Researchers need methods for evaluating whether statistical results are worthy of interpretation. Likelihood functions contain large amounts of information regarding the support for differing estimates. However, maximum likelihood estimates (MLE) are typically the only set of estimates interpreted. Previous research has indicated that these alternative estimates can often be computed and represent data approximately as well as their MLE counterparts. The close fit between these alternative estimates are said to make them fungible. While similar in fit, fungible estimates are in some cases different enough (from the MLE) that they would support alternative substantive interpretations of the data. By calculating fungible parameter estimates (FPEs) one can either strengthen or weaken one’s inference by exploring the degree in which diverging estimates are supported.

This dissertation has two contributions. First, it proposes a new method for generating FPEs under a broader definition of what should constitute fungible parameter estimates. This method allows for flexible computation of FPEs. Second, this method allows for an exploration of research inquiries that have been largely unexplored. What are the circumstances in which FPEs would convey uncertainty in the parameter estimates? That is, what are the causes of uncertainty that are measured by FPEs. Understanding the causes of this uncertainty are important for utilizing FPEs in practice.

This dissertation uses a simulation study in order to investigate several factors that might be encountered in applied data analytic scenarios and affect the range of fungible parameter estimates including model misfit. The results of this study indicate the importance of interactions when examining FPEs. For some conditions, FPE ranges indicate that there was less uncertainty when the model was correctly specified. Under alternative conditions, FPE ranges suggest greater uncertainty for the correctly specified model. This example is mirrored in several results that suggest that a simple prediction of the level of uncertainty is difficult for likelihoods characterizing real world modeling scenarios.