Regularized MIMIC

Joshua Pritikin and Ross Jacobucci and Timothy R. Brick

2024-08-16

Regularized MIMIC model

This example uses the immortal Holzinger Swineford data set.

library(OpenMx)
data(HS.ability.data)

The OpenMx model looks like this:

HS.ability.data$ageym <- HS.ability.data$agey*12 + HS.ability.data$agem
HS.ability.data$male <- as.numeric(HS.ability.data$Gender == 'Male')

# Specify variables
indicators <- c('visual','cubes','paper','flags','paperrev','flagssub',
                'general','paragrap','sentence','wordc','wordm')
covariates <- c("male","ageym","grade")
latents = c("g", covariates)

# Build the model
mimicModel <- mxModel(
  "MIMIC", type="RAM",
  manifestVars = indicators, latentVars = latents,

  # Set up exogenous predictors
  mxPath("one", covariates, labels=paste0('data.',covariates), free=FALSE),

  # Fix factor variance
  mxPath('g', arrows=2, free=FALSE, values=1),

  # Error variances:
  mxPath(from=c(indicators), arrows=2, free=TRUE, values=10),

  # Means (saturated means model):
  mxPath(from="one", to=indicators, values=rep(5, length(indicators))),

  # Loadings:
  mxPath(from="g", to=indicators, values=.5),

  # Covariate paths
  mxPath(covariates, "g", labels=covariates),

  # Data
  mxData(observed = HS.ability.data, type = "raw"))

# Get some good starting values for regularization. This
# saves 2-3 minutes on my laptop.
mimicModel <- mxRun(mimicModel)
## Running MIMIC with 36 parameters

Add the penalty:

mimicModel <- mxModel(
  mimicModel,
  mxMatrix('Full',1,1,free=TRUE,values=0,labels="lambda",name="hparam"),
  # Set scale to ML estimates for adaptive lasso
  mxPenaltyLASSO(what=covariates, name="LASSO",
                    scale = coef(mimicModel)[covariates],
                    lambda =  0, lambda.max =2, lambda.step=.04)
)

Run the regularization. With only three covariates, the plot of results is not very exciting. We learn that sex is not a good predictor of this factor.

regMIMIC <- mxPenaltySearch(mimicModel)
## Running MIMIC with 37 parameters
## Warning: In model 'MIMIC' Optimizer returned a non-zero status code 6. The
## model does not satisfy the first-order optimality conditions to the required
## accuracy, and no improved point for the merit function could be found during
## the final linesearch (Mx status RED)
detail <- regMIMIC$compute$steps$PS$output$detail

library(reshape2)
library(ggplot2)

est <- detail[,c(covariates, 'lambda')]
ggplot(melt(est, id.vars = 'lambda')) +
  geom_line(aes(x=lambda, y=value, color=variable)) +
  geom_vline(aes(xintercept=coef(regMIMIC)['lambda']),
             linetype="dashed", alpha=.5)

The regularized factor loadings can be found here,

detail[detail$EBIC == min(detail$EBIC), covariates]
##            male       ageym    grade
## 27 3.372766e-07 -0.02815012 1.063449

The regularization causes a lot of bias. One way to deal with this is to fix zerod parameters to zero, discard the regularization penalty, and re-fit model.

