Stat Computing > MPlus > Textbook Examples > Applied Longitudinal Data Analysis

HLM Textbook Examples
Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence
by Judith D. Singer and John B. Willett
Supplement to Chapter 5:  Treating time more flexibly

• Supplement 5.1 - Relaxing assumption of equality of variances across time.

Supplement 5.1 - Relaxing assumption of equality of variances across time.

This example uses the data file reading_pp.ssm with the raw data files reading_pp1_l1.dta and reading_pp1_l2.dta) AGEGRPi-6.5 is used as a temporal predictor, called cagegrpi (i.e., cagegrp1, cagegrp2 and cagegrp3).  These were created before making the data file.

• On page 145 in Table 5.2 Model A, the assumption is made that the residual variances are the same across all 3 time points.  This example shows how you can relax that assumption, allowing them to vary freely.
• HLM computes a likelihood ratio test comparing more restricted model (shown in Chapter 5) by taking the differences in the Deviances for the 2 models, yielding  = 7.598.  When compared to a chi-square table with 2 degrees of freedom, the p value is .022.

This data file includes dummy variables for each of the time points (called dum1, dum2, dum3).  We chose Optional Specifications and then Heterogenous Sima^2 and then included dum1 and dum2 as predictors of the level 1 heterogeneity.  The results are then shown below.

Summary of Model Fit

 -------------------------------------------------------------------
 Model                                Number of         Deviance
                                      Parameters
 -------------------------------------------------------------------
 1. Homogeneous sigma_squared              6           1818.11142
 2. Heterogeneous sigma_squared            8           1810.51286
 -------------------------------------------------------------------
 Model Comparison                 Chi-square       df    P-value
 -------------------------------------------------------------------
 Model 1 vs Model 2                   7.59855       2     0.022

 Tau
 INTRCPT1,B0     24.07699      -3.17255 
  CAGEGRP,B1     -3.17255       6.59741 


 Standard Errors of Tau
 INTRCPT1,B0      8.36033       2.57727 
  CAGEGRP,B1      2.57727       1.59046 


Tau (as correlations)
 INTRCPT1,B0  1.000 -0.252
  CAGEGRP,B1 -0.252  1.000

 ----------------------------------------------------
  Random level-1 coefficient   Reliability estimate
 ----------------------------------------------------
  INTRCPT1, B0                        0.776
   CAGEGRP, B1                        0.848
 ----------------------------------------------------

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00          20.771504   0.590235    35.192        88    0.000
 For  CAGEGRP slope, B1
    INTRCPT2, G10           5.052929   0.295599    17.094        88    0.000
 ----------------------------------------------------------------------------

 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        4.90683      24.07699    88     398.20496    0.000
  CAGEGRP slope, U1        2.56854       6.59741    88     586.78159    0.000
 -----------------------------------------------------------------------------

 Statistics for current covariance components model
 --------------------------------------------------
 Deviance                       = 1810.512864
 Number of estimated parameters = 8

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