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Article Excerpt
1. INTRODUCTION
Setting confidence intervals for linear combinations of variance components related to a normal random vector plays an important role in many statistical applications. The simplest example is to compare the variances of two normal random variables or, more generally, the variances of two linear combinations of independent normal random variables. Comparing two variances, [[sigma].sub.1.sup.2] and [[sigma].sub.2.sup.2], is usually done in terms of the ratio [[sigma].sub.1.sup.2]/[[sigma].sub.2.sup.2]. If testing hypotheses such as [[sigma].sub.1.sup.2]/[[sigma].sub.2.sup.2] [greater than or equal to] [[delta].sub.0] versus [[sigma].sub.1.sup.2]/[[sigma].sub.2.sup.2] < [[delta].sub.0] is concerned, where [[delta].sub.0] is a given margin, then the problem may be solved by constructing a confidence bound for the linear combination [[sigma].sub.1.sup.2] - [[delta].sub.0][[sigma].sub.2.sup.2], because [[sigma].sub.1.sup.2]/[[sigma].sub.2.sup.2] [greater than or equal to] [[delta].sub.0] is the same as [[sigma].sub.1.sup.2] - [[delta].sub.0][[sigma].sub.2.sup.2] [greater than or equal to] 0. In clinical trials, comparing variability is often carried out together with comparing the mean response values of drug products. When the mean response values of the primary endpoints from two treatments are close, the value with smaller variability may be considered clinically superior than the other. In assessing bioequivalence between two drug products, the U.S. Food and Drug Administration (FDA) recommends the assessment of individual bioequivalence and population bioequivalence, which is equivalent to setting a confidence bound for some variance components. Other examples of setting confidence intervals for linear combinations of variance components include various testing hypothesis problems in balanced k-way analysis of variance models and k-factor nested variance components models (for more details, see Burdick and Graybill 1992; Graybill and Wang 1980; Ting, Burdick, Graybill, Jeyaratnam, and Lu 1990).
Except for a few special cases, there are no exact confidence intervals (i.e., confidence intervals with confidence coefficients exactly equal to the specified nominal level) available for a given linear combination of variance components. Various approximate confidence intervals (i.e., confidence intervals with confidence coefficients close to the nominal confidence coefficients in some sense) have been proposed (see, e.g., Burdick and Graybill 1992; Graybill and Wang 1980; Ting et al. 1990). One method that is very easy to use and has better finite-sample performance than many other methods in empirical studies (Graybill and Wang 1980; Ting et al. 1990; Hyslop, Hsuan, and Holder 2000) is the modified large-sample (MLS) method, which was first proposed by Howe (1974) for the sum of two variance components, later generalized by Graybill and Wang (1980) to nonnegative linear combinations of variance components, and then generalized by Ting et al. (1990) to general linear combinations of variance components.
A key condition assumed in all previously published works on MLS is that the estimated variance components are independent. In some applications, however, variance component estimators are dependent. This occurs in particular when the study design is a crossover design, which is chosen by the FDA for bioequivalence studies. In fact, in the 2001 FDA guidelines for bioequivalence studies (FDA 2001), the MLS method is applied by ignoring the dependence among variance component estimators, which results in a test with incorrect size for assessing population bioequivalence. The first purpose of this article is to extend the MLS method to the cases where variance component estimators are dependent. After a detailed description of the MLS method, the extension is introduced in Section 2. The key idea in our extension is a chi-squared representation; that is, under some conditions, a given quadratic form of a normal random vector has the same distribution as that of a linear combination of independent chi-squared random variables.
From their derivation (see Sec. 2), the MLS confidence intervals are asymptotically correct; that is, coverage probability converges to the nominal level as the sample size increases. However, little is known about their other asymptotic properties. For example, are the MLS confidence intervals second-order asymptotically accurate? Note that a second-order asymptotically accurate confidence interval can be derived using Edgeworth and Cornish-Fisher expansions; however, simulation results have shown that sometimes the MLS confidence intervals are even better than the intervals based on Edgeworth and Cornish-Fisher expansions. The second purpose of this article is to study second-order asymptotic properties of the MLS confidence bounds. Using the chi-squared representation and Edgeworth and Cornish-Fisher expansions, we explicitly derive in Section 3 the second-order asymptotic coverage error of the MLS confidence bounds. Although the MLS confidence bounds are not second-order asymptotically accurate (except for a few special cases), our result shows that the MLS confidence bounds are better than the confidence bounds based on normal approximation and in some cases are nearly second-order accurate, which explains its nice empirical performance in simulation studies. Our result also indicates how to construct an MLS confidence bound that is second-order accurate.
As an application, in Section 4 we apply the MLS method to population bioequivalence testing. The MLS confidence bounds are derived under commonly used 2 X 2 and 2 X 4 crossover designs. We also present some simulation results and an example.