regMIMIC <- mxPenaltyZap(regMIMIC)
## Zapping 'male'
## Fixing 'lambda'
## Tip: Use
##   model = mxRun(model)
## to re-estimate the model without any penalty terms.
regMIMIC <- mxRun(regMIMIC)
## Running MIMIC with 35 parameters
summary(regMIMIC)
## Summary of MIMIC 
##  
## free parameters:
##              name matrix      row      col    Estimate   Std.Error A
## 1   MIMIC.A[1,12]      A   visual        g  2.61817594 0.359298787  
## 2   MIMIC.A[2,12]      A    cubes        g  0.93493904 0.249165023  
## 3   MIMIC.A[3,12]      A    paper        g  0.70084632 0.148850149  
## 4   MIMIC.A[4,12]      A    flags        g  1.57351184 0.488495921  
## 5   MIMIC.A[5,12]      A paperrev        g  0.99010502 0.241890553  
## 6   MIMIC.A[6,12]      A flagssub        g  3.33349725 0.636982509  
## 7   MIMIC.A[7,12]      A  general        g  9.23700795 0.531325575  
## 8   MIMIC.A[8,12]      A paragrap        g  2.53491899 0.153265387  
## 9   MIMIC.A[9,12]      A sentence        g  3.96091919 0.218337618  
## 10 MIMIC.A[10,12]      A    wordc        g  3.80400282 0.254812615  
## 11 MIMIC.A[11,12]      A    wordm        g  5.73048123 0.332521021  
## 12          ageym      A        g    ageym -0.02870565 0.005116147  
## 13          grade      A        g    grade  1.08144484 0.147319560  
## 14   MIMIC.S[1,1]      S   visual   visual 40.27118644 3.350709894  
## 15   MIMIC.S[2,2]      S    cubes    cubes 21.00803435 1.721514961  
## 16   MIMIC.S[3,3]      S    paper    paper  7.36561018 0.605048154  
## 17   MIMIC.S[4,4]      S    flags    flags 78.47382780 6.425595339 !
## 18   MIMIC.S[5,5]      S paperrev paperrev  8.35235110 0.994617503  
## 19   MIMIC.S[6,6]      S flagssub flagssub 56.56097630 6.796716392 !
## 20   MIMIC.S[7,7]      S  general  general 45.64539430 4.716316397  
## 21   MIMIC.S[8,8]      S paragrap paragrap  4.06597950 0.402134862  
## 22   MIMIC.S[9,9]      S sentence sentence  6.80451253 0.749512037  
## 23 MIMIC.S[10,10]      S    wordc    wordc 13.88563458 1.285502142  
## 24 MIMIC.S[11,11]      S    wordm    wordm 17.27857521 1.794717772 !
## 25   MIMIC.M[1,1]      M        1   visual 21.29329866 2.947678352  
## 26   MIMIC.M[1,2]      M        1    cubes 21.38065555 1.263730076  
## 27   MIMIC.M[1,3]      M        1    paper 12.00174359 0.891365009  
## 28   MIMIC.M[1,4]      M        1    flags 13.00224785 2.420589129  
## 29   MIMIC.M[1,5]      M        1 paperrev 12.12977939 1.369287374  
## 30   MIMIC.M[1,6]      M        1 flagssub 24.45733301 4.211407211  
## 31   MIMIC.M[1,7]      M        1  general 11.26671178 9.773539364  
## 32   MIMIC.M[1,8]      M        1 paragrap  1.12601996 2.684513304  
## 33   MIMIC.M[1,9]      M        1 sentence  4.77318252 4.172379784  
## 34  MIMIC.M[1,10]      M        1    wordc 14.03602611 4.045727959  
## 35  MIMIC.M[1,11]      M        1    wordm -2.91412180 6.049740939  
## 
## Model Statistics: 
##                |  Parameters  |  Degrees of Freedom  |  Fit (-2lnL units)
##        Model:             35                   2964              17843.68
##    Saturated:             77                   2922                    NA
## Independence:             22                   2977                    NA
## Number of observations/statistics: 301/2999
## 
## Information Criteria: 
##       |  df Penalty  |  Parameters Penalty  |  Sample-Size Adjusted
## AIC:     11915.6773               17913.68                 17923.19
## BIC:       927.8025               18043.43                 17932.43
## To get additional fit indices, see help(mxRefModels)
## timestamp: 2024-08-16 20:56:56 
## Wall clock time: 9.974365 secs 
## optimizer:  SLSQP 
## OpenMx version number: 2.21.12 
## Need help?  See help(mxSummary)