sa_sizePCM | R Documentation |

Returns sample size for Wald (W), likelihood ratio (LR), Rao score (RS)
and gradient (GR) test given probabilities of errors of first and second kinds `\alpha`

and
`\beta`

as well as a deviation from the hypothesis to be tested. The hypothesis to be tested
assumes equal item-category parameters in the partial credit model between two predetermined groups of persons.
The alternative assumes that at least one parameter differs between the two groups.

```
sa_sizePCM(
alpha = 0.05,
beta = 0.05,
persons1 = rnorm(10^6),
persons2 = rnorm(10^6),
local_dev
)
```

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

`beta` |
Probability of the error of second kind. |

`persons1` |
A vector of person parameters for group 1 (drawn from a specified distribution). By default |

`persons2` |
A vector of person parameters for group 2 (drawn from a specified distribution). By default |

`local_dev` |
A list consisting of two lists. One list refers to group 1, the other to group 2. Each of the two lists contains a numerical vector per item, i.e., each list contains as many vectors as items. Each vector contains the free item-cat. parameters of the respective item. The number of free item-cat. parameters per item equals the number of categories of the item minus 1. |

In general, the sample size 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 = l`

,
where `l`

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

. The latter is, inter alia, a function of the sample size. Hence, the sample size can be
determined from the condition `\lambda = \lambda_0`

, where `\lambda_0`

is a predetermined constant
which depends on the probabilities of the errors of the first and second kinds `\alpha`

and `\beta`

(or power). 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, the determination of `\lambda`

and the sample size, respectively, is based on a simple Monte Carlo
approach. As regards the concept of sample size a distinction between informative and total sample size has
to be made. In the conditional maximum likelihood context, the responses of persons with minimum or maximum
person score are completely uninformative. They do not contribute to the value of the test statistic. Thus,
the informative sample size does not include these persons. The total sample size is composed of all persons.
The Monte Carlo approach used in the present problem to determine `\lambda`

and informative (and total) sample size
can briefly be described as follows. Data (responses of a large number of persons to a number of items) are
generated given a user-specified scenario of a deviation from the hypothesis to be tested. The hypothesis
to be tested assumes equal item-category parameters between the two groups of persons. A scenario of a
deviation is given by a choice of the item-cat. parameters and the person parameters (to be drawn randomly
from a specified distribution) for each of the two groups. Such a scenario may be called local deviation
since deviations can be specified locally for each item-category. The relative group sizes are determined
by the choice of the number of person parameters for each of the two groups. For instance, by default
`10^6`

person parameters are selected randomly for each group. In this case, it is implicitly assumed that
the two groups of persons are of equal size. The user can specify the relative groups sizes by choosing
the length of the arguments `persons1`

and `persons2`

appropriately. Note that the relative group sizes do
have an impact on power and sample size of the tests. The next step is to compute a test statistic `T`

(Wald, LR, score, or gradient) from the simulated data. The observed value `t`

of the test statistic is
then divided by the informative sample size `n_{infsim}`

observed in the simulated data. This yields the
so-called global deviation `e = t / n_{infsim}`

, i.e., the chosen scenario of a deviation from the
hypothesis to be tested being represented by a single number. Let the informative sample size sought
be denoted by `n_{inf}`

(thus, this is not the informative sample size observed in the sim. data). The
noncentrality parameter `\lambda`

can be expressed by the product `n_{inf} * e`

. Then, it follows from the
condition `\lambda = \lambda_0`

that

`n_{inf} * e = \lambda_0`

and

`n_{inf} = \lambda_0 / e.`

Note that the sample of size `n_{inf}`

is assumed to be composed only of persons with informative person scores in both groups,
where the relative frequency distribution of these informative scores in each of both groups is considered to be
equal to the observed relative frequency distribution of informative scores in each of both groups in the simulated
data. Note also that the relative sizes of the two person groups are
assumed to be equal to the relative sizes of the two groups in the simulated data. By default, the two
groups are equal-sized in the simulated data, i.e., one yields `n_{inf} / 2`

persons (with informative scores)
in each of the two groups. The total sample size `n_{total}`

is obtained from the relation `n_{inf} = n_{total} * pr`

,
where `pr`

is the proportion or relative frequency of persons observed in the simulated data with a minimum or
maximum score. Basing the tests given a level `\alpha`

on an informative sample of size `n_{inf}`

the
probability of rejecting the hypothesis to be tested will be at least `1 - \beta`

if the true global deviation
`\ge e`

.

Note that in this approach the data have to be generated only once. There are no replications needed. Thus, the procedure is computationally not very time-consuming.

