The Use of Classical Quadratic Test Statistics for Testing Hypothesis with Complex Survey Data: an Application to Testing Informativeness of Sampling Weights in Multiple Linear Regression Analysis

1. INTRODUCTION k-L-p+l. Thus, the Wald statistic X W =-01V-10 can have very low power (Korn Complex surveys usually involve and Graubard 1990). multistage cluster sampling where the selection of primary sampling units (PSU's) is within Alternative test procedures the Wald well-defined strata at the first stage of sampling, test, along with their asymptotic properties, are The standard error of an estimator can be presented in the next section of this paper. computed from the variability of the estimator These procedures are based upon the work of using data from different subsets of the sampled Rao and Scott (1981) and Fay (1985). An PSU's. For parameters involving nonlinear application to testing whether sampling weights combinations of population means, methods are informative (i.e., reduce bias) in a multiple including Taylor series linearization, the linear regression analysis is given in the third jackknife, balanced half sample replication, and section. In the last section we discuss related the bootstrap have been developed to estimate work. The beauty of these replication 2. TEST STATISTICS AND THEIR methods is that they accommodate quite ASYMPTOTIC PROPERTIES complicated sampling and estimation at the second and further stages of sampling. 2.1 Framework for Hypothesis Testing and Construction of Test Statistics In this paper, we are interested in testing hypotheses about an infinite population Let 0 be a p-dimensional vector of parameter 0-g(p) where # is a mean vector, parameters that is equal to zero under the null Following Krewski and Rao (1981), we do not hypothesis, i.e., Ho: 0-0. We assume that 0 specify the infinite population. Instead, it is can be expressed as a (nonlinear)function of a q-assumed that there is sequence of increasing dimensional mean vector #, i.e., 0-g(p). finite populations whose means converge to #. Additionally, we assume the existence of a Another aspect of the inferential approach of quadratic test statistic that would be utilized if this paper is that we utilize replicated estimates we had a simple random sample (srs) of size n of standard errors instead of model-based from the population namely, nT(Ysrs)-n O~ srs estimates for hypothesis testing. It can be M(Ysrs) 0sr s , where Ysrs is the sample mean, difficult to model the complex variance structure 0srs-g(Ysrs) and M(Ysrs) is a pxp matrix. from data that comes from a multistage For example, nT could be a Wald statistic for stratified cluster sample, and even if modeled, testing a vector …