A weighted generalized score statistic for comparison of predictive values of diagnostic tests

Positive and negative predictive values are important measures of a medical diagnostic test performance. We consider testing equality of two positive or two negative predictive values within a paired design in which all patients receive two diagnostic tests. The existing statistical tests for testing equality of predictive values are either Wald tests based on the multinomial distribution or the empirical Wald and generalized score tests within the generalized estimating equations (GEE) framework. As presented in the literature, these test statistics have considerably complex formulas without clear intuitive insight. We propose their re‐formulations that are mathematically equivalent but algebraically simple and intuitive. As is clearly seen with a new re‐formulation we presented, the generalized score statistic does not always reduce to the commonly used score statistic in the independent samples case. To alleviate this, we introduce a weighted generalized score (WGS) test statistic that incorporates empirical covariance matrix with newly proposed weights. This statistic is simple to compute, always reduces to the score statistic in the independent samples situation, and preserves type I error better than the other statistics as demonstrated by simulations. Thus, we believe that the proposed WGS statistic is the preferred statistic for testing equality of two predictive values and for corresponding sample size computations. The new formulas of the Wald statistics may be useful for easy computation of confidence intervals for difference of predictive values. The introduced concepts have potential to lead to development of the WGS test statistic in a general GEE setting. Copyright © 2012 John Wiley & Sons, Ltd.     

A method for combining matched and unmatched binary data. Application to randomized, controlled trials of photocoagulation in the treatment of diabetic retinopathy

A method for combining matched and unmatched data is described and was applied to the results of randomized, controlled trials of photocoagulation in the treatment of diabetic retinopathy. A pooled estimate from the matched and unmatched studies was obtained by adaptation of the Mantel-Haenszel method, where the strata were unmatched studies and matched pairs within studies. A test of significance was based on the Mantel-Haenszel chi-square statistic, the latter also being used to calculate test-based confidence intervals. A test of homogeneity was performed by combining Mantel-Haenszel chi-square statistics from the matched and unmatched studies. By these methods, the combined estimate of the risk of deterioration of visual acuity for those receiving photocoagulation (relative to a risk of unity for those not receiving photocoagulation) was 0.37 (95 per cent confidence interval 0.29-0.46). The chi-square statistic (1 df) for an effect of treatment was 87.75 (p less than 0.0001). The chi-square statistic for homogeneity of relative risk among studies was 11.20 (4 df, p less than 0.05). However, this result was influenced disproportionately by one small matched study.     

A comparison of tests of homogeneity of odds ratios in K 2 x 2 tables

We conducted a simulation study to determine the performance of nine procedures for testing the homogeneity of odds ratios in K 2 x 2 contingency tables. We recommend Tarone's approximate score test, based on the Mantel-Haenszel estimator of the common odds ratio, for use in practice. We also recommend a non-iterative statistic developed by Gart and based on the modified Woolf estimator of the common odds ratio for very large samples in balanced or mildly unbalanced designs. We base our recommendation of a statistic on its performance in terms of size and power in comparison with the other statistics considered.     

Comparison of predictive values of two diagnostic tests from the same sample of subjects using weighted least squares

Screening and diagnostic tests are important in disease prevention or control. The predictive values of positive and negative (PPV and NPV) test results are two of four operational characteristics of a screening test. We review an existing method based on the generalized estimating equation (GEE) methodology for comparing predictive values from the same sample of subjects and propose two Wald test statistics derived from the weighted least squares (WLS) method for the analysis of categorical data. Using these results, we propose sample size calculation formulae for this problem. Simulation studies are conducted to compare the performances of the two Wald test statistics (one based on the difference of two PPVs or NPVs, another based on the logarithm of the ratio of two PPVs or NPVs) and the score/Wald test statistic derived from GEE. We recommend using the difference-based WLS approach.     

