -37.232 0.000 -4.2838e+13 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
Class_1 202 202 0.68291 0.68291 0.98665 0.98665 1.53705
Class_2 589 589 0.85142 0.84951 0.98091 0.98091 1.16150
MTR_JLMS
Class_1 1.53705
Class_2 1.16424
Overall:
TE_group_BC=0.7672 TE_group_JLMS=0.7662
TE_meta_BC=0.9838 TE_meta_JLMS=0.9838
MTR_BC=1.3493 MTR_JLMS=1.3506
------------------------------------------------------------
Posterior Class Membership (pooled LCM):
------------------------------------------------------------
% assigned Mean post. prob.
Class 1 25.5 0.354
Class 2 74.5 0.646
------------------------------------------------------------
Total Log-likelihood: 2010.521
AIC: -3983.041 BIC: -3894.249 HQIC: -3948.913
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:12 Section 3: Sample Selection SFA Group Frontier (groupType = "sfaselectioncross")
When the observed sample is not random (e.g., only firms above a revenue threshold are surveyed), sample selection bias can distort frontier estimates. sfaselectioncross corrects for this using the two-step approach of Greene (2010). Only selected observations (d == 1) participate in the frontier and metafrontier; efficiency estimates for non-selected observations are NA. Here is a simulated example (adapted from sfaR):
Data Preparation (Simulated)
N <- 2000; set.seed(12345)
z1 <- rnorm(N); z2 <- rnorm(N)
v1 <- rnorm(N); v2 <- rnorm(N)
g <- rnorm(N)
e1 <- v1
e2 <- 0.7071 * (v1 + v2)
ds <- z1 + z2 + e1
d <- ifelse(ds > 0, 1, 0) # binary selection indicator: 1 = selected
group <- ifelse(g > 0, 1, 0) # two technology groups
u <- abs(rnorm(N))
x1 <- abs(rnorm(N)); x2 <- abs(rnorm(N))
y <- abs(x1 + x2 + e2 - u)
dat <- as.data.frame(cbind(y=y, x1=x1, x2=x2, z1=z1, z2=z2, d=d, group=group))
# About 50% of observations are selected:
table(dat$d)Toggle to see the output
0 1
1013 987 3a. sfaselectioncross + LP Metafrontier
meta_sel_lp <- smfa(
formula = log(y) ~ log(x1) + log(x2),
selectionF = d ~ z1 + z2,
data = dat,
group = "group",
S = 1L,
udist = "hnormal",
groupType = "sfaselectioncross",
modelType = "greene10",
lType = "kronrod",
Nsub = 100,
uBound = Inf,
method = "bfgs",
itermax = 2000,
metaMethod = "lp"
)Toggle to see the output
First step probit model...
Second step Frontier model...
First step probit model...
Second step Frontier model...
summary(meta_sel_lp)Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Linear Programming (LP) Metafrontier
Stochastic Production/Profit Frontier, e = v - u
Group approach : Sample Selection Stochastic Frontier Analysis
Group estimator : sfaselectioncross
Group optim solver : BFGS maximization
Groups ( 2 ): 0, 1
Total observations : 2000
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: 0 (N = 994) Log-likelihood: -799.94522
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 349
Log likelihood value: -799.94522
Log likelihood gradient norm: 3.41571e+01
Estimation based on: N = 489 of 994 obs. and K = 6
Inf. Cr: AIC = 1611.9 AIC/N = 3.296
BIC = 1637.0 BIC/N = 3.348
HQIC = 1621.8 HQIC/N = 3.317
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05814
Sigma(v) = 0.05814
Sigma-squared(u) = 2.29587
Sigma(u) = 2.29587
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.53428
Gamma = sigma(u)^2/sigma^2 = 0.97530
Lambda = sigma(u)/sigma(v) = 6.28402
Var[u]/{Var[u]+Var[v]} = 0.93485
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.20897
Average efficiency E[exp(-ui)] = 0.40883
