Package 'tcl'

Tests in context of measurement of change using LLTM.

Description

Computes gradient (GR), likelihood ratio (LR), Rao score (RS) and Wald (W) test statistics for hypotheses on parameters expressing change between two time points.

Usage

change_test(X)

Arguments

X	Data matrix containing the responses of n persons to 2k binary items. Columns 1 to k contain the responses to k items at time point 1, and columns (k+1) to 2k the responses to the same k items at time point 2.

Details

Assume all items be presented twice (2 time points) to the same persons. The data matrix X has n rows (number of persons) and 2k columns considered as virtual items. Assume a constant shift of item difficulties of each item between the 2 time points represented by one parameter. The shift parameter is the only parameter of interest. Of interest is the test of the hypothesis that the shift parameter equals 0 against the two-sided alternative that it is not equal to zero.

Value

A list of test statistics, degrees of freedom, and p-values.

test	A numeric vector of gradient (GR), likelihood ratio (LR), Rao score (RS), and Wald test statistics.
df	Degrees of freedom.
pvalue	A vector of corresponding p-values.
call	The matched call.

References

Fischer, G. H. (1995). The Linear Logistic Test Model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, Recent Developments, and Applications (pp. 131-155). New York: Springer.

Fischer, G. H. (1983). Logistic Latent Trait Models with Linear Constraints. Psychometrika, 48(1), 3-26.

See Also

invar_test, and LLTM_test.

Examples

## Not run: 
# Numerical example with 400 persons and 4 items
# presented twice, thus 8 virtual items

# Data y generated under the assumption that shift parameter equals 0
# (no change from time point 1 to 2)

# design matrix W used only for example data generation
#     (not used for estimating in change_test function)
W <- rbind(c(1,0,0,0,0),
  c(0,1,0,0,0),
  c(0,0,1,0,0),
  c(0,0,0,1,0),
  c(1,0,0,0,1),
  c(0,1,0,0,1),
  c(0,0,1,0,1),
  c(0,0,0,1,1))

# eta Parameter, first 4 are nuisance, i.e. , easiness parameters of the 4 items
# at time point 1, last one is the shift parameter.
eta <- c(-2,-1,1,2,0)

y <- eRm::sim.rasch(persons = rnorm(400), items = colSums(eta * t(W)))

res <- change_test(X = y)

res$test # test statistics
res$df # degrees of freedoms
res$pvalue # p-values


## End(Not run)

---

Testing item discriminations

Description

Computes gradient (GR), likelihood ratio (LR), Rao score (RS) and Wald (W) test of hypothesis of equal item discriminations against the alternative that at least one item discriminates differently (only for binary data).

Usage

discr_test(X)

Arguments

X	Data matrix.

Details

The tests are based on the following model suggested in Draxler, Kurz, Gürer, and Nolte (2024)

$\text{logit} \big( E(Y) \big ) = \tau + \alpha + \delta (r - 1),$

where $E(Y)$ ist the expected value of a binary response (of a person to an item), $r = 1, \dots, k - 1$ is the person score, i.e., number of correct responses of that person when responding to $k$ items, $\tau$ is the respective person parameter and $\alpha$ and $\delta$ are two parameters referring to the respective item. The parameter $\alpha$ represents a baseline, i.e., the easiness or attractiveness of the respective item in person score group $r = 1$. The parameter $\delta$ denotes the constant change of the attractiveness of that item between successive person score groups. Thus, the model assumes a linear effect of the person score $r$ on the logit of the probability of a correct response.

The four test statistics are derived from a conditional likelihood function in which the $\tau$ parameters are eliminated by conditioning on the observed person scores. The hypothesis to be tested is formally given by setting all $\delta$ parameters equal to $0$. The alternative assumes that at least one $\delta$ parameter is not equal to $0$.

Value

A list of test statistics, degrees of freedom, and p-values.

test	A numeric vector of gradient (GR), likelihood ratio (LR), Rao score (RS), and Wald test statistics.
df	A numeric vector of corresponding degrees of freedom.
pvalue	A vector of corresponding p-values.
call	The matched call.

References

Draxler, C., Kurz. A., Gürer, C., & Nolte, J. P. (2024). An improved inferential procedure to evaluate item discriminations in a conditional maximum likelihood framework. Manuscript submitted for publication.

See Also

invar_test, change_test, and LLTM_test.

Examples

## Not run: 
##### Dataset PISA Mathematics data.pisaMath {sirt} #####

library(sirt)
data(data.pisaMath)
y <- data.pisaMath$data[, grep(names(data.pisaMath$data), pattern = "M" )]

res <- discr_test(X = y)
# $test
# GR     LR     RS      W
# 72.430 73.032 76.725 73.470
#
# $df
# GR LR RS  W
# 10 10 10 10
#
# $pvalue
#       GR        LR        RS         W
# "< 0.001" "< 0.001" "< 0.001" "< 0.001"
#
# $call
# discr_test(X = y)


## End(Not run)

---

Test of invariance of item parameters between two groups.

Description

Computes gradient (GR), likelihood ratio (LR), Rao score (RS) and Wald (W) test statistics for hypothesis of equality of item parameters between two groups of persons against a two-sided alternative that at least one item parameter differs between the two groups.

