Joseph L. Gastwirth

Trend Tests for Case-Control Studies of Genetic Markers: Power, Sample Size and Robustness

The Cochran-Armitage trend test is commonly used as a genotype-based test for candidate gene association. Corresponding to each underlying genetic model there is a particular set of scores assigned to the genotypes that maximizes its power. When the variance of the test statistic is known, the formulas for approximate power and associated sample size are readily obtained. In practice, however, the variance of the test statistic needs to be estimated. We present formulas for the required sample size to achieve a prespecified power that account for the need to estimate the variance of the test statistic. When the underlying genetic model is unknown one can incur a substantial loss of power when a test suitable for one mode of inheritance is used where another mode is the true one. Thus, tests having good power properties relative to the optimal tests for each model are useful. These tests are called efficiency robust and we study two of them: the maximin efficiency robust test is a linear combination of the standardized optimal tests that has high efficiency and the MAX test, the maximum of the standardized optimal tests. Simulation results of the robustness of these two tests indicate that the more computationally involved MAX test is preferable.

A Scale-Free Goodness-of-Fit Test for the Exponential Distribution Based on the Lorenz Curve

Abstract The sample Lorenz curve provides a powerful, easily computed goodness-of-fit test for exponentiality which does not depend on the unknown scale parameter. Exact critical values and asymptotic results are given, which can also be used to test the related hypothesis of randomness on an interval. General limit theorems show that the sample Lorenz curve converges almost surely to the population Lorenz curve and that the standardized Lorenz statistic converges to normality provided the underlying random variable has finite variance. Asymptotic relative efficiencies and Monte Carlo power estimates are also given.

ON ROBUST PROCEDURES

Abstract This paper discusses a procedure for finding robust estimators of the location parameter of symmetric unimodal distributions. The estimators are based on robust rank tests and the methods used are applicable to other one parameter problems. To every density function there corresponds an asymptotically most powerful rank test (a.m.p.r.t.). For a set , of density functions the maximin rank test, R, maximizes the minimum limiting Pitmans efficiency of R relative to the a.m.p.r.t. for each member of . This maximin test, R, can be used to construct estimators according to the proposal of Hodges and Lehman; it generates another estimator T in the following manner. If the test based on R is the a.m.p.r.t. for samples from a density function g(x − θ), then the estimator T will be the best linear unbiased estimate (b.l.u.e.) of the location parameter for samples from g(x). Unfortunately, the estimator T is not necessarily consistent for all members of . A class of rank tests which generate linear combina...

The Use of Maximin Efficiency Robust Tests in Combining Contingency Tables and Survival Analysis

Abstract A test is the maximin efficiency robust test for a family of possible models underlying the data if no other test has higher minimum efficiency relative to the asymptotically optimum test for each model. Methods used to examine the efficiency robustness of rank tests for the two-sample problem are adapted to obtain maximin efficiency robust procedures for testing the equality of proportions across several 2×2 tables, for combining the results of tests for trend in several 2 x J tables in which the dose-response function is one of a set of possible monotone functions, and to analyze censored survival data when either the Wilcoxon or log-rank may be appropriate. In the survival setting the robust test has maximin efficiency 93.3% relative to the Wilcoxon or long-rank when each is optimum, in contrast to the 75% relative efficiency each statistic has when the other is optimum.

The Impact of Levene's Test of Equality of Variances on Statistical Theory and Practice.

In many applications, the underlying scientific question concerns whether the variances of k samples are equal. There are a substantial number of tests for this problem. Many of them rely on the assumption of normality and are not robust to its violation. In 1960 Professor Howard Levene proposed a new approach to this problem by applying the F -test to the absolute deviations of the observations from their group means. Levene’s approach is powerful and robust to nonnormality and became a very popular tool for checking the homogeneity of variances. This paper reviews the original method proposed by Levene and subsequent robust modifications. A modification of Levene-type tests to increase their power to detect monotonic trends in variances is discussed. This procedure is useful when one is concerned with an alternative of increasing or decreasing variability, for example, increasing volatility of stocks prices or “open or closed gramophones” in regression residual analysis. A major section of the paper is devoted to discussion of various scientific problems where Levene-type tests have been used, for example, economic anthropology, accuracy of medical measurements, volatility of the price of oil, studies of the consistency of jury awards in legal cases and the effect of hurricanes on ecological systems.

