The Effect of Missing Data on Repeated Measures Models

Researchers involved with longitudinal studies are faced with the problem of trying to get study subjects to return for every follow-up visit. There is always some amount of missing data when looking at these types of studies. The MIXED procedure of the SAS enables examination of correlational structures and variability changes between repeated measurements on experimental units across time. While PROC MIXED has the capacity to handle unbalanced data when the data are missing at random, a question arises as to when the degree of sparseness jeopardizes inference. Simulation is a tool that can be used to answer these types of questions. This paper shows the application of simulation to determine inference problems in a data set with a specific pattern of missing data. This technique is also applied to the topic of initial sample size determination. INTRODUCTION Researchers at the Medical College of Georgia have been collecting data on and studying children from families with a history of hypertension for a number of years. A measurement of interest is the systolic blood pressure (SBP) measurement obtained from a monitor that the child wears for 24 hours. SBP measurements are obtained every 20 minutes from 6am to 10pm and every 30 minutes during the night. Daytime and nighttime means are calculated and used in analysis. Because of the nature of these measurements not all children in the study consent to wear an ambulatory BP monitor and those that consent to wear the monitor do not do so every year of the study. When they do wear the monitor, there may be technical problems which result in an insufficient number of readings for analysis. This resulted in a small and sparse data set when four consecutive years of data were looked at. Table 1 shows the frequency and percent of the 92 children who had at least two of four measurements and the years in which the measurements were obtained. The data set is only 57% complete. Table 1. Unbalanced data structure _______________________________________________ Y_1 Y_2 Y_3 Y_4 Frequency Percent 1 . . 4 14 15.2 1 . 3 . 24 26.1 1 . 3 4 4 4.3 1 2 . . 29 31.5 1 2 . 4 6 6.5 1 2 3 . 13 14.1 1 2 3 4 2 2.2 ________________________________________ When these data were analyzed using PROC MIXED the preferred error variance-covariance (V-C) structure was determined to be compound symmetry (CS). Toeplitz (TOEP) or autoregressive (AR(1)) would seem to more intuitive given the nature of a year separation between measurements. Correlations between measurements separated by more time are expected to be lower than those between measurements obtained closer together. However, since 67 of 92 (73%) of the individuals were represented by only 2 data points it is not surprising that a determination of CS was returned since that is all that could be estimated. PROC MIXED does not impute missing data but uses what data is present for estimation of the error structure of the random effects. We thought that an objective example for the researcher was needed to show the inherent problems associated with such incomplete data. That motivation along with the desire to determine the effect of more reasonable patterns of missing data on error structure estimation prompted us to undertake a simulation project. Simulation was used to investigate the possibility that the inferences obtained from the MIXED analyses might be due to the small sample size and/or sparseness of the data. It was also used to investigate the sample sizes needed to make correct determinations of the assumed underlying structure. SIMULATION The simulation problem was to generate samples of various sizes from a 4-variable multivariate normal distribution with specified mean vector and variance-covariance matrix. In a separate study of SBP in children it was determined that the mean±SD for SBP in each of 4 years was 110±10 mmHg. The correlation between measurements separated by 1 year was .70, 2 years was .60 and 3 years was .48, following a TOEP covariance structure. Thus, the samples to be generated are of the form