First Advisor

Jong Sung Kim

Date of Publication

Summer 7-16-2019

Document Type


Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences






Longitudinal method, Statistics



Physical Description

1 online resource (xi, 72 pages)


In longitudinal studies, we observe the subjects who are likely to progress to a new state during the study time. For example, in clinical trials the stage of a progressing disease is recorded at each follow-up visit. The primary goal is to estimate the relationship between the attributes and the subject's progression state. In such studies, some subjects complete all their follow-up visits and their progression state are observed without any missingness. However, others miss their follow-up visits and when they come back, they learn that they have progressed to a new state. In this case, not only are their progression states at each follow-up interval-censored, but their time-dependent covariates are incomplete. In such studies, the observations are missing at random (MAR).

The event of interest, i.e., progression, may have several possible patterns. In some studies, we might be studying progression to only one new state. For example, we are interested in studying the attributes that affect an individual's progression from being a non-smoker to a frequent smoker. Another example would be the patients who are believed to have high risk for developing diabetes, are monitored for advancing to type 2 diabetes. In other studies, the event of interest involves multiple stages. Examples of these studies include several stages of cancer, or different stages of smoking (nonsmoker, light smoker, intermittent smoker, heavy smoker, etc.). These states are chronological.

The times of observation, i.e., follow-up interview visits, are pre-specified for these studies. At each time point, the attributes are measured and recorded. Since the study continues over time, it is common for some subjects to miss their follow-up visits. In this case not only the outcome (event of interest) is censored, but their time-dependent attributes are incomplete. In this case, both outcome and attributes need to be estimated for the missed visits.

We are interested in studying the time-dependent covariates' effect on the progression. Expectation-maximization (EM) algorithm is used for estimating the parameters. The variance-covariance matrix of the maximum likelihood estimator (MLE) is calculated using the missing information principles. Simulation studies revealed that the proposed method works well in terms of variance, bias, and power in the samples of moderate sizes.

When we are estimating the association between longitudinal covariates and an event, we may run into the large number of attributes, which are explanatory but could be highly correlated. Using the usual maximum likelihood estimation method leads to inaccurate parameter estimates. Additionally, the estimators have large variance. Elliot, et al., [14] proposed Mixed Ridge Regression when the outcome of the process is continuous. This method applies ridge regression to a linear mixed effects longitudinal model. In our proposed model, the longitudinal outcome is binary. We apply ridge penalization (based on the L2 norm) to our model to get more accurate parameter estimates.

Another important aspect in building a good predictive model is variable selection. Sometimes there are many attributes in a dataset. These attributes are not necessarily correlated. We are interested in choosing a smallest best subset of them for inference. We perform the variable selection by adding the LASSO penalization (based on the L1 norm) to the likelihood to be able to simultaneously choose the appropriate covariates and estimate the covariate effects.

Lastly, the preliminary model is extended to the case when there are more than one progression states in the model. These progressions are chronological and assumed to be non-time-reversible. Missing pattern is more complex than that for one progression state case, but the rest of the procedures are pretty similar to those for one progression state case.


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