Date of Award


Document Type


Degree Name

Master of Science (MS)


College of Science and Mathematics


Mathematical Sciences

Thesis Sponsor/Dissertation Chair/Project Chair

Haiyan Su

Committee Member

Andrew McDougall

Committee Member

Diana Thomas


Longitudinal method, HIV infections--Treatment--Longitudinal studies, Medicine--Research--Statistical methods


Longitudinal and survival data are frequently collected in biomedical studies. The research questions of interest in these studies often require separate analysis of the outcomes. But in many occasions interest also lies in studying their association structures, such as in biomarker research, where the clinical studies are designed to identify biomarkers with strong prognostic capabilities for event time outcomes. In the separate analyses, a linear mixed-effects model is used for modeling the longitudinal data to study the changing trend of the response overtime when controlling some covariates and a survival model is used to model the time-to-event data. A common issue in longitudinal studies is that informative dropout in the data can cause bias in the analysis. Associations between longitudinal and survival data can occur in the explanatory variables or through stochastic dependence between the subject-specific random effect component of the longitudinal model and the survival model. Ignoring the association between the longitudinal and survival data can result in biased inference. The joint model can account for these issues and simultaneously analyze the longitudinal and time-to-event data. This approach enables researchers to obtain more accurate inference regarding the survival probability to certain event when the longitudinal responses associated with the survival response or outcome-dependent study dropout.

In an HIV/AIDS study, our primary interest is to compare the survival for the patients with two antiretroviral drugs, Didanosine (ddl) and Zalcitabine (ddC) with some other risk factors. We also want to determine how the biomarker-CD4 lymphocyte cell counts changed over the period of the study. We use separate analysis and the joint model to analyze the survival and longitudinal outcome and then compare the two analysis results. In the longitudinal analysis, we used a linear mixed-effects model to fit the CD4 cell counts using a random intercept and slope for the observation time. In the survival analysis, we compared the survival between the two treatment groups by using a cox-proportional hazard model. Then a joint model was fitted by using the fitted longitudinal and survival objects. To compare the separate analysis and the joint analysis, we use the Akaike’s Information Criteria (AIC). The joint model was shown to be better than the separate analyses of the longitudinal models and survival models with a smaller AIC value. Using the joint model for inference on the HIV study, Zalcitabine (ddC) was significantly effective in reducing a person’s risk of death. The risk of death was 1.44 times as likely for patients assigned to ddl as compared to the patients assigned to ddC. The previous diagnosis result and observation time were significant predictors of the change in CD4 cell count at a 0.05 significance level. A patient having a previous diagnosis of AIDS at the study entry led to a decrease in CD4 cell counts thus, a patient was more likely to die or the disease progressed. The joint model showed a significant association between the CD4 count and survival: with higher CD4 count, the survival probability is also significantly higher (or the hazard of death is lower). The joint model approach provided more accurate inference than the separate approaches for the HIV study.

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