Some goodness-of-fit methods for the Poisson plus added zeros distribution.

Methods for making inferences about the Poisson plus added zeros distribution and the truncated Poisson distribution are presented and illustrated with bacteriological data. Some of the methods are designed for testing the compatibility of the zero frequency with the Poisson distribution, whereas others are given for testing the goodness of fit for the truncated Poisson. In particular, a modified form of the Fisher index of dispersion is presented which is suitable for the truncated case. It is shown that the use of the usual expression of the index of dispersion for testing the adequacy of the truncated Poisson is not correct and leads to accepting inadequate fits more frequently than expected on the basis of test of significance. Furthermore, three test statistics are presented for testing the compatability of the zero frequency with the Poisson distribution. The results of the simulation show that two test statistics, one due to Cochran (W. G. Cochran, Biometrics 10:417-451, 1954) and the other to Rao and Chakravarti (C. R. Rao and I. M. Chakravarti, Biometrics 12:264-282, 1956), are preferable to those from the likelihood ratio test.     

On the First-Order Rao—Scott Correction of the Umesh—Loughin—Scherer Statistic

Decady and Thomas (2000, Biometrics 56, 893-896) propose a first-order corrected Umesh-Loughin-Scherer statistic to test for association in an r x c contingency table with multiple column responses. Agresti and Liu (1999, Biometrics 55, 936-943) point out that such statistics are not invariant to the arbitrary designation of a zero or one to a positive response. This paper shows that, in addition, the proposed testing procedure does not hold the correct size when there are strong pairwise associations between responses.     

The equivalence of two tests and models for HLA data with no observed double blanks

Gart and Nam (1984, Biometrics 40, 887-894) consider the analysis of an ABO-like model based on the Hardy-Weinberg law in the commonly occurring case of human leukocyte antigen (HLA) data where there are no double blanks, that is, no recessive homozygotes. They derive a score test, based on the truncated likelihood, of the hypothesis that the true recessive gene (or allele) frequency is zero. Yasuda (1968, Biometrics 24, 915-935) considers a similar codominant system wherein the true recessive gene frequency is assumed zero, but the Hardy-Weinberg law does not hold. In particular, he considers the possibility of a nonzero inbreeding coefficient. We show that the two models are equivalent; each likelihood can be shown to be a reparameterization of the other. Furthermore, the score test of the zero gene frequency in Gart and Nam is identical to the score test for a zero inbreeding coefficient given by Yasuda. The results are applied to an example wherein it appears that the Hardy-Weinberg model is appropriate. Thus, it is not possible in this population to identify homozygous individuals without error from phenotypic data alone.     

The ML estimation and testing of generalized ABO-like data with no observed double recessives

The general method of the discrepancy or heterogeneity chi-square is applied to ABO-like data in which there are no observed double blanks in either the disease or the control group. When the recessive gene frequency is assumed zero, this method leads to an approximate chi-square test identical to that suggested by Smouse and Williams (1982, Biometrics 38, 757-768). When this assumption is relaxed, there arise two cases which are determined by whether the maximum likelihood estimate of this frequency is zero or not. It is shown that the value of the simple score statistic of Gart and Nam (1984, Biometrics 40, 887-894) discriminates between the two cases. The various omnibus test statistics for comparing groups are shown to differ little in several practical examples. However, under the more general assumption the appropriate degrees of freedom is one more than the number previously suggested.     

Estimators and tests in the analysis of multiple nonindependent 2 x 2 tables with partially missing observations

We present methods for the analysis of a K-variate binary measure for two independent groups where some observations may be incomplete, as in the case of K repeated measures in a comparative trial. For the K 2 X 2 tables, let theta = (theta 1,..., theta K) be a vector of association parameters where theta k is a measure of association that is a continuous function of the probabilities pi ik in each group (i = 1, 2; k = 1,..., K), such as the log odds ratio or log relative risk. The asymptotic distribution of the estimates theta = (theta 1,..., theta K) is derived. Under the assumption that theta k = theta for all k, we describe the maximally efficient linear estimator theta of the common parameter theta. Tests of contrasts on the theta are presented which provide a test of homogeneity Ha: theta k = theta l for all k not equal to l. We then present maximally efficient tests of aggregate association Hb: theta = theta 0, where theta 0 is a given value. It is shown that the test of aggregate association Hb is asymptotically independent of the preliminary test of homogeneity Ha. These methods generalize the efficient estimators of Gart (1962, Biometrics 18, 601-610), and the Cochran (1954, Biometrics 10, 417-451), Mantel-Haenszel (1959, Journal of the National Cancer Institute 22, 719-748), and Radhakrishna (1965, Biometrics 21, 86-98) tests to nonindependent tables. The methods are illustrated with an analysis of repeated morphologic evaluations of liver biopsies obtained in the National Cooperative Gallstone Study.     

