ABSTRACT
Most models for the odds ratio typically assume an additive model on the log scale so that effects of covariates are multiplicative on the odds ratio. Standard models for the hazard ratio, rate ratio, or odds ratio are typically of the form e x T β $ e^{\mathbf{x}^{ \mathrm T } \boldsymbol{\beta }} $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315154084/5e5abef9-a934-4c3a-a7bb-f3e6e5296d03/content/inline-math11_1.tif"/> that implies a multiplicative effect among risk factors denoted by the vector included in the model. Multiplicative models are simple to interpret and behave well in model fitting when data are not sparse. However, epidemiologists argue that such models do not reflect public health impact or underlying biology (i.e., they might lack biological plausibility) and may not adequately describe the interactions of two risk factors on outcome (Walker and Rothman, 1982); Greenland:83; FigueroaEtAl:14; Moonsinghe:11; HanEtAl:12. Consequently, alternative models for risk, risk ratios, or odds ratios have been suggested to represent more realistic models for excess risk associated with risk factors or their interaction. The simplest alternative model is the linear model for risk of the form 1 + x T β $ 1+ \mathbf{x}^{ \mathrm T } \boldsymbol{\beta } $ https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315154084/5e5abef9-a934-4c3a-a7bb-f3e6e5296d03/content/inline-math11_2.tif"/> so that each risk factor acts additively on the risk. However, such models have constraints on estimability as well as poor statistical properties. In this chapter, we consider alternative models for the odds ratio, rate ratio, hazard ratio, or probability of an event. We explore linear models as well as hybrid mixture models that address some of the concerns. Estimation of absolute risk is possible from stratified case-control studies when the sampling fraction is known. As an example we consider results from a full cohort estimation of breast cancer risk following a screening mammogram. We show that sampling controls with known sampling fraction can 208produce comparable absolute risk predictions, risk differences, or odds ratios depending on the model form chosen.
