Latent Class SFA Metafrontier (groupType = "sfalcmcross")
Source:vignettes/sfalcmcross-metafrontier.Rmd
sfalcmcross-metafrontier.RmdOverview
In many applications, the technology groups that firms belong to are unobserved — we cannot directly observe which firms operate under which technology type. The latent class model (LCM) addresses this by:
- Fitting a pooled latent class SFA on the entire
dataset using the
sfaRimplementation based on Dakpo et al. (2021), simultaneously estimating class-specific frontier parameters and class membership probabilities for each firm. - Assigning each firm to a class based on its highest posterior class probability.
- Using these assignments as the technology groups for the metafrontier.
This approach is appropriate when: - No observed group variable is available. - Technology heterogeneity is suspected but not directly measurable. - A priori group boundaries are unclear or arbitrary.
Data Preparation
We use the utility dataset from sfaR, which
contains 791 observations from US electricity utilities. We estimate a
cost frontier with no explicit group variable.
library(smfa)
#> Loading required package: sfaR
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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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#> 88888P' 888 888 888 888 "Y888888
#> 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("utility", package = "sfaR")Method 1: LCM + LP Metafrontier
Fit a 2-class latent class pooled SFA, then estimate the LP deterministic envelope over the inferred class frontiers.
meta_lcm_lp <- smfa(
formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
data = utility,
S = -1, # cost frontier (S = -1)
groupType = "sfalcmcross",
lcmClasses = 2, # number of latent classes
metaMethod = "lp"
)
#> Initialization: SFA + halfnormal - normal distributions...
#> LCM 2 Classes Estimation...
#> Warning: hessian is singular for 'qr.solve' switching to 'ginv'
summary(meta_lcm_lp)
#> ============================================================
#> 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: 61.35324
#> ------------------------------------------------------------
#>
#> -- Latent Class 1 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -1.4472e+00 4.0975e-05 -35320 < 2.2e-16 ***
#> log(y) 8.4541e-01 2.4537e-06 344539 < 2.2e-16 ***
#> log(wl/wf) 3.5408e-01 4.7006e-06 75327 < 2.2e-16 ***
#> log(wk/wf) 4.2883e-01 1.4333e-05 29919 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029717 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.8640e+01 3.6587e-13 -1.0561e+14 < 2.2e-16 ***
#> Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=101648736.5567
#>
#> -- Latent Class 2 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -2.0490e+00 2.5608e-05 -80011.10 < 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 -408.92 < 2.2e-16 ***
#> log(wk/wf) 8.8450e-01 6.9315e-05 12760.73 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653955 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219984 < 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 -1192457 < 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: 61.35324
#> AIC: -96.70649 BIC: -35.95362 HQIC: -73.35552
#> ------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14Note: The
groupargument is not needed whengroupType = "sfalcmcross"— the latent classes are identified automatically by the LCM. ThelcmClassesargument controls the number of classes.
Method 2: 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"
)
#> Initialization: SFA + halfnormal - normal distributions...
#> LCM 2 Classes Estimation...
#> Warning: hessian is singular for 'qr.solve' switching to 'ginv'
summary(meta_lcm_qp)
#> ============================================================
#> 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: 61.35324
#> ------------------------------------------------------------
#>
#> -- Latent Class 1 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -1.4472e+00 4.0975e-05 -35320 < 2.2e-16 ***
#> log(y) 8.4541e-01 2.4537e-06 344539 < 2.2e-16 ***
#> log(wl/wf) 3.5408e-01 4.7006e-06 75327 < 2.2e-16 ***
#> log(wk/wf) 4.2883e-01 1.4333e-05 29919 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029717 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.8640e+01 3.6587e-13 -1.0561e+14 < 2.2e-16 ***
#> Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=101648736.5567
#>
#> -- Latent Class 2 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -2.0490e+00 2.5608e-05 -80011.10 < 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 -408.92 < 2.2e-16 ***
#> log(wk/wf) 8.8450e-01 6.9315e-05 12760.73 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653955 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219984 < 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 -1192457 < 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) -1.3923609 0.0310122 -44.897 < 2.2e-16 ***
#> log(y) 0.8649986 0.0012419 696.520 < 2.2e-16 ***
#> log(wl/wf) 0.2909729 0.0047965 60.664 < 2.2e-16 ***
#> log(wk/wf) 0.5028977 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: 61.35324
#> AIC: -88.70649 BIC: -9.26042 HQIC: -58.17061
#> ------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14Method 3: LCM + SFA (Huang)
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"
)
#> Initialization: SFA + halfnormal - normal distributions...