A weighted partial likelihood approach for zero‐truncated models

Zero‐truncated data arises in various disciplines where counts are observed but the zero count category cannot be observed during sampling. Maximum likelihood estimation can be used to model these data; however, due to its nonstandard form it cannot be easily implemented using well‐known software packages, and additional programming is often required. Motivated by the Rao–Blackwell theorem, we develop a weighted partial likelihood approach to estimate model parameters for zero‐truncated binomial and Poisson data. The resulting estimating function is equivalent to a weighted score function for standard count data models, and allows for applying readily available software. We evaluate the efficiency for this new approach and show that it performs almost as well as maximum likelihood estimation. The weighted partial likelihood approach is then extended to regression modelling and variable selection. We examine the performance of the proposed methods through simulation and present two case studies using real data.     

Bootstrap tests for overdispersion in a zero-inflated Poisson regression model

Ridout, Hinde, and Demétrio (2001, Biometrics 57, 219-223) derived a score test for testing a zero-inflated Poisson (ZIP) regression model against zero-inflated negative binomial (ZINB) alternatives. They mentioned that the score test using the normal approximation might underestimate the nominal significance level possibly for small sample cases. To remedy this problem, a parametric bootstrap method is proposed. It is shown that the bootstrap method keeps the significance level close to the nominal one and has greater power uniformly than the existing normal approximation for testing the hypothesis.     

Testing for simple independent action between two factors for dichotomous response data

This paper compares three statistics for testing simple independent action between two dichotomous factors with respect to the occurrence of a dichotomous outcome. Sizes and powers are examined for the statistics proposed, under a variety of model parameterizations. The results suggest that a test based on the ratio of the nonresponse probability estimates [considered originally by Wahrendorf, Zentgraf, and Brown (1981, Biometrics 37, 45-54)] has proper size and acceptable power, and is recommended for use in this setting.     

Zero-inflated generalized Poisson regression mixture model for mapping quantitative trait loci underlying count trait with many zeros

Phenotypes measured in counts are commonly observed in nature. Statistical methods for mapping quantitative trait loci (QTL) underlying count traits are documented in the literature. The majority of them assume that the count phenotype follows a Poisson distribution with appropriate techniques being applied to handle data dispersion. When a count trait has a genetic basis, “naturally occurring” zero status also reflects the underlying gene effects. Simply ignoring or miss-handling the zero data may lead to wrong QTL inference. In this article, we propose an interval mapping approach for mapping QTL underlying count phenotypes containing many zeros. The effects of QTLs on the zero-inflated count trait are modelled through the zero-inflated generalized Poisson regression mixture model, which can handle the zero inflation and Poisson dispersion in the same distribution. We implement the approach using the EM algorithm with the Newton-Raphson algorithm embedded in the M-step, and provide a genome-wide scan for testing and estimating the QTL effects. The performance of the proposed method is evaluated through extensive simulation studies. Extensions to composite and multiple interval mapping are discussed. The utility of the developed approach is illustrated through a mouse F₂ intercross data set. Significant QTLs are detected to control mouse cholesterol gallstone formation.     

Multipoint linkage detection in the presence of heterogeneity

Linkage heterogeneity is common for complex diseases. It is well known that loss of statistical power for detecting linkage will result if one assumes complete homogeneity in the presence of linkage heterogeneity. To this end, Smith (1963, Annals of Human Genetics 27, 175-182) proposed an admixture model to account for linkage heterogeneity. It is well known that for this model, the conventional chi-squared approximation to the likelihood ratio test for no linkage does not apply even when the sample size is large. By dealing with nuclear families and one marker at a time for genetic diseases with simple modes of inheritance, score-based test statistics (Liang and Rathouz, 1999, Biometrics 55, 65-74) and likelihood-ratio-based test statistics (Lemdani and Pons, 1995, Biometrics 51, 1033-1041) have been proposed which have a simple large-sample distribution under the null hypothesis of linkage. In this paper, we extend their work to more practical situations that include information from multiple markers and multi-generational pedigrees while allowing for a class of general genetic models. Three different approaches are proposed to eliminate the nuisance parameters in these test statistics. We show that all three approaches lead to the same asymptotic distribution under the null hypothesis of no linkage. Simulation results show that the proposed test statistics have adequate power to detect linkage and that the performances of these two classes of test statistics are quite comparable. We have applied the proposed method to a family study of asthma (Barnes et al., 1996), in which the score-based test shows evidence of linkage with p-value <0.0001 in the region of interest on chromosome 12. Additionally, we have implemented this score-based test within the frequently used computer package GENEHUNTER.     

A score test for testing a zero-inflated Poisson regression model against zero-inflated negative binomial alternatives

Count data often show a higher incidence of zero counts than would be expected if the data were Poisson distributed. Zero-inflated Poisson regression models are a useful class of models for such data, but parameter estimates may be seriously biased if the nonzero counts are overdispersed in relation to the Poisson distribution. We therefore provide a score test for testing zero-inflated Poisson regression models against zero-inflated negative binomial alternatives.     

