estimation alternatives using the LMS approach
estimation_lms.RmdAccelerated EM and Adaptive Quadrature
By default the LMS approach uses a standard Expectation-Maximization (EM) algorithm to estimate the model parameters, along with a fixed quadrature.
However, it is possible to use an accelerated EM procedure
("EMA") that uses Quasi-Newton and Fisher Scoring
optimization steps when needed. It is also possible to use an adaptive
quadrature instead of a fixed quadrature. Using the EMA and adaptive
quadrature might yield estimates that are closer to results from
Mplus.
If the model struggles to converge, you can try using the accelerated
EM procedure by setting method = "EMA", and
adaptive.quad = TRUE in the modsem() function.
Additionally it is possible to tweak these parameters:
-
max.iter: Maximum number of iterations for the EM algorithm (default is 500). -
max.step: Maximum number of steps used in the Maximization step of the EM algorithm (default is 1). -
convergence.rel: Relative convergence criterion for the EM algorithm. -
convergence.abs: Absolute convergence criterion for the EM algorithm. -
nodes: Number of nodes for numerical integration (default is 24). Increasing this number can improve the accuracy of the estimates, especially for complex models. -
quad.range: Integration range for quadrature. Smaller ranges means that the integral is more focused. -
adaptive.quad.tol: Relative tolerance for determining whether a sub-interval of the adaptive quadrature is accurate enough.
Here we can see an example using the TPB_UK dataaset,
which is more troublesome to estimate than the simulated
TPB dataset.
tpb_uk <- "
# Outer Model (Based on Hagger et al., 2007)
ATT =~ att3 + att2 + att1 + att4
SN =~ sn4 + sn2 + sn3 + sn1
PBC =~ pbc2 + pbc1 + pbc3 + pbc4
INT =~ int2 + int1 + int3 + int4
BEH =~ beh3 + beh2 + beh1 + beh4
# Inner Model (Based on Steinmetz et al., 2011)
INT ~ ATT + SN + PBC
BEH ~ INT + PBC
BEH ~ INT:PBC
"
fit <- modsem(tpb_uk,
data = TPB_UK,
method = "lms",
nodes = 32, # Number of nodes for numerical integration
adaptive.quad = TRUE, # Use adaptive quadrature
algorithm ="EMA", # Use accelerated EM algorithm
convergence.abs = 1e-4, # Relative convergence criterion
convergence.rel = 1e-10, # Relative convergence criterion
max.iter = 500, # Maximum number of iterations
max.step = 1, # Maximum number of steps in the maximization step
adaptive.quad.tol = 1e-4) # Tolerance when building adaptive quadrature
summary(fit)
#> Estimating baseline model (H0)
#>
#> modsem (version 1.0.9):
#> Estimator LMS
#> Optimization method EMA-NLMINB
#> Number of observations 1169
#> Number of iterations 102
#> Loglikelihood -33386.83
#> Akaike (AIC) 66911.65
#> Bayesian (BIC) 67261.06
#>
#> Numerical Integration:
#> Points of integration (per dim) 32
#> Dimensions 1
#> Total points of integration 32
#>
#> Fit Measures for H0:
#> Loglikelihood -35523
#> Akaike (AIC) 71181.74
#> Bayesian (BIC) 71526.09
#> Chi-square 5519.01
#> Degrees of Freedom (Chi-square) 162
#> P-value (Chi-square) 0.000
#> RMSEA 0.168
#>
#> Comparative fit to H0 (no interaction effect)
