Binary and Ordinal Outcomes: Considerations for the Generalized Linear Model with the Log Link and with the Identity Link
Abstract
There are gaps in the current literature on Generalized Linear Models (GLM) for binary outcomes with the log link. This dissertation explores a number of these gaps and presents specific results:
(1) Uniqueness considerations for the Maximum Likelihood Estimate (MLE) are established from the conditions for the strict concavity of the log-likelihood. The full column rank of certain subsets of the covariate matrix is shown to be a condition for the strict concavity of the loglikelihood.
(2) Conditions are established for the finiteness of components of the MLE. A method is proposed to address the possibility of non-finite components for the MLE, and it is based on determining directions of recession of the log-likelihood. In addition, it is established when the MLE will be in the interior of the parameter space and when the MLE will possibly be on a boundary of the parameter space.
(3) Examples are presented of closed form expressions for the MLE. For a number of models with indicator variables and measured variables, closed form expressions for the MLE are presented.
(4) There are considerations for the construction of confidence intervals when the MLE is close to a boundary of the parameter space. A new metric, called the “fraction within the parameter space”, is introduced for assessing intervals for MLEs close to a boundary. A simulation study is provided that offers support for Bootstrap Percentile Intervals having larger fractions when compared to Relative Likelihood Intervals and Normal Confidence Intervals.
This dissertation continues by developing a proportional probability model using the log link for ordinal outcomes. For this model, similar results are presented for topics (1) and (3) above. In addition, there is the introduction of a score test to assess proportionality.
The dissertation concludes with a discussion of future work. In particular, this discussion includes some preliminary work with the identity link GLM for binary and ordinal outcomes.
Throughout this dissertation, there are many practical considerations and illustrations presented. The use of the log link and the identity link for binary and ordinal outcomes should now become a viable modeling option for researchers.
Description
Keywords
Biostatistics, Mathematics, Statistics
Citation
Singh, G. (2017). Binary and Ordinal Outcomes: Considerations for the Generalized Linear Model with the Log Link and with the Identity Link (Doctoral thesis, University of Calgary, Calgary, Canada). Retrieved from https://prism.ucalgary.ca. doi:10.11575/PRISM/25118