An application of longitudinal analysis with skewed outcomes

Abstract

Background: Longitudinal designs are indispensable to the study of change in outcomes over time and have an important role in health, social, and behavioral sciences. However, these designs present statistical challenges particularly related to accounting for the variance and covariance of the repeated measurements on the same participants and to modeling outcomes that are not normally distributed.

Objectives: The purpose of this study was to introduce a general methodology for longitudinal designs to address these statistical challenges and to present an example of an analysis conducted with data collected in a randomized clinical trial. In this example, the outcome of interest-monthly health-related out-of-pocket expenses incurred by breast cancer survivors-had a skewed distribution.

Methods: Common statistical approaches are for longitudinal analysis using linear and generalized linear mixed models are reviewed, and the discussed methods are applied to analyze monthly health-related out-of-pocket expenses.

Discussion: Although standard statistical software is available to conduct longitudinal analyses, training is necessary to understand and to take advantage of the various options available for model fitting. However, knowledge of the basics of the methodology allows assimilation and incorporation into practice of evidence from the numerous studies that use these designs.

Figure 1

Illustration of the use of the three basic parameters: time, group, and time by group, in a longitudinal model comparing two groups over time. (A) The two groups begin at similar levels, but separation occurs over time. Significance is expected in the time by group interaction parameter alone. (B) Both groups change over time, but there is no separation between them. Significance is expected in the time parameter alone. (C) The groups do not change for the duration of the study, but separation is constant over time. Significance is expected in the group parameter alone. (D) Separation is present at the beginning of the study and remains over time. Group 1 changes over time, while Group 2 remains constant. Significance is expected for the group and time by group parameters.

Figure 2. Histograms of monthly health-related out of pocket expenses in dollars by time point (n = 121)

Note the highly right-skewed distribution of the data at each time point

Figure 3. Assorted shapes of the Gamma distribution

The form of the distribution’s curve depends on shape and scale parameters. The mean and variance are also functions of these two parameters. In a Generalized Linear Model using the Gamma distribution, the shape of the curve would be estimated through maximum likelihood methods.