#> LCM 2 Classes Estimation...
#> Warning: hessian is singular for 'qr.solve' switching to 'ginv'
summary(meta_lcm_huang)
#> ============================================================
#> 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: 61.35324
#> ------------------------------------------------------------
#>
#> -- Latent Class 1 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -1.4472e+00 4.0975e-05 -35320 < 2.2e-16 ***
#> log(y) 8.4541e-01 2.4537e-06 344539 < 2.2e-16 ***
#> log(wl/wf) 3.5408e-01 4.7006e-06 75327 < 2.2e-16 ***
#> log(wk/wf) 4.2883e-01 1.4333e-05 29919 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029717 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.8640e+01 3.6587e-13 -1.0561e+14 < 2.2e-16 ***
#> Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=101648736.5567
#>
#> -- Latent Class 2 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -2.0490e+00 2.5608e-05 -80011.10 < 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 -408.92 < 2.2e-16 ***
#> log(wk/wf) 8.8450e-01 6.9315e-05 12760.73 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653955 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219984 < 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 -1192457 < 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) -2.2495090 0.0686629 -32.7617 < 2.2e-16 ***
#> log(y) 0.9909917 0.0024687 401.4295 < 2.2e-16 ***
#> log(wl/wf) 0.0399592 0.0106166 3.7638 0.0001673 ***
#> log(wk/wf) 0.7890982 0.0143800 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: 759.62958
#> Log likelihood gradient norm: 2.06038e-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) = 588.59019
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 342.07878
#> 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 -32.7617 < 2.2e-16 ***
#> .X2 0.99099 0.00247 401.4295 < 2.2e-16 ***
#> .X3 0.03996 0.01062 3.7638 0.0001673 ***
#> .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 : Apr Fri 24, 2026 at 15:14
#> 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: 820.9828
#> AIC: -1603.966 BIC: -1515.173 HQIC: -1569.837
#> ------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14Method 4: LCM + SFA (O’Donnell)
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"
)
#> 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.
#> ============================================================
#> 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: 61.35324
#> ------------------------------------------------------------
#>
#> -- Latent Class 1 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -1.4472e+00 4.0975e-05 -35320 < 2.2e-16 ***
#> log(y) 8.4541e-01 2.4537e-06 344539 < 2.2e-16 ***
#> log(wl/wf) 3.5408e-01 4.7006e-06 75327 < 2.2e-16 ***
#> log(wk/wf) 4.2883e-01 1.4333e-05 29919 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.7658e+00 5.8803e-08 -30029717 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.8640e+01 3.6587e-13 -1.0561e+14 < 2.2e-16 ***
#> Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=101648736.5567
#>
#> -- Latent Class 2 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -2.0490e+00 2.5608e-05 -80011.10 < 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 -408.92 < 2.2e-16 ***
#> log(wk/wf) 8.8450e-01 6.9315e-05 12760.73 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653955 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219984 < 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 -1192457 < 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 1.4795e-05 -99060 < 2.2e-16 ***
#> log(y) 8.5052e-01 2.0492e-06 415057 < 2.2e-16 ***
#> log(wl/wf) 3.4196e-01 8.1363e-06 42028 < 2.2e-16 ***
#> log(wk/wf) 4.4325e-01 1.1627e-05 38121 < 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: 1703
#> Log likelihood value: 1949.16842
#> Log likelihood gradient norm: 6.64643e+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) = 4078429.86815
#> 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) = 1595.36072
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 707.61541
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = 30.29501
#> 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 -99060 < 2.2e-16 ***
#> .X2 0.85052 0.00000 415057 < 2.2e-16 ***
#> .X3 0.34196 0.00001 42028 < 2.2e-16 ***
#> .X4 0.44325 0.00001 38121 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -6.38 0.00 -7.3289e+13 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -36.822 0.000 -4.5151e+13 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14
#> 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: 2010.522
#> AIC: -3983.043 BIC: -3894.251 HQIC: -3948.915
#> ------------------------------------------------------------
#> Model was estimated on : Apr Fri 24, 2026 at 15:14Choosing the Number of Classes
The number of latent classes (lcmClasses) should be
guided by economic theory and information criteria. You can compare
models with different numbers of classes:
meta_lcm_2 <- smfa(
formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
data = utility, S = -1,
groupType = "sfalcmcross", lcmClasses = 2, metaMethod = "lp"
)
#> Initialization: SFA + halfnormal - normal distributions...
#> LCM 2 Classes Estimation...
