JangDong Seo


Shafie, Khalil

Committee Member

Schaffer, Jay Ryan, 1969-

Committee Member

Lalonde, Trent L.


Applied Statistics and Research Methods


University of Northern Colorado

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Greeley (Colo.)


University of Northern Colorado

Date Created





140 pages

Digital Origin

Born digital


In longitudinal data analyses, it is commonly assumed that time intervals for collecting outcomes are predetermined -- the same across all subjects -- and have no information regarding the measured variables. However, in practice researchers might occasionally have irregular time intervals and informative time, which violate the above assumptions. Hence, if traditional statistical methods are used for this situation, the results would be biased. In this study, as a solution, joint models of longitudinal outcomes and informative time are presented by using joint probability distributions, incorporating the relationships between outcomes and time. The joint models are designed to handle outcome distributions from a member of the exponential family of distributions with informative time following an exponential distribution. For instance, the Poisson probability density function is combined with the exponential distribution for count data, as well as the relations between outcomes and time; the Bernoulli probability density function is combined for binary data; and the Gamma probability density function is combined when the outcome is waiting time or survival time. The maximum likelihood parameter estimates of the joint model are found by using a nonlinear optimization method, and the asymptotic behaviors of the estimators are studied. Moreover, the likelihood ratio test statistic is computed for comparing nested models, and the model selection criteria, such as AIC, AICc, BIC, are found as well. Through simulation studies, the maximum likelihood parameter estimates of the joint models appeared to be multivariate normal as the number of observations increased. As a result, the likelihood ratio test statistic could be utilized for model comparisons since the asymptotic normality of the maximum likelihood estimators has been varied. Also, AIC, AICc, and BIC scores were calculated as model selection criteria. Furthermore, the computing package using R was developed to handle the joint models and used to analyze the bladder cancer data for demonstration purposes.


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Copyright is held by author.