Statistical analysis is a vital component of an assay. In an immunoassay, the diagnosis of an illness or the determination of a treatment may be at stake. In this and other assays, it is essential that the assay be analyzed with the greatest accuracy. Standard models for assays tend to have several complicating characteristics which have led to approximate rather than exact evaluation of inferences. The main focus of this thesis is the development of methods that do not compromise the accuracy of the statistical analysis of an assay. For the most part, a Bayesian view is taken. There are many philosophical arguments in favour of the Bayesian approach. However, the involvement of the Bayesian paradigm in this thesis is through necessity rather than philosophy. The frequentist paradigm provides no apparent means of evaluating many of the calculations involved in the analysis of an assay. On the contrary, there is a formal procedure for solving all inference and decision making problems under the Bayesian paradigm. Heterogeneity in the variance of the response is one of the complicating characteristics of assay data. Inferences for heteroscedastic regression models are strongly dependent on the fitted variance function. The estimation of the variance function for an assay is addressed in this thesis. Fitting the assay model and drawing inferences about the parameters is only one side of the statistical analysis of an assay. The other endeavour is the assessment of the quality of the assay as a measuring device. This is needed in order to maintain the quality of the assay over time and potentially for use as a criterion for the determination of an assay's optimal analytical and statistical (i.e. experimental) designs. The assessment of the quality of an assay has also been compromised by frequentist approximations. The development and analysis of a Bayesian model for an assay along with the assessment of the quality of an assay join variance function estimation as the focal points of this thesis.