Usage

invar_test(X, splitcr = "median", model = "RM")

Arguments

X	Data matrix.
splitcr	Split criterion which is either "mean", "median" or a numeric vector x. "mean" Corresponds to division of the sample according to the mean of the person score. "median" Corresponds to division of the sample according to the median of the person score. x Has length equal to number of persons and contains zeros and ones. It indicates group membership for every person.
model	RM, PCM, RSM

Details

Note that items are excluded for the computation of GR, LR, and W due to inappropriate response patterns within subgroups and for computation of RS due to inappropriate response patterns in the total data. If the model is identified from the total data but not from one or both subgroups only RS will be computed. If the model is not identified from the total data, no test statistic is computable.

Value

A list of test statistics, degrees of freedom, and p-values.

test	A numeric vector of gradient (GR), likelihood ratio (LR), Rao score (RS), and Wald test statistics.
df	A numeric vector of corresponding degrees of freedom.
pvalue	A vector of corresponding p-values.
deleted_items	A list with numeric vectors of item numbers that were excluded before computing corresponding test statistics.
call	The matched call.

References

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.

See Also

change_test, and LLTM_test.

Examples

## Not run: 
##### Rasch Model #####
y <- eRm::sim.rasch(persons = rnorm(400), c(0,-3,-2,-1,0,1,2,3))
x <- c(rep(1,200),rep(0,200))

res <- invar_test(y, splitcr = x, model = "RM")

res$test # test statistics
res$df # degrees of freedoms
res$pvalue # p-values
res$deleted_items # excluded items

$test
  GR    LR    RS     W
14.492 14.083 13.678 12.972

$df
GR LR RS  W
7  7  7  7

$pvalue
  GR    LR    RS     W
"0.043" "0.050" "0.057" "0.073"

$deleted_items
 $deleted_items$GR
 [1] "none"

 $deleted_items$LR
 [1] "none"

 $deleted_items$RS
 [1] "none"

 $deleted_items$W
 [1] "none"


$call
invar_test(X = y, splitcr = x, model = "RM")


## End(Not run)

---

Testing linear restrictions on parameter space of item parameters of RM.

Description

Computes gradient (GR), likelihood ratio (LR), Rao score (RS) and Wald (W) test statistics for hypotheses defined by linear restrictions on parameter space of the item parameters of RM.

Usage

LLTM_test(X, W)

Arguments

X	Data matrix.
W	Design matrix of LLTM.

Details

The RM item parameters are assumed to be linear in the LLTM parameters. The coefficients of the linear functions are specified by a design matrix W. In this context, the LLTM is considered as a more parsimonious model than the RM. The LLTM parameters can be interpreted as the difficulties of certain cognitive operations needed to respond correctly to psychological test items. The item parameters of the RM are assumed to be linear combinations of these cognitive operations. These linear combinations are defined in the design matrix W.

Value

A list of test statistics, degrees of freedom, and p-values.

test	A numeric vector of gradient (GR), likelihood ratio (LR), Rao score (RS), and Wald test statistics.
df	Degrees of freedom.
pvalue	A vector of corresponding p-values.
call	The matched call.

References

Fischer, G. H. (1995). The Linear Logistic Test Model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, Recent Developments, and Applications (pp. 131-155). New York: Springer.

Fischer, G. H. (1983). Logistic Latent Trait Models with Linear Constraints. Psychometrika, 48(1), 3-26.

See Also

change_test, and invar_test.

Examples

## Not run: 
# Numerical example assuming no deviation from linear restriction

# design matrix W defining linear restriction
W <- rbind(c(1,0), c(0,1), c(1,1), c(2,1))

# assumed eta parameters of LLTM for data generation
eta <- c(-0.5, 1)

# assumed vector of item parameters of RM
b <- colSums(eta * t(W))

y <- eRm::sim.rasch(persons = rnorm(400), items = b - b[1])  # sum0 = FALSE

res <- LLTM_test(X = y, W = W )

res$test # test statistics
res$df # degrees of freedoms
res$pvalue # p-values


## End(Not run)

---

Power analysis of tests in context of measurement of change using LLTM

Description

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 states that the shift parameter quantifying the constant change for all items between time points 1 and 2 equals 0. The alternative states that the shift parameter is not equal to 0. It is assumed that the same items are presented at both time points. See function change_test.

Usage

post_hocChange(data, alpha = 0.05)

Arguments

data	Data matrix as required for function change_test.
alpha	Probability of error of first kind.

Details

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 $df = 1$ 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 = 1. Then,

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

where $F_{df, \lambda}$ is the cumulative distribution function of the noncentral $\chi^2$ distribution with $df = 1$ and $\lambda$ equal to the observed value of the test statistic.

Value

A list of results.

test	A numeric vector of Wald (W), likelihood ratio (LR), Rao score (RS), and gradient (GR) test statistics.
power	Posthoc power value for each test.
observed deviation	CML estimate of shift parameter expressing observed deviation from hypothesis to be tested.
person score distribution	Relative frequencies of person scores. 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 $df$.
noncentrality parameter	Noncentrality parameter $\lambda$ of $\chi^2$ distribution from which power is determined. It equals observed value of test statistic.
call	The matched call.

References

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.