#> Warning: hessian is singular for 'qr.solve' switching to 'ginv'
meta_lcm_3 <- smfa(
formula = log(tc/wf) ~ log(y) + log(wl/wf) + log(wk/wf),
data = utility, S = -1,
groupType = "sfalcmcross", lcmClasses = 3, metaMethod = "lp"
)
#> Initialization: SFA + halfnormal - normal distributions...
#> LCM 3 Classes Estimation...
#> Warning: hessian is singular for 'qr.solve' switching to 'ginv'
# Compare information criteria
ic(meta_lcm_2)
#> AIC BIC HQIC
#> 1 -96.70649 -35.95362 -73.35552
ic(meta_lcm_3)
#> AIC BIC HQIC
#> 1 -116.4994 -23.03341 -80.5748Prefer the model with the lower AIC/BIC.
Extracting Efficiencies and Posterior Probabilities
For LCM models, efficiencies() returns extra columns for
posterior class membership probabilities, which can be used for
robustness checks or classification:
eff_lcm <- efficiencies(meta_lcm_lp)
head(eff_lcm)
#> id Group_c u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal
#> 1 1 2 0.15842767 0.8534847 0.8557685 1.174838
#> 2 2 2 0.11418786 0.8920904 0.8939941 1.123406
#> 3 3 2 0.08540301 0.9181422 0.9196135 1.090950
#> 4 4 2 0.08020650 0.9229257 0.9243035 1.085176
#> 5 5 2 0.05774143 0.9438940 0.9448244 1.060519
#> 6 6 2 0.08181804 0.9214396 0.9228469 1.086964
#> PosteriorProb_c PosteriorProb_c1 PriorProb_c1 u_c1 teBC_c1
#> 1 0.7249993 0.2750007 0.3539713 0.19428008 0.8234272
#> 2 0.7334018 0.2665982 0.3539713 0.16086081 0.8514106
#> 3 0.7039783 0.2960217 0.3539713 0.10301513 0.9021133
#> 4 0.6909122 0.3090878 0.3539713 0.07745685 0.9254670
#> 5 0.5808759 0.4191241 0.3539713 0.05410185 0.9473356
#> 6 0.6927820 0.3072180 0.3539713 0.05781993 0.9438199
#> teBC_reciprocal_c1 PosteriorProb_c2 PriorProb_c2 u_c2 teBC_c2
#> 1 1.214436 0.7249993 0.6460287 0.15842767 0.8557685
#> 2 1.174521 0.7334018 0.6460287 0.11418786 0.8939941
#> 3 1.108508 0.7039783 0.6460287 0.08540301 0.9196135
#> 4 1.080536 0.6909122 0.6460287 0.08020650 0.9243035
#> 5 1.055592 0.5808759 0.6460287 0.05774143 0.9448244
#> 6 1.059524 0.6927820 0.6460287 0.08181804 0.9228469
#> teBC_reciprocal_c2 ineff_c1 ineff_c2 effBC_c1 effBC_c2 ReffBC_c1 ReffBC_c2
#> 1 1.174838 NA 0.15842767 NA 0.8557685 NA 1.174838
#> 2 1.123406 NA 0.11418786 NA 0.8939941 NA 1.123406
#> 3 1.090950 NA 0.08540301 NA 0.9196135 NA 1.090950
#> 4 1.085176 NA 0.08020650 NA 0.9243035 NA 1.085176
#> 5 1.060519 NA 0.05774143 NA 0.9448244 NA 1.060519
#> 6 1.086964 NA 0.08181804 NA 0.9228469 NA 1.086964
#> u_meta TE_meta_JLMS TE_meta_BC MTR_JLMS MTR_BC
#> 1 0.15842767 0.8534847 0.8557685 1 1
#> 2 0.11418786 0.8920904 0.8939941 1 1
#> 3 0.08540301 0.9181422 0.9196135 1 1
#> 4 0.08020650 0.9229257 0.9243035 1 1
#> 5 0.05774143 0.9438940 0.9448244 1 1
#> 6 0.08181804 0.9214396 0.9228469 1 1
# Key LCM-specific columns:
# Group_c — most likely class assignment
# PosteriorProb_c — posterior probability of assigned class
# PosteriorProb_c1 — posterior probability of Class 1
# PosteriorProb_c2 — posterior probability of Class 2Key Difference from sfacross
| Feature | sfacross |
sfalcmcross |
|---|---|---|
| Group variable | Required | Not required |
| Group estimation | Separate SFA per group | Pooled LCM simultaneously |
| Output | Group-level SFA summaries | Pooled LCM summary with class-specific parameters |
| Extra efficiency columns | Confidence bounds | Posterior class probabilities |