Through 4 chapters of Applied Longitudinal Data Analysis (ALDA), the data sets have had the following constraints:

• Balanced – all subjects have the same number of measurements.
• Time structured – all subjects measured at the same time.
• Time-invariant predictors – predictors that do not change over time, such as gender or treatment group.

In chapter 5 these constraints are relaxed. We work with unbalanced datasets with variably-spaced measurements and time-varying predictors. As usual, the UCLA stats consulting site replicates the chapter’s examples in 18 different stats programs. I won’t redo their work, but I will give you my boiled-down-most-important-points that I took away from this chapter. I’ll also show a couple of examples using the lmer() function from the lme4 package.

Section 5.1 Variably Spaced Measurement Occasions
Analyzing data sets with variably spaced measurement occasions is no different than analyzing data sets with identical occasions across individuals (time structured).

Example with unstructured data set (variably spaced measurements)
Data: reading scores recorded at three different times (i.e., 3 waves of data)
Fit two unconditional growth models

reading <- read.table("http://www.ats.ucla.edu/stat/r/examples/
                      alda/data/reading_pp.txt", header=T, sep=",")
mat2 <- reading[ ,3:4]-6.5
dimnames(mat2)[[2]] <- c("agegrp.c", "age.c")  
reading <- cbind(reading, mat2)

library(lme4)
# forcing structure on data
lmer.agegrp <- lmer(piat ~ agegrp.c + (agegrp.c | id), reading, REML = FALSE)
summary(lmer.agegrp)
# using unstructured data
lmer.age <- lmer(piat ~ age.c + (age.c | id), reading, REML = FALSE)
summary(lmer.age)

The first model treats the data as structured. Instead of using child’s precise age, we are using their age classification group (6.5, 8.5, 10.5). The second model uses the child’s precise age. Notice the second model’s lower deviance: 1803 versus 1820. “Treating the unstructured data as though it is time-structured introduces error in the analysis – error that we can reduce by using the child’s age at testing as the temporal predictor.” (p. 145)

Lesson: never force an unstructured data set to be structured.

Section 5.2 Varying Numbers of Measurement Occasions
Section 5.1 concerned varying spacing of measurements. This section concerns varying number of measurements . AKA Unbalanced data. Multilevel modeling allows analysis of data sets with varying numbers of waves of data.

All subjects can contribute to a multilevel model regardless of how many waves of data they contribute. No special procedures are needed to fit a multilevel model to unbalanced data, provided it’s not too unbalanced (i.e., too many people with too few waves with respect to the complexity of your specified model).

Potential Problems with unbalanced data

• The iterative estimation algorithms may not converge. This affects variance components, not fixed effects. “Estimation of variance components requires that enough people have sufficient data to allow quantification of within-person residual variation.” (p. 152)
• Exceeding boundary constraints, such as negative variance components. Your output may have an estimate of 0 to indicate this. Simplifying your model by removing random effects is usually the fix.
• Nonconvergence. This can result from poorly specified models and insufficient data. Can also result from the outcome variable’s scale (too small, make larger) or the temporal predictor’s variable scale (too brief, make longer)

5.3 Time-Varying Predictors

A time-varying predictor is a variable whose values may differ over time. Examples: hours worked per week, money earned per year, employment status. No special strategies are needed to include a time-varying predictor in a multilevel model.

Examples with time-varying predictor
Data: depression scores (cesd) for unemployed; status of employment (unemp; 0 or 1) changes over time

unemployment <- read.table("http://www.ats.ucla.edu/stat/r/examples/
                           alda/data/unemployment_pp.txt", header=T, sep=",")

# time-varying predictor is unemp
lmer.unb <- lmer(cesd ~ months + unemp + (months | id), 
                 unemployment, REML = FALSE)
summary(lmer.unb)

# allow effect of time-varying predictor (unemp) to vary over time
lmer.unc <- lmer(cesd ~ months + unemp*months + (months | id), 
                 unemployment, REML = FALSE)
summary(lmer.unc)

# constant slope for unemp=0, changing slope for unemp=1
lmer.und <- lmer(cesd ~ unemp + unemp:months + (unemp + unemp:months | id), 
                 unemployment, REML = FALSE)
summary(lmer.und)

5.3 Recentering the Effect of Time

Recentering time can produce interpretive advantages such as an intercept that represents initial status. Time can also be recentered in such a way to produce an intercept that represents final status. This is useful when final status is of special concern. Changes in recentering produce different intercept parameters but leave slope and deviance statistics unchanged. It can also lead to an intercept being significant when it previously was not (and vice versa).