# 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)