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:
sfaR implementation based on Dakpo et
al. (2021), simultaneously estimating class-specific frontier parameters
and class membership probabilities for each firm.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.
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.
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 5.0799e-05 -28489 < 2.2e-16 ***
#> log(y) 8.4541e-01 2.8752e-06 294038 < 2.2e-16 ***
#> log(wl/wf) 3.5408e-01 5.6877e-06 62254 < 2.2e-16 ***
#> log(wk/wf) 4.2883e-01 1.7424e-05 24612 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.7658e+00 5.8804e-08 -30029297 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.7713e+01 7.2436e-13 -5.2064e+13 < 2.2e-16 ***
#> Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=63950436.4742
#>
#> -- Latent Class 2 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -2.0490e+00 2.5608e-05 -80011.01 < 2.2e-16 ***
#> log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
#> log(wl/wf) -2.5913e-02 6.3375e-05 -408.88 < 2.2e-16 ***
#> log(wk/wf) 8.8450e-01 6.9315e-05 12760.72 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653953 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219961 < 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 -1192446 < 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: -96.70648 BIC: -35.9536 HQIC: -73.35551
#> ------------------------------------------------------------
#> Model was estimated on : Jul Tue 14, 2026 at 21:42Note: The
groupargument is not needed whengroupType = "sfalcmcross"— the latent classes are identified automatically by the LCM. ThelcmClassesargument controls the number of classes.
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 5.0799e-05 -28489 < 2.2e-16 ***
#> log(y) 8.4541e-01 2.8752e-06 294038 < 2.2e-16 ***
#> log(wl/wf) 3.5408e-01 5.6877e-06 62254 < 2.2e-16 ***
#> log(wk/wf) 4.2883e-01 1.7424e-05 24612 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.7658e+00 5.8804e-08 -30029297 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.7713e+01 7.2436e-13 -5.2064e+13 < 2.2e-16 ***
#> Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=63950436.4742
#>
#> -- Latent Class 2 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -2.0490e+00 2.5608e-05 -80011.01 < 2.2e-16 ***
#> log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
#> log(wl/wf) -2.5913e-02 6.3375e-05 -408.88 < 2.2e-16 ***
#> log(wk/wf) 8.8450e-01 6.9315e-05 12760.72 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653953 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219961 < 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 -1192446 < 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) -1.3923615 0.0310119 -44.898 < 2.2e-16 ***
#> log(y) 0.8649984 0.0012419 696.525 < 2.2e-16 ***
#> log(wl/wf) 0.2909738 0.0047964 60.665 < 2.2e-16 ***
#> log(wk/wf) 0.5028970 0.0069500 72.360 < 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: -88.70648 BIC: -9.26041 HQIC: -58.17059
#> ------------------------------------------------------------
#> Model was estimated on : Jul Tue 14, 2026 at 21:42meta_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 5.0799e-05 -28489 < 2.2e-16 ***
#> log(y) 8.4541e-01 2.8752e-06 294038 < 2.2e-16 ***
#> log(wl/wf) 3.5408e-01 5.6877e-06 62254 < 2.2e-16 ***
#> log(wk/wf) 4.2883e-01 1.7424e-05 24612 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.7658e+00 5.8804e-08 -30029297 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.7713e+01 7.2436e-13 -5.2064e+13 < 2.2e-16 ***
#> Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=63950436.4742
#>
#> -- Latent Class 2 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -2.0490e+00 2.5608e-05 -80011.01 < 2.2e-16 ***
#> log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
#> log(wl/wf) -2.5913e-02 6.3375e-05 -408.88 < 2.2e-16 ***
#> log(wk/wf) 8.8450e-01 6.9315e-05 12760.72 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653953 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219961 < 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 -1192446 < 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) -2.2495068 0.0686626 -32.7618 < 2.2e-16 ***
#> log(y) 0.9909912 0.0024686 401.4310 < 2.2e-16 ***
#> log(wl/wf) 0.0399614 0.0106166 3.7641 0.0001672 ***
