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.
powerChange(n_total, eta, alpha = 0.05, persons = rnorm(10^6))A list of results of class tcl_power.
Power value for each test.
Monte Carlo error of power computation for each test.
Shift parameter estimated from the simulated data, representing the constant shift of item parameters between time points 1 and 2.
Relative frequencies of person scores observed in simulated data. Uninformative scores, i.e., minimum and maximum scores, are omitted. Note that the person score distribution also has an influence on the power of the tests.
Degrees of freedom (\(df\)).
Noncentrality parameter (\(\lambda\)) of the \(\chi^2\) distribution from which power is determined.
The matched call.
Total sample size for which power shall be determined.
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.
Probability of the error of first kind.
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.
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 \(q_{1- \alpha}\) be the \(1 - \alpha\) quantile of the central \(\chi^2\) distribution with \(df = 1\). Then,
$$power = 1 - F_{df, \lambda} (q_{1- \alpha}),$$
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.
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.
sa_sizeChange, and post_hocChange.
if (FALSE) {
# 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_err_power #`MC error of power`
# W LR RS GR
# 0.002 0.002 0.002 0.002
#
# $dev_est_shift #`deviation (estimate of shift parameter)`
# [1] 0.499
#
# $score_dist #`person score distribution`
#
# 1 2 3 4 5 6 7
# 0.034 0.093 0.181 0.249 0.228 0.147 0.068
#
# $df #`degrees of freedom`
# [1] 1
#
# $ncp #`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))
#
}
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