Fischer, G. H. (1995). The Linear Logistic Test Model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, Recent Developments, and Applications (pp. 131-155). New York: Springer.

Fischer, G. H. (1983). Logistic Latent Trait Models with Linear Constraints. Psychometrika, 48(1), 3-26.

See Also

sa_sizeChange, and powerChange.

Examples

## Not run: 
# Numerical example with 200 persons and 4 items
# presented twice, thus 8 virtual items

# Data y generated under the assumption that shift parameter equals 0.5
# (change from time point 1 to 2)

# design matrix W used only for exmaple data generation
#     (not used for estimating in change_test function)
W <- rbind(c(1,0,0,0,0), c(0,1,0,0,0), c(0,0,1,0,0), c(0,0,0,1,0),
           c(1,0,0,0,1), c(0,1,0,0,1), c(0,0,1,0,1), c(0,0,0,1,1))

# eta parameter vector, first 4 are nuisance, i.e., item parameters at time point 1.
# (easiness parameters of the 4 items at time point 1),
# last one is the shift parameter
eta <- c(-2,-1,1,2,0.5)

y <- eRm::sim.rasch(persons=rnorm(150), items=colSums(-eta*t(W)))

res <- post_hocChange(data = y, alpha = 0.05)

# > res
# $test
#     W     LR     RS     GR
# 9.822 10.021  9.955 10.088
#
# $power
#     W    LR    RS    GR
# 0.880 0.886 0.884 0.888
#
# $`observed deviation (estimate of shift parameter)`
# [1] 0.504
#
# $`person score distribution`
#
#     1     2     3     4     5     6     7
# 0.047 0.047 0.236 0.277 0.236 0.108 0.047
#
# $`degrees of freedom`
# [1] 1
#
# $`noncentrality parameter`
#     W     LR     RS     GR
# 9.822 10.021  9.955 10.088
#
# $call
# post_hocChange(alpha = 0.05, data = y)

## End(Not run)

---

Power analysis of tests of invariance of item parameters between two groups of persons in partial credit model

Description

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.

Usage

post_hocPCM(data, x, alpha = 0.05)

Arguments

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.

Details

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.

Value

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 $df$.
noncentrality parameter	Noncentrality parameter $\lambda$ of $\chi^2$ distribution from which power is determined. It equals observed value of test statistic.
call	The matched call.

References

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.

See Also

sa_sizePCM, and powerPCM.

Examples

## 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)

---

Power analysis of tests of invariance of item parameters between two groups of persons in binary Rasch model

Description

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 parameters between two predetermined groups of persons. The alternative states that at least one of the parameters differs between the two groups.

Usage

post_hocRM(data, x, alpha = 0.05)

Arguments

data	Binary data matrix.
x	A numeric vector of length equal to number of persons containing zeros and ones indicating group membership of the persons.
alpha	Probability of error of first kind.

Details

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 items minus 1 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 items minus 1. 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 items reduced by 1 and $\lambda$ equal to the observed value of the test statistic.

Value

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.
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.
local deviation	CML estimates of free item parameters in both groups of persons (first item parameter set to 0 in both groups) representing observed deviation from hypothesis to be tested locally per item.
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 $df$.
noncentrality parameter	Noncentrality parameter $\lambda$ of $\chi^2$ distribution from which power is determined. It equals observed value of test statistic.
call	The matched call.

References

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.

See Also

sa_sizeRM, and powerRM.

Examples

## Not run: 
# Numerical example for post hoc power analysis for Rasch Model

y <- eRm::raschdat1
n <- nrow(y) # sample size
x <- c( rep(0,n/2), rep(1,n/2) ) # binary covariate

res <-  post_hocRM(data = y, x = x, alpha = 0.05)

# > res
# $test
#      W     LR     RS     GR
# 29.241 29.981 29.937 30.238
#
# $power
#     W    LR    RS    GR
# 0.890 0.900 0.899 0.903
#
# $`observed global deviation`
#     W    LR    RS    GR
# 0.292 0.300 0.299 0.302
#
# $`observed local deviation`
#           I2    I3    I4    I5    I6    I7    I8    I9   I10   I11
# group1 1.039 0.693 2.790 2.404 1.129 1.039 0.864 1.039 2.790 2.244
# group2 2.006 0.945 2.006 3.157 1.834 0.690 0.822 1.061 2.689 2.260
#          I12   I13   I14   I15   I16   I17   I18   I19   I20   I21
# group1 1.412 3.777 3.038 1.315 2.244 1.039 1.221 2.404 0.608 0.608
# group2 0.945 2.962 4.009 1.171 2.175 1.472 2.091 2.344 1.275 0.690
#          I22   I23   I24   I25   I26   I27   I28   I29   I30
# group1 0.438 0.608 1.617 3.038 0.438 1.617 2.100 2.583 0.864
# group2 0.822 1.275 1.565 2.175 0.207 1.746 1.746 2.260 0.822
#
# $`person score distribution in group 1`
#
#    1    2    3    4    5    6    7    8    9   10   11   12   13
# 0.02 0.02 0.02 0.06 0.02 0.10 0.10 0.06 0.10 0.12 0.08 0.12 0.12
#   14   15   16   17   18   19   20   21   22   23   24   25   26
# 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
#   27   28   29
# 0.00 0.00 0.00
#
# $`person score distribution in group 2`
#
#    1    2    3    4    5    6    7    8    9   10   11   12   13
# 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
#   14   15   16   17   18   19   20   21   22   23   24   25   26
# 0.08 0.12 0.10 0.16 0.06 0.04 0.10 0.12 0.08 0.02 0.02 0.02 0.08
#   27   28   29
# 0.00 0.00 0.00
#
# $`degrees of freedom`
# [1] 29
#
# $`noncentrality parameter`
#      W     LR     RS     GR
# 29.241 29.981 29.937 30.238
#
# $call
# post_hocRM(alpha = 0.05, data = y, x = x)