#> log(wk/wf) 0.7890981 0.0143800 54.8748 < 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: 759.63171
#> Log likelihood gradient norm: 2.06994e-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) = 6.31589
#> 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) = 588.59175
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 342.07992
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = 10.23624
#> 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 -32.7618 < 2.2e-16 ***
#> .X2 0.99099 0.00247 401.4310 < 2.2e-16 ***
#> .X3 0.03996 0.01062 3.7641 0.0001672 ***
#> .X4 0.78910 0.01438 54.8748 < 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) -7.29335 0.12966 -56.25 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Jul Tue 14, 2026 at 21:42
#> 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: 820.985
#> AIC: -1603.97 BIC: -1515.177 HQIC: -1569.842
#> ------------------------------------------------------------
#> Model was estimated on : Jul Tue 14, 2026 at 21:42meta_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 5.0799e-05 -28489 < 2.2e-16 ***
#> log(y) 8.4541e-01 2.8752e-06 294038 < 2.2e-16 ***
#> log(wl/wf) 3.5408e-01 5.6877e-06 62254 < 2.2e-16 ***
#> log(wk/wf) 4.2883e-01 1.7424e-05 24612 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -1.7658e+00 5.8804e-08 -30029297 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -3.7713e+01 7.2436e-13 -5.2064e+13 < 2.2e-16 ***
#> Sigma_u=0.4136 Sigma_v=0.0000 Sigma=0.4136 Gamma=1.0000 Lambda=63950436.4742
#>
#> -- Latent Class 2 --
#> Frontier:
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -2.0490e+00 2.5608e-05 -80011.01 < 2.2e-16 ***
#> log(y) 1.0079e+00 4.1082e-04 2453.37 < 2.2e-16 ***
#> log(wl/wf) -2.5913e-02 6.3375e-05 -408.88 < 2.2e-16 ***
#> log(wk/wf) 8.8450e-01 6.9315e-05 12760.72 < 2.2e-16 ***
#> Var(u):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -3.0117e+00 1.1348e-06 -2653953 < 2.2e-16 ***
#> Var(v):
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -4.9663e+00 5.3865e-07 -9219961 < 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 -1192446 < 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 1.6691e-05 -87805 < 2.2e-16 ***
#> log(y) 8.5052e-01 2.3139e-06 367577 < 2.2e-16 ***
#> log(wl/wf) 3.4196e-01 9.2564e-06 36943 < 2.2e-16 ***
#> log(wk/wf) 4.4325e-01 1.3052e-05 33960 < 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: 1946
#> Log likelihood value: 1949.17638
#> Log likelihood gradient norm: 3.82419e+03
#> Estimation based on: N = 791 and K = 6
#> Inf. Cr: AIC = -3886.4 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) = 3453499.37923
#> 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) = 1595.36741
#> LR statistic:
#> Chisq = 2*[LogL(H0)-LogL(H1)] = 707.61794
#> Kodde-Palm C*: 95%: 2.70554 99%: 5.41189
#> Coelli (1995) skewness test on OLS residuals
#> M3T: z = 30.29517
#> M3T: p.value = 0.00000
#> Final maximum likelihood estimates
#> --------------------------------------------------------------------------------
#> Deterministic Component of SFA
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> (Intercept) -1.46554 0.00002 -87805 < 2.2e-16 ***
#> .X2 0.85052 0.00000 367577 < 2.2e-16 ***
#> .X3 0.34196 0.00001 36943 < 2.2e-16 ***
#> .X4 0.44325 0.00001 33960 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameter in variance of u (one-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zu_(Intercept) -6.38 0.00 -5.404e+14 < 2.2e-16 ***
#> --------------------------------------------------------------------------------
#> Parameters in variance of v (two-sided error)
#> --------------------------------------------------------------------------------
#> Coefficient Std. Error z value Pr(>|z|)
#> Zv_(Intercept) -36.49 0.00 -3.1769e+13 < 2.2e-16 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> --------------------------------------------------------------------------------
#> Model was estimated on : Jul Tue 14, 2026 at 21:42
#> 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: 2010.53
#> AIC: -3983.059 BIC: -3894.267 HQIC: -3948.931
#> ------------------------------------------------------------
#> Model was estimated on : Jul Tue 14, 2026 at 21:42The 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 -96.70648 -35.9536 -73.35551#> AIC BIC HQIC
#> 1 -116.0641 -22.59817 -80.13956Prefer the model with the lower AIC/BIC.
