Extension of the Peters-Belson method to estimate health disparities among multiple groups using logistic regression with survey data.

Determining the extent of a disparity, if any, between groups of people, for example, race or gender, is of interest in many fields, including public health for medical treatment and prevention of disease. An observed difference in the mean outcome between an advantaged group (AG) and disadvantaged group (DG) can be due to differences in the distribution of relevant covariates. The Peters-Belson (PB) method fits a regression model with covariates to the AG to predict, for each DG member, their outcome measure as if they had been from the AG. The difference between the mean predicted and the mean observed outcomes of DG members is the (unexplained) disparity of interest. We focus on applying the PB method to estimate the disparity based on binary/multinomial/proportional odds logistic regression models using data collected from complex surveys with more than one DG. Estimators of the unexplained disparity, an analytic variance-covariance estimator that is based on the Taylor linearization variance-covariance estimation method, as well as a Wald test for testing a joint null hypothesis of zero for unexplained disparities between two or more minority groups and a majority group, are provided. Simulation studies with data selected from simple random sampling and cluster sampling, as well as the analyses of disparity in body mass index in the National Health and Nutrition Examination Survey 1999-2004, are conducted. Empirical results indicate that the Taylor linearization variance-covariance estimation is accurate and that the proposed Wald test maintains the nominal level.

Power and related statistical properties of conditional likelihood score tests for association studies in nuclear families with parental genotypes.

Both population based and family based case control studies are used to test whether particular genotypes are associated with disease. While population based studies have more power, cryptic population stratification can produce false-positive results. Family-based methods have been introduced to control for this problem. This paper presents the full likelihood function for family-based association studies for nuclear families ascertained on the basis of their number of affected and unaffected children. The likelihood of a family factors into the probability of parental mating type, conditional on offspring phenotypes, times the probability of offspring genotypes given their phenotypes and the parental mating type. The first factor can be influenced by population stratification, whereas the latter factor, called the conditional likelihood, is not. The conditional likelihood is used to obtain score tests with proper size in the presence of population stratification (see also Clayton (1999) and Whittemore & Tu (2000)). Under either the additive or multiplicative model, the TDT is known to be the optimal score test when the family has only one affected child. Thus, the class of score tests explored can be considered as a general family of TDT-like procedures. The relative informativeness of the various mating types is assessed using the Fisher information, which depends on the number of affected and unaffected offspring and the penetrances. When the additive model is true, families with parental mating type Aa x Aa are most informative. Under the dominant (recessive) model, however, a family with mating type Aa x aa(AA x Aa) is more informative than a family with doubly heterozygous (Aa x Aa) parents. Because we derive explicit formulae for all components of the likelihood, we are able to present tables giving required sample sizes for dominant, additive and recessive inheritance models.

On the power of affected relative pair designs for linkage studies.

Using the recurrence risk ratio (lambda), Risch (1990b) indicated that affected pairs of distant relatives are preferable to affected sib-pairs for linkage analysis when lambda is large and the mode of inheritance is additive. By using the optimum test for affected sib-pairs instead of the test used by Risch (1990b), the range of values of lambda for which the affected sib-pair design has larger or smaller power than other pairs is clarified. Risch's conclusion remains true when lambdaO > 2.5, however, sib-pairs have larger power for lower values. As affected sib-pairs occur smore frequently than other relative pairs, when ascertainment costs are non-negligible, they may be the most cost-effective relative pairs to use.

Robust TDT-type candidate-gene association tests.

In studies of association between genetic markers and a disease, the transmission disequilibrium test (TDT) has become a standard procedure. It was introduced originally as a test for linkage in the presence of association and can be used as a test for association under appropriate assumptions. The power of the TDT test for association between a candidate gene and disease depends on the underlying genetic model and the TDT is the optimal test if the additive model holds. Related methods have been obtained for a given mode of inheritance (e.g. dominant or recessive). Quite often, however, the true model is unknown and selection of a single method of analysis is problematic, since use of a test optimal for one genetic model usually leads to a substantial loss of power if another genetic model is the true one. The general approach of efficiency robustness has suggested two types of robust procedures, which we apply to TDT-type association tests. When the plausible range of alternative models is wide (e.g. dominant through recessive) our results indicate that the maximum (MAX) of several test statistics, each of which is optimal for quite different models, has good power under all genetic models. In situations where the set of possible models can be narrowed (e.g. dominant through additive) a simple linear combination also performs well. In general, the MAX has better power properties than the TDT for the study of candidate genes when the mode of inheritance is unknown.

