Abstract
A Bayes factor is introduced for the normal linear regression model, which can be used to estimate bounds of the treatment effect on the dependent variable, from the data. This is done while accounting for hidden omitted-variable bias, due to an unobserved covariate, and adjusting for any other observed covariates. The Bayes factor measures how much the data have changed the odds for some specified hidden bias versus no hidden bias, and is defined by a ratio of residual sums-of-squares raised to a power proportional to half the sample size. Therefore, the estimated bounds for the treatment effect can be determined by values of the hidden bias parameter that attain non-small Bayes factors, while the Bayes factor can be quickly computed in closed-form. The Bayes factor is illustrated through the analysis of real data and simulated data sets. Software code for the Bayes factor method is provided as Supplemental Material (available upon request of the author).
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