Two Approaches for Consolidating Results from Genome Scans of Complex Traits: Selection Methods and Scan Statistics

This work has two purposes: (i) empirically selecting levels of significance that maximize the fraction of markers close to a gene (hit rate) when performing linkage analyses of simulated data and (ii) evaluating the utility of a previously reported scan statistic on the same data. Genotype data were simulated from a trait model of seven susceptibility genes. For purpose (i), five statistics were evaluated on all marker loci in fifty replicates; two‐point lod and heterogeneity lod scores maximized over dominance (mlod, mhlod), a multi‐allelic TDT test, an affected sib‐pair test (ASP), and a model‐free test on all sib‐pairs (ALL_SIBS). Within each replicate the fraction of markers (hit rate) significant at specified levels of significance and also (a) within fifty markers of, or (b) on the same chromosome as a major gene was calculated. For purpose (ii), scan statistics of length 15 were calculated for each chromosome and their empirical significance levels estimated on the basis of 500 replicates generated under no linkage. The scan statistic was applied to the mhlod scores from one replicate (Replicate 5). Empirical p‐values for the scan statistic were determined by computing mhlod scores on 500 replicates of simulated null data. For purpose (i), significance levels between 0.001 and 0.01 had the greatest hit rate for all five methods and both criteria. For criterion (a) at the 0.001 level of significance, both mlod and mhlod displayed the highest hit rates, approximately 0.4 for each. For criterion (b), all methods but ALL_SIBS and ASP had hit rates ranging between 0.4 and 0.5. For purpose (ii), the scan statistic proved equally or more powerful than the single‐locus statistic for two of the seven susceptibility genes while the remaining five genes were not detected. © 2001 Wiley‐Liss, Inc.     

Elementary methods for the analysis of dichotomous outcomes in unselected samples of twins.

This paper presents an elementary statistical method for analyzing dichotomous outcomes in unselected samples of twin pairs using stratified estimators of the odds ratio. The methodology begins by first randomly designating one member of each twin pair as an "index" twin and the other member as the "co-twin." Stratifying on zygosity, odds ratios are used to measure the association between disease in the index twin and disease in the co-twin. From these zygosity-specific tables we calculate the Woolf-Haldane estimator of the common odds ratio (psi F, the weighted average of the zygosity-specific odds ratios), the Mantel-Haenszel test statistic (chi 2M-H) for the common odds ratio, and a test (chi 2G) for the difference in the zygosity-specific odds ratios. In this application, psi F provides an estimate of the familial association for disease and the accompanying chi 2M-H provides a test of the null hypothesis, psi F = 1 (i.e., there is no evidence for a familial influence on disease). The chi 2G is a test of the null hypothesis that psi MZ = psi DZ; a significant value for chi 2G suggests a genetic influence on disease (assuming that the observed odds ratios follow a pattern where psi MZ greater than psi DZ). A new test statistic (chi 2c) is proposed that incorporates the expectation that psi MZ = psi 2DZ under a purely additive genetic model with no common environmental effects. A significant value of chi c2 indicates that the different odds ratios across zygosity are partly due to common environmental influences. Conversely, a nonsignificant value of chi 2c is an indication that the zygosity-specific odds ratios are due solely to additive genetic effects and not to common environment. This basic approach is extended to examine the effects of measured indicators of the specific environment and the assessment of certain forms of gene by environment interaction. All of the methods are easily understood, highly flexible, readily computed using a hand calculator, and incorporate the inherent genetic information contained within twin samples.     

On tests for equality of predictive values for t diagnostic procedures

This paper concerns comparisons of the efficiency of several diagnostic tests, as characterized by the measures of sensitivity (xi), specificity (eta) and predictive value (rho). We show that hypotheses concerning the equality of predictive values relate only to hypotheses concerning xi and eta and that we can test these by approximate chi 2 statistics. Data for the cases of t = 2 or 3 diagnostic tests illustrate the method.     

Analysis of the ratio of marginal probabilities in a matched-pair setting

Statistical methods for testing and interval estimation of the ratio of marginal probabilities in the matched-pair setting are considered in this paper. We are especially interested in the situation where the null value is not one, as in one-sided equivalence trials. We propose a Fieller-type statistic based on constrained maximum likelihood (CML) estimation of nuisance parameters. For a series of examples, the significance level of the CML test is satisfactorily close to the nominal level, while a Wald-type test is anticonservative for reasonable sample sizes. We present formulae for approximate power and sample size for the CML and Wald tests. The matched design is seen to have a clear advantage over the unmatched design in terms of asymptotic efficiency when the two responses of the pair are highly positively correlated. We recommend the CML method over the Wald method, especially for small or moderate sample sizes.     

