# Bayesian Loglinear Regression

• Ton J. Cleophas
• Aeilko H. Zwinderman
Chapter

## Abstract

In studies with both a binary outcome (for example event yes/no) and binary predictor variable (for example treatment group 1 or 2) for traditional analysis, a 2 × 2 interaction matrix can be drawn with the predictor in two rows and the outcome in two columns. Instead of a 2 × 2 chi-square test also a Bayesian loglinear regression is possible. The mathematical model of a linear regression and loglinear regression are respectively

$$\begin{array}{l}\mathrm{y}=\mathrm{a}+\mathrm{bx},\mathrm{and}\\ {}\log \kern0.5em \mathrm{y}=\mathrm{a}+\mathrm{bx},\end{array}}$$
where log is usually the natural logarithm of the variable y. A traditional 2 × 2 analysis provided a chi-square value of 5.304, with a p-value 0.021. The null hypothesis of no difference between two treatment groups could be rejected. A Bayesian loglinear regression provided support in favor of the traditional test with a Bayes factor (BF) of 0.161. The BF was slightly closer to 0.0 (or very small) than the p-value of 0.021 was to 0.0, and, so, the BF seemed to provide a slightly better test statistic than the p-value did. The confidence intervals of three approaches are underneath.
1. 1.

Gaussian 95% confidence interval 0.148 and 2.624

2. 2.

Bootstrap 95% confidence interval 0.176 and 3.043

3. 3.

Bayesian 95% credible interval 0.176 and 2.528.

Obviously, the Gaussian confidence interval, the bootstrap confidence interval, and the Bayesian credible interval had very much similarly sized intervals. Overfitting was not obvious.

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© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Ton J. Cleophas
• 1
• Aeilko H. Zwinderman
• 2
1. 1.Department Medicine Albert Schweitzer HospitalAlbert Schweitzer HospitalSliedrechtThe Netherlands
2. 2.Department Biostatistics and EpidemiologyAcademic Medical Center Department Biostatistics and EpidemiologyAmsterdamThe Netherlands