Abstract
In this issue, the author of the article entitled, “Assessing the Longitudinal Change in Low Vision: A Test of Competing Hypotheses,” uses a statistical approach that has not been discussed before in a Statistical Sidebar: growth curve modeling. In reading through the article, some readers might be a bit nonplussed by the equations involved, but this author does a nice job of helping readers parse the meanings of the different equation components.
Before diving into the details, let us take a broad look at growth curve modeling first. It is a useful approach for looking at changes in data over time, especially when that change is not constant or when something changes gradually. In a research design, where data are taken before and after an intervention, change is also measured, but such a design is not sensitive to fluctuations in change. This sort of pre–post design might be fine for a situation in which a researcher expects a relatively sudden change due to the introduction of an experimental variable (e.g., before jumping in a pool you are dry and after you are wet). However, when trying to study changes in a measures over time, all sorts of potentially complicating variables intrude that also change over time and might impact the study.
Rather than grouping participants together and measuring something before and after an intervention in a study, growth curve modeling makes a model of each participant’s data over time, resulting in a trajectory of their data, like a smoothed regression line. This model, then, becomes the unit of measurement in the analysis, instead of a single measure in time or two measures in time. Each participant in a group within a study would then contribute a “growth curve,” and these curves would be grouped together to form the dataset. One reason why using growth curves as a dependent measure is useful (if enough points of data allow curves to be constructed) is that if individuals have missing data, their data will still allow a curve to be constructed. In a pretest–posttest design looking at change over time, if an individual participant is missing either of the two data points, their data are unusable.
This broad description applies more to the type of data collected than the analysis, since there are a host of approaches to analyzing growth curve data. Although there are more complicated analytical approaches, the basic approach to analyzing growth curves is to look at fixed and random effects in the models. A fixed effect represents a value that is in the population (e.g., the average weight of adult men), and a random effect represents the variation around this value. In a study sample, the fixed effect would be the average trajectory of the individual curves, and the random effects would be the variability around this average. So, in a group of growth curves, there would be an average curve fit to the group with individual differences to that average curve.
In the article that used growth curve modeling in this issue, an average curve was fit to different groups of participants, allowing group differences to be studied as well as differences within a group. The dependent variable was the level of vision an individual had. The authors used data from a large-scale national survey that contained information related to vision for many individuals between the years of 2002 and 2014, allowing a longitudinal growth curve to be mapped. These growth curves were fit using a regression model, so that the likelihood of having low vision was predicted by whatever fixed and random effects were entered into the model (e.g., race or ethnicity, sex, and education). These regression models were compared across groups of participants (e.g., those born between 1900 and 1945, versus those born between 1946 and 1964). Using growth curves as dependent measures that were then fit with growth curve models allowed the authors to more precisely track changes in the dependent variable (vision) over time and more precisely account for the impact of different predictor variables on the dependent variable.