Bayesian Statistics for Experimental Scientists: A General Introduction Using Distribution-Free Methods by Richard A. Chechile

Author:Richard A. Chechile [Chechile, Richard A.]
Language: eng
Format: epub
Tags: Mathematics, Probability & Statistics, Bayesian Analysis, General
ISBN: 9780262044585
Google: Ur34DwAAQBAJ
Publisher: MIT Press
Published: 2020-09-08T00:26:14.018551+00:00

Six people were correct for both recall tests; whereas 43 participants were incorrect for both tests. The two cell frequencies in the contingency table where the participants were correct for only one of the two trials are the basis for the Bayesian analysis (i.e., the frequency counts of 21 and 2). Given a uniform prior distribution for the two cells associated with the McNemar test, the posterior distribution for the ϕRB parameter is a beta distribution with the shape parameters of a′′ = 22 and b′′ = 3. The posterior mean RB is . Thus the estimated effect size = | − .5| is equal to .38. By using the R qbeta() command, the two-equal-tail 95-percent interval is found to be (.7300, .9734). For the null hypothesis of H0: ϕRB ≤.5 along with the alternative hypothesis of H1: ϕRB >.5, the Bayes factor BF10 is found via equation (2.14) to be 55, 737. Thus the alternative hypothesis is overwhelmingly favored. Clearly, the repeated use of a common word category results in proactive interference.

Although Bayesian statisticians typically eschew testing a point-null hypothesis, there may be occasions where either the client or a persistent journal editor insists that the point-null hypothesis must be statistically assessed. Thus for the 2 × 2 RB contingency table, the null hypothesis of H0: ϕRB = .5 might be of interest to some parties. Of course, this hypothesis has a probability measure of 0, but as shown in chapter 2, it nonetheless has a Bayes factor. For the above memory experiment, the Bayes factor BF10 for the testing of this sharp-null hypothesis is computed via theorem (2.6). The result of this calculation is 1, 382. One might ask: what does this large Bayes factor in support of the alternative hypothesis mean. A sardonic statistician might give the following answer to that question: “The probability of the null hypothesis that ϕRB is equal to .5 had initially a probability of 0 for being correct. It was dead from the outset. But now that data have been collected, and there was a formal statistical test of the hypothesis, it is really dead! In fact, it is in a state that is worse than death.”

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