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.27249 0.05191 24.5133 < 2.2e-16 ***
log(x1) 0.13811 0.02289 6.0339 1.601e-09 ***
log(x2) 0.18668 0.02252 8.2892 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.83111 0.06438 12.91 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.84491 0.33126 -8.5882 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.89550 0.28696 3.1207 0.001804 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:13
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: 1 (N = 1006) Log-likelihood: -851.19119
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 75
Log likelihood value: -851.19119
Log likelihood gradient norm: 7.36354e-06
Estimation based on: N = 498 of 1006 obs. and K = 6
Inf. Cr: AIC = 1714.4 AIC/N = 3.443
BIC = 1739.6 BIC/N = 3.493
HQIC = 1724.3 HQIC/N = 3.462
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.06106
Sigma(v) = 0.06106
Sigma-squared(u) = 2.36720
Sigma(u) = 2.36720
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.55829
Gamma = sigma(u)^2/sigma^2 = 0.97485
Lambda = sigma(u)/sigma(v) = 6.22643
Var[u]/{Var[u]+Var[v]} = 0.93372
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.22760
Average efficiency E[exp(-ui)] = 0.40470
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.33000 0.06335 20.9937 < 2.2e-16 ***
log(x1) 0.18342 0.02349 7.8098 5.727e-15 ***
log(x2) 0.09987 0.01918 5.2061 1.928e-07 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.86171 0.06328 13.618 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.79590 0.33567 -8.3292 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.40516 0.35322 1.147 0.2514
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:13
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (lp):
(LP: deterministic envelope - no estimated parameters)
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
0 994 489 0.38742 0.38227 0.35455 0.34983 0.91286
1 1006 498 0.41484 0.40585 0.41112 0.40220 0.99226
MTR_JLMS
0 0.91286
1 0.99226
Overall:
TE_group_BC=0.4011 TE_group_JLMS=0.3941
TE_meta_BC=0.3828 TE_meta_JLMS=0.3760
MTR_BC=0.9526 MTR_JLMS=0.9526
------------------------------------------------------------
Total Log-likelihood: -1651.136
AIC: 3326.273 BIC: 3393.484 HQIC: 3350.951
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:13
# Efficiencies: non-selected observations have NA
ef_sel_lp <- efficiencies(meta_sel_lp)
head(ef_sel_lp)Toggle to see the output
id group u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal u_meta
1 1 0 NA NA NA NA NA
2 2 0 3.0260520 0.04850676 0.04908457 20.872707 3.2369372
3 3 1 NA NA NA NA NA
4 4 0 NA NA NA NA NA
5 5 1 0.9405135 0.39042730 0.40073806 2.629065 0.9405135
6 6 0 NA NA NA NA NA
TE_meta_JLMS TE_meta_BC MTR_JLMS MTR_BC
1 NA NA NA NA
2 0.03928403 0.03975198 0.8098671 0.8098671
3 NA NA NA NA
4 NA NA NA NA
5 0.39042730 0.40073806 1.0000000 1.0000000
6 NA NA NA NA
# Selected observations in group 0:
#head(ef_sel_lp[ef_sel_lp$group == 0 & !is.na(ef_sel_lp$TE_group_BC), ])3b. sfaselectioncross + QP Metafrontier
meta_sel_qp <- smfa(
formula = log(y) ~ log(x1) + log(x2),
selectionF = d ~ z1 + z2,
data = dat,
group = "group",
S = 1L,
udist = "hnormal",
groupType = "sfaselectioncross",
modelType = "greene10",
lType = "kronrod",
Nsub = 100,
uBound = Inf,
method = "bfgs",
itermax = 2000,
metaMethod = "qp"
)Toggle to see the output
First step probit model...
Second step Frontier model...
First step probit model...
Second step Frontier model...
