# Bayesian Regressions

## Abstract

In studies with two unpaired samples of continuous data as outcome and a binary predictor like treatment modality the difference of the two means and their pooled standard error is usually compared to zero. Instead of the above tests a Bayesian regression analysis is possible.

A traditional regression analysis with a single binary predictor provided a linear correlation coefficient of 0.643, p-value = 0.002, and a linear regression coefficient of −1.720, t = −3.558, p-value = 0.002. A Bayesian regression of these data did not compute “supporting null versus alternative hypothesis” but, in contrast, the other ways around, and it provided support in favor of the traditional test with a Bayes factor of 17.329.

A traditional regression analysis with multiple binary predictors provided a linear correlation coefficient of 0.669, p-value = 0.021, and only one significant linear regression coefficients of −1.642, t = −3.223, p-value = 0.005. A Bayesian regression of these data did again not compute “supporting null versus alternative hypothesis” but, in contrast, the other ways around, and it provided support in favor of the traditional test with a Bayes factor of 1.626.

A traditional regression analysis with a single continuous predictor provided a linear correlation coefficient of 0.975, p-value = 0.000, and a linear regression coefficient of 0.093, t = 18.443, p-value = 0.000. A Bayesian regression of these data again did not compute “supporting null versus alternative hypothesis” but, in contrast, the other ways around, and it provided support in favor of the traditional test with a Bayes factor of 2.284^{10}.

A disadvantage of Bayesian analysis may be an increased risk of overfitting, but in the regression examples in this chapter this was virtually not observed.

## Suggested Reading^{1}^{,}^{2}

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