Since e is determined from the value of the test statistic observed in the simulated data it has to be
treated as a realization of a random variable `E`

. Consequently, `n_{inf}`

is also a realization of a random
variable `N_{inf}`

. Thus, the (realized) value `n_{inf}`

need not be equal to the exact value of the informative
sample size that follows from the user-specified (predetermined) `\alpha`

,`\beta`

, and scenario of a deviation
from the hypothesis to be tested, i.e., the selected item-category parameters used for the simulation of
the data. If the CML estimates of these parameters computed from the simulated data are close to the
predetermined parameters `n_{inf}`

will be close to the exact value. This will generally be the case if
the number of person parameters used for simulating the data, i.e., the lengths of the vectors persons1
and persons2, is large, e.g., `10^5`

or even `10^6`

persons. In such cases, the possible random error of the
computation procedure of `n_{inf}`

based on the sim. data may not be of practical relevance any more. That is
why a large number (of persons for the simulation process) is generally recommended.

For theoretical reasons, the random error involved in computing n_inf can be pretty well approximated.
A suitable approach is the well-known delta method. Basically, it is a Taylor polynomial of first order,
i.e., a linear approximation of a function. According to it the variance of a function of a random variable
can be linearly approximated by multiplying the variance of this random variable with the square of the first
derivative of the respective function. In the present problem, the variance of the test statistic `T`

is
(approximately) given by the variance of a noncentral `\chi^2`

distribution. Thus,
`Var(T) = 2 (df + 2 \lambda)`

, with `df = l`

and
`\lambda = t`

. Since the global deviation `e = (1 / n_{infsim}) * t`

it follows for the variance of the
corresponding random variable `E`

that `Var(E) = (1 / n_{infsim})^2 * Var(T)`

. Since
`n_{inf} = f(e) = \lambda_0 / e`

one obtains by the delta method (for the variance of the corresponding
random variable `N_{inf}`

)

`Var(N_{inf}) = Var(E) * (f'(e))^2,`

where `f'(e) = - \lambda_0 / e^2`

is the derivative of `f(e)`

. The square root of `Var(N_{inf})`

is then used to
quantify the random error of the suggested Monte Carlo computation procedure. It is called Monte Carlo
error of informative sample size.

A list of results.

`informative sample size` |
Informative sample size for each test, omitting persons with min. and max score. |

`MC error of sample size` |
Monte Carlo error of informative sample size for each test. |

`global deviation` |
Global deviation computed from simulated data. See Details. |

`local deviation` |
CML estimates of free item-category parameters in both group of persons obtained from the simulated data expressing a deviation from the hypothesis to be tested locally per item and response category. |

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

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

`degrees of freedom` |
Degrees of freedom |

`noncentrality parameter` |
Noncentrality parameter |

`total sample size in group 1` |
Total sample size in group 1 for each test. See Details. |

`total sample size in group 1` |
Total sample size in group 2 for each test. See Details. |

`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.

`powerPCM`

, and `post_hocPCM`

.

```
## Not run:
##### Sample size of PCM Model #####
# free item-category parameters for group 1 and 2 with 5 items, with 3 categories each
local_dev <- list ( list(c( 0, 0), c( -1, 0), c( 0, 0), c( 1, 0), c( 1, 0.5)) ,
list(c( 0, 0), c( -1, 0), c( 0, 0), c( 1, 0), c( 0, -0.5)) )
res <- sa_sizePCM(alpha = 0.05, beta = 0.05, persons1 = rnorm(10^6),
persons2 = rnorm(10^6), local_dev = local_dev)
# > res
# $`informative sample size`
# W LR RS GR
# 234 222 227 217
#
# $`MC error of sample size`
# W LR RS GR
# 1.105 1.018 1.053 0.988
#
# $`global deviation`
# W LR RS GR
# 0.101 0.107 0.104 0.109
#
# $`local deviation`
# I1-C2 I2-C1 I2-C2 I3-C1 I3-C2 I4-C1 I4-C2 I5-C1 I5-C2
# group1 -0.001 -1.000 -1.001 -0.003 -0.011 0.997 0.998 0.996 1.492
# group2 0.001 -0.998 -0.996 -0.007 -0.007 0.991 1.001 0.004 -0.499
#
# $`person score distribution in group 1`
#
# 1 2 3 4 5 6 7 8 9
# 0.111 0.130 0.133 0.129 0.122 0.114 0.101 0.091 0.070
#
# $`person score distribution in group 2`
#
# 1 2 3 4 5 6 7 8 9
# 0.090 0.109 0.117 0.121 0.121 0.121 0.116 0.111 0.093
#
# $`degrees of freedom`
# [1] 9
#
# $`noncentrality parameter`
# [1] 23.589
#
# $`total sample size in group 1`
# W LR RS GR
# 132 125 128 123
#
# $`total sample size in group 2`
# W LR RS GR
# 133 126 129 123
#
# $call
# sa_sizePCM(alpha = 0.05, beta = 0.05, persons1 = rnorm(10^6),
# persons2 = rnorm(10^6), local_dev = local_dev)
## End(Not run)
```

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