STATISTICAL ANALYSIS OF INTRACLASS CORRELATIONS FROM MULTIPLE SAMPLES WITH APPLICATIONS TO ARTERIAL BLOOD PRESSURE DATA

We consider inference procedures on intraclass correlations for unbalanced data from several multivariate normal populations. We derive several tests, including ones based on Fisher's variance stabilizing transformation and Neyman's score functions, to test the homogeneity of intraclass correlations. We illustrate the methodology with an example that uses arterial blood pressure data collected by Miall and Oldham and we compare the procedures in terms of their empirical levels and powers with a Monte Carlo simulation study. We recommend the use of Neyman's C(α) test and a test based on the ANOVA estimators of the intraclass correlations as they hold their significance levels and give consistently higher powers. © 1997 John Wiley & Sons, Ltd.     

The statistical performance of an MCF-7 cell culture assay evaluated using generalized linear mixed models and a score test

Biological assays often utilize experimental designs where observations are replicated at multiple levels, and where each level represents a separate component of the assay's overall variance. Statistical analysis of such data usually ignores these design effects, whereas more sophisticated methods would improve the statistical power of assays. This report evaluates the statistical performance of an in vitro MCF-7 cell proliferation assay (E-SCREEN) by identifying the optimal generalized linear mixed model (GLMM) that accurately represents the assay's experimental design and variance components. Our statistical assessment found that 17beta-oestradiol cell culture assay data were best modelled with a GLMM configured with a reciprocal link function, a gamma error distribution, and three sources of design variation: plate-to-plate; well-to-well, and the interaction between plate-to-plate variation and dose. The gamma-distributed random error of the assay was estimated to have a coefficient of variation (COV) = 3.2 per cent, and a variance component score test described by X. Lin found that each of the three variance components were statistically significant. The optimal GLMM also confirmed the estrogenicity of five weakly oestrogenic polychlorinated biphenyls (PCBs 17, 49, 66, 74, and 128). Based on information criteria, the optimal gamma GLMM consistently out-performed equivalent naive normal and log-normal linear models, both with and without random effects terms. Because the gamma GLMM was by far the best model on conceptual and empirical grounds, and requires only trivially more effort to use, we encourage its use and suggest that naive models be avoided when possible.     

Confidence intervals of the difference between areas under two ROC curves in matched-pair experiments

In a matched-pair study, when outcomes of two diagnostic tests are ordinal/continuous, the difference between two correlated areas under ROC curves (AUCs) is usually used to compare the overall discriminatory ability of two diagnostic tests. This article considers confidence interval (CI) construction problems of difference between two correlated AUCs in a matched-pair experiment, and proposes 13 hybrid CIs based on variance estimates recovery with the maximum likelihood estimation, Delong's statistic, Wilson score statistic (WS) and WS with continuity correction, the modified Wald statistic (MW) and MW with continuity correction and Agresti-Coull statistic, and three Bootstrap-resampling-based CIs. For comparison, we present traditional parametric and nonparametric CIs. Simulation studies are conducted to assess the performance of the proposed CIs in terms of empirical coverage probabilities, empirical interval widths, and ratios of the mesial noncoverage probabilities to the noncoverage probabilities. Two examples from clinical studies are illustrated by the proposed methodologies. Empirical results evidence that the hybrid Agresti-Coull CI with the empirical estimation (EAC) behaved most satisfactorily because its coverage probability was quite close to the prespecified confidence level with short interval width. Hence, we recommend the usage of the EAC CI in applications.     

CLUMPHAP: a simple tool for performing haplotype-based association analysis

The completion of the HapMap Project and the development of high-throughput single nucleotide polymorphism genotyping technologies have greatly enhanced the prospects of identifying and characterizing the genetic variants that influence complex traits. In principle, association analysis of haplotypes rather than single nucleotide polymorphisms may better capture an underlying causal variant, but the multiple haplotypes can lead to reduced statistical power due to the testing of (and need to correct for) a large number of haplotypes. This paper presents a novel method based on clustering similar haplotypes to address this issue. The method, implemented in the CLUMPHAP program, is an extension of the CLUMP program designed for the analysis of multi-allelic markers (Sham and Curtis [1995] Ann. Hum. Genet. 59(Pt1):97-105). CLUMPHAP performs a hierarchical clustering of the haplotypes and then computes the chi(2) statistic between each haplotype cluster and disease; the statistical significance of the largest of the chi(2) statistics is obtained by permutation testing. A significant result suggests that the presence of a disease-causing variant in the haplotype cluster is over-represented in cases. Using simulation studies, we have compared CLUMPHAP and more widely used approaches in terms of their statistical power to identify an untyped susceptibility locus. Our results show that CLUMPHAP tends to have greater power than the omnibus haplotype test and is comparable in power to multiple regression locus-coding approaches.     

