Blending the New Statistics with Mixture Modeling -- A ROPE-based single-block Gibbs sampler for Bayesian t-tests
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Testing the difference of means between two groups is one of the oldest problems in hypothesis testing. Usual solutions like the two-sample t-test rely on p-values. Bayesian alternatives were introduced recently by Gonen et al. (2005), building on the Bayes factor as introduced by Jeffreys (1939). These were improved further by Rouder et al. (2009), Wetzels et al. (2009), Ly et al. (2016a) with a further generalisation from Wang and Liu (2016). All of these solutions are based on Bayes factors, the problems of which have already been detailed in Kamary et al. (2014) and Robert (2016). By blending the shift of hypothesis testing to estimation under uncertainty -- titled the Bayesian New Statistics by Kruschke and Liddell (2018) -- with mixture estimation, this paper introduces a new Bayesian t-test by interpreting the underlying model as a two-component Gaussian mixture in which the effect size is the quantity of interest. A Gibbs sampler for the posterior distribution of δ is derived and combined with the ROPE-criterion for acceptance and rejection of parameter values in Bayesian estimation. In a simulation study the performance of the Gibbs sampler is tested for simulated data with small, medium and large effect sizes, showing promising results.
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