# dfba_mcnemar

    library(DFBA)

# 1 Introduction to the `dfba_mcnemar()` Function

Researchers are sometimes interested in the detection of a change in the response rate pre- and post-treatment. The frequentist McNemar procedure 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 \(0.5\). That is, the frequentist analysis is a \(\chi^2\) test to assess if there is a significant departure from the change rate of \(.5\). Chechile (2020) pointed out that the subset of the change cases is a Bernoulli process, so the Bayesian analysis can be done for the population response-switching rate \(\phi_{rb}\) in the same way as with binomial data. The \(rb\) subscript on the \(\phi\) parameter denotes randomized block, which is a statistical term for the fact that the respondents where randomly sampled but each respondent was measured twice (i.e., within a block). The Bayesian analysis of all Bernoulli processes has a prior and posterior that are beta distributions (see the vignettes on the `dfba_beta_descriptive()`, `dfba_binomial()`, and `dfba_beta_bayes_factor()` functions).

The McNemar test is a change-detection assessment of a binary response. To illustrate, please consider the hypothetical case of a sample of \(50\) people who are evaluating two political candidates before and after a debate. Suppose \(26\) people prefer Candidate A both before and after the debate, \(14\) people prefer Candidate B both before and after the debate, \(9\) people switched their preference from Candidate A to Candidate B, and \(1\) person switched their preference from Candidate B to Candidate A. Despite the fact that this sample has \(50\) participants, it is only the \(10\) people who switched 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 \(50\) (\(54\%\)). Candidate A still has a majority of support among the sample, but by a smaller margin than the one that Candidate A enjoyed prior to the debate. Thus, the statistical inference for the McNemar test is centered on the population change-rate \(\phi_{rb}\) rather than the overall proportional preferences.

# 2 Using the `dfba_mcnemar()` Function

The `dfba_mcnemar()` function has the following five arguments (default values are included for arguments that have them):

- `n_01`

- `n_10`

- `a0 = 1`

- `b0 = 1`

- `prob_interval = .95`

The `n_01` argument is the number of respondents that were initially scored as a \(0\) but switched to a \(1\) score after some treatment or experience. The `n_10` argument is the frequency of respondents that switch from \(1\) to \(0\) All of respondents who did not switch their response are ignored. The analytic focus on only those observations of category change is a central feature of both the frequentist and the Bayesian forms of the McNemar test.

The `a0` and `b0` arguments are the two shape parameters for the prior beta distribution for the \(\phi_{rb}\) parameter. The uniform distribution (i.e., \(a0=b0=1\)) is the default prior for the change-rate parameter. Finally, the `prob_interval` argument is the value for the interval estimates for \(\phi_{rb}\). The default value for is \(.95\).

## 2.1 Example

For example, in a repeated-measures design, the responses \(n=17\) participants indicate category change from \(0\) to \(1\), and the responses of \(2\) participants indicate category change from \(1\) to \(0\). Using the default values of the prior beta shape parameters (`a0 = 1, b0 = 1`) and of the desired probability interval (`prob_interval = .95`):

    dfba_mcnemar(n_01 = 17,
                 n_10 = 2)
    #> Descriptive Statistics
    #> ========================
    #>   Frequencies of a change in 0/1 response between the two tests
    #>   0 to 1 shift        1 to 0 shift
    #>   17                  2
    #>
    #>   Bayesian Analysis
    #> ========================
    #>   Posterior Beta Shape Parameters for Phi_rb
    #>   a_post          b_post
    #>   18              3
    #>   Posterior Point Estimates for Phi_rb
    #>   Mean            Median
    #>   0.857143        0.8685263
    #>   Equal-tail 95% Probability Interval
    #>   Lower Limit     Upper Limit
    #>   0.683017        0.9679291
    #>   Point Bayes factor against null of phi_rb = .5:
    #>   153.3006
    #>   Interval Bayes factor against the null that phi_rb less than or equal to .5:
    #>   4968.555
    #>   Posterior Probability that Phi_rb > .5:
    #>   0.9997988

For this example, there is a high probability – \(.9997988\) – for the hypothesis that \(\phi_{rb}> .5\). Point and interval estimates for \(\phi_{rb}\) are also provided. The Bayes factor against the point-null hypothesis that \(\phi_{rb}=.5\) is \(BF_{10}=153.3006\) (see the vignette for the `dfba_beta_bayes_factor()` function). The output also provides a Bayes factor in support of the interval alternative hypothesis that \(H_1:\,\phi_{rb}>.5\) (i.e., \(BF_{10}=4,968.55\)).

The `plot()` method produces a visualization of the prior and posterior distributions for the \(\phi_{rb}\) parameter. Note: a plot of the posterior distribution without the prior distribution is given by including the argument `plot.prior = FALSE` (the default is `plot.prior = TRUE`).

    plot(dfba_mcnemar(n_01 = 17,
                      n_10 = 2))

In conclusion, the `dfba_mcnemar()` function computes centrality estimates and interval estimates for the population change-rate parameter. Bayes factors for tests of hypotheses about the population parameter are also provided. The function further enables plots of the prior and posterior distributions.

# 3 References

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

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