Optimal Designs for the Rasch Model

Abstract

In this paper, optimal designs will be derived for estimating the ability parameters of the Rasch model when difficulty parameters are known. It is well established that a design is locally D-optimal if the ability and difficulty coincide. But locally optimal designs require that the ability parameters to be estimated are known. To attenuate this very restrictive assumption, prior knowledge on the ability parameter may be incorporated within a Bayesian approach. Several symmetric weight distributions, e.g., uniform, normal and logistic distributions, will be considered. Furthermore, maximin efficient designs are developed where the minimal efficiency is maximized over a specified range of ability parameters.

Appendix: Proofs

For the proofs we may assume throughout that θ ₀=0, which can be achieved by a location shift without loss of generality.

Proof of Lemma 1

Let F(z)=1/(1+exp(−z)) be the distribution function of the standard logistic distribution. We will employ that F′(z)=f(z)=F(z)(1−F(z)) and, hence, f′(z)=f(z)(1−2F(z)).

• (i) We set ψ(σ)=Ψ ₁(ξ _σ ).

For the derivatives of ψ we obtain ψ′(σ)=∫(1−2F(θ−σ))f(θ−σ)⁻¹  Π(dθ), which results in ψ′(0)=∫(1−2F(θ))f(θ)⁻¹  Π(dθ), where the integration with respect to θ and the differentiation with respect to σ may be interchanged under suitable regularity conditions on the weight distribution Π. Since Π is symmetric around θ ₀=0 and since F(z)=1−F(−z), we have $\int F(\theta) f(\theta)^{-1}\, \varPi( {\mathrm{d}} \theta)=\int (1-F(\theta)) f(\theta)^{-1}\, \varPi( {\mathrm{d}} \theta)=\frac{1}{2} \int f(\theta)^{-1}\, \varPi( {\mathrm{d}} \theta)$, and, hence, ψ′(0)=0. Further ψ″(σ)=ψ(σ)−2>0 implies that ψ is convex, such that ψ(σ)=Ψ ₁(ξ _σ ) is minimized for σ=θ ₀.

• (ii) For Ψ ₂ we obtain $\frac{\partial}{\partial\sigma} \varPsi_{2}(\xi_{\sigma})= -\int\frac {\partial}{\partial\sigma} \log(f(\theta-\sigma))\, \varPi( {\mathrm{d}} \theta)=1-2 \int F(\theta-\sigma)\, \varPi( {\mathrm{d}} \theta)$.

Similar arguments as used in (i) lead to $\int F(\theta)\, \varPi( {\mathrm{d}} \theta )=\int F(-\theta)\, \varPi( {\mathrm{d}} \theta)=\frac{1}{2}$ and, consequently, $\frac{\partial}{\partial\sigma}\varPsi_{2}(\xi_{\sigma })|_{\sigma=0} =0$.

As the second derivative, $\frac{\partial^{2} }{\partial\sigma^{2}} \varPsi_{2}(\xi_{\sigma})=-\int\frac {\partial^{2} }{\partial\sigma^{2}}\log(f(\theta-\sigma))\, \varPi( {\mathrm{d}} \theta)= 2 \int f(\theta-\sigma)\, \varPi( {\mathrm{d}} \theta)\allowbreak >0$, is positive for all σ, the criterion Ψ ₂(ξ _σ ) is convex in σ and, hence, Ψ ₂(ξ _σ ) is minimized for σ=θ ₀. □

Proof of Theorem 1

(a) For the sensitivity function d ₁(σ;ξ ₀), the derivatives are $d_{1}'(\sigma;\xi_{0})= -c \int(1-2\, F(\theta-\sigma)) f(\theta-\sigma) f(\theta)^{-2}\, \varPi( {\mathrm{d}} \theta)$ and $d_{1}''(\sigma;\xi_{0})=c \int(1-6 f(\theta-\sigma)) f(\theta-\sigma ) f(\theta)^{-2}\, \varPi( {\mathrm{d}} \theta)$, where c=(∫f(θ)⁻¹  Π(dθ))⁻¹. For the point σ=0 we obtain $d_{1}'(0;\xi_{0})=0$ and $d_{1}''(0;\xi_{0})= 1-6 c$. Thus, the necessary condition $d_{2}''(0;\xi_{0})\leq0$ is equivalent to c ⁻¹=∫f(θ)⁻¹  Π(dθ)≤6.

