Distributional Characterization of ICV-Normalized Regional Brain Volumes in Alzheimer’s Disease

Introduction: Alzheimer’s disease (AD) has progressive structural brain changes that can be described using magnetic resonance imaging (MRI). Previous work has used several statistical approaches to compare the volume of brain regions such as the hippocampus, entorhinal cortex, and amygdala. However, many of these approaches assume a Normal or symmetric distribution of regional volumes, which may not fully capture the variance and regional morphological variability of observed brain volumes. This study compares different probability distributions to assess the distributional flexibility needed to characterize brain volumes normalized to intracranial volume (ICV). Method: Six candidate probability distributions were evaluated across 21 FreeSurfer-derived, ICV-normalized regional brain volumes classified by sex and diagnosis group (Female-CN, Male-CN, Female-AD, and Male-AD). After ICV quality control, data were drawn from 3409 T1-weighted 3T MRI scans of 1343 participants in the Alzheimer’s Disease Neuroimaging Initiative (ADNI); the primary analysis used one baseline scan per participant. The resulting 84 region-by-group combinations (21 regions × 4 groups) were fitted using maximum likelihood, with all four parameters of the four-parameter Exponentiated Weibull (Weibull4P) estimated by multistart, ridge-stabilized maximum likelihood with identifiability safeguards. Distributions were compared with the Akaike information criterion (AIC), its small-sample correction (AICc) and the Bayesian information criterion (BIC). A distribution was regarded as adequate relative to other candidates when the AIC was within 10 units of the lowest AIC. Absolute fit was assessed with parametric bootstrap goodness-of-fit tests and robustness was assessed with a participant-level bootstrap, five-fold cross-validation of all six distributions and age-stratified refits. Results: The Normal distribution gave the lowest AIC in 40 of the 84 combinations, Gamma in 21, Lognormal in 12, Weibull4P in 10 and the two-parameter Weibull in one. Two or more distributions were within 2 AIC units of the lowest AIC in 51 combinations. Weibull4P was adequate by the relative AIC (ΔAIC < 10) in all 84 combinations, compared with 74 for Gamma, 68 for Normal and 66 for Lognormal; its largest difference from the lowest AIC was 4.9. In the 12 ventricular combinations, Normal was adequate in only one, compared with all 12 for Weibull4P. Weibull4P retained relative AIC support in at least 95% of the 500 participant-bootstrap replicates in 83 combinations and in all three age bands in every combination, including refits that did not meet StablePass. It met the descriptive BIC threshold in 53 combinations and the cross-validation threshold in 55. In the parametric-bootstrap goodness-of-fit tests, Weibull4P was not rejected by the uncorrected Anderson–Darling test in 71 of the 84 combinations, more than any other candidate (Gamma: 60; Normal: 57). Normality was rejected after FDR correction in 21 combinations, including 10 ventricular combinations. Of the 42 sex-specific AD–CN comparisons, 41 were significant after age adjustment and FDR correction. Conclusions: The Normal distribution most often had the lowest AIC and had relative support in most tissue combinations in this ADNI dataset, but rarely in ventricular combinations. The four-parameter Exponentiated Weibull had the lowest AIC in fewer combinations but was the only candidate with relative AIC support across all regions and groups, including the right-skewed ventricular volumes. It was also rejected less often by the Anderson–Darling goodness-of-fit test than any other candidate. However, its additional parameters were less supported by BIC, and its estimated lower bound restricted its out-of-sample performance. These findings support an empirical approach to evaluating distributional form rather than assuming normality, with Exponentiated Weibull as an alternative to consider alongside simpler distributions.