## End(Not run)

---

Power analysis of tests in context of measurement of change using LLTM

Description

Returns power of Wald (W), likelihood ratio (LR), Rao score (RS) and gradient (GR) test given probability of error of first kind $\alpha$, sample size, and a deviation from the hypothesis to be tested. The latter states that the shift parameter quantifying the constant change for all items between time points 1 and 2 equals 0. The alternative states that the shift parameter is not equal to 0. It is assumed that the same items are presented at both time points. See function change_test.

Usage

powerChange(n_total, eta, alpha = 0.05, persons = rnorm(10^6))

Arguments

n_total	Total sample size for which power shall be determined.
eta	A vector of eta parameters of the LLTM. The last element represents the constant change or shift for all items between time points 1 and 2. The other elements of the vector are the item parameters at time point 1. A choice of the eta parameters constitutes a scenario of deviation from the hypothesis of no change.
alpha	Probability of the error of first kind.
persons	A vector of person parameters (drawn from a specified distribution). By default $10^6$ parameters are drawn at random from the standard normal distribution. The larger this number the more accurate are the computations. See Details.

Details

In general, the power of the tests is determined from the assumption that the approximate distributions of the four test statistics are from the family of noncentral $\chi^2$ distributions with $df = 1$ and noncentrality parameter $\lambda$. The latter depends on a scenario of deviation from the hypothesis to be tested and a specified sample size. Given the probability of the error of the first kind $\alpha$ the 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).

As regards the concept of sample size a distinction between informative and total sample size has to be made since the power of the tests depends only on the informative sample size. 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.

In particular, the determination of $\lambda$ and the power of the tests, respectively, is based on a simple Monte Carlo approach. Data (responses of a large number of persons to a number of items presented at two time points) are generated given a user-specified scenario of a deviation from the hypothesis to be tested. The hypothesis to be tested assumes no change between time points 1 and 2. A scenario of a deviation is given by a choice of the item parameters at time point 1 and the shift parameter, i.e., the LLTM eta parameters, as well as the person parameters (to be drawn randomly from a specified distribution). The shift parameter represents a constant change of all item parameters from time point 1 to time point 2. A test statistic $T$ (Wald, LR, score, or gradient) is computed 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. The power of the tests can be determined given a user-specified total sample size denoted by $n_{total}$. The noncentrality parameter $\lambda$ can then be expressed by $\lambda = n_{total}* (n_{infsim} / n_{totalsim}) * e$, where $n_{totalsim}$ denotes the total number of persons in the simulated data and $n_{infsim} / n_{totalsim}$ is the proportion of informative persons in the sim. data. Let $q_{\alpha}$ be the $1 - \alpha$ quantile of the central $\chi^2$ distribution with $df = 1$. Then,

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

where $F_{df, \lambda}$ is the cumulative distribution function of the noncentral $\chi^2$ distribution with $df = 1$ and $\lambda = n_{total} * (n_{infsim} / n_{totalsim}) * e$. Thereby, it is assumed that $n_{total}$ is composed of a frequency distribution of person scores that is proportional to the observed distribution of person scores in the simulated data.

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 realized value of a random variable $E$. The same holds true for $\lambda$ as well as the power of the tests. Thus, the power is a realized value of a random variable that shall be denoted by $P$. Consequently, the (realized) value of the power of the tests need not be equal to the exact power that follows from the user-specified $n_{total}$, $\alpha$, and the chosen item parameters and shift parameter 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 the power of the tests will be close to the exact value. This will generally be the case if the number of person parameters used for simulating the data is large, e.g., $10^5$ or even $10^6$ persons. In such cases, the possible random error of the computation procedure 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 the power of the tests 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 $df = 1$ and $\lambda = t$. Since the global deviation $e = (1 / n_{infsim})* t$ it follows for the variance of the corresponding random variable $E$ that $Var(E) = (1 / n_{infsim})^2 * Var(T)$. The power of the tests is a function of $e$ which is given by $F_{df, \lambda} (q_{\alpha})$, where $\lambda = n_{total} * (n_{infsim} / n_{totalsim}) * e$ and $df = 1$. Then, by the delta method one obtains (for the variance of $P$)

$Var(P) = Var(E) * (F'_{df, \lambda} (q_{\alpha}))^2,$

where $F'_{df, \lambda}$ is the derivative of $F_{df, \lambda}$ with respect to $e$. This derivative is determined numerically and evaluated at $e$ using the package numDeriv. The square root of $Var(P)$ is then used to quantify the random error of the suggested Monte Carlo computation procedure. It is called Monte Carlo error of power.

