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Sample Size Determination Within the Scope of Conditional Maximum Likelihood Estimation with Special Focus on Testing the Rasch Model

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Abstract

This paper refers to the exponential family of probability distributions and the conditional maximum likelihood (CML) theory. It is concerned with the determination of the sample size for three groups of tests of linear hypotheses, known as the fundamental trinity of Wald, score, and likelihood ratio tests. The main practical purpose refers to the special case of tests of the class of Rasch models. The theoretical background is discussed and the formal framework for sample size calculations is provided, given a predetermined deviation from the model to be tested and the probabilities of the errors of the first and second kinds.

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Authors and Affiliations

  1. The Health and Life Sciences University, EWZ 1, 6060 , Hall, Austria

    Clemens Draxler & Rainer W. Alexandrowicz

  2. University of Klagenfurt, Psychology Institute, Universitaetsstrasse 65-67, 9020, Klagenfurt, Austria

    Clemens Draxler & Rainer W. Alexandrowicz

Authors
  1. Clemens Draxler
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  2. Rainer W. Alexandrowicz
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Correspondence to Clemens Draxler.

Appendix

Appendix

This appendix provides a discussion of the fundamentals of the asymptotic properties of the CML estimator with reference to the concept of a sequence of local alternative hypotheses and a fixed alternative. It refers to the result that a joint limiting distribution for Wald, score, and likelihood ratio test statistics does not exist under a fixed alterative whereas it does exist under a sequence of local alternative hypotheses.

Let \(\varvec{\beta }_{0} \) denote a vector satisfying (4) and assume that \(\varvec{\beta }=\varvec{\beta }_{0} \) holds. Then, according to Andersen’s (Andersen, 1970) results,

$$\begin{aligned} \sqrt{n} \left( {\hat{\varvec{{\beta }}}-\varvec{\beta }_{0}} \right) \xrightarrow {d}N\left[ {\varvec{0},n\varvec{{I}}\left( {\varvec{\beta }_{0}} \right) ^{-1}} \right] , \end{aligned}$$

for \(n\rightarrow \infty \), the Fisher information matrix \(\varvec{{I}}\left( {\varvec{\beta }_{0}} \right) \) being evaluated using \(\varvec{\beta }_{0} \). The covariance \(n\varvec{{I}}\left( {\varvec{\beta }_{0}} \right) ^{-1}\) of \(\sqrt{n} \left( {{\hat{\varvec{{\beta }}}-\varvec{\beta } }_{0}} \right) \) remains bounded as \(n\rightarrow \infty \) since the Fisher information is additive and may be rewritten as

$$\begin{aligned} \varvec{{I}}\left( {\varvec{\beta }_{0}} \right) = - E\left[ {\frac{{\partial ^{2} L\left( {\varvec{\beta }_{0}} \right) }}{{\partial \varvec{\beta }_{0} \partial \varvec{\beta } ^{\prime }_{0}}}} \right] = - \sum \limits _{{v = 1}}^{n} {E\left\{ {\frac{{\partial ^{2} \log \left[ {h_{{\varvec{\beta }_{0}}} \left( {\varvec{y}_{v} \left| {\varvec{T}_{v} = \varvec{t}_{v}} \right. } \right) } \right] }}{{\partial \varvec{\beta }_{0} \partial \varvec{\beta } ^{{\prime }}_{0}}}} \right\} }, \end{aligned}$$

i.e., the information contained in the conditional distribution of \(\varvec{Y}_{1} \left| {\varvec{T}_{1} =\varvec{t}_{1}} \right. ,\ldots ,\varvec{Y}_{n} \left| {\varvec{T}_{n} =\varvec{t}_{n}} \right. \) is obtained by summation of the information contained in the conditional distribution of each \(\varvec{Y}\left| {\varvec{T}=\varvec{t}} \right. \).

Let \(\varvec{\beta }_{a} =\varvec{\beta }_{0} +\varvec{\delta }, \varvec{\delta }\ne \varvec{0}\), be a vector not satisfying (4), i.e., a fixed alternative, a fixed deviation independent of n. If \(\varvec{\beta }=\varvec{\beta }_{a} \) holds, a limiting distribution of \(\sqrt{n} \left( {\hat{\varvec{{\beta }}}-\varvec{\beta }_{0}} \right) \) will not exist since, according to Andersen’s (1970) and Pfanzagl’s (1993) consistency theorems, \(\hat{\varvec{{\beta }}}\) will converge to \(\varvec{\beta }_{a} \) so that

$$\begin{aligned} \sqrt{n} \left( {\hat{\varvec{{\beta }}}-\varvec{\beta }_{0}} \right) =\sqrt{n} \left[ {\hat{\varvec{{\beta }}}-\left( {\varvec{\beta }_{a} \varvec{-\delta }} \right) } \right] \end{aligned}$$

will grow unboundedly with \(n\rightarrow \infty \), i.e., the mean vector amounts to \(\sqrt{n} \varvec{\delta }\). Hence, a limiting distribution for the three test statistics Wald, score, and likelihood ratio will also not exist if the fixed alternative \(\varvec{\beta }=\varvec{\beta }_{a} \) holds.