summary(meta_sel_qp)Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: Quadratic Programming (QP) Metafrontier
Stochastic Production/Profit Frontier, e = v - u
Group approach : Sample Selection Stochastic Frontier Analysis
Group estimator : sfaselectioncross
Group optim solver : BFGS maximization
Groups ( 2 ): 0, 1
Total observations : 2000
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: 0 (N = 994) Log-likelihood: -799.94522
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 349
Log likelihood value: -799.94522
Log likelihood gradient norm: 3.41571e+01
Estimation based on: N = 489 of 994 obs. and K = 6
Inf. Cr: AIC = 1611.9 AIC/N = 3.296
BIC = 1637.0 BIC/N = 3.348
HQIC = 1621.8 HQIC/N = 3.317
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05814
Sigma(v) = 0.05814
Sigma-squared(u) = 2.29587
Sigma(u) = 2.29587
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.53428
Gamma = sigma(u)^2/sigma^2 = 0.97530
Lambda = sigma(u)/sigma(v) = 6.28402
Var[u]/{Var[u]+Var[v]} = 0.93485
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.20897
Average efficiency E[exp(-ui)] = 0.40883
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.27249 0.05191 24.5133 < 2.2e-16 ***
log(x1) 0.13811 0.02289 6.0339 1.601e-09 ***
log(x2) 0.18668 0.02252 8.2892 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.83111 0.06438 12.91 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.84491 0.33126 -8.5882 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.89550 0.28696 3.1207 0.001804 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:14
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: 1 (N = 1006) Log-likelihood: -851.19119
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 75
Log likelihood value: -851.19119
Log likelihood gradient norm: 7.36354e-06
Estimation based on: N = 498 of 1006 obs. and K = 6
Inf. Cr: AIC = 1714.4 AIC/N = 3.443
BIC = 1739.6 BIC/N = 3.493
HQIC = 1724.3 HQIC/N = 3.462
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.06106
Sigma(v) = 0.06106
Sigma-squared(u) = 2.36720
Sigma(u) = 2.36720
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.55829
Gamma = sigma(u)^2/sigma^2 = 0.97485
Lambda = sigma(u)/sigma(v) = 6.22643
Var[u]/{Var[u]+Var[v]} = 0.93372
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.22760
Average efficiency E[exp(-ui)] = 0.40470
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.33000 0.06335 20.9937 < 2.2e-16 ***
log(x1) 0.18342 0.02349 7.8098 5.727e-15 ***
log(x2) 0.09987 0.01918 5.2061 1.928e-07 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.86171 0.06328 13.618 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.79590 0.33567 -8.3292 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.40516 0.35322 1.147 0.2514
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:14
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (qp):
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.33523296 0.00074356 1795.74 < 2.2e-16 ***
log(x1) 0.16970553 0.00054744 310.00 < 2.2e-16 ***
log(x2) 0.10714467 0.00050675 211.44 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
0 994 489 0.38742 0.38227 0.35331 0.34860 0.90889
1 1006 498 0.41484 0.40585 0.41052 0.40162 0.98932
MTR_JLMS
0 0.90889
1 0.98932
Overall:
TE_group_BC=0.4011 TE_group_JLMS=0.3941
TE_meta_BC=0.3819 TE_meta_JLMS=0.3751
MTR_BC=0.9491 MTR_JLMS=0.9491
------------------------------------------------------------
Total Log-likelihood: -1651.136
AIC: 3332.273 BIC: 3416.286 HQIC: 3363.121
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:14 3c. sfaselectioncross + Two-stage SFA Metafrontier — Huang et al. (2014)
meta_sel_huang <- smfa(
formula = log(y) ~ log(x1) + log(x2),
selectionF = d ~ z1 + z2,
data = dat,
group = "group",
S = 1L,
udist = "hnormal",
groupType = "sfaselectioncross",
modelType = "greene10",
lType = "kronrod",
Nsub = 100,
uBound = Inf,
simType = "halton",
Nsim = 300,
prime = 2L,
burn = 10,
antithetics = FALSE,
seed = 12345,
method = "bfgs",
itermax = 2000,
metaMethod = "sfa",
sfaApproach = "huang"
)Toggle to see the output
First step probit model...
Second step Frontier model...
First step probit model...
Second step Frontier model...
