This dissertation develops statistical methods for time-conditional survival probability and for equally spaced count data. Time-conditional survival probabilities are an alternative measure of future survival by accounting for time elapsed from diagnosis and are estimated as a ratio of survival probabilities. In Chapter 2, we derive the asymptotic distribution of a vector of nonparametric estimators and use weighted least squares methodology for the analysis of time-conditional survival probabilities. We show that the proposed test statistics for evaluating the relationship between time-conditional survival probabilities and additional time survived have central Chi-Square distributions under the null hypotheses. Further, we conducted simulation studies to assess the empirical probability of making a type I error for one of the hypotheses tests developed and to assess the power of the various models and statistics proposed. Additionally, we used weighted least squares techniques to fit regression models for the log time-conditional survival probabilities as a function of time survived after diagnosis to address clinically relevant questions. In Chapter 3, we derive the asymptotic distribution of time-conditional survival probability estimators from a Weibull parametric regression model and from a Logistic-Weibull cure model, adjusting for continuous covariates. We implement the weighted least squares methodology to assess relevant hypotheses. We create a statistical framework for investigating time-conditional survival probability by developing additional methodological approaches to address the relationship between estimated time-conditional survival probabilities, time survived, and patient prognostic factors. Over-dispersed count data are often encountered in longitudinal studies. In Chapter 4, we implement a maximum-likelihood based method for the analysis of equally spaced longitudinal count data with over-dispersion. The key features of this approach are first-order antedependence and linearity of the conditional expectations. We also assume a Markovian model of first order, implying that the value of an outcome on a subject at a specific measurement occasion only depends on the value at the previous measurement occasion. Our maximum likelihood approach using the Poisson model for count data benefits from a simple interpretation of regression parameters, like that in GEE analysis of count data.