Value

A list of results.

power	Power value for each test.
MC error of power	Monte Carlo error of power computation for each test.
deviation	Shift parameter estimated from the simulated data representing the constant shift of item parameters between time points 1 and 2.
person score distribution	Relative frequencies of person scores 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 power of the tests.
degrees of freedom	Degrees of freedom $df$.
noncentrality parameter	Noncentrality parameter $\lambda$ of $\chi^2$ distribution from which power is determined.
call	The matched call.

References

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.

Fischer, G. H. (1995). The Linear Logistic Test Model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, Recent Developments, and Applications (pp. 131-155). New York: Springer.

Fischer, G. H. (1983). Logistic Latent Trait Models with Linear Constraints. Psychometrika, 48(1), 3-26.

See Also

sa_sizeChange, and post_hocChange.

Examples

## Not run: 

# Numerical example: 4 items presented twice, thus 8 virtual items

# eta Parameter, first 4 are nuisance
# (easiness parameters of the 4 items at time point 1),
# last one is the shift parameter
eta <- c(-2,-1,1,2,0.5)
res <- powerChange(n_total = 150, eta = eta, persons=rnorm(10^6))

# > res
# $power
#     W    LR    RS    GR
# 0.905 0.910 0.908 0.911
#
# $`MC error of power`
#     W    LR    RS    GR
# 0.002 0.002 0.002 0.002
#
# $`deviation (estimate of shift parameter)`
# [1] 0.499
#
# $`person score distribution`
#
#     1     2     3     4     5     6     7
# 0.034 0.093 0.181 0.249 0.228 0.147 0.068
#
# $`degrees of freedom`
# [1] 1
#
# $`noncentrality parameter`
#      W     LR     RS     GR
# 10.692 10.877 10.815 10.939
#
# $call
# powerChange(alpha = 0.05, n_total = 150, eta = eta, persons = rnorm(10^6))
#

## End(Not run)

---

Power analysis of tests of invariance of item parameters between two groups of persons in partial credit model

Description

Returns power of Wald (W), likelihood ratio (LR), Rao score (RS) and gradient (GR) test given probability of error of first kind $\alpha$, sample size, and a deviation from the hypothesis to be tested. 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.

Usage

powerPCM(
  n_total,
  local_dev,
  alpha = 0.05,
  persons1 = rnorm(10^6),
  persons2 = rnorm(10^6)
)

Arguments

n_total	Total sample size for which power shall be determined.
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 numeric 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.
alpha	Probability of error of first kind.
persons1	A vector of person parameters in group 1 (drawn from a specified distribution). By default $10^6$ parameters are drawn at random from the standard normal distribution. The larger this number the more accurate are the computations. See Details.
persons2	A vector of person parameters in group 2 (drawn from a specified distribution). By default $10^6$ parameters are drawn at random from the standard normal distribution. The larger this number the more accurate are the computations. See Details.

Details

In general, the power of the tests 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 and noncentrality parameter $\lambda$. The latter depends on a scenario of deviation from the hypothesis to be tested and a specified sample size. Given the probability of the error of the first kind $\alpha$ the 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).

As regards the concept of sample size a distinction between informative and total sample size has to be made since the power of the tests depends only on the informative sample size. 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.

In particular, the determination of $\lambda$ and the power of the tests, respectively, is based on a simple Monte Carlo approach. 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. 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 group 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. The power of the tests can be determined given a user-specified total sample size denoted by n_total. The noncentrality parameter $\lambda$ can then be expressed by $\lambda = n_{total}* (n_{infsim} / n_{totalsim}) * e$, where $n_{totalsim}$ denotes the total number of persons in the simulated data and $n_{infsim} / n_{totalsim}$ is the proportion of informative persons in the sim. data. 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 $\lambda = n_{total} (n_{infsim} / n_{totalsim}) * e$. Thereby, it is assumed that $n_{total}$ is composed of a frequency distribution of person scores that is proportional to the observed distribution of person scores in the simulated data. The same holds true in respect of the relative group sizes, i.e., the relative frequencies of the two person groups in a sample of size $n_{total}$ are assumed to be equal to the relative frequencies of the two groups in the simulated data.

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 realized value of a random variable $E$. The same holds true for $\lambda$ as well as the power of the tests. Thus, the power is a realized value of a random variable that shall be denoted by $P$. Consequently, the (realized) value of the power of the tests need not be equal to the exact power that follows from the user-specified $n_{total}$, $\alpha$, and the chosen 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 the power of the tests will be close to the exact value. This will generally be the case if the number of person parameters used for simulating the data is large, e.g., $10^5$ or even $10^6$ persons. In such cases, the possible random error of the computation procedure 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 the power of the tests 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 with $df$ equal to the number of free item-category parameters and noncentrality parameter $\lambda$. Thus, $Var(T) = 2 (df + 2 \lambda)$, with $\lambda = t$. Since the global deviation $e = (1 / n_{infsim}) * t$ it follows for the variance of the corresponding random variable $E$ that $Var(E) = (1 / n_{infsim})^2 * Var(T)$. The power of the tests is a function of $e$ which is given by $F_{df, \lambda} (q_{\alpha})$, where $\lambda = n_{total} * (n_{infsim} / n_{totalsim}) * e$ and $df$ equal to the number of free item-category parameters. Then, by the delta method one obtains (for the variance of P).