(b) For the one-point design $\xi_{\sigma^{*}}=\xi_{0}$ at σ ^∗=θ ₀=0 the first and the second derivatives of d ₂ with respect to σ can be calculated as $d_{2}'(\sigma;\xi_{0})=-\int(1-2 F(\theta-\sigma)) \frac{f(\theta-\sigma)}{f(\theta)}\, \varPi( {\mathrm{d}} \theta)$ and $d_{2}''(\sigma;\xi_{0})=\int(1-6 f(\theta-\sigma)) \frac{f(\theta-\sigma)}{f(\theta)}\, \varPi( {\mathrm{d}} \theta)$.

For σ=0 this gives $d_{2}'(0;\xi_{0})=0$ and $d_{2}''(0;\xi _{0})=\int(1-6 f(\theta))\, \varPi( {\mathrm{d}} \theta)$. Hence, the necessary condition $d_{2}''(0;\xi_{0})\leq0$ for a maximum is equivalent to $\int f(\theta)\, \varPi( {\mathrm{d}} \theta)\geq\frac {1}{6}$. □

Proof of Theorem 2

(i) Let γ ₁(τ)=∫f(τθ)⁻¹  Π(dθ) and note that ∫f(θ)⁻¹  Π _τ (dθ)=γ ₁(τ). The integrand f(τθ)⁻¹ in γ ₁ is strictly continuously increasing in τ for all θ. Therefore, γ ₁ is a strictly increasing continuous function in τ with γ ₁(0)=4 and γ ₁(τ)→∞ for τ→∞. Then τ ₁ is the unique solution of γ ₁(τ)=6.

Similarly, γ ₂(τ)=∫f(τθ) Π(dθ)=∫f(θ) Π _τ (dθ) is a strictly decreasing function with $\gamma_{2} (0) = \frac{1}{4}$ and $\gamma_{2} (\tau) \to\frac{1}{4} \pi _{0}$ for τ→∞, where π ₀ is the weight of Π in θ ₀=0. Hence, τ ₂ can be chosen as the unique solution of $\gamma_{2} (\tau) = \frac{ 1}{6}$, if $\pi_{0} < \frac{ 2}{3}$, and τ ₂=∞ otherwise. As γ ₁(τ)≥1/γ ₂(τ), by Jensen’s inequality, we may conclude that τ ₁≤τ ₂.

(ii) For fixed σ let $g_{\tau} (\theta) = \frac{f(\tau \theta + \sigma) + f (\tau \theta - \sigma)}{ f (\tau \theta)}$. Denote by d _2,τ the sensitivity function when the weight distribution Π _τ is used. Then, $d_{2,\tau} (\sigma; \xi_{0}) = \int^{\infty}_{0} g_{\tau} (\theta) \, \varPi( {\mathrm{d}} \theta)$ due to symmetry considerations.

Now, $\frac{\partial}{\partial\tau} g_{\tau} (\theta) \geq\theta g_{\tau} (\theta) [( F (\tau \theta) - F( - \tau \theta)) - (F (\tau \theta- \sigma) - F ( -\tau, \theta- \sigma))]$, and the term within the brackets is positive for σ≠σ ^∗=0 due to the strict unimodality of f=F′ at zero. Hence, $d_{2,\tau} (\sigma; \xi_{0}) = \int^{\infty}_{0} g_{\tau} (\theta) \, \varPi( {\mathrm{d}} \theta)$ is increasing continuously in τ for each σ. Thus, max _σ d _2,τ (σ;ξ ₀) is an increasing continuous function in τ, as long as it is bounded by 1 and $\tilde{\tau}_{2}$ may be chosen as the largest value for τ satisfying max _σ d _2,τ (σ;ξ ₀)=1. Obviously, $\tilde{\tau}_{2}$ cannot exceed τ ₂ obtained from the necessary condition. □