#> Loglikelihood change 2136.04
#> Difference test (D) 4272.09
#> Degrees of freedom (D) 1
#> P-value (D) 0.000
#>
#> R-Squared:
#> INT 0.901
#> BEH 0.920
#> R-Squared Null-Model (H0):
#> INT 0.896
#> BEH 0.867
#> R-Squared Change:
#> INT 0.005
#> BEH 0.053
#>
#> Parameter Estimates:
#> Coefficients unstandardized
#> Information observed
#> Standard errors standard
#>
#> Latent Variables:
#> Estimate Std.Error z.value P(>|z|)
#> PBC =~
#> pbc2 1.000
#> pbc1 0.857 0.021 41.28 0.000
#> pbc3 0.934 0.017 55.38 0.000
#> pbc4 0.817 0.021 39.72 0.000
#> ATT =~
#> att3 1.000
#> att2 0.965 0.011 86.33 0.000
#> att1 0.812 0.017 47.19 0.000
#> att4 0.870 0.019 45.46 0.000
#> SN =~
#> sn4 1.000
#> sn2 1.313 0.041 32.30 0.000
#> sn3 1.350 0.041 32.72 0.000
#> sn1 1.000 0.038 26.60 0.000
#> INT =~
#> int2 1.000
#> int1 0.970 0.011 92.12 0.000
#> int3 0.984 0.010 98.44 0.000
#> int4 0.992 0.009 104.57 0.000
#> BEH =~
#> beh3 1.000
#> beh2 0.986 0.013 77.73 0.000
#> beh1 0.814 0.019 42.72 0.000
#> beh4 0.803 0.019 41.48 0.000
#>
#> Regressions:
#> Estimate Std.Error z.value P(>|z|)
#> INT ~
#> PBC 1.038 0.036 29.11 0.000
#> ATT -0.059 0.029 -2.03 0.042
#> SN 0.046 0.032 1.44 0.151
#> BEH ~
#> PBC 0.391 0.052 7.48 0.000
#> INT 0.576 0.049 11.80 0.000
#> PBC:INT 0.141 0.008 17.81 0.000
#>
#> Intercepts:
#> Estimate Std.Error z.value P(>|z|)
#> pbc2 3.936 0.054 73.54 0.000
#> pbc1 3.915 0.054 72.51 0.000
#> pbc3 3.678 0.052 71.25 0.000
#> pbc4 3.723 0.053 70.83 0.000
#> att3 3.653 0.055 66.49 0.000
#> att2 3.771 0.053 70.80 0.000
#> att1 4.153 0.054 76.80 0.000
#> att4 3.628 0.059 61.79 0.000
#> sn4 4.463 0.048 92.12 0.000
#> sn2 4.299 0.050 86.73 0.000
#> sn3 4.337 0.050 87.19 0.000
#> sn1 4.432 0.049 90.11 0.000
#> int2 3.638 0.054 67.09 0.000
#> int1 3.786 0.054 70.04 0.000
#> int3 3.657 0.054 67.47 0.000
#> int4 3.700 0.054 68.39 0.000
#> beh3 2.572 0.066 38.87 0.000
#> beh2 2.491 0.066 37.85 0.000
#> beh1 2.468 0.066 37.43 0.000
#> beh4 2.612 0.066 39.60 0.000
#> INT 0.000
#> BEH 0.000
#> PBC 0.000
#> ATT 0.000
#> SN 0.000
#>
#> Covariances:
#> Estimate Std.Error z.value P(>|z|)
#> PBC ~~
#> ATT 3.719 0.145 25.73 0.000
#> SN 1.962 0.104 18.78 0.000
#> ATT ~~
#> SN 1.698 0.101 16.90 0.000
#>
#> Variances:
#> Estimate Std.Error z.value P(>|z|)
#> pbc2 0.690 0.043 16.17 0.000
#> pbc1 1.459 0.069 21.03 0.000
#> pbc3 0.799 0.042 18.86 0.000
#> pbc4 1.464 0.069 21.27 0.000
#> att3 0.296 0.023 12.81 0.000
#> att2 0.306 0.022 13.81 0.000
#> att1 1.285 0.057 22.58 0.000
#> att4 1.583 0.071 22.42 0.000
#> sn4 1.363 0.065 21.03 0.000
#> sn2 0.491 0.032 15.40 0.000
#> sn3 0.377 0.032 11.89 0.000
#> sn1 1.445 0.068 21.13 0.000
#> int2 0.237 0.014 16.98 0.000
#> int1 0.404 0.020 20.25 0.000
#> int3 0.334 0.017 19.35 0.000
#> int4 0.271 0.015 17.92 0.000
#> beh3 0.456 0.030 15.27 0.000
#> beh2 0.513 0.031 16.54 0.000
#> beh1 1.836 0.082 22.30 0.000
#> beh4 1.917 0.086 22.40 0.000
#> PBC 4.421 0.171 25.83 0.000
#> ATT 4.487 0.178 25.21 0.000
#> SN 1.727 0.117 14.79 0.000
#> INT 0.496 0.037 13.37 0.000
#> BEH 0.447 0.034 13.20 0.000