Bayesian Loglinear Regression

  • Ton J. Cleophas
  • Aeilko H. Zwinderman


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

$$ {\displaystyle \begin{array}{l}\mathrm{y}=\mathrm{a}+\mathrm{bx},\mathrm{and}\\ {}\log \kern0.5em \mathrm{y}=\mathrm{a}+\mathrm{bx},\end{array}} $$
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.

Suggested Reading1,2

  1. Statistics applied to clinical studies 5th edition, 2012Google Scholar
  2. Machine learning in medicine a complete overview, 2015Google Scholar
  3. SPSS for starters and 2nd levelers 2nd edition, 2015Google Scholar
  4. Clinical data analysis on a pocket calculator 2nd edition, 2016Google Scholar
  5. Understanding clinical data analysis from published research, 2016Google Scholar
  6. Modern Meta-analysis, 2017Google Scholar
  7. Regression Analysis in Clinical Research, 2018Google Scholar

Copyright information

© 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

Personalised recommendations