dfba_mcnemar {DFBA} | R Documentation |

## 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 |

`b0` |
The second shape parameter for the prior beta distribution for the |

`prob_interval` |
Desired probability for interval estimates for |

### 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 |

`a0` |
The first shape parameter for the prior beta distribution for the |

`b0` |
The second shape parameter for the prior beta distribution for the |

`a_post` |
First shape parameter for the posterior beta distribution for the |

`b_post` |
Second shape parameter for the posterior beta distribution for the |

`post_mean` |
Posterior mean for |

`post_median` |
Posterior median for |

`eti_lower` |
Lower limit for the posterior equal-tail interval estimate for |

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

`BF10point` |
The Bayes factor against the point null hypothesis that |

`BF10interval` |
The Bayes factor against the interval null hypothesis that |

`postH1` |
The posterior probability that |

### 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)
```

*DFBA*version 0.1.0 Index]