What is smfa?
smfa is an R package for implementing
metafrontier analysis — a framework for productivity
and performance benchmarking of firms (or farms, utilities, hospitals,
etc.) that operate under different technologies. This is also
referred to as “heterogeneous technology” analysis.
The classic stochastic frontier analysis (SFA) estimates a single frontier for all firms. Metafrontier analysis extends this by:
- Estimating group-specific frontiers for each technology sub-group.
- Estimating a common metafrontier that envelops all group frontiers.
- Computing metatechnology ratios (MTR), which quantify how far each group’s frontier lies below the best-practice metafrontier.
This allows researchers to disentangle:
- Technical efficiency relative to the group’s own frontier (TE_group).
- Metafrontier efficiency relative to the common best-practice frontier (TE_meta).
- Metatechnology ratio (MTR = TE_meta / TE_group), which captures how technologically advanced a group’s production possibilities are.
Conceptual Framework
A MTR close to 1 means the group’s technology is near the metafrontier (advanced technology). A MTR far below 1 means the group operates under a less advanced technology.
Methods Supported
smfa supports four metafrontier estimation methods
(metaMethod):
| Method | Description |
|---|---|
"lp" |
Linear programming deterministic envelope — Battese, Rao & O’Donnell (2004) |
"qp" |
Quadratic programming deterministic envelope |
"sfa" (sfaApproach = "huang") |
Two-stage stochastic metafrontier — Huang, Huang & Liu (2014) |
"sfa" (sfaApproach = "ordonnell") |
Two-stage SFA on LP envelope — O’Donnell, Rao & Battese (2008) |
And three group frontier types (groupType):
| Group Type | When to Use |
|---|---|
"sfacross" |
Technology groups are observed (e.g., a group variable exists) |
"sfalcmcross" |
Technology groups are unobserved — latent class model identifies them (Dakpo et al. 2021) |
"sfaselectioncross" |
Sample selection bias is present (Dakpo et al. 2022) |
Installation
# Install smfa from CRAN
install.packages("smfa")
# Or install the development version from GitHub
# if (!require("devtools")) install.packages("devtools")
# devtools::install_github("SulmanOlieko/smfa")Note:
sfaRis automatically installed as a dependency — you do not need to install it separately.
Quick-Start Example
This minimal example demonstrates the core workflow using the
ricephil dataset from the sfaR package, with
three Filipino rice farm-size groups (small, medium, large).
Step 1: Load data and create groups
library(smfa)
#> Loading required package: sfaR
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#> //***** /** ******* /**///**
#> /////** /** **////** /** //**
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#> ////// // //////// // // version 1.0.1
#>
#> * Please cite the 'sfaR' package as:
#> Dakpo KH., Desjeux Y., Henningsen A., and Latruffe L. (2024). sfaR: Stochastic Frontier Analysis Using R. R package version 1.0.1.
#>
#> See also: citation("sfaR")
#>
#> * For any questions, suggestions, or comments on the 'sfaR' package, you can contact directly the authors or visit: https://github.com/hdakpo/sfaR/issues
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#> version 1.0.0
#>
#> * Please cite the 'smfa' package as:
#> Owili, S. O. (2026). smfa: Stochastic Metafrontier Analysis. R package version 1.0.0.
#>
#> See also: citation("smfa")
#>
#> * For any questions, suggestions, or comments on the 'smfa' package, you can contact the authors directly or visit:
#> https://github.com/SulmanOlieko/smfa/issues
data("ricephil", package = "sfaR")
# Create technology groups based on farm area terciles
ricephil$group <- cut(
ricephil$AREA,
breaks = quantile(ricephil$AREA, probs = c(0, 1 / 3, 2 / 3, 1), na.rm = TRUE),
labels = c("small", "medium", "large"),
include.lowest = TRUE
)
table(ricephil$group)
#>
#> small medium large
#> 125 104 115Step 3: Summarise results
summary(meta_lp)
#> ============================================================
#> 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 : Apr Fri 24, 2026 at 15:13
#> 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 : Apr Fri 24, 2026 at 15:13
#> 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 : Apr Fri 24, 2026 at 15:13
#> 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
#> 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 : Apr Fri 24, 2026 at 15:13Step 4: Extract firm-level efficiencies
eff <- efficiencies(meta_lp)
head(eff)
#> 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 0.8774139Key output columns:
| Column | Description |
|---|---|
TE_group_BC |
Group TE (Battese & Coelli 1988) |
TE_meta_BC |
Metafrontier TE |
MTR_BC |
Metatechnology ratio |
What Next?
-
Standard SFA groups → see
vignette("sfacross-metafrontier") -
Unobserved/latent groups → see
vignette("sfalcmcross-metafrontier") -
Sample selection → see
vignette("sfaselectioncross-metafrontier") -
Extracting all outputs → see
vignette("efficiency-extraction")