Author: Joshua Pritikin and Ross Jacobucci and Timothy R. Brick
Date: 2022-02-21

# Regularized MIMIC

#### Joshua Pritikin and Ross Jacobucci and Timothy R. Brick

#### 2022-02-21

# 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
    ## 35 3.551246e-07 -0.02792019 1.060184

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  2.61817539 0.359734332
    ## 2   MIMIC.A[2,12]      A    cubes        g  0.93495892 0.249140391
    ## 3   MIMIC.A[3,12]      A    paper        g  0.70084647 0.148839958
    ## 4   MIMIC.A[4,12]      A    flags        g  1.57351355 0.485114112
    ## 5   MIMIC.A[5,12]      A paperrev        g  0.99010508 0.241840086
    ## 6   MIMIC.A[6,12]      A flagssub        g  3.33338049 0.638126581
    ## 7   MIMIC.A[7,12]      A  general        g  9.23700633 0.532147467
    ## 8   MIMIC.A[8,12]      A paragrap        g  2.53491888 0.154443963
    ## 9   MIMIC.A[9,12]      A sentence        g  3.96091929 0.218438121
    ## 10 MIMIC.A[10,12]      A    wordc        g  3.80400196 0.254970074
    ## 11 MIMIC.A[11,12]      A    wordm        g  5.73048429 0.332716170
    ## 12          ageym      A        g    ageym -0.02870552 0.005178867
    ## 13          grade      A        g    grade  1.08144998 0.144108407
    ## 14   MIMIC.S[1,1]      S   visual   visual 40.27135796 3.352951308 !
    ## 15   MIMIC.S[2,2]      S    cubes    cubes 21.00804335 1.720709660
    ## 16   MIMIC.S[3,3]      S    paper    paper  7.36561698 0.605079121
    ## 17   MIMIC.S[4,4]      S    flags    flags 78.47431024 6.427213845
    ## 18   MIMIC.S[5,5]      S paperrev paperrev  8.35238391 0.994207854 !
    ## 19   MIMIC.S[6,6]      S flagssub flagssub 56.56060067 6.793459063
    ## 20   MIMIC.S[7,7]      S  general  general 45.64552429 4.717436705 !
    ## 21   MIMIC.S[8,8]      S paragrap paragrap  4.06597962 0.402767938
    ## 22   MIMIC.S[9,9]      S sentence sentence  6.80450465 0.749101684
    ## 23 MIMIC.S[10,10]      S    wordc    wordc 13.88565834 1.285595220 !
    ## 24 MIMIC.S[11,11]      S    wordm    wordm 17.27850943 1.798287305
    ## 25   MIMIC.M[1,1]      M        1   visual 21.29314806 2.978783300
    ## 26   MIMIC.M[1,2]      M        1    cubes 21.38053062 1.281420562
    ## 27   MIMIC.M[1,3]      M        1    paper 12.00170261 0.885460579
    ## 28   MIMIC.M[1,4]      M        1    flags 13.00215459 2.293226807
    ## 29   MIMIC.M[1,5]      M        1 paperrev 12.12972056 1.350150836
    ## 30   MIMIC.M[1,6]      M        1 flagssub 24.45758218 4.270239961
    ## 31   MIMIC.M[1,7]      M        1  general 11.26617876 9.736136751
    ## 32   MIMIC.M[1,8]      M        1 paragrap  1.12587264 2.698409922
    ## 33   MIMIC.M[1,9]      M        1 sentence  4.77295161 4.165463544
    ## 34  MIMIC.M[1,10]      M        1    wordc 14.03580671 4.049673917
    ## 35  MIMIC.M[1,11]      M        1    wordm -2.91446624 6.048415610
    ##
    ## 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: 2022-02-21 08:50:02
    ## Wall clock time: 1.383407 secs
    ## optimizer:  SLSQP
    ## OpenMx version number: 2.20.6
    ## Need help?  See help(mxSummary)