Analysis of unbalanced multi-outcome longitudinal data using quasi-least squares in SAS
Many studies consider multiple outcomes that are measured on subjects over time. This manuscript describes our user-written SAS macro, %QLS version 2, which can be used to analyze longitudinal data for multiple outcomes in the framework of general- ized estimating equations (GEE) via application of quasi-least squares (QLS). A spe- cial feature of our software is that it can handle unbalanced data, which result when study investigators planned for an equal number of measurements to be collected on several outcomes on each participant, but some measurements were missed on some sub- jects. For unbalanced data, the working correlation structure for subjects with miss- ing measurements is represented by a sub-matrix of a larger Kronecker product struc- ture that describes the pattern of association among measurements on subjects with complete data. %QLS version 2 is an extended version of our previous macro %QLS version 1 for implementing QLS for a single outcome that can be downloaded from http://www.cceb.upenn.edu/~sratclif/QLSproject.html. %QLS version 2 can im- plement various working correlations structures including the first-order autoregressive (AR(1)); exchangeable; Markov; and tri-diagonal structures for single outcomes; and an additional four correlation structures that are formed by taking Kronecker products be- tween the exchangeable structure and the AR1, exchangeable, Markov, or tri-diagonal structures for multiple outcomes.