GMM logistic regression models for longitudinal data with time-dependent covariates and extended classifications

Abstract

When analyzing longitudinal data, it is essential to account both for the correlation inherent from the repeated measures of the responses as well as the correlation realized on account of the feedback created between the responses at a particular time and the predictors at other times. As such one can analyze these data using generalized estimating equation with the independent working correlation. However, because it is essential to include all the appropriate moment conditions as you solve for the regression coefficients, we explore an alternative approach using a generalized method of moments for estimating the coefficients in such data. We develop an approach that makes use of all the valid moment conditions necessary with each time-dependent and time-independent covariate. This approach does not assume that feedback is always present over time, or if present occur at the same degree. Further, we make use of continuously updating generalized method of moments in obtaining estimates. We fit the generalized method of moments logistic regression model with time-dependent covariates using SAS PROC IML and also in R. We used p-values adjusted for multiple correlated tests to determine the appropriate moment conditions for determining the regression coefficients. We examined two datasets for illustrative purposes. We looked at re-hospitalization taken from a Medicare database. We also revisited data regarding the relationship between the body mass index and future morbidity among children in the Philippines. We conducted a simulated study to compare the performances of extended classifications.

Improved methods for the marginal analysis of longitudinal data in the presence of time‐dependent covariates

“…However, because of the inadequacies we found with this approach, we feel further study is warranted. Additionally, Lalonde et al [15] proposed an alternative testing approach for assessing the validity of moment conditions based on tests of correlation between moment conditions at different time points.…”

“…Lalonde et al. defined a type IV time‐dependent covariate, which is the opposite of a type II covariate in that it satisfies Specifically, Y i j given x i j does affect the future time‐dependent covariate process, x i , j +1 ,…, x i T , but the previous covariates have no impact on future outcomes and therefore the feedback cycle is ruled out.…”

“…such that there does exist a feedback cycle in which the covariate value affects the outcome, and that outcome influences future covariate values. Lalonde et al [15] defined a type IV time-dependent covariate, which is the opposite of a type II covariate in that it satisfies…”

Partitioned GMM logistic regression models for longitudinal data

“…As an alternative to the grouping of moments based on covariate type, Lalonde et al introduced a method to ignore the classification and to instead look at the validity of each moment separately. In their individual approach to identifying valid moments, they relied on bivariate correlations to determine validity of the corresponding moment condition.…”

“…Lai and Small and Lalonde et al, among others, have proposed generalized method of moments (GMM) models to account for time‐dependent covariates. These approaches employed various methods for identifying valid moment conditions to be used in the estimation and provide gains in relative efficiency as compared to GEE with the independent working correlation structure .…”