Statistical properties of Teng and Risch's sibship type tests for detecting an association between disease and a candidate allele.

Risch and Teng [Genome Res 1998;8:1273-1288] and Teng and Risch [Genome Res 1999;9:234-241] proposed a class of transmission/disequilibrium test-like statistical tests based on the difference between the estimated allele frequencies in the affected and control populations. They evaluated the power of a variety of family-based and nonfamily-based designs for detecting an association between a candidate allele and disease. Because they were concerned with diseases with low penetrances, their power calculations assumed that unaffected individuals can be treated as a random sample from the population. They predicted that this assumption rendered their sample size calculations slightly conservative. We generalize their partial ascertainment conditioning by including the status of the unaffected sibs in the calculations of the distribution and power of the statistic used to compare the allele frequency in affected offspring to the estimated frequency in the parents, based on sibships with genotyped affected and unaffected sibs. Sample size formulas for our full ascertainment methods are presented. The sample sizes for our procedure are compared to those of Teng and Risch. The numerical results and simulations indicate that the simplifying assumption used in Teng and Risch can produce both conservative and anticonservative results. The magnitude of the difference between the sample sizes needed by their partial ascertainment approximation and the full ascertainment is small in the circumstances they focused on but can be appreciable in others, especially when the baseline penetrances are moderate. Two other statistics, using different estimators for the variance of the basic statistic comparing the allele frequencies in the affected and unaffected sibs are introduced. One of them incorporates an estimate of the null variance obtained from an auxiliary sample and appears to noticeably decrease the sample sizes required to achieve a prespecified power.

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.

Screening without a "gold standard": the Hui-Walter paradigm revisited.

The authors consider screening populations with two screening tests but where a definitive "gold standard" is not readily available. They discuss a recent article in which a Bayesian approach to this problem is developed based on data that are sampled from a single population. It was subsequently pointed out that such inferences will not necessarily be accurate in the sense that standard errors for parameters may not decrease as n increases. This problem will generally occur when the data are insufficient to estimate all of the parameters as is the case when screening a single population with two tests. If both tests are applied to units sampled from two populations, however, this particular difficulty disappears. In this article the authors further examine this issue and develop an approach based on sampling two populations that yields increasingly accurate inferences as the sample size increases.

A weighted test using both extreme discordant and concordant sib pairs for detecting linkage.

Statistical procedures using extremely discordant and concordant sib-pairs have been developed for mapping quantitative trait loci in humans. To improve the power of the existing methods, test statistics placing greater weight on the more discordant or more concordant pairs are proposed. Because the optimum choice of weights would depend on the underlying genetic model, which is not usually known, a test with simple weights is suggested. This test is shown to have greater power than the currently available ones for a variety of genetic models.

Use of weighted p-values in regional inference procedures.

Our previous studies have demonstrated that the power to detect linkage was improved by calculating a moving average of consecutive p-values in a small region as compared with testing all single p-values. The goal of this study was to test whether the power can be improved further with an alternative method whereby the middle p-values in the sequence were given more weight than the others. We also wanted to compare the moving average tests with multipoint linkage tests. The simulated extended pedigree data from the general population was analyzed to identify two major genes (MG1 and MG5) underlying two quantitative traits (Q1 and Q5). We used the variance components method implemented in the GENEHUNTER program to test for linkage of 14-marker regions each on chromosome 19 and chromosome 1 to the adjusted quantitative traits Q1 and Q5, respectively, in all 50 replicates. As before, we found that the moving average test was more powerful than a test based on single p-values. In some cases, the weighting procedure increased the power further and was similar to that of multipoint analysis, but this was not consistently found. In addition, all methods had low power and it is not possible to make a general conclusion that some weighting schemes are better than others.

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 Kaposi's 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 kappa, 0.05-0.34). We used latent class analysis to model this lack of agreement, estimating each test's sensitivity and specificity and each population's 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 Kaposi's sarcoma incidence there. Prevalence was also high in Botswana (87% in Sans, an indigenous group, and 76% in Bantus), though Kaposi's sarcoma is not common, suggesting that additional co-factors besides HHV-8 are needed for development of Kaposi's sarcoma.

