Multiplicative Effect Modeling

Abstract

Generalized linear models, such as logistic regression, are widely used to model the association between a treatment and a binary outcome as a function of baseline covariates. However, the coefficients of a logistic regression model correspond to log odds ratios, while subject-matter scientists are often interested in relative risks. Although odds ratios are sometimes used to approximate relative risks, this approximation is appropriate only when the outcome of interest is rare for all levels of the covariates. Poisson regressions do measure multiplicative treatment effects including relative risks, but with a binary outcome not all combinations of parameters lead to fitted means that are between zero and one. Enforcing this constraint makes the parameters variation dependent, which is undesirable for modeling, estimation and computation. Focusing on the special case where the treatment is also binary, Richardson et al. (2017) propose a novel binomial regression model, that allows direct modeling of the relative risk. The model uses a log odds-product nuisance model leading to variation independent parameter spaces. However, their method is restricted to binary treatments. My research presents general approaches to modeling the multiplicative effect of a continuous or categorical treatment on a binary outcome. We also attempt to develop a method to estimate relative risks in a meta-analysis including cohort and case-control studies. In Chapter 1, we introduce the relative risk modeling of the binary treatment case in Richardson et al. (2017). In their work, they have proposed a nuisance model for the log of the odds product, which is the product of two odds of the treatments. The odds product is variation independent from the primary of interest, relative risk, thus this leads to a valid probability distribution. In Chapter 2, we introduce a new approach which imposes an assumption that the relative risk is a monotone function of an ordinal treatment. This assumption is reasonable in many real-life situations, such as the recovery probability in the arm receiving full-dosage is often higher than it in the small-dosage arm. Furthermore, having the relative risks and the odds product of the lowest and highest level of treatment, we are able to have a valid probability distribution too. We also provide simulations to demonstrate our proposed method and compare the performance with other methods. In Chapter 3, we propose another new method for the case where the relative risk is not monotonic in treatment. We introduce the generalized odds product as the nuisance model, which is the product of odds for all the levels of the treatments. Simulation and model comparison are also provided. In Chapter 4, we illustrate the use of our proposed methods in Chapter 2 and 3 by studying the association between the passenger class and death/survival status in the Titanic data. We also compare the results from our proposed models with those obtained from the generalized linear models: Poisson and logistic regression. In Chapter 5, we propose a novel method to estimate the relative risk in a meta-analysis including cohort and case-control studies. A case-control is often conducted for rare diseases, however it cannot determine a relative risk because the prevalence of disease is set by the study design. Cohort studies can provide information on prevalence, while for rare events they cannot collect sufficient cases. In this chapter, we combine cohort and case-control studies to have a better estimation of the relative risk. Monte Carlo simulations demonstrate the superior performance of our proposed method. We also apply the method to the National Longitudinal Survey of Youth Data to study the relative risk of smoking on people's heath.