$Var(P) = Var(E) * (F'_{df, \lambda} (q_{\alpha}))^2,$

where $F'_{df, \lambda}$ is the derivative of $F_{df, \lambda}$ with respect to $e$. This derivative is determined numerically and evaluated at $e$ using the package numDeriv. The square root of $Var(P)$ is then used to quantify the random error of the suggested Monte Carlo computation procedure. It is called Monte Carlo error of power.

Value

A list of results.

power	Power value for each test.
MC error of power	Monte Carlo error of power computation for each test.
global deviation	Global deviation computed from simulated data for each test. See Details.
local deviation	CML estimates of free item-category parameters in both groups 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 power of the tests.
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 power of the tests.
degrees of freedom	Degrees of freedom $df$.
noncentrality parameter	Noncentrality parameter $\lambda$ of $\chi^2$ distribution from which power is determined.
call	The matched call.

References

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.

See Also

sa_sizePCM, and post_hocPCM.

Examples

## Not run: 
# Numerical example

# 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 <-  powerPCM(n_total = 200, local_dev = local_dev)

# > res
# $power
#     W    LR    RS    GR
# 0.863 0.885 0.876 0.892
#
# $`MC error of power`
#     W    LR    RS    GR
# 0.002 0.002 0.002 0.002
#
# $`global deviation`
#     W    LR    RS    GR
# 0.102 0.107 0.105 0.109
#
# $`local deviation`
#         I1-C2  I2-C1  I2-C2  I3-C1  I3-C2 I4-C1 I4-C2  I5-C1  I5-C2
# group1  0.002 -0.997 -0.993  0.006  0.012 1.002 1.007  1.006  1.508
# group2 -0.007 -1.005 -1.007 -0.006 -0.009 0.993 0.984 -0.006 -0.510
#
# $`person score distribution in group 1`
#
#     1     2     3     4     5     6     7     8     9
# 0.112 0.130 0.131 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
# 0.091 0.108 0.117 0.122 0.122 0.121 0.115 0.110 0.093
#
# $`degrees of freedom`
# [1] 9
#
# $`noncentrality parameter`
#      W     LR     RS     GR
# 18.003 19.024 18.596 19.403
#
# $call
# powerPCM(alpha = 0.05, n_total = 200, persons1 = rnorm(10^6),
#          persons2 = rnorm(10^6), local_dev = local_dev)

## End(Not run)

---

Power analysis of tests of invariance of item parameters between two groups of persons in binary Rasch model

Description

Returns power of Wald (W), likelihood ratio (LR), Rao score (RS) and gradient (GR) test given probability of error of first kind $\alpha$, sample size, and a deviation from the hypothesis to be tested. The latter assumes equality of the item parameters in the Rasch model between two predetermined groups of persons. The alternative states that at least one of the parameters differs between the two groups.

Usage

powerRM(
  n_total,
  local_dev,
  alpha = 0.05,
  persons1 = rnorm(10^6),
  persons2 = rnorm(10^6)
)

Arguments

n_total	Total sample size for which power shall be determined.
local_dev	A list of two vectors containing item parameters for the two person groups representing a deviation from the hypothesis to be tested locally per item.
alpha	Probability of error of first kind.
persons1	A vector of person parameters in group 1 (drawn from a specified distribution). By default $10^6$ parameters are drawn at random from the standard normal distribution. The larger this number the more accurate are the computations. See Details.
persons2	A vector of person parameters in group 2 (drawn from a specified distribution). By default $10^6$ parameters are drawn at random from the standard normal distribution. The larger this number the more accurate are the computations. See Details.

Details

In general, the power of the tests 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 items minus 1 and noncentrality parameter $\lambda$. The latter depends on a scenario of deviation from the hypothesis to be tested and a specified sample size. Given the probability of the error of the first kind $\alpha$ the 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).

As regards the concept of sample size a distinction between informative and total sample size has to be made since the power of the tests depends only on the informative sample size. 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.

In particular, the determination of $\lambda$ and the power of the tests, respectively, is based on a simple Monte Carlo approach. 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. A scenario of a deviation is given by a choice of the item 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. 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 group 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. The power of the tests can be determined given a user-specified total sample size denoted by n_total. The noncentrality parameter $\lambda$ can then be expressed by $\lambda = n_{total}* (n_{infsim} / n_{totalsim}) * e$, where $n_{totalsim}$ denotes the total number of persons in the simulated data and $n_{infsim} / n_{totalsim}$ is the proportion of informative persons in the sim. data. Let $q_{\alpha}$ be the $1 - \alpha$ quantile of the central $\chi^2$ distribution with df equal to the number items minus 1. 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 items minus 1 and $\lambda = n_{total} (n_{infsim} / n_{totalsim}) * e$. Thereby, it is assumed that $n_{total}$ is composed of a frequency distribution of person scores that is proportional to the observed distribution of person scores in the simulated data. The same holds true in respect of the relative group sizes, i.e., the relative frequencies of the two person groups in a sample of size $n_{total}$ are assumed to be equal to the relative frequencies of the two groups in the simulated data.