On power and efficiency robust linkage tests for affected sibs.

For diseases that do not follow a clear Mendelian pattern of inheritance nonparametric tests applied to affected sibs have been shown to be robust to the inherent uncertainty about the precise underlying genetic model. It is known that the weights optimizing the power of tests using IBD alleles shared by affected sib pairs or triples depend on the underlying model. We show how efficiency robustness techniques, used in other areas of statistics, provide a systematic approach for constructing a robust linear combination of the statistics that are optimal for the individual members of a family of plausible genetic models. The method depends on the correlation matrix of the optimal tests as these correlations reflect how different the models are. When the minimal correlation is less than 0.5, an alternate robust procedure is proposed. The methods apply to combining data from sibships of different sizes.

Efficiency robust tests for survival or ordered categorical data.

The selection of a single method of analysis is problematic when the data could have been generated by one of several possible models. We examine the properties of two tests designed to have high power over a range of models. The first one, the maximum efficiency robust test (MERT), uses the linear combination of the optimal statistics for each model that maximizes the minimum efficiency. The second procedure, called the MX, uses the maximum of the optimal statistics. Both approaches yield efficiency robust procedures for survival analysis and ordinal categorical data. Guidelines for choosing between them are provided.

Efficiency robust tests of independence in contingency tables with ordered classifications.

Ordered categorical data occur frequently in biomedical research. The linear by linear association test for ordered R x C tables permits the investigator to specify row and column scores for analysis. When an investigator believes that there may be more than one set of reasonable scores or when more than one investigator proposes scores, we need a method to decide upon a single procedure to use. We show how to use efficiency robustness principles to combine tests from two or more sets of scores into one robust test for analysis. This test minimizes the worst possible efficiency loss over all the sets of scores. We illustrate the methodology for the R x C case and, in detail, for the important special 2 x C case.

Biostatistical concepts and methods in the legal setting.

Biostatistical concepts and methods apply to various problems arising in actual U.S. legal cases. These involve: measures of association, assessing the potential effect of omitted variables and the Peters-Belson approach to regression. In particular, we present the inapplicability of Fisher's exact test in the case where the process determining the marginal sample sizes is not independent of the hypothesis under study by the 2 x 2 table. We adapt Cornfield's procedure to assess whether the omission of another factor could have contributed to a non-significant finding of the effect of the major factor under investigation. Finally, we demonstrate the relevance and utility of the Peters-Belson regression methodology to equal pay and promotion cases.

A cautionary note on applying scores in stratified data.

When rank tests are used to analyze stratified data, three methods for assigning scores to the observations have been proposed: (S) independently within each stratum (see Lehmann, 1975, Nonparametrics: Statistical Methods Based on Ranks; San Francisco: Holden-Day); (A) after aligning the observations within each stratum and then pooling the aligned observations (Hodges and Lehmann, 1962, Annals of Mathematical Statistics 33, 482-497); and (P) after pooling the observations across all strata (that is, without alignment) (Mantel, 1963, Journal of the American Statistical Association 58, 690-700; Mantel and Ciminera, 1979, Cancer Research 39, 4308-4315). Test statistics are formed for each method by combining the stratum-specific linear rank tests using the assigned scores. We show that method P is sensitive to the score function used in the case of two moderately sized strata. In general, we recommend methods S and A for use with moderate to large-sized strata.

On non-parametric and generalized tests for the two-sample problem with location and scale change alternatives.

Various tests have been proposed for the two-sample problem when the alternative is more general than a simple shift in location: non-parametric tests; O'Brien's generalized t and rank sum tests; and other tests related to the t. We show that the generalized tests are directly related to non-parametric tests proposed by Lepage. As a result, we obtain a wider, more flexible class of O'Brien-type procedures which inherit the level robustness property of non-parametric tests. We have also computed the tests' empirical sizes and powers under several models. The non-parametric procedures and the related O'Brien-type tests are valid and yield good power in the settings investigated. They are preferable to the t-test and related procedures whose type I errors differ noticeably from nominal size for skewed and long-tailed distributions.