Conditional Linear Mixed Models
Verbeke, Geert, Spiessens, Bart, Lesaffre, Emmanuel, The American Statistician
The main advantage of longitudinal studies is that they can distinguish changes over time within individuals (longitudinal effects) from differences among subjects at the start of the study (cross-sectional effects). In observational studies, however, longitudinal changes need to be studied after correction for potential important cross-sectional differences between subjects. It will be shown that, in the context of linear mixed models, the estimation of longitudinal effects may be highly influenced by the assumptions about cross-sectional effects. Furthermore, aspects from conditional and mixture inference will be combined, yielding so-called conditional linear mixed models that allow estimation of longitudinal effects (average trends as well as subject-specific trends), independent of any cross-sectional assumptions. These models will be introduced and justified, and extensively illustrated in the analysis of longitudinal data from 680 participants in the Baltimore Longitudinal Study of Aging.
KEY WORDS: Conditional inference; Cross-sectional effects; Longitudinal effects; Linear mixed model; Longitudinal data; Mixture inference.
In medical science, studies are often designed to investigate changes in a specific parameter which is measured repeatedly over time in the participating subjects. Such studies are in contrast to cross-sectional studies where the response of interest is measured only once for each individual. As pointed out by Diggle, Liang, and Zeger (1994, sec. L4), the main advantage of longitudinal studies is that they can distinguish changes over time within individuals (longitudinal effects) from differences among people in their baseline values (cross-sectional effects).
Consider a randomized longitudinal clinical trial, where subjects are first randomly assigned to one out of a set of treatments, and then followed for a certain period of time during which measurements are taken at prespecified time-points. Treatment effects are then completely represented by differences in evolutions over time; that is, by interactions of treatment with time. The randomization then assures that, at least in large trials, the treatment groups are completely comparable at baseline with respect to factors which potentially influence change afterwards. Hence, a statistical model for such data does not need a cross-sectional model component.
In observational studies, however, subjects may be very heterogeneous at baseline such that longitudinal changes need to be studied after correction for potential confounders such as age, gender, and so on. For example, Diggle, Liang, and Zeger (1994, sec. 9.3) used longitudinal data on 250 children to investigate the evolution of the risk for respiratory infection, and its relation to vitamin A deficiency. They hereby adjusted for factors like gender, season, and age at entry in the study. Pearson et al. (1994) performed a retrospective longitudinal study, the aim of which was to compare evolutions of prostate specific antigen between males with no clinical signs of prostate disease, males with benign prostatic hyperplasia (BPH), and males with local or metastatic prostate cancer. Due to the high prevalence of BPH in men over age 50, it was difficult to find age-matched controls with no evidence of prostate disease. In fact, the control group remained significantly younger on average than the BPH cases, at first visit as well as at the time of diagnosis. They therefore corrected all analyses for age at diagnosis. Brant, Pearson, Morrell, and Verbeke (1992) analyzed repeated measures of systolic blood pressure from 955 healthy males. Their models included cross-sectional effects for age at first visit (linear as well as quadratic effect), obesity, and birth cohort.
All these authors assumed that their response of interest could be well described by a model in which some parameters are common to all participants in the study, while other parameters are subject-specific, representing the natural heterogeneity in the population. …