dfba_wilcoxon {DFBA} | R Documentation |

## Repeated-Measures Test (Wilcoxon Signed-Ranks Test)

### Description

Given two continuous, paired variates `Y1`

and `Y2`

,
computes the sample `T_pos`

and `T_neg`

statistics for the Wilcoxon
signed-rank test and provides a Bayesian analysis for the population
sign-bias parameter `phi_w`

, which is the population proportion of
positive differences.

### Usage

```
dfba_wilcoxon(
Y1,
Y2,
a0 = 1,
b0 = 1,
prob_interval = 0.95,
samples = 30000,
method = NULL,
hide_progress = FALSE
)
```

### Arguments

`Y1` |
Numeric vector for one continuous variate |

`Y2` |
Numeric vector for values paired with Y1 variate |

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

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

`prob_interval` |
Desired probability for interval estimates of the sign bias parameter |

`samples` |
When |

`method` |
(Optional) The method option is either |

`hide_progress` |
(Optional) If |

### Details

The Wilcoxon signed-rank test is the frequentist nonparametric counterpart to
the paired *t*-test. The procedure is based on the rank of the difference
scores *d* = `Y1`

- `Y2`

.
The ranking is initially done on the absolute value of the nonzero `d`

values,
and each rank is then multiplied by the sign of the difference. Differences
equal to zero are dropped. Since the procedure is based on only ranks of the
differences, it is robust with respect to outliers in either the `Y1`

or
`Y2`

measures. The procedure does not depend on the assumption of a
normal distribution for the two continuous variates.

The sample `T_pos`

statistic is the sum of the ranks that have a positive
sign, whereas `T_neg`

is the positive sum of the ranks that have a
negative value. Given *n* nonzero *d* scores, `T_pos`

+
`T_neg`

= *n*(*n* + 1)/2. Tied ranks are possible, especially
when there are `Y1`

and `Y2`

values that have low precision. In
such cases, the Wilcoxon statistics are rounded to the nearest integer.

The Bayesian analysis is based on a parameter `phi_w`

, which is the
population proportion for positive *d* scores. The default prior for `phi_w`

is a flat beta distribution with shape parameters `a0`

= `b0`

=1,
but the user can stipulate their preferred beta prior by assigning values for
`a0`

and `b0`

. The `prob_interval`

input, which has a default
value of .95, is the value for interval estimates for the `phi_w`

parameter, but the user can alter this value if they prefer.

There are two cases for the Bayesian analysis - one for a small number of
pairs and another for when there is a large number of pairs. The `method = small`

sample algorithm uses a discrete approximation where there are 200 candidate
values for phi_w, which are .0025 to .9975 in steps of .005. For each
candidate value for `phi_w`

, there is a prior and posterior probability.
The posterior probability is based on Monte Carlo sampling to approximate the
likelihood for obtaining the observed Wilcoxon statistics. That is, for each
candidate value for `phi_w`

, thousands of Monte Carlo samples are
generated for the signs on the numbers (1,2, ..., n) where each number is
multiplied by the sign. The proportion of the samples that result in the
observed Wilcoxon statistics is an estimate for the likelihood value for that
candidate `phi_w`

. The likelihood values along with the prior result in
a discrete posterior distribution for `phi_w`

. The default for the
number of Monte Carlo samples per candidate `phi_w`

is the input
quantity called `samples`

. The default value for samples is 30000,
but this quantity can be altered by the user.

Chechile (2018) empirically found that for large *n* there was a beta
distribution that approximated the quantiles of the discrete, small sample
approach. This approximation is reasonably accurate for *n* > 24, and is
used when `method = "large"`

.

If the `method`

argument is omitted, the function employs the method
that is appropriate given the sample size. Note: the `method = "small"`

algorithm is slower than the algorithm for `method = "large"`

; for cases
where *n* > 24, `method = "small"`

and `method = "large"`

will
produce similar estimates but the former method requires increased
processing time.

### Value

A list containing the following components:

`T_pos` |
Sum of the positive ranks in the pairwise comparisons |

`T_neg` |
Sum of the negative ranks in the pairwise comparisons |

`n` |
Number of nonzero differences for differences |

`prob_interval` |
User-defined probability for interval estimates for phi_w |

`samples` |
The number of Monte Carlo samples per candidate phi_w for |

`method` |
A character string that is either |

`a0` |
The first shape parameter for the beta prior distribution (default is 1) |

`b0` |
The second shape parameter for the beta distribution prior (default is 1) |

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

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

`phiv` |
The 200 candidate values for phi_w for |

`phipost` |
The discrete posterior distribution for phi_w when |

`priorprH1` |
The prior probability that phi_w > .5 |

`prH1` |
The posterior probability for phi_w > .5 |

`BF10` |
Bayes factor for the relative increase in the posterior odds for the alternative hypothesis that phi_w > .5 over the null model for phi_w <= .5 |

`post_mean` |
The posterior mean for phi_w |

`cumulative_phi` |
The posterior cumulative distribution for phi_w when |

`hdi_lower` |
The lower limit for the posterior highest-density interval estimate for phi_w |

`hdi_upper` |
The upper limit for the posterior highest-density interval estimate for phi_w |

`a_post` |
The first shape parameter for a beta distribution model for phi_w when |

`b_post` |
The second shape parameter for a beta distribution model for phi_w when |

`post_median` |
The posterior median for phi_w when |

`eti_lower` |
The equal-tail lower limit for phi_w |

`eti_upper` |
The equal-tail upper limit for phi_w |

### References

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

Chechile, R. A. (2018) A Bayesian analysis for the Wilcoxon signed-rank statistic. Communications in Statistics - Theory and Methods, https://doi.org/10.1080/03610926.2017.1388402

### Examples

```
## Examples with a small number of pairs
conditionA <- c(1.49, 0.64, 0.96, 2.34, 0.78, 1.29, 0.72, 1.52, 0.62, 1.67,
1.19, 0.86)
conditionB <- c(0.53, 0.55, 0.58, 0.97, 0.60, 0.22, 0.05, 13.14, 0.63, 0.33,
0.91, 0.37)
dfba_wilcoxon(Y1 = conditionA,
Y2 = conditionB,
samples = 250,
hide_progress = TRUE)
# Examples with large sample size
E <- c(6.45, 5.65, 4.34, 5.92, 2.84, 13.06, 6.61, 5.47, 4.49, 6.39, 6.63,
3.55, 3.76, 5.61, 7.45, 6.41, 10.16, 6.26, 8.46, 2.29, 3.16, 5.68,
4.13, 2.94, 4.87, 4.44, 3.13, 8.87)
C <- c(2.89, 4.19, 3.22, 6.50, 3.10, 4.19, 5.13, 3.77, 2.71, 2.58, 7.59,
2.68, 4.98, 2.35, 5.15, 8.46, 3.77, 8.83, 4.06, 2.50, 5.48, 2.80,
8.89, 3.19, 9.36, 4.58, 2.94, 4.75)
BW<-dfba_wilcoxon(Y1 = E,
Y2 = C)
BW
plot(BW)
```

*DFBA*version 0.1.0 Index]