Let the assumption of a sequence of local alternative hypotheses \(\varvec{\beta }_{an} =\varvec{\beta }_{0} +\varvec{\delta }n^{-0.5}\) be introduced so that \(\varvec{T\beta }_{an} \rightarrow \varvec{c}\) as \(n\rightarrow \infty \). Then, one obtains

$$\begin{aligned} \sqrt{n} \left( {\hat{\varvec{{\beta }}}-\varvec{\beta }_{0}} \right) =\sqrt{n} \left[ {\hat{\varvec{{\beta }}}-\left( {\varvec{\beta }_{an} \varvec{-\delta }n^{-0.5}} \right) } \right] \xrightarrow {d}N\left[ {\varvec{\delta },n\varvec{{I}}\left( {\varvec{\beta }_{0}} \right) ^{-1}} \right] \end{aligned}$$

as \(n\rightarrow \infty \), since both \(\hat{\varvec{{\beta }}}\rightarrow \varvec{\beta }_{0} \) and \(\varvec{\beta }_{an} \rightarrow \varvec{\beta }_{0} \). With this and given technical regularity conditions, the results regarding the joint limiting distribution of Wald, score, and likelihood ratio test statistics derived in the papers cited in 3.2 apply. The non-central \(\chi ^{2}\) distribution obviously follows from the multivariate normal distribution of \(\sqrt{n} \left( {\hat{\varvec{{\beta }}}-\varvec{\beta }_{0}} \right) \) with mean vector \(\varvec{\delta }\ne \varvec{0}\). It might also be interesting to note that the statistic divided by n has a positive limit which may provide a measure of model error comparable to the well-known effect measures suggested by Cohen (1988).

In 4.1 it is assumed that the sequence of nuisance parameters \(\varvec{\theta }_{1} ,\ldots ,\varvec{\theta }_{n} \) is independently and identically distributed with joint probability density function \(\varphi \left( {\varvec{\theta }} \right) \) so that the common probability distribution of the sequence \(\varvec{T}_{1} ,\ldots ,\varvec{T}_{n} \) is given by (9). Under this assumption it holds for the asymptotic covariance matrix of \(\sqrt{n} \left( {\hat{\varvec{{\beta }}}-\varvec{\beta }_{0}} \right) \) that, according to (10),

$$\begin{aligned} n\varvec{{I}}\left( {\varvec{\beta }_{0}} \right) ^{-1}=nn^{-1}{\varvec{\varGamma }}\left( {\varvec{\beta }_{0}} \right) ^{-1}={\varvec{\varGamma }}\left( {\varvec{\beta }_{0}} \right) ^{-1}, \end{aligned}$$

with \({\varvec{\varGamma }}\left( {\varvec{\beta }_{0}} \right) \) given by the second factor (the integral) of (10) times \(-1\) evaluated at \(\varvec{\beta }_{0} \). Hence, for \(n\rightarrow \infty \), if \(\varvec{\beta }=\varvec{\beta }_{0} \) holds

$$\begin{aligned} \sqrt{n} \left( {\hat{\varvec{{\beta }}}-\varvec{\beta }_{0}} \right) \xrightarrow {d}N\left[ {\varvec{0},{\varvec{\varGamma }}\left( {\varvec{\beta }_{0}} \right) ^{-1}} \right] \end{aligned}$$

and under the sequence of local alternative hypotheses

$$\begin{aligned} \sqrt{n} \left( {\hat{\varvec{{\beta }}}-\varvec{\beta }_{0}} \right) \xrightarrow {d}N\left[ {\varvec{\delta },{\varvec{\varGamma }}\left( {\varvec{\beta }_{0}} \right) ^{-1}} \right] . \end{aligned}$$

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Draxler, C., Alexandrowicz, R.W. Sample Size Determination Within the Scope of Conditional Maximum Likelihood Estimation with Special Focus on Testing the Rasch Model. Psychometrika 80, 897–919 (2015). https://doi.org/10.1007/s11336-015-9472-y

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