We used longitudinal structural MRI data and a subset of available covariates collected by the ADNI over the course of a two year period from Alzheimer’s Disease (AD) patients and healthy controls (HC). Associations of interest, between brain structure and the diagnostic group by age interaction, were examined by fitting a MA-GMM model to the response measure (the RAVENS value) at each voxel of the ventricle template. Our goal was to show that our method could lend support for a robust finding shown in these subjects, namely, between-group differences in increased ventricular expansion over time. While previous methods have examined this question, none have considered time-varying covariate classification in modeling and few have utilized data from more than two time points.

To carry out our analysis, we included in the model the following variables as predictors: intracranial volume (ICV) at each scan, education, sex, age at each scan, and diagnostic status (AD or HC), as well as the two-way interaction between age and diagnostic status. In this set-up, the time-varying covariates included ICV, age, and the interaction between age and diagnostic status, while the time-independent covariates included education, sex, and diagnostic status. As done in previous research, the covariate of ICV was included to correct for individual subjects’ varying brain sizes (e.g.,

Risacher et al. (2010)). By specifying ICV as a time-varying covariate, we were able to examine the longitudinal change of local volume relative to any change in ICV measurement over time. Further information regarding our classification of ICV as time-varying can be found in

Web Appendix G.

The time-dependent covariate of age was clearly a Type I time-varying covariate (i.e., for age, (

*x*_{i1},

*x*_{i2},

*x*_{i3},

*x*_{i4}) was a function of

*x*_{ij} for each of the

*j* = 1, 2, 3, 4 time points) and thus we treated the age by diagnosis interaction as a Type I time-dependent covariate as well. We investigated ICV’s covariate classification using the approach described in Section 3.5 in two steps. First, at each voxel in both the right and left lateral ventricle ROIs, we tested the null hypothesis that ICV was a Type II covariate versus the alternative that it was a Type III covariate. To do so, we obtained two sets of unweighted GMM estimates of model parameters at each voxel using the estimating equations considered valid for the covariates in the two respective models under consideration (one model classified ICV as Type III and the second classified ICV as Type II; the latter utilized an additional 12 moment conditions in estimation). We computed minimands corresponding to both models and obtained a statistic map of

*Q*(

*d*) statistics (see

equation (8)), then thresholded the resulting statistic map according to a

*χ*^{2} critical value with 12 degrees of freedom. In both ROIs, the majority of hypothesis tests were not rejected, indicating that the Type II classification was more appropriate for this covariate than a Type III classification. Subsequently, we tested the null hypothesis that ICV was a Type I covariate against the alternative that it was a Type II. Again, in each of the two ROIs, we failed to reject the null at over 95% of the voxels. Thus, we treated ICV as a Type I covariate for the remainder of our analysis. Our initial assessment of our covariate categorization choice can be found in

Web Appendix H.

After specifying covariate type, we applied the iterative MA-GMM procedure to each voxel within the unsmoothed RAVENS maps separately, as is typical in statistical brain mapping. We obtained estimates for the model’s coefficients and their standard errors at each voxel at the last iteration of the procedure, from which we calculated multivariate Wald test statistics and associated

*p*-values, showing the significance of covariate effects at each voxel. We limited our focus to only those Wald statistic maps which tested for the presence of an age by diagnostic group interaction and generated

*p*-value maps which corresponded to the hypothesis test that the parameter associated with the age by diagnosis interaction at that voxel was equal to zero. In a cluster-level inference approach (as done in

Skup et al. (2011)), we thresholded the maps at

*p*-value ≤ 0.05 with an extent threshold of ≥ 20 contiguous voxels. The primary threshold was set to a significance level of 0.05 in order find a compromise between sensitivity to cluster extent and separation of maximal voxel results. The anatomical localization of each cluster was established by comparing the coordinates of the cluster to a labeled template (

Kabani et al., 1998).

Using a MA-GMM model and inference procedure where ICV was specified as a Type I time-varying covariate, we identified two clusters (a 100 voxel cluster and a 20 voxel cluster) in the right lateral ventricle ROI and one cluster (a 135 voxel cluster) in the left lateral ventricle ROI that resulted in a significant Wald test of interest. Post-hoc analyses at a representative peak voxel within the clusters that spanned the ROIs that showed evidence of an age by diagnosis interaction revealed that AD subjects demonstrated increased ventricle volume expansion compared to HC subjects, thereby confirming that our method could lend support for one of the robust findings reported by previous researchers (e.g.,

Weiner (2008)). depicts the uncorrected −log

_{10}(

*p*) maps corresponding to the two-way interaction in the two ROIs. shows the fitted brain volume average trajectories of AD and HC subjects at the voxel of peak difference.

To investigate the impact that covariate classification had on our final results, we conducted two additional analyses in an identical fashion, but treated ICV as a Type II and a Type III time-varying covariate, respectively. We present these results in

Web Appendix I, which shows that MA-GMM with appropriately classified covariates can detect larger significant activation areas compared to methods that incorrectly classify or do not consider time-varying covariates.