summary(meta_sel_huang)Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [Huang et al. (2014), two-stage]
Stochastic Production/Profit Frontier, e = v - u
SFA approach : huang
Group approach : Sample Selection Stochastic Frontier Analysis
Group estimator : sfaselectioncross
Group optim solver : BFGS maximization
Groups ( 2 ): 0, 1
Total observations : 2000
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: 0 (N = 994) Log-likelihood: -799.94522
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 349
Log likelihood value: -799.94522
Log likelihood gradient norm: 3.41571e+01
Estimation based on: N = 489 of 994 obs. and K = 6
Inf. Cr: AIC = 1611.9 AIC/N = 3.296
BIC = 1637.0 BIC/N = 3.348
HQIC = 1621.8 HQIC/N = 3.317
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05814
Sigma(v) = 0.05814
Sigma-squared(u) = 2.29587
Sigma(u) = 2.29587
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.53428
Gamma = sigma(u)^2/sigma^2 = 0.97530
Lambda = sigma(u)/sigma(v) = 6.28402
Var[u]/{Var[u]+Var[v]} = 0.93485
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.20897
Average efficiency E[exp(-ui)] = 0.40883
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.27249 0.05191 24.5133 < 2.2e-16 ***
log(x1) 0.13811 0.02289 6.0339 1.601e-09 ***
log(x2) 0.18668 0.02252 8.2892 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.83111 0.06438 12.91 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.84491 0.33126 -8.5882 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.89550 0.28696 3.1207 0.001804 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:15
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Group: 1 (N = 1006) Log-likelihood: -851.19119
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 75
Log likelihood value: -851.19119
Log likelihood gradient norm: 7.36354e-06
Estimation based on: N = 498 of 1006 obs. and K = 6
Inf. Cr: AIC = 1714.4 AIC/N = 3.443
BIC = 1739.6 BIC/N = 3.493
HQIC = 1724.3 HQIC/N = 3.462
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.06106
Sigma(v) = 0.06106
Sigma-squared(u) = 2.36720
Sigma(u) = 2.36720
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.55829
Gamma = sigma(u)^2/sigma^2 = 0.97485
Lambda = sigma(u)/sigma(v) = 6.22643
Var[u]/{Var[u]+Var[v]} = 0.93372
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.22760
Average efficiency E[exp(-ui)] = 0.40470
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.33000 0.06335 20.9937 < 2.2e-16 ***
log(x1) 0.18342 0.02349 7.8098 5.727e-15 ***
log(x2) 0.09987 0.01918 5.2061 1.928e-07 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.86171 0.06328 13.618 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.79590 0.33567 -8.3292 < 2.2e-16 ***
--------------------------------------------------------------------------------
Selection bias parameter
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
rho 0.40516 0.35322 1.147 0.2514
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:16
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.3636175 0.0018936 720.12 < 2.2e-16 ***
log(x1) 0.1610981 0.0014891 108.19 < 2.2e-16 ***
log(x2) 0.1094264 0.0010363 105.59 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Meta-frontier model details:
--------------------------------------------------------------------------------
Normal-Half Normal SF Model
Dependent Variable: group_fitted_values
Log likelihood solver: BFGS maximization
Log likelihood iter: 130
Log likelihood value: 1332.99378
Log likelihood gradient norm: 1.60687e-04
Estimation based on: N = 987 and K = 5
Inf. Cr: AIC = -2656.0 AIC/N = -2.691
BIC = -2631.5 BIC/N = -2.666
HQIC = -2646.7 HQIC/N = -2.682
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.00025
Sigma(v) = 0.00025
Sigma-squared(u) = 0.01270
Sigma(u) = 0.01270
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.11381
Gamma = sigma(u)^2/sigma^2 = 0.98050
Lambda = sigma(u)/sigma(v) = 7.09083
Var[u]/{Var[u]+Var[v]} = 0.94811
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 0.08992
Average efficiency E[exp(-ui)] = 0.91607
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = 1227.26017
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = 211.46722
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = -1.05865
M3T: p.value = 0.28976
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.36362 0.00189 720.12 < 2.2e-16 ***
.X2 0.16110 0.00149 108.19 < 2.2e-16 ***
.X3 0.10943 0.00104 105.59 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -4.36616 0.05277 -82.743 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameters in variance of v (two-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -8.28377 0.17628 -46.992 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
--------------------------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:16
Log likelihood status: successful convergence
--------------------------------------------------------------------------------
Log likelihood status: successful convergence
------------------------------------------------------------
Efficiency Statistics (group means):
------------------------------------------------------------
N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
0 994 489 0.38742 0.38227 0.34425 0.33963 0.88461
1 1006 498 0.41484 0.40585 0.39861 0.38993 0.95990
MTR_JLMS
0 0.88451
1 0.95980
Overall:
TE_group_BC=0.4011 TE_group_JLMS=0.3941
TE_meta_BC=0.3714 TE_meta_JLMS=0.3648
MTR_BC=0.9223 MTR_JLMS=0.9222
------------------------------------------------------------
Total Log-likelihood: -318.1426
AIC: 670.2853 BIC: 765.5006 HQIC: 705.2464
------------------------------------------------------------
Model was estimated on : Jul Wed 15, 2026 at 00:16 3d. sfaselectioncross + O’Donnell et al. (2008) Stochastic Metafrontier
meta_sel_odonnell <- smfa(
formula = log(y) ~ log(x1) + log(x2),
selectionF = d ~ z1 + z2,
data = dat,
group = "group",
S = 1L,
udist = "hnormal",
groupType = "sfaselectioncross",
modelType = "greene10",
lType = "kronrod",
Nsub = 100,
uBound = Inf,
method = "bfgs",
itermax = 2000,
metaMethod = "sfa",
sfaApproach = "ordonnell"
)Toggle to see the output
First step probit model...