Simultaneous confidence intervals using ordinal effect measures for ordered categorical outcomes

When outcomes are ordered categorical, a model using an ordinal effect size measure is a good alternative of the cumulative logit model to compare several independent group differences. We present a method of constructing simultaneous confidence intervals for the ordinal effect size measures, using the studentized range distribution with the score test statistic. A simulation study shows that the proposed method performs well in terms of coverage probability, and it seems better than the method using a Bonferroni correction for Wald‐type statistics and methods that account for the dependencies among pairwise ordinal effect size measures using the multivariate normal distribution (or the multivariate t‐distribution for small samples). Copyright © 2009 John Wiley & Sons, Ltd.     

Optimal methods for meta-analysis of genome-wide association studies

Meta-analysis of genome-wide association studies involves testing single nucleotide polymorphisms (SNPs) using summary statistics that are weighted sums of site-specific score or Wald statistics. This approach avoids having to pool individual-level data. We describe the weights that maximize the power of the summary statistics. For small effect-sizes, any choice of weights yields summary Wald and score statistics with the same power, and the optimal weights are proportional to the square roots of the sites' Fisher information for the SNP's regression coefficient. When SNP effect size is constant across sites, the optimal summary Wald statistic is the well-known inverse-variance-weighted combination of estimated regression coefficients, divided by its standard deviation. We give simple approximations to the optimal weights for various phenotypes, and show that weights proportional to the square roots of study sizes are suboptimal for data from case-control studies with varying case-control ratios, for quantitative trait data when the trait variance differs across sites, for count data when the site-specific mean counts differ, and for survival data with different proportions of failing subjects. Simulations suggest that weights that accommodate intersite variation in imputation error give little power gain compared to those obtained ignoring imputation uncertainties. We note advantages to combining site-specific score statistics, and we show how they can be used to assess effect-size heterogeneity across sites. The utility of the summary score statistic is illustrated by application to a meta-analysis of schizophrenia data in which only site-specific P-values and directions of association are available.     

Testing for genetic association: a powerful score test

To study the association between a candidate gene and a complex genetic disease, Pearson's chi(2) statistic can be applied to an m x 2 contingency table, where the m categories correspond to m haplotypes or marker alleles. For m>2, two alternative approaches for Pearson's chi(2) can be followed, which are more powerful if one haplotype or marker allele is associated. For the first approach, various 2 x 2 tables are formed by combining various categories and the maximum of the corresponding chi-square statistics is considered as the final statistic. The second approach takes the average over the possible associated categories by writing down an overall likelihood. For the latter approach, we propose a new score statistic, which gives more weight to haplotypes or marker alleles that are common. Since the disease allele is often not observed, the power of the various statistics depends on both the linkage disequilibrium pattern and the frequencies of the associated haplotype or marker allele in the cases and the controls. We heuristically compare various statistics within the two approaches and present the results of a simulation that compares the performance of all considered statistics. Finally, we apply the statistics to a case-control study on the association between COL2A1 gene and radiographic osteoarthritis. Our conclusion is that overall the new proposed score statistic has good power. Copyright (c) 2008 John Wiley & Sons, Ltd.     

Testing homogeneity of proportion ratios for stratified correlated bilateral data in two‐arm randomized clinical trials

Stratified data analysis is an important research topic in many biomedical studies and clinical trials. In this article, we develop five test statistics for testing the homogeneity of proportion ratios for stratified correlated bilateral binary data based on an equal correlation model assumption. Bootstrap procedures based on these test statistics are also considered. To evaluate the performance of these statistics and procedures, we conduct Monte Carlo simulations to study their empirical sizes and powers under various scenarios. Our results suggest that the procedure based on score statistic performs well generally and is highly recommended. When the sample size is large, procedures based on the commonly used weighted least square estimate and logarithmic transformation with Mantel–Haenszel estimate are recommended as they do not involve any computation of maximum likelihood estimates requiring iterative algorithms. We also derive approximate sample size formulas based on the recommended test procedures. Finally, we apply the proposed methods to analyze a multi‐center randomized clinical trial for scleroderma patients. Copyright © 2014 John Wiley & Sons, Ltd.     

