Using Exponential Families for Equating

Abstract

In common equipercentile equating methods such as the percentile-rank method or kernel equating (von Davier, Holland, & Thayer, 2004b), sample distributions of test scores are approximated by continuous distributions with positive density functions on intervals that include all possible scores. The use of continuous distributions with positive densities facilitates the equating process, for such distributions have continuous and strictly increasing distribution functions on intervals of interest, so conversion functions can be constructed based on the principles of equipercentile equating. When the density functions are also continuous, as is the case in kernel equating, the further gain is achieved that the conversion functions are differentiable. This gain permits derivation of normal approximations for the distribution of the conversion function, so estimated asymptotic standard deviations (EASDs) can be derived.

Chapter 8 Appendix

1.1 A.1 Computation of Orthogonal Polynomials

Computation of the orthogonal polynomials ${v_k}(C)$, $k \geq 0$, is rather straightforward given standard properties of Legendre polynomials (Abramowitz & Stegun, 1965, Ch. 8, 22). The Legendre polynomial of degree 0 is ${P_0}(x) = 1$; the Legendre polynomial of degree 1 is ${P_1}(x) = x$; and the Legendre polynomial ${P_{k + 1}}(x)$ of degree $k + 1$, $k \geq 1$, is determined by the recurrence relationship

$${P_{k + 1}}(x) = {(k + 1)^{ - 1}}[(2k + 1)x{P_k}(x) - k{P_{k - 1}}(x)],$$

(8.A.1)

so that ${P_2}(x) = (3{x^2} - 1)/2$. For nonnegative integers i and k, the integral $v \int_{ - 1}^1\,{P_i}{P_k}$ is 0 for $i \ne k$ and $1/(2k + 1)$ for $i = k$. Use of elementary rules of integration shows that one may let

$${v_k}(x,C) = {(2k + 1)^{1/2}}{P_k}([2x - {\rm inf}(C) - {\rm sup}(C)]/[{\rm sup}(C) - {\rm inf}(C)]).$$

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