Due to potential violation of standard constraints for the correlation for binary data, it has been argued recently that the working correlation matrix should be viewed as a weight matrix that should not be confused with the true correlation structure. We propose two arguments to support our view to the contrary for the first-order autoregressive AR(1) correlation matrix. First, we prove that the standard constraints are not unduly restrictive for the AR(1) structure that is plausible for longitudinal data; furthermore, for the logit link function the upper boundary value only depends on the regression parameter and the change in covariate values between successive measurements. In addition, for given marginal means and parameter $\alpha$, we provide a general proof that satisfaction of the standard constraints for consecutive marginal means will guarantee the existence of a compatible multivariate distribution with an AR(1) structure. The relative laxity of the standard constraints for the AR(1) structure coupled with the existence of a simple model that yields data with an AR(1) structure bolsters our view that for the AR(1) structure at least, it is appropriate to view this model as a correlation structure versus a weight matrix.
Longitudinal Data Analysis and Time Series
Kim, Hanjoo; Hilbe, Joseph M.; and Shults, Justine, "On the designation of the patterned associations for longitudinal Bernoulli data: weight matrix versus true correlation structure?" (June 2008). UPenn Biostatistics Working Papers. Working Paper 26.