Linear Mixed Model with Random Effect Distribution as a Mixture

in many applications, this may not be accurate. A mixture of two or more normal distributions may more accurately describe the distributions of one or both of these terms. By specifying a mean model in which none of the fixed effects differ by latent class, that all subjects be fit with the same mixture, and that the random effect and/or error variances be permitted to vary by latent class, the LCLMM allows this extension of the LMM to be fit efficiently.

As an example, consider a model in which there are 1,000 subjects. Each subject is measured for some continuous trait ten times. A possible LMM to describe this is as follows:

yij=µ+αi+ij

where:

yij is the value of the trait for subject i at observation j µis the mean value for the trait

αi is the random intercept for subject i

ij is the error term associated with subject i at observation j

αi∼ Normal 0, σ2r

that the distribution of the random intercepts is instead a mixture of two normal distributions.

αi∼ 50 percent Normal 0, σ2rn

+ 50 percent Normal 0, σ_ro2

For purposes of illustration, the following values were used to generate data:

µ = 100

σ2rn = 20 σ_ro2 = 200

σ2_e = 40

In order to run the model through the SASR macro, the user would specify the files in the format presented in Figures 2.2 and 2.3. Note that it is not necessary to specify D Corr, D Structure, and R Structure since there is only one random effect (no correlation and default for D Structure is to have a separate variance for each random effect) and since all observations are assumed to have the same residual error variance (also the default). Also, since class membership is determined based on the logistic model, having the same intercept-only specification for each subject in the class-membership model assures that all subjects will be fit with the same mixture.

The usual LMM was run as well as the LCLMM assuming that all subjects have the same mixture. Figure 2.4 displays the options that need to be set to run each of the models. Note that when only one class is run (LMM), the macro variable ’PieMethod’ must be set to ’UNSTRUCTURED’ and ’Method’ cannot be set to ’EM’. The parameter estimates from each model are shown in Table 2.1. Since the LMM effectively fits a combined variance to reflectσ2

rn andσ2ro, the variableσro2 is populated in the table only for the LCLMM. It is interesting to note that the random effect variance estimate from the LMM is very close to the weighted average of the random effect variance estimates from the two latent classes in the structured model.

As seen in Figure 2.5, the predictions of the random effects from the LCLMM are very similar to those from the LMM. However, note that the distribution of these random effects is NOT in fact normal. The histograms of the actual random effects generated in creating the data as well as the predicted random effects from each of the models are presented in Figure 2.6. Notice that, while the

Figure 2.2: Screenshot of SASR Datasets Needed to Run Example 1

Table 2.1: Random Effect Distribution as a Mixture - Fitted Values

(a) Linear Mixed Model

(b) Same Mixture LCLMM

LMM assumes that the random effects are distributed normally, the predictions of the random effects need not be normally distributed. Each of the models has a similar distribution of random effects and all have a greater percentage of area in the center and heavier tails than a normal distribution.

Random Intercepts - LCLMM -50 -40 -30 -20 -10 0 10 20 30 40 50 Random Intercepts - LMM -50 -40 -30 -20 -10 0 10 20 30 40 50

Figure 2.5: Comparison of Random Intercepts - LCLMM vs. Linear Mixed Model

Given the above, where do the models differ? The answer relates mainly to standard errors. The standard errors of the random effects for the LMM are determined based on the assumption that the random effects are normally distributed. In this example, the random intercepts from the LMM had a variance of 3.86. As noted in Section 1.4.5, the standard errors of the random effects for the LCLMM reflect information about each subject’s structured class probabilities as well as the relative fit of the underlying LMMs. For the LCLMM, the standard errors of the random effects for each subject are graphed in Figure 2.7 versus the bayesian subject probabilities of being in each latent class (defined in the vectorcin Section 1.4.1). A horizontal line is included at 3.86 to indicate the random effect standard error computed by the LMM. Note that since the standard error calculation for the LCLMM incorporates information as to which of the underlying distributions is most appropriate, each subject’s

-50 -40 -30 -20 -10 0 10 20 30 40 50 0 5 10 15 20 25 30 35 Percent

Random Intercepts - Actual

(a) Actual 0 5 10 15 20 25 30 35 Percent 0 5 10 15 20 25 30 35 Percent

random intercept could potentially have a different standard error. In Figure 2.7, subjects at the left side (with low probability of being in the higher-standard-error class) are those that are in the very center of the random intercept distribution, while those at the right side are in the tails. Since there is a much greater percentage of random intercepts clumped in the middle of the mixture distribution, the standard errors of the random intercepts for these subjects would be expected to be smaller than under the LMM. However, since the tails in the mixture distribution are more spread out than under the LMM, the standard errors of the random intercepts out in the tails have a higher standard error than under the LMM.

Standard Error (Random Intercept)

3.4 3.5 3.6 3.7 3.8 3.9 4.0

Subject Class Probability (Higher SE Class) 0% 20% 40% 60% 80% 100%

Figure 2.7: Plot of Random Effect Standard Errors from the LCLMM