2. THE MLS METHOD AND ITS EXTENSION
Throughout this article, we denote the distribution of a random quantity x by L(x). Hence L(x) = L(y) means that x and y have the same distribution. We let N([mu], [SIGMA]) denote the normal distribution with mean [mu] and covariance matrix [SIGMA] and let [[chi square].sub.r] denote the central chi-squared distribution with r degrees of freedom. We denote the [alpha]th quantiles of N(0, 1), [[chi square].sub.r], and the central t distribution with degrees of freedom r by [z.sub.[alpha]], [[chi square].sub.[alpha],r], and [t.sub.[alpha],r]; the transpose of the vector x by x'; and the Kronecker product of two matrices A and B by A [cross product] B.
2.1 The MLS Method
Consider the problem of setting a 1 - [alpha] upper confidence bound for [eta] = [c.sub.1][[sigma].sub.1.sup.2] + ... + [c.sub.p][[sigma].sub.p.sup.2], where [[sigma].sub.1.sup.2],..., [[sigma].sub.p.sup.2] are unknown variance components and [c.sub.1],..., [c.sub.p] are known nonzero constants. Let [^.[sigma].sub.i.sup.2] be an unbiased estimator of [[sigma].sub.i.sup.2] such that L([^.[sigma].sub.i.sup.2]) = L([[sigma].sub.i.sup.2][n.sub.i.sup.-1][X.sub.i]), where L([X.sub.i]) = [[chi square].sub.n.sub.i], i = 1,..., p. In applications, [^.[sigma].sub.i.sup.2] is typically a quadratic form of an observed normal random vector (e.g., a sample variance). For any fixed i, an exact 1 - [alpha] upper confidence bound for [[sigma].sub.i.sup.2] is
[[n.sub.i]/[[chi square].sub.[alpha],[n.sub.i]]][^.[sigma].sub.i.sup.2] = [^.[sigma].sub.i.sup.2] + [square root of ([^.[sigma].sub.i.sup.4]([[n.sub.i]/[[chi square].sub.[alpha],[n.sub.i]]] - 1)[.sup.2])]. (1)
When p [greater than or equal to] 2, an exact upper confidence bound for [eta] usually does not exist.
Suppose that [^.[sigma].sub.1.sup.2],..., [^.[sigma].sub.p.sup.2] are independent. Then
var([c.sub.1][^.[sigma].sub.1.sup.2] + ... + [c.sub.p][^.[sigma].sub.p.sup.2]) = [c.sub.1.sup.2]var([^.[sigma].sub.1.sup.2]) + ... + [c.sub.p.sup.2]var([^.[sigma].sub.p.sup.2]) = [c.sub.1.sup.2][[sigma].sub.1.sup.4][n.sub.1.sup.-2]var([X.sub.1]) + ... + [c.sub.p.sup.2][[sigma].sub.p.sup.4][n.sub.p.sup.-2]var([X.sub.p]) = [c.sub.1.sup.2][[sigma].sub.1.sup.4]2[n.sub.1.sup.-1] + ... + [c.sub.p.sup.2][[sigma].sub.p.sup.4]2[n.sub.p.sup.-1]. (2)
An estimator of the variance in (2) is then obtained by replacing [[sigma].sub.i.sup.4] in (2) by its estimator [^.[sigma].sub.i.sup.4]. For large [n.sub.i]'s, these results and the central limit theorem lead to the following approximate 1 - [alpha] upper confidence bound for [eta]:
[c.sub.1][^.[sigma].sub.1.sup.2] + ... + [c.sub.p][^.[sigma].sub.p.sup.2] + [z.sub.1-[alpha]][square root of ([c.sub.1.sup.2][^.[sigma].sub.1.sup.4]2[n.sub.1.sup.-1] + ... + [c.sub.p.sup.2][^.[sigma].sub.p.sup.4]2[n.sub.p.sup.-1])] = [c.sub.1][^.[sigma].sub.1.sup.2] + ... + [c.sub.p][^.[sigma].sub.p.sup.2] + [square root of ([c.sub.1.sup.2][^.[sigma].sub.1.sup.4]2[z.sub.1-[alpha].sup.2][n.sub.1.sup.-1] + ... + [c.sub.p.sup.2][^.[sigma].sub.p.sup.4]2[z.sub.1-[alpha].sup.2][n.sub.p.sup.-1])]. (3)
Consider first the case where [c.sub.i] > for all i. In view of (1), Graybill and Wang (1980) proposed replacing 2[z.sub.1-[alpha].sup.2][n.sub.i.sup.-1] in (3) by ([n.sub.i]/[[chi square].sub.[alpha],[n.sub.i]] - 1)[.sup.2], i = 1,..., p, and termed this method the MLS...
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