dfba_mcnemar {DFBA}

Bayesian Repeated-Measures McNemar Test for Change

Description

Given a randomized-block or repeated-measures design where the response is coded as either 0 or 1, examines the subset of cases where there is a change in the response between the two measurements and provides a Bayesian analysis of the population change rate phi_rb (\phi_{rb}) between the two measurements.

Usage

dfba_mcnemar(n_01, n_10, a0 = 1, b0 = 1, prob_interval = 0.95)

Arguments

n_01

The number of cases where the first response is 0 and the second response is 1.

n_10

The number of cases where the first response is 1 and the second response is 0.

a0

The first shape parameter for the prior beta distribution for the phi_rb parameter. Must be positive and finite.

b0

The second shape parameter for the prior beta distribution for the phi_rb parameter. Must be positive and finite.

prob_interval

Desired probability for interval estimates for phi_rb (default is .95).

Details

Sometimes, researchers are interested in the detection of a change in the response rate pre- and post-treatment. The frequentist McNemar test is a nonparametric test that examines the subset of binary categorical responses where the response changes between the two tests (Siegel & Castellan, 1988). The frequentist test assumes the null hypothesis that the change rate is 0.5. Chechile (2020) pointed out that the subset of change cases are binomial data, so a Bayesian analysis can be done for the population response-switching rate \phi_{rb} (styled phi_rb elsewhere in the documentation for this function). Both the prior and posterior distribution for \phi_{rb} are beta distributions.

The user should be aware that the McNemar test is a change-detection assessment of a binary response. To illustrate this fact, consider the hypothetical case of a sample of 50 people who evaluate two political candidates before and after a debate. Suppose 26 people prefer Candidate A both before and after the debate and 14 people prefer Candidate B both before and after the debate, but 9 people switch their preference from Candidate A to Candidate B and 1 person switches their preference from Candidate B to Candidate A. Despite the fact that this sample has 50 participants, it is only the 10 people who switch their preference that are being analyzed with the McNemar test. Among this subset, there is evidence that Candidate B did better on the debate. Overall, support for Candidate A in the whole sample fell from 35 out of 50 (70%) to 27 out of 50 (54%): still a majority, but a smaller one than Candidate A enjoyed prior to the debate.

The dfba_mcnemar() function requires two inputs, n_01 and n_10, which are, respectively, the number of 0 \to 1 changes and the number of 1 \to 0 switches in the binary responses between the two tests. Since the cases where there is a switch are binomial trials, the prior and posterior distributions for \phi_{rb} are beta distributions. The prior distribution shape parameters are a0 and b0. The default prior is a uniform distribution (i.e., a0 = b0 = 1). The prob_interval argument stipulates the probability within the equal-tail interval limits for \phi_{rb}. The default value for that argument is prob_interval =.95.

Besides computing the posterior mean, posterior median, equal-tail interval limits, and the posterior probability that \phi_{rb} > .5, the function also computes two Bayes factor values. One is the point Bayes factor BF10 against the null hypothesis that phi_rb = 0.5. The second Bayes factor BF10 is the interval Bayes factor against the null hypothesis that \phi_{rb} \le 0.5. If the interval Bayes factor BF10 is very low, then there is support to some degree for the null hypothesis that \phi_{rb} < 0.5. In this case the Bayes factor BF01 in support of the interval null hypothesis is given by BF01 = 1/BF10.

Value

A list containing the following components:

n_01

The number of cases where the first response is 0 and the second response is 1

n_10

The number of cases where the first response is 1 and the second response is 0

prob_interval

Desired posterior probability within the equal-tail interval limits for phi_rb

a0

The first shape parameter for the prior beta distribution for the phi_rb parameter

b0

The second shape parameter for the prior beta distribution for the phi_rb parameter

a_post

First shape parameter for the posterior beta distribution for the phi_rb parameter

b_post

Second shape parameter for the posterior beta distribution for the phi_rb parameter

post_mean

Posterior mean for phi_rb

post_median

Posterior median for phi_rb

eti_lower

Lower limit for the posterior equal-tail interval estimate for phi_rb that contains the probability defined in prob_interval

eti_upper

Upper limit for the posterior equal-tail interval estimate for phi_rb that contains the probability defined in prob_interval

BF10point

The Bayes factor against the point null hypothesis that phi_rb = .5

BF10interval

The Bayes factor against the interval null hypothesis that phi_rb is less than or equal to .5

postH1

The posterior probability that phi_rb > .5

References

Chechile, R.A. (2020). Bayesian Statistics for Experimental Scientists: A General Introduction Using Distribution-Free Methods. Cambridge: MIT Press.

Siegel, S., & Castellan, N. J. (1988) Nonparametric Statistics for the Behavioral Sciences. New York: McGraw Hill.

See Also

dfba_beta_bayes_factor for further documentation about the Bayes factor and its interpretation.

Examples

## Examples with default value for a0, b0 and prob_interval

dfba_mcnemar(n_01 = 17,
             n_10 = 2)

## Using the Jeffreys prior and .99 equal-tail interval

dfba_mcnemar(n_01 = 17,
             n_10 = 2,
             a0 = .5,
             b0 = .5,
             prob_interval = .99)