Second step Frontier model...
First step probit model...
Second step Frontier model...Warning: The residuals of the OLS are right-skewed. This may indicate the absence of inefficiency or
model misspecification or sample 'bad luck'
summary(meta_sel_odonnell)
Warning: 987 MTR value(s) > 1 detected in O'Donnell SFA approach. This
typically occurs when the second-stage SFA estimates near-zero inefficiency
(sigma_u -> 0), causing TE_meta ~= 1 and MTR = TE_meta/TE_group > 1. Consider
using metaMethod='lp' or sfaApproach='huang' instead.Toggle to see the output
============================================================
Stochastic Metafrontier Analysis
Metafrontier method: SFA Metafrontier [O'Donnell et al. (2008), envelope]
Stochastic Production/Profit Frontier, e = v - u
SFA approach : ordonnell
Group approach : Sample Selection Stochastic Frontier Analysis
Group estimator : sfaselectioncross
Group optim solver : BFGS maximization
Groups ( 2 ): 0, 1
Total observations : 2000
Distribution : hnormal
============================================================
------------------------------------------------------------
Group: 0 (N = 994) Log-likelihood: -799.94522
------------------------------------------------------------
--------------------------------------------------------------------------------
Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 349
Log likelihood value: -799.94522
Log likelihood gradient norm: 3.41571e+01
Estimation based on: N = 489 of 994 obs. and K = 6
Inf. Cr: AIC = 1611.9 AIC/N = 3.296
BIC = 1637.0 BIC/N = 3.348
HQIC = 1621.8 HQIC/N = 3.317
--------------------------------------------------------------------------------
Variances: Sigma-squared(v) = 0.05814
Sigma(v) = 0.05814
Sigma-squared(u) = 2.29587
Sigma(u) = 2.29587
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.53428
Gamma = sigma(u)^2/sigma^2 = 0.97530
Lambda = sigma(u)/sigma(v) = 6.28402
Var[u]/{Var[u]+Var[v]} = 0.93485
--------------------------------------------------------------------------------
Average inefficiency E[ui] = 1.20897
Average efficiency E[exp(-ui)] = 0.40883
--------------------------------------------------------------------------------
Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
--------------------------------------------------------------------------------
Deterministic Component of SFA
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.27249 0.05191 24.5133 < 2.2e-16 ***
log(x1) 0.13811 0.02289 6.0339 1.601e-09 ***
log(x2) 0.18668 0.02252 8.2892 < 2.2e-16 ***
--------------------------------------------------------------------------------
Parameter in variance of u (one-sided error)
--------------------------------------------------------------------------------
Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.83111 0.06438 12.91 < 2.2e-16 ***
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Parameters in variance of v (two-sided error)
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Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.84491 0.33126 -8.5882 < 2.2e-16 ***
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Selection bias parameter
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Coefficient Std. Error z value Pr(>|z|)
rho 0.89550 0.28696 3.1207 0.001804 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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Model was estimated on : Jul Wed 15, 2026 at 00:17
Log likelihood status: successful convergence
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------------------------------------------------------------
Group: 1 (N = 1006) Log-likelihood: -851.19119
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Sample Selection Correction Stochastic Frontier Model
Dependent Variable: log(y)
Log likelihood solver: BFGS maximization
Log likelihood iter: 75
Log likelihood value: -851.19119
Log likelihood gradient norm: 7.36354e-06
Estimation based on: N = 498 of 1006 obs. and K = 6
Inf. Cr: AIC = 1714.4 AIC/N = 3.443
BIC = 1739.6 BIC/N = 3.493
HQIC = 1724.3 HQIC/N = 3.462
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Variances: Sigma-squared(v) = 0.06106
Sigma(v) = 0.06106
Sigma-squared(u) = 2.36720
Sigma(u) = 2.36720
Sigma = Sqrt[(s^2(u)+s^2(v))] = 1.55829
Gamma = sigma(u)^2/sigma^2 = 0.97485