Latent class analysis of human herpesvirus 8 assay performance and infection prevalence in sub-Saharan Africa and Malta

Human herpesvirus 8 (HHV‐8) is thought to be highly prevalent in Mediterranean countries and sub‐Saharan Africa, where it causes Kaposis sarcoma in a small proportion of infected immunocompetent persons. However, the lack of serological tests with established accuracy has hindered our understanding of the prevalence, risk factors and natural history of HHV‐8 infection. We tested 837 subjects from Congo, Botswana (mostly young adults) and Malta (elderly adults), using an immunofluorescence assay and 2 enzyme immunoassays (EIAs, to viral proteins K8.1 and orf65). Each assay found HHV‐8 seroprevalence to be high (49–87%) in the African populations and generally lower (9–54%) in Malta. However, there was only modest agreement among tests regarding which subjects were seropositive (3‐way κ, 0.05–0.34). We used latent class analysis to model this lack of agreement, estimating each tests sensitivity and specificity and each populations HHV‐8 prevalence. Using this approach, the K8.1 EIA had consistently high sensitivity (91–100%) and specificity (92–100%) across populations, suggesting that it might be useful for epidemiological studies. Compared with the K8.1 EIA, both the immunofluorescence assay and the orf65 EIA had more variable sensitivity (80–100% and 58–87%, respectively) and more variable specificity (57–100% and 48–85%, respectively). HHV‐8 prevalence was 7% among elderly Maltese adults. Prevalence was much higher (82%) in Congo, consistent with very high Kaposis sarcoma incidence there. Prevalence was also high in Botswana (87% in Sans, an indigenous group, and 76% in Bantus), though Kaposis sarcoma is not common, suggesting that additional co‐factors besides HHV‐8 are needed for development of Kaposis sarcoma. Int. J. Cancer 88:1003–1008, 2000.

Screening with Cost-Effective Quality Control: Potential Applications to HIV and Drug Testing

Abstract This article investigates the applicability of group testing as a quality control procedure to monitor the sensitivity of screening tests used to check the blood supply for infective agents or to check employees for drug use. The problem is important, as the accuracy of screening tests in the field may deteriorate over time. In the blood screening application, our results demonstrate that group testing the screened negatives provides a procedure with high power to detect a decline of .02 in the sensitivity of the original test when the prevalence in the population is quite low (.0001). Moreover, the procedure is cost-effective in the sense that the expected cost per human immunodeficiency virus (HIV) infection avoided could be less than

Estimation of the prevalence of a rare disease, preserving the anonymity of the subjects by group testing: application to estimating the prevalence of aids antibodies in blood donors

1 million, in contrast to much higher economic valuations of life that are used in regulatory analyses. The statistical properties of estimates of the prevalence, as well as those for the sensitivity and specificity of the screening test using the extra informati...

Percentile Modifications of Two Sample Rank Tests

Abstract In order to preserve individual anonymity when estimating the prevalence of a disease or trait possesing a possible social stigma, we group the sample to be tested in batches of size k. Each batch is first given a screening test. Those classified as positive are given a confirmatory test. The classifications of the batches follow a trinomial model. Two estimators of the prevalence including the maximum likelihood are determined and are shown to yield better estimates (at small prevalence) for a fixed cost than estimates from the procedure used to test individuals. While mathematically related to the group screening procedures originated by Dorfman and developed by Sobel, Kotz, and Johnson, the objective is different. We specifically do not want to identify individuals in order to protect their civil liberties. The methods are illustrated on AIDS test data obtained from Canadian subjects. The available data contains individual test results. To compare our procedure we must obtain the distribution of the number of batches, X1, that contain a positive sample. The corresponding prevalence estimates are found for each x1. The expected value of these prevalence estimates is shown to essentially equal the estimate obtained from individual testing.

Effect of Dependence on the Level of Some One-Sample Tests

Abstract This paper presents a simple method for increasing the limiting Pitman efficiency of rank tests relative to the best tests for samples normal distributions without using complicated scoring systems. Our proposal is to select two numbers p and r (0 < p, r < 1) and then score, with integer weights, the data in the top p th and bottom r th fractions of the combined sample. The percentile modified tests for scale are quite effective. When p = r = 1/8 (i.e., we score only the extreme quarter of the combined sample) the A.R.E. of the test to the F test is .85.