This thesis is concerned with quantification of dose-response relationships for continuous exposures using causal inference methods based on the potential outcomes framework. The thesis applies and develops causal techniques in a longitudinal case study of the effect of area-based deprivation on the incidence of Child Pedestrian Casualties (CPCs) in Britain. There are four main contributory chapters of the thesis. The first investigates the link between socioeconomic deprivation and the incidence of CPCs using spatial longitudinal Generalized Linear Mixed Models (GLMMs) estimated using frequentist and Bayesian approaches. The chapter addresses issues of confounding, spatial dependence, and transmission of deprivation effects across zones (i.e. interference). The second analytical chapter develops a flexible mixed model propensity score (PS) approach to estimate average treatment effects (ATEs). Analytical results and simulations show that such an approach can be useful in capturing unobserved confounding but can also involve more extensive conditioning than is strictly necessary for causal comparison, with implications for viable sample selection and for the efficiency of ATE estimation. The third contribution constructs a semiparametric double robust (DR) estimator for continuous treatment effects. This is achieved by augmenting an outcome regression (OR) model with inverse PS covariates to obtain DR point estimates of ATEs at various doses. A penalized spline regression is then fitted to these point estimates to derive a semiparametric approximation to the dose-response. Simulations show that this DR model can provide a good approximation to linear or nonlinear dose-response functions under various sources of misspecification of the OR or PS models. Finally, the thesis proposes an approximate Bayesian version of the DR estimator. This is achieved by specifying Dirichlet prior weights for the parameters of the PS augmented OR model, and by repeatedly estimating the weighted model to build up posterior predictive distributions that are shown to be DR for ATEs.