Gene-dropping vs. empirical variance estimation for allele-sharing linkage statistics

In this study, we compare the statistical properties of a number of methods for estimating P-values for allele-sharing statistics in non-parametric linkage analysis. Some of the methods are based on the normality assumption, using different variance estimation methods, and others use simulation (gene-dropping) to find empirical distributions of the test statistics. For variance estimation methods, we consider the perfect variance approximation and two empirical variance estimates. The simulation-based methods are gene-dropping with and without conditioning on the observed founder alleles. We also consider the Kong and Cox linear and exponential models and a Monte Carlo method modified from a method for finding genome-wide significance levels. We discuss the analytical properties of these various P-value estimation methods and then present simulation results comparing them. Assuming that the sample sizes are large enough to justify a normality assumption for the linkage statistic, the best P-value estimation method depends to some extent on the (unknown) genetic model and on the types of pedigrees in the sample. If the sample sizes are not large enough to justify a normality assumption, then gene-dropping is the best choice. We discuss the differences between conditional and unconditional gene-dropping.     

Testing the equality of two Poisson means using the rate ratio

In this article, we investigate procedures for comparing two independent Poisson variates that are observed over unequal sampling frames (i.e. time intervals, populations, areas or any combination thereof). We consider two statistics (with and without the logarithmic transformation) for testing the equality of two Poisson rates. Two methods for implementing these statistics are reviewed. They are (1) the sample-based method, and (2) the constrained maximum likelihood estimation (CMLE) method. We conduct an empirical study to evaluate the performance of different statistics and methods. Generally, we find that the CMLE method works satisfactorily only for the statistic without the logarithmic transformation (denoted as W(2)) while sample-based method performs better for the statistic using the logarithmic transformation (denoted as W(3)). It is noteworthy that both statistics perform well for moderate to large Poisson rates (e.g. > or =10). For small Poisson rates (e.g. <10), W(2) can be liberal (e.g. actual type I error rate/nominal level > or =1.2) while W(3) can be conservative (e.g. actual type I error rate/nominal level < or =0.8). The corresponding sample size formulae are provided and valid in the sense that the simulated powers associated with the approximate sample size formulae are generally close to the pre-chosen power level. We illustrate our methodologies with a real example from a breast cancer study.     

Sample size and optimal design for logistic regression with binary interaction

There is no consensus on what test to use as the basis for sample size determination and power analysis. Some authors advocate the Wald test and some the likelihood-ratio test. We argue that the Wald test should be used because the Z-score is commonly applied for regression coefficient significance testing and therefore the same statistic should be used in the power function. We correct a widespread mistake on sample size determination when the variance of the maximum likelihood estimate (MLE) is estimated at null value. In our previous paper, we developed a correct sample size formula for logistic regression with single exposure (Statist. Med. 2007; 26(18):3385-3397). In the present paper, closed-form formulas are derived for interaction studies with binary exposure and covariate in logistic regression. The formula for the optimal control-case ratio is derived such that it maximizes the power function given other parameters. Our sample size and power calculations with interaction can be carried out online at www.dartmouth.edu/ approximately eugened.     

大学生体重指数影响因素分析

目的找出大学生体重指数(body mass index,BMI)的影响因素,为进一步保持大学生健康体重采取干预措施提供依据。方法对某高校大学生的2012年体质健康测试数据进行分析,主要是将大学生的身高、体重数据及相应的BMI与相关因素进行Logistic回归分析。本研究采取SAS9.1进行统计分析。结果 Logistic回归分析发现,变量x1(性别)的回归系数的Wald卡方值为22.925 3(P0.05),其OR=0.710;变量x2(民族)的回归系数的Wald卡方值为20.025 4(P0.05),其OR=0.696;变量x5(年级)的回归系数的Wald卡方值为62.339 4(P0.05),其OR=1.512,所以变量x1、x2、x5对大学生的BMI有影响。而变量x3(年龄)、x4(系别)的回归系数的Wald卡方检验的P0.05,对大学生的BMI无影响。进一步分析发现,变量x1、x2、x5对BMI的影响主要表现在体重正常组,体重正常组不同性别学生BMI秩和检验P0.05(男生21.10 kg/m2,女生20.73 kg/m2),体重正常组不同民族学生BMI秩和检验P0.05(汉族20.74 kg/m2,少数民族21.22 kg/m2),体重正常组不同年级学生BMI秩和检验P0.05(2010级20.50 kg/m2,2011级20.84 kg/m2,2012级20.92 kg/m2)。结论为保持大学生的健康体重,需要加强大学生的健康教育,使其养成正确的健康观,同时加强高校食堂的营养学知识水平和建立科学的学生体质监控体系也是必要的。