Proof of Corollary 1

If Π is symmetric around θ ₀=0, it may be represented as a mixture of symmetric two-point weight distributions Π _θ , which assign equal weights 1/2 to each of ±θ. Denote by $\tilde{\varPi}$ the corresponding mixing distribution, which is the image of Π under the mapping of the absolute value, and denote by Ψ _i,θ (ξ) the value of the criterion function Ψ _i for design ξ when the weight distribution Π _θ is used. From Example 4.1 we obtain Ψ _i,θ (ξ)≥Ψ _i,θ (ξ ₀) for every design ξ, if $|\theta|\leq\tilde{\tau}=\log (2+\sqrt{3}\,)$. As Π is supported within $[-\tilde{\tau},\tilde{\tau }]$ we may conclude

$$\varPsi_i(\xi)=\int \varPsi_{i,\theta}(\xi)\, \varPi( {\mathrm{d}} \theta) \geq \int \varPsi_{i,\theta}(\xi_0)\, \varPi( {\mathrm{d}} \theta)=\varPsi_i(\xi_0), $$

which establishes the optimality of $\xi_{\sigma^{*}}$ for σ ^∗=θ ₀=0 with respect to Ψ _i . □

Proof of Theorem 3

The assumption $\theta_{0}=\frac{1}{2} (\theta_{1} + \theta_{2})=0$ implies that the range is centered, i.e. θ ₁=−θ ₂. Due to invariance considerations we may restrict the search for maximin efficient designs to those which are symmetric around θ ₀=0, as for any symmetric region |θ|≤θ ₂ we have $\min_{|\theta| \leq\theta_{2}} \operatorname{eff}(\xi; \theta) \leq \min_{|\theta| \leq\theta_{2}} \operatorname{eff} ( \bar{\xi}; \theta )$ for every design ξ, where $\bar{\xi} = \frac{1}{2} (\xi+ \xi^{-})$ is its symmetrization and ξ ⁻ is the image of ξ under sign change (see Graßhoff & Schwabe, 2008).

First we consider symmetric two-point designs $\bar{\xi}_{\sigma} = \frac{1}{2} (\xi_{\sigma} + \xi_{-\sigma})$, which assign equal weights 1/2 to ±σ each. The corresponding information $M (\bar{\xi}_{\sigma} ; \theta) = \frac{1}{2} (f ( \theta-\sigma) + f ( \theta+\sigma))$ equals the Bayesian criterion $\varPsi_{1} (\xi_{ \sigma} ) = \frac{1}{2} (f ( \theta-\sigma) + f( -\theta-\sigma))$ for the one-point design ξ _σ , when the symmetric two-point prior on ±θ is used. According to Example 1 of Section 4 this criterion is majorized by Ψ ₁(ξ ₀)=f(θ)=M(ξ ₀;θ) as long as $|\theta| \leq\tilde{\tau}= \log(2 + \sqrt{3}\,)$.

Hence, $M ( \bar{\xi}_{\sigma} ; \theta) \leq M ( \xi_{0} ; \theta)$ uniformly in $|\theta| \leq\tilde{\tau}$. Note that also ξ ₀ is, of course, symmetric around 0. Now every design $\bar{\xi}$, which is symmetric around 0, can be written as a mixture of symmetric two-point designs $\bar{\xi}_{\sigma}$ and, hence, $M( \bar{\xi} ; \theta) = \int M ( \bar{\xi}_{\sigma} ; \theta) \tilde{\xi} ({\mathrm{d}} \sigma)$, where $\bar{\xi}$ is a suitable mixing measure. This implies $M ( \bar{\xi} ; \theta) \leq M (\xi_{0} ; \theta )$ for all $|\theta| \leq\tilde{\tau}$.

Combining the above results yields $\min_{|\theta| \leq\theta _{2}} \operatorname{eff}(\xi; \theta) \leq\min_{|\theta| \leq\theta _{2}} \operatorname{eff} (\xi_{\sigma^{*}} ; \theta)$ for every $\theta_{2} \leq\nobreak \tilde{\tau}$, which establishes the result. □