Lambda = sigma(u)/sigma(v) = 6.22643
Var[u]/{Var[u]+Var[v]} = 0.93372
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Average inefficiency E[ui] = 1.22760
Average efficiency E[exp(-ui)] = 0.40470
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Stochastic Production/Profit Frontier, e = v - u
Estimator is 2 step Maximum Likelihood
Final maximum likelihood estimates
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Deterministic Component of SFA
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Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.33000 0.06335 20.9937 < 2.2e-16 ***
log(x1) 0.18342 0.02349 7.8098 5.727e-15 ***
log(x2) 0.09987 0.01918 5.2061 1.928e-07 ***
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Parameter in variance of u (one-sided error)
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Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) 0.86171 0.06328 13.618 < 2.2e-16 ***
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Parameters in variance of v (two-sided error)
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Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -2.79590 0.33567 -8.3292 < 2.2e-16 ***
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Selection bias parameter
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Coefficient Std. Error z value Pr(>|z|)
rho 0.40516 0.35322 1.147 0.2514
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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Model was estimated on : Jul Wed 15, 2026 at 00:17
Log likelihood status: successful convergence
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------------------------------------------------------------
Metafrontier Coefficients (sfa):
Meta-optim solver : BFGS maximization
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.33534799 0.00637899 209.34 < 2.2e-16 ***
log(x1) 0.16970553 0.00054661 310.47 < 2.2e-16 ***
log(x2) 0.10714469 0.00050598 211.76 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Meta-frontier model details:
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Normal-Half Normal SF Model
Dependent Variable: lp_envelope
Log likelihood solver: BFGS maximization
Log likelihood iter: 336
Log likelihood value: 2562.82127
Log likelihood gradient norm: 7.45486e-02
Estimation based on: N = 987 and K = 5
Inf. Cr: AIC = -5115.6 AIC/N = -5.183
BIC = -5091.2 BIC/N = -5.158
HQIC = -5106.3 HQIC/N = -5.174
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Variances: Sigma-squared(v) = 0.00033
Sigma(v) = 0.00033
Sigma-squared(u) = 0.00000
Sigma(u) = 0.00000
Sigma = Sqrt[(s^2(u)+s^2(v))] = 0.01803
Gamma = sigma(u)^2/sigma^2 = 0.00006
Lambda = sigma(u)/sigma(v) = 0.00799
Var[u]/{Var[u]+Var[v]} = 0.00002
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Average inefficiency E[ui] = 0.00012
Average efficiency E[exp(-ui)] = 0.99988
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Stochastic Production/Profit Frontier, e = v - u
-----[ Tests vs. No Inefficiency ]-----
Likelihood Ratio Test of Inefficiency
Deg. freedom for inefficiency model 1
Log Likelihood for OLS Log(H0) = 2562.82131
LR statistic:
Chisq = 2*[LogL(H0)-LogL(H1)] = -0.00007
Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
Coelli (1995) skewness test on OLS residuals
M3T: z = 25.86056
M3T: p.value = 0.00000
Final maximum likelihood estimates
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Deterministic Component of SFA
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Coefficient Std. Error z value Pr(>|z|)
(Intercept) 1.33535 0.00638 209.34 < 2.2e-16 ***
.X2 0.16971 0.00055 310.47 < 2.2e-16 ***
.X3 0.10714 0.00051 211.76 < 2.2e-16 ***
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Parameter in variance of u (one-sided error)
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Coefficient Std. Error z value Pr(>|z|)
Zu_(Intercept) -17.689 110.175 -0.1606 0.8724
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Parameters in variance of v (two-sided error)
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Coefficient Std. Error z value Pr(>|z|)
Zv_(Intercept) -8.03105 0.04509 -178.12 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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Model was estimated on : Jul Wed 15, 2026 at 00:17