For LCM models, efficiencies() returns extra columns for
posterior class membership probabilities, which can be used for
robustness checks or classification:
#> id Group_c u_g TE_group_JLMS TE_group_BC TE_group_BC_reciprocal
#> 1 1 2 0.15842872 0.8534838 0.8557677 1.174839
#> 2 2 2 0.11418869 0.8920896 0.8939934 1.123407
#> 3 3 2 0.08540348 0.9181418 0.9196131 1.090951
#> 4 4 2 0.08020699 0.9229253 0.9243031 1.085176
#> 5 5 2 0.05774182 0.9438936 0.9448240 1.060519
#> 6 6 2 0.08181852 0.9214392 0.9228464 1.086964
#> PosteriorProb_c PosteriorProb_c1 PriorProb_c1 u_c1 teBC_c1
#> 1 0.7249980 0.2750020 0.353972 0.19428009 0.8234272
#> 2 0.7334012 0.2665988 0.353972 0.16086083 0.8514106
#> 3 0.7039784 0.2960216 0.353972 0.10301514 0.9021133
#> 4 0.6909126 0.3090874 0.353972 0.07745687 0.9254669
#> 5 0.5808787 0.4191213 0.353972 0.05410186 0.9473356
#> 6 0.6927824 0.3072176 0.353972 0.05781994 0.9438199
#> teBC_reciprocal_c1 PosteriorProb_c2 PriorProb_c2 u_c2 teBC_c2
#> 1 1.214436 0.7249980 0.646028 0.15842872 0.8557677
#> 2 1.174521 0.7334012 0.646028 0.11418869 0.8939934
#> 3 1.108508 0.7039784 0.646028 0.08540348 0.9196131
#> 4 1.080536 0.6909126 0.646028 0.08020699 0.9243031
#> 5 1.055592 0.5808787 0.646028 0.05774182 0.9448240
#> 6 1.059524 0.6927824 0.646028 0.08181852 0.9228464
#> teBC_reciprocal_c2 ineff_c1 ineff_c2 effBC_c1 effBC_c2 ReffBC_c1 ReffBC_c2
#> 1 1.174839 NA 0.15842872 NA 0.8557677 NA 1.174839
#> 2 1.123407 NA 0.11418869 NA 0.8939934 NA 1.123407
#> 3 1.090951 NA 0.08540348 NA 0.9196131 NA 1.090951
#> 4 1.085176 NA 0.08020699 NA 0.9243031 NA 1.085176
#> 5 1.060519 NA 0.05774182 NA 0.9448240 NA 1.060519
#> 6 1.086964 NA 0.08181852 NA 0.9228464 NA 1.086964
#> u_meta TE_meta_JLMS TE_meta_BC MTR_JLMS MTR_BC
#> 1 0.15842872 0.8534838 0.8557677 1 1
#> 2 0.11418869 0.8920896 0.8939934 1 1
#> 3 0.08540348 0.9181418 0.9196131 1 1
#> 4 0.08020699 0.9229253 0.9243031 1 1
#> 5 0.05774182 0.9438936 0.9448240 1 1
#> 6 0.08181852 0.9214392 0.9228464 1 1sfacross| 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 |