post_hocPCM | R Documentation |

Returns post hoc power of Wald (W), likelihood ratio (LR), Rao score (RS)
and gradient (GR) test given data and probability of error of first kind `\alpha`

.
The hypothesis to be tested assumes equal item-category parameters of the partial
credit model between two predetermined groups of persons. The alternative states that
at least one of the parameters differs between the two groups.

```
post_hocPCM(data, x, alpha = 0.05)
```

`data` |
Data matrix with item responses (in ordered categories starting from 0). |

`x` |
A numeric vector of length equal to number of persons that contains zeros and ones indicating group membership of the persons. |

`alpha` |
Probability of error of first kind. |

The power of the tests (Wald, LR, score, and gradient) is determined from the assumption
that the approximate distributions of the four test statistics are from the family of
noncentral `\chi^2`

distributions with `df`

equal to the number of free item-category
parameters in the partial credit model and noncentrality parameter `\lambda`

. In case of evaluating
the post hoc power, `\lambda`

is assumed to be given by the observed value of the test statistic.
Given the probability of the error of the first kind `\alpha`

the post hoc power of the tests
can be determined from `\lambda`

. More details about the distributions of the test statistics and the
relationship between `\lambda`

, power, and sample size can be found in Draxler and Alexandrowicz (2015).

In particular, let `q_{\alpha}`

be the `1- \alpha`

quantile of the central `\chi^2`

distribution
with `df`

equal to the number of free item-category parameters. Then,

`power = 1 - F_{df, \lambda} (q_{\alpha}),`

where `F_{df, \lambda}`

is the cumulative distribution function of the noncentral `\chi^2`

distribution with `df`

equal to the number of free item-category parameters and `\lambda`

equal to the
observed value of the test statistic.

A list of results.

`test` |
A numeric vector of Wald (W), likelihood ratio (LR), Rao score (RS), and gradient (GR) test statistics. |

`power` |
Post hoc power value for each test. |

`observed global deviation` |
Observed global deviation from hypothesis to be tested represented by a single number. It is obtained by dividing the test statistic by the informative sample size. The latter does not include persons with minimum or maximum person score. |

`observed local deviation` |
CML estimates of free item-category parameters in both groups of persons representing observed deviation from hypothesis to be tested locally per item and response category. |

`person score distribution in group 1` |
Relative frequencies of person scores in group 1. Uninformative scores, i.e., minimum and maximum score, are omitted. Note that the person score distribution does also have an influence on the power of the tests. |

`person score distribution in group 2` |
Relative frequencies of person scores in group 2. Uninformative scores, i.e., minimum and maximum score, are omitted. Note that the person score distribution does also have an influence on the power of the tests. |

`degrees of freedom` |
Degrees of freedom |

`noncentrality parameter` |
Noncentrality parameter |

`call` |
The matched call. |

Draxler, C. (2010). Sample Size Determination for Rasch Model Tests. Psychometrika, 75(4), 708–724.

Draxler, C., & Alexandrowicz, R. W. (2015). Sample Size Determination Within the Scope of Conditional Maximum Likelihood Estimation with Special Focus on Testing the Rasch Model. Psychometrika, 80(4), 897–919.

Draxler, C., Kurz, A., & Lemonte, A. J. (2020). The Gradient Test and its Finite Sample Size Properties in a Conditional Maximum Likelihood and Psychometric Modeling Context. Communications in Statistics-Simulation and Computation, 1-19.

Glas, C. A. W., & Verhelst, N. D. (1995a). Testing the Rasch Model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch Models: Foundations, Recent Developments, and Applications (pp. 69–95). New York: Springer.

Glas, C. A. W., & Verhelst, N. D. (1995b). Tests of Fit for Polytomous Rasch Models. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch Models: Foundations, Recent Developments, and Applications (pp. 325-352). New York: Springer.

`sa_sizePCM`

, and `powerPCM`

.

```
## Not run:
# Numerical example for post hoc power analysis for PCM
y <- eRm::pcmdat2
n <- nrow(y) # sample size
x <- c( rep(0,n/2), rep(1,n/2) ) # binary covariate
res <- post_hocPCM(data = y, x = x, alpha = 0.05)
# > res
# $test
# W LR RS GR
# 11.395 11.818 11.628 11.978
#
# $power
# W LR RS GR
# 0.683 0.702 0.694 0.709
#
# $`observed global deviation`
# W LR RS GR
# 0.045 0.046 0.045 0.047
#
# $`observed local deviation`
# I1-C2 I2-C1 I2-C2 I3-C1 I3-C2 I4-C1 I4-C2
# group1 2.556 0.503 2.573 -2.573 -2.160 -1.272 -0.683
# group2 2.246 0.878 3.135 -1.852 -0.824 -0.494 0.941
#
# $`person score distribution in group 1`
#
# 1 2 3 4 5 6 7
# 0.016 0.097 0.137 0.347 0.121 0.169 0.113
#
# $`person score distribution in group 2`
#
# 1 2 3 4 5 6 7
# 0.015 0.083 0.136 0.280 0.152 0.227 0.106
#
# $`degrees of freedom`
# [1] 7
#
# $`noncentrality parameter`
# W LR RS GR
# 11.395 11.818 11.628 11.978
#
# $call
# post_hocPCM(alpha = 0.05, data = y, x = x)
## End(Not run)
```

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