Log likelihood status: successful convergence
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Log likelihood status: successful convergence
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Efficiency Statistics (group means):
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N_obs N_valid TE_group_BC TE_group_JLMS TE_meta_BC TE_meta_JLMS MTR_BC
0 994 489 0.38742 0.38227 0.99989 0.99988 8.54748
1 1006 498 0.41484 0.40585 0.99988 0.99988 8.59832
MTR_JLMS
0 8.65155
1 8.82540
Overall:
TE_group_BC=0.4011 TE_group_JLMS=0.3941
TE_meta_BC=0.9999 TE_meta_JLMS=0.9999
MTR_BC=8.5729 MTR_JLMS=8.7385
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Total Log-likelihood: 911.6848
AIC: -1789.37 BIC: -1694.154 HQIC: -1754.409
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Model was estimated on : Jul Wed 15, 2026 at 00:17 Output: Efficiency and Metatechnology Ratio Extraction
The efficiencies() function returns a data frame with one row per observation containing group-specific and metafrontier efficiency estimates and MTRs. The columns present depend on groupType:
| Column | Description |
|---|---|
id |
Observation identifier |
group / Group_c
|
Technology group identifier |
u_g |
Group-specific inefficiency — Jondrow et al. (1982) |
TE_group_JLMS |
Group TE — Jondrow et al. (1982): exp(−u) |
TE_group_BC |
Group TE — Battese & Coelli (1988): E[exp(−u)|ε] |
TE_group_BC_reciprocal |
Reciprocal of Battese & Coelli (1988) group TE |
uLB_g, uUB_g
|
Confidence bounds for u (sfacross only) |
m_g, TE_group_mode
|
Mode-based inefficiency and TE (sfacross only) |
PosteriorProb_c, PosteriorProb_c1 … |
Posterior class probabilities (sfalcmcross only) |
u_meta |
Metafrontier technology gap U |
TE_meta_JLMS |
Metafrontier TE (JLMS basis): TE_group_JLMS × MTR |
TE_meta_BC |
Metafrontier TE (BC basis): TE_group_BC × MTR |
MTR_JLMS |
Metatechnology ratio (JLMS basis) |
MTR_BC |
Metatechnology ratio (BC basis) |
# Example: extract and print for group 1 selected farms only
ef_sel_lp <- efficiencies(meta_sel_lp)
sel_grp1 <- ef_sel_lp[ef_sel_lp$group == 1 & !is.na(ef_sel_lp$TE_group_BC), ]
summary(sel_grp1[, c("TE_group_BC", "TE_meta_BC", "MTR_BC")])Toggle to see the output
TE_group_BC TE_meta_BC MTR_BC
Min. :0.001211 Min. :0.001144 Min. :0.8454
1st Qu.:0.199471 1st Qu.:0.199471 1st Qu.:1.0000
Median :0.413689 Median :0.412915 Median :1.0000
Mean :0.414836 Mean :0.411120 Mean :0.9923
3rd Qu.:0.630272 3rd Qu.:0.620927 3rd Qu.:1.0000
Max. :0.924528 Max. :0.924528 Max. :1.0000 References
- Battese, G. E., & Coelli, T. J. (1988). Prediction of firm-level technical efficiencies with a generalized frontier production function and panel data. Journal of Econometrics, 38(3), 387–399. https://doi.org/10.1016/0304-4076(88)90053-X
- Battese, G. E., Rao, D. S. P., & O’Donnell, C. J. (2004). A metafrontier production function for estimation of technical efficiencies and technology gaps for firms operating under different technologies. Journal of Productivity Analysis, 21(1), 91–103. https://doi.org/10.1023/B:PROD.0000012454.06094.29
- Dakpo, K. H., Desjeux, Y., & Latruffe, L. (2023). sfaR: Stochastic Frontier Analysis using R. R package version 1.0.1. https://CRAN.R-project.org/package=sfaR
- Greene, W. (2010). A stochastic frontier model with correction for sample selection. Journal of Productivity Analysis, 34(1), 15–24. https://doi.org/10.1007/s11123-009-0159-1
- Huang, C. J., Huang, T.-H., & Liu, N.-H. (2014). A new approach to estimating the metafrontier production function based on a stochastic frontier framework. Journal of Productivity Analysis, 42(3), 241–254. https://doi.org/10.1007/s11123-014-0402-2
- Jondrow, J., Lovell, C. A. K., Materov, I. S., & Schmidt, P. (1982). On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics, 19(2–3), 233–238. https://doi.org/10.1016/0304-4076(82)90004-5
- O’Donnell, C. J., Rao, D. S. P., & Battese, G. E. (2008). Metafrontier frameworks for the study of firm-level efficiencies and technology ratios. Empirical Economics, 34(2), 231–255. https://doi.org/10.1007/s00181-007-0119-4
- Orea, L., & Kumbhakar, S. C. (2004). Efficiency measurement using a latent class stochastic frontier model. Empirical Economics, 29(1), 169–183. https://doi.org/10.1007/s00181-003-0184-2