sa_sizeRM: Sample size planning for tests of invariance of item...
In tcl: Testing in Conditional Likelihood Context
sa_sizeRM R Documentation
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(
alpha = 0.05,
beta = 0.05,
persons1 = rnorm(10^6),
persons2 = rnorm(10^6),
local_dev
)
Arguments
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.
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.
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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| sa_sizeRM | R Documentation |
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(
alpha = 0.05,
beta = 0.05,
persons1 = rnorm(10^6),
persons2 = rnorm(10^6),
local_dev
)
Arguments
alpha |
Probability of the error of first kind. |
beta |
Probability of the error of second kind. |
persons1 |
A vector of person parameters for group 1 (drawn from a specified distribution). By default
|
persons2 |
A vector of person parameters for group 2 (drawn from a specified distribution). By default
|
local_dev |
A list consisting of two vectors containing item parameters for the two person groups representing a deviation from the hypothesis to be tested locally per item. |
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 |
noncentrality parameter |
Noncentrality parameter |
total sample size in group 1 |
Total sample size in group 1 for each test. See Details. |
total sample size in group 1 |
Total sample size in group 2 for each test. See Details. |
call |
The matched call. |
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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