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 realized value of a random variable $E$. The same holds true for $\lambda$ as well as the power of the tests. Thus, the power is a realized value of a random variable that shall be denoted by $P$. Consequently, the (realized) value of the power of the tests need not be equal to the exact power that follows from the user-specified $n_{total}$, $\alpha$, and the chosen item 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 the power of the tests will be close to the exact value. This will generally be the case if the number of person parameters used for simulating the data is large, e.g., $10^5$ or even $10^6$ persons. In such cases, the possible random error of the computation procedure 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 the power of the tests 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 with $df$ equal to the number of free item parameters and noncentrality parameter $\lambda$. Thus, $Var(T) = 2 (df + 2 \lambda)$, with $\lambda = t$. Since the global deviation $e = (1 / n_{infsim}) * t$ it follows for the variance of the corresponding random variable $E$ that $Var(E) = (1 / n_{infsim})^2 * Var(T)$. The power of the tests is a function of $e$ which is given by $F_{df, \lambda} (q_{\alpha})$, where $\lambda = n_{total} * (n_{infsim} / n_{totalsim}) * e$ and $df$ equal to the number of free item parameters. Then, by the delta method one obtains (for the variance of P).

$Var(P) = Var(E) * (F'_{df, \lambda} (q_{\alpha}))^2,$

where $F'_{df, \lambda}$ is the derivative of $F_{df, \lambda}$ with respect to $e$. This derivative is determined numerically and evaluated at $e$ using the package numDeriv. The square root of $Var(P)$ is then used to quantify the random error of the suggested Monte Carlo computation procedure. It is called Monte Carlo error of power.

Value

A list of results.

power	Power value for each test.
MC error of power	Monte Carlo error of power computation for each test.
global deviation	Global deviation computed from simulated data for each test. See Details.
local deviation	CML estimates of item parameters in both groups of persons obtained from the simulated data expressing a deviation from the hypothesis to be tested locally per item.
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 power of the tests.
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 power of the tests.
degrees of freedom	Degrees of freedom $df$.
noncentrality parameter	Noncentrality parameter $\lambda$ of $\chi^2$ distribution from which power is determined.
call	The matched call.

References

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.

See Also

sa_sizeRM, and post_hocRM.

Examples

## Not run: 
# Numerical example

res <-  powerRM(n_total = 130, local_dev = list( c(0, -0.5, 0, 0.5, 1) , c(0, 0.5, 0, -0.5, 1)))

# > res
# $power
#     W    LR    RS    GR
# 0.824 0.840 0.835 0.845
#
# $`MC error of power`
#     W    LR    RS    GR
# 0.002 0.002 0.002 0.002
#
# $`global deviation`
#     W    LR    RS    GR
# 0.118 0.122 0.121 0.124
#
# $`local deviation`
#         Item2 Item3  Item4 Item5
# group1 -0.499 0.005  0.500 1.001
# group2  0.501 0.003 -0.499 1.003
#
# $`person score distribution in group 1`
#
#     1     2     3     4
# 0.249 0.295 0.269 0.187
#
# $`person score distribution in group 2`
#
#     1     2     3     4
# 0.249 0.295 0.270 0.186
#
# $`degrees of freedom`
# [1] 4
#
# $`noncentrality parameter`
#      W     LR     RS     GR
# 12.619 13.098 12.937 13.264
#
# $call
# powerRM(n_total = 130, local_dev = list(c(0, -0.5, 0, 0.5, 1),
#                                         c(0, 0.5, 0, -0.5, 1)))


## End(Not run)

---

Sample size planning for tests in context of measurement of change using LLTM

Description

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 states that the shift parameter quantifying the constant change for all items between time points 1 and 2 equals 0. The alternative states that the shift parameter is not equal to 0. It is assumed that the same items are presented at both time points. See function change_test.

Usage

sa_sizeChange(eta, alpha = 0.05, beta = 0.05, persons = rnorm(10^6))

Arguments

eta	A vector of eta parameters of the LLTM. The last element represents the constant change or shift for all items between time points 1 and 2. The other elements of the vector are the item parameters at time point 1. A choice of the eta parameters constitutes a scenario of deviation from the hypothesis of no change.
alpha	Probability of error of first kind.
beta	Probability of error of second kind.
persons	A vector of person parameters (drawn from a specified distribution). By default $10^6$ parameters are drawn at random from the standard normal distribution. The larger this number the more accurate are the computations. See Details.

Details

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 $df = 1$ 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 presented at two time points) are generated given a user-specified scenario of a deviation from the hypothesis to be tested. The hypothesis to be tested assumes no change between time points 1 and 2. A scenario of a deviation is given by a choice of the item parameters at time point 1 and the shift parameter, i.e., the LLTM eta parameters, as well as the person parameters (to be drawn randomly from a specified distribution). The shift parameter represents a constant change of all item parameters from time point 1 to time point 2. A test statistic $T$ (Wald, LR, score, or gradient) is computed 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, where the relative frequency distribution of these informative scores is considered to be equal to the observed relative frequency distribution of the informative scores in the simulated data. The total sample size $n_{total}$ is then 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 $\geq e$.

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 realized value 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 parameters and shift parameter 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 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 $df = 1$ 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.

