Data showing a trend that characterizes a change due to a shock to the system are a type of changepoint data, and may be referred to as shock-through data. As a result of the shock, this type of data may exhibit one of two types of transitions: gradual or abrupt. Although shock-through data are of particular interest in many areas of study such as biological, medical, health and environmental applications, previous research has shown that statistical inference from modeling the trend is challenging in the presence of discontinuous derivatives. Further complications arise when we have (1) longitudinal data, and/or (2) samples which come from two potential populations: one with a gradual transition, and the other abrupt. Bent-cable regression is an appealing statistical tool to model shock-through data due to the model's flexibility while being parsimonious with greatly interpretable regression coefficients. It comprises two linear segments (incoming and outgoing) joined by a quadratic bend. In this thesis, we develop extended bent-cable methodology for longitudinal data in a Bayesian framework to account for both types of transitions; inference for the transition type is driven by the data rather than a presumption about the nature of the transition. We describe explicitly the computationally intensive Bayesian implementation of the methodology. Moreover, we describe modeling only one type of transition, which is a special case of this more general model. We demonstrate our methodology by a simulation study, and with two applications: (1) assessing the transition to early hypothermia in a rat model, and (2) understanding CFC-11 trends monitored globally. Our methodology can be further extended at the cost of both theoretical and computational extensiveness. For example, we assume that the two populations mentioned above share common intercept and slopes in the incoming and outgoing phases, an assumption that can be relaxed for instances when intercept and slope parameters could behave differently between populations. In addition to this, we discuss several other directions for future research out of the proposed methodology presented in this thesis.

Data showing a trend that characterizes a change due to a shock to the system are a type of changepoint data, and may be referred to as shock-through data. As a result of the shock, this type of data may exhibit one of two types of transitions: gradual or abrupt. Although shock-through data are of particular interest in many areas of study such as biological, medical, health and environmental applications, previous research has shown that statistical inference from modeling the trend is challenging in the presence of discontinuous derivatives. Further complications arise when we have (1) longitudinal data, and/or (2) samples which come from two potential populations: one with a gradual transition, and the other abrupt. Bent-cable regression is an appealing statistical tool to model shock-through data due to the model's flexibility while being parsimonious with greatly interpretable regression coefficients. It comprises two linear segments (incoming and outgoing) joined by a quadratic bend. In this thesis, we develop extended bent-cable methodology for longitudinal data in a Bayesian framework to account for both types of transitions; inference for the transition type is driven by the data rather than a presumption about the nature of the transition. We describe explicitly the computationally intensive Bayesian implementation of the methodology. Moreover, we describe modeling only one type of transition, which is a special case of this more general model. We demonstrate our methodology by a simulation study, and with two applications: (1) assessing the transition to early hypothermia in a rat model, and (2) understanding CFC-11 trends monitored globally. Our methodology can be further extended at the cost of both theoretical and computational extensiveness. For example, we assume that the two populations mentioned above share common intercept and slopes in the incoming and outgoing phases, an assumption that can be relaxed for instances when intercept and slope parameters could behave differently between populations. In addition to this, we discuss several other directions for future research out of the proposed methodology presented in this thesis.

Segmented regression is a type of nonlinear regression that allows differing functional forms to be fit over different ranges of the explanatory variable. This paper considers the simple segmented regression case of two linear segments that are constrained to meet, often called the linear-linear model. Parameter estimation in the case where the joinpoint between the regimes is unknown can be tricky. Using a simulation study, four estimators for the parameters of the linear-linear model are evaluated. The bias and mean squared error of the estimators are considered under differing parameter combinations and sample sizes. Parameters estimated in the model are the location of the change-point, the slope and intercept of the first segment, the change in slope from the first segment to the second, and the variance over both segments.