dfba_median_test {DFBA} | R Documentation |

## Bayesian Median Test

### Description

Given two independent groups of continuous variables, performs a Bayesian analysis of the likelihood of observing an above-median value from one of the groups relative to expectation.

### Usage

```
dfba_median_test(E, C, a0 = 1, b0 = 1)
```

### Arguments

`E` |
Numeric vector of values for the continuous measurements for group 1 (generically denoted |

`C` |
Numeric vector of values for the continuous measurements for group 2 (generically denoted |

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

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

### Details

Given continuous measurements `E`

and `C`

from two separate and
independent groups, a combined sample median value can be computed. For the
frequentist median test, a 2x2 table is created. Row 1 consists of the
frequencies of the above-median responses in terms of the two groups (*i.e.*,
`nEabove`

and `nCabove`

). Row 2 has the respective frequencies for the
values that are at or below the combined median (*i.e.*, `nEbelow`

and
`nCbelow`

). See Siegel & Castellan (1988) for the details concerning the
frequentist median test.

Chechile (2020) provided an alternative Bayesian analysis for the median-test
procedure of examining continuous data in terms of the categorization of the
values as being either above the combined median or not. The frequencies in
row 1 (above median response) are binomial frequencies in terms of the group
origin (*i.e.*, `E`

versus `C`

). From a Bayesian perspective, a
population-level `\phi`

parameter can be defined for the population
proportion of `E`

values that are above the combined sample median.
Similarly, the frequencies for the scores at or below the combined sample
median can also be examined; in that case, the corresponding population
proportion in the E condition must be `1-\phi`

. Thus, it is sufficient only
to examine the above-median frequencies to make an inference about the `\phi`

parameter. Since this is a binomial problem, the prior and posterior
distributions for the population `\phi`

parameter belong to the beta family
of distributions. The default prior for this function is the uniform
distribution, *i.e*, `a0 = b0 = 1`

. The posterior shape parameters
for `\phi`

are `a_post = a0 + nEabove`

and
`b_post = b0 + nCabove`

.

Because the number of scores in groups `E`

and `C`

might be very
different, it is important to examine the `\phi`

parameter relative to an
expected base-rate value from the sample. For example, suppose that there are
`nE = 90`

values from the `E`

group and `nC = 10`

values from
the `C`

group. In this example, there are 50 scores that are above the
combined median (and no ties that would result in fewer than half of the
scores being greater than the median) that should be examined to see if `\phi`

is greater than 0.9. If there were no difference between the `E`

and `C`

conditions whatsoever in this hypothetical example, then about 90 percent of
the above-median values would be from the `E`

group. If the posterior
`\phi`

parameter were substantially above the group `E`

base rate,
then that would support the hypothesis that group `E`

has larger values
than group `C`

in the population.

The `dfba_median_test()`

provides the descriptive sample information for
the combined median as well as the entries for a table for the frequencies
for the `E`

and `C`

scores that are above the median, as well as the
frequencies for the `E`

and `C`

scores at or below the median. The
function also provides the prior and posterior probabilities that the `E`

and `C`

groups exceeding their respective base rates for a value being
above the median. The function also evaluates the hypotheses that the `E`

and `C`

response rates for the above-median responses exceeding their
base rate. Bayes factors are provided for these hypothesis.

Because the Bayesian median test ignores the available rank-order information, this procedure has less power than the Bayesian Mann-Whitney analysis that can be computed for the same data. Nonetheless, sometimes researchers are interested if condition differences are so strong that even a lower power median test can detect the difference.

### Value

A list containing the following components:

`median` |
The sample combined median for the |

`nE` |
The number of scores from group |

`nC` |
The number of scores from group |

`Ebaserate` |
The proportion nE/(nE+nC) |

`Cbaserate` |
The proportion nC/(nE+nC) |

`nEabove` |
Number of |

`nCabove` |
Number of |

`nEbelow` |
Number of |

`nCbelow` |
Number of |

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

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

`a_post` |
Posterior first shape parameter for the beta distribution for the probability that an above-median response is from the |

`b_post` |
Posterior second shape parameter for the beta distribution for the probability that an above-median response is from the |

`postEhi` |
Posterior probability that an above-median response exceeds the |

`postChi` |
Posterior probabilty that an above-median response exceeds the |

`priorEhi` |
The probability that a beta prior distribution would exceed the |

`priorChi` |
The probability that a beta prior distribution would exceed the |

`BF10E` |
The Bayes factor in favor of the hypothesis that an above-median response from the |

`BF10C` |
The Bayes factor in favor of the hypothesis that an above-median response from the |

### 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.

`dfba_mann_whitney`

for a more powerful alternative Bayesian
analysis of the `E`

and `C`

values that use rank order information.

### Examples

```
## Example with the default uniform prior
group1 <- c(12.90, 10.84, 22.67, 10.64, 10.67, 10.79, 13.55, 10.95, 12.19,
12.76, 10.89, 11.02, 14.27, 13.98, 11.52, 13.49, 11.22, 15.07,
15.74, 19.00)
group2 <- c(4.63, 58.64, 5.07, 4.66, 4.13, 3.92, 3.39, 3.57, 3.56, 3.39)
dfba_median_test(E = group1,
C = group2)
## Example with the Jeffreys prior
dfba_median_test(group1,
group2,
a0 = .5,
b0 = .5)
```

*DFBA*version 0.1.0 Index]