Value

A list 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 sample size computation for each test.
deviation	Shift parameter estimated from the simulated data representing the constant shift of item parameters between time points 1 and 2.
person score distribution	Relative frequencies of person scores 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 $df$.
noncentrality parameter	Noncentrality parameter $\lambda$ of $\chi^2$ distribution from which sample size is determined.
total sample size	Total sample size for each test. See Details.
call	The matched call.

References

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.

Fischer, G. H. (1995). The Linear Logistic Test Model. In G. H. Fischer & I. W. Molenaar (Eds.), Rasch models: Foundations, Recent Developments, and Applications (pp. 131-155). New York: Springer.

Fischer, G. H. (1983). Logistic Latent Trait Models with Linear Constraints. Psychometrika, 48(1), 3-26.

See Also

powerChange, and post_hocChange.

Examples

## Not run: 
# Numerical example 4 items presented twice, thus 8 virtual items

# eta Parameter, first 4 are nuisance
# (easiness parameters of the 4 items at time point 1),
# last one is the shift parameter
eta <- c(-2,-1,1,2,0.5)

res <- sa_sizeChange(eta = eta)

# > res
# $`informative sample size`
#   W  LR  RS  GR
# 177 174 175 173
#
# $`MC error of sample size`
#     W    LR    RS    GR
# 1.321 1.287 1.299 1.276
#
# $`deviation (estimate of shift parameter)`
# [1] 0.501
#
# $`person score distribution`
#
#     1     2     3     4     5     6     7
# 0.034 0.094 0.181 0.249 0.227 0.147 0.068
#
# $`degrees of freedom`
# [1] 1
#
# $`noncentrality parameter`
# [1] 12.995
#
# $`total sample size`
#   W  LR  RS  GR
# 182 179 180 178
#
# $call
# sa_sizeChange(alpha = 0.05, beta = 0.05, eta = eta, persons = rnorm(10^6))

## End(Not run)

---

Sample size planning for tests of invariance of item-category parameters between two groups of persons in partial credit model

Description

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.

Usage

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

Arguments

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.
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 $10^6$ parameters are drawn at random from the standard normal distribution. The larger this number the more accurate are the computations. See Details. .
persons2	A vector of person parameters for group 2 (drawn from a specified distribution). By default $10^6$ parameters are drawn at random from the standard normal distribution. The larger this number the more accurate are the computations. See Details.

Details

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.

Value

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 $df$.
noncentrality parameter	Noncentrality parameter $\lambda$ of $\chi^2$ distribution from which sample size is determined.
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.

References

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.

See Also

powerPCM, and post_hocPCM.

Examples

## 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)

---

Sample size planning for tests of invariance of item parameters between two groups of persons in binary Rasch model

Description

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 parameters between two predetermined groups of persons. The alternative assumes that at least one parameter differs between the two groups.

Usage

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

Arguments

local_dev	A list consisting of two vectors containing item parameters for the two person groups representing a deviation from the hypothesis to be tested locally per item.
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 $10^6$ parameters are drawn at random from the standard normal distribution. The larger this number the more accurate are the computations. See Details.
persons2	A vector of person parameters for group 2 (drawn from a specified distribution). By default $10^6$ parameters are drawn at random from the standard normal distribution. The larger this number the more accurate are the computations. See Details.

Details

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 $df$ equal to the number of items minus 1, 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 o f 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 parameters between the two groups of persons. A scenario of a deviation is given by a choice of the item 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. 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 lengths 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 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 $df$ equal to the number of items minus 1 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.

Value

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 parameters in both groups obtained from the simulated data. First item parameter set 0 in both groups.
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 $df$.
noncentrality parameter	Noncentrality parameter $\lambda$ of $\chi^2$ distribution from which sample size is determined.
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.

References

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.

See Also

powerRM, and post_hocRM.

Examples

## Not run: 
##### Sample size of Rasch Model #####

res <-  sa_sizeRM(local_dev = list( c(0, -0.5, 0, 0.5, 1) , c(0, 0.5, 0, -0.5, 1)))

# > res
# $`informative sample size`
#   W  LR  RS  GR
# 159 153 155 151
#
# $`MC error of sample size`
#     W    LR    RS    GR
# 0.721 0.682 0.695 0.670
#
# $`global deviation`
#     W    LR    RS    GR
# 0.117 0.122 0.120 0.123
#
# $`local deviation`
#         Item2  Item3  Item4 Item5
# group1 -0.502 -0.005  0.497 1.001
# group2  0.495 -0.006 -0.501 0.994
#
# $`person score distribution in group 1`
#
#     1     2     3     4
# 0.249 0.295 0.268 0.188
#
# $`person score distribution in group 2`
#
#     1     2     3     4
# 0.249 0.295 0.270 0.187
#
# $`degrees of freedom`
# [1] 4
#
# $`noncentrality parameter`
# [1] 18.572
#
# $`total sample size in group 1`
#  W LR RS GR
# 97 93 94 92
#
# $`total sample size in group 2`
#  W LR RS GR
# 97 93 94 92
#
# $call
# sa_sizeRM(local_dev = list(c(0, -0.5, 0, 0.5, 1),
#                            c(0, 0.5, 0, -0.5, 1)))

## End(Not run)

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