Weighted least-squares approach for comparing correlated kappa

In the medical sciences, studies are often designed to assess the agreement between different raters or different instruments. The kappa coefficient is a popular index of agreement for binary and categorical ratings. Here we focus on testing for the equality of two dependent kappa coefficients. We use the weighted least-squares (WLS) approach of Koch et al. (1977, Biometrics 33, 133-158) to take into account the correlation between the estimated kappa statistics. We demonstrate how the SAS PROC CATMOD can be used to test for the equality of dependent Cohen's kappa coefficients and dependent intraclass kappa coefficients with nominal categorical ratings. We also test for the equality of dependent Cohen's kappa and dependent weighted kappa with ordinal ratings. The major advantage of the WLS approach is that it allows the data analyst a way of testing dependent kappa with popular SAS software. The WLS approach can handle any number of categories. Analyses of three biomedical studies are used for illustration.     

Testing for conditional multiple marginal independence

Survey respondents are often prompted to pick any number of responses from a set of possible responses. Categorical variables that summarize this kind of data are called pick any/c variables. Counts from surveys that contain a pick any/c variable along with a group variable (r levels) and stratification variable (q levels) can be marginally summarized into an r x c x q contingency table. A question that may naturally arise from this setup is to determine if the group and pick any/c variable are marginally independent given the stratification variable. A test for conditional multiple marginal independence (CMMI) can be used to answer this question. Since subjects may pick any number out of c possible responses, the Cochran (1954, Biometrics 10, 417-451) and Mantel and Haenszel (1959, Journal of the National Cancer Institute 22, 719-748) tests cannot be used directly because they assume that units in the contingency table are independent of each other. Therefore, new testing methods are developed. Cochran's test statistic is extended to r x 2 x q tables, and a modified version of this statistic is proposed to test CMMI. Its sampling distribution can be approximated through bootstrapping. Other CMMI testing methods discussed are bootstrap p-value combination methods and Bonferroni adjustments. Simulation findings suggest that the proposed bootstrap procedures and the Bonferroni adjustments consistently hold the correct size and provide power against various alternatives.     

Comparison of maximum statistics for hypothesis testing when a nuisance parameter is present only under the alternative

In many practical problems, a hypothesis testing involves a nuisance parameter which appears only under the alternative hypothesis. Davies (1977, Biometrika 64, 247-254) proposed the maximum of the score statistics over the whole range of the nuisance parameter as a test statistic for this type of hypothesis testing. Freidlin, Podgor, and Gastwirth (1999, Biometrics 55, 883-886) studied two other simpler maximum test statistics, the maximum of the score statistics at two extreme points of the nuisance parameter, and the maximum of the score statistics at three points of the nuisance parameter including the two extreme points. In this article, we compare the powers of these three maximum-type statistics in the context of three genetic problems.     

Confidence interval for rate ratio in a 2 x 2 table with structural zero: an application in assessing false-negative rate ratio when combining two diagnostic tests

In this article, we consider problems with correlated data that can be summarized in a 2 x 2 table with structural zero in one of the off-diagonal cells. Data of this kind sometimes appear in infectious disease studies and two-step procedure studies. Lui (1998, Biometrics54, 706-711) considered confidence interval estimation of rate ratio based on Fieller-type, Wald-type, and logarithmic transformation statistics. We reexamine the same problem under the context of confidence interval construction on false-negative rate ratio in diagnostic performance when combining two diagnostic tests. We propose a score statistic for testing the null hypothesis of nonunity false-negative rate ratio. Score test-based confidence interval construction for false-negative rate ratio will also be discussed. Simulation studies are conducted to compare the performance of the new derived score test statistic and existing statistics for small to moderate sample sizes. In terms of confidence interval construction, our asymptotic score test-based confidence interval estimator possesses significantly shorter expected width with coverage probability being close to the anticipated confidence level. In terms of hypothesis testing, our asymptotic score test procedure has actual type I error rate close to the pre-assigned nominal level. We illustrate our methodologies with real examples from a clinical laboratory study and a cancer study.     

An inferential procedure for the Poisson intervention parameter.

To describe a health chance mechanism whose incidence rate is altered in the middle of a data collection period due to preventive treatments taken by health service agencies, a model called Intervened Poisson Distribution (cf. Shanmugam, 1985, Biometrics 41, 1025-1029) was introduced. In this article, we derive a statistic to test the hypothesis concerning the effectiveness of such preventive treatments, and compare its power with another suggested test (cf. Streit, 1987, Biometrics 43, 999-1000).     

The analysis of group truncated binary data with random effects: injury severity in motor vehicle accidents

The analysis of group truncated binary data has been previously considered by O'Neill and Barry (1995b, Biometrics 51, 533-541), where the analysis assumed that responses within each group were independent. In this paper, we consider the analysis of such data when there is group-level heterogeneity. A generalized linear mixed model is hypothesized to model the response and maximum likelihood estimates are derived for the truncated case. A score test is derived to test for heterogeneity. Finally, the method is applied to a set of traffic accident data.     

Identifying genetic marker sets associated with phenotypes via an efficient adaptive score test

In recent years, genome-wide association studies (GWAS) and gene-expression profiling have generated a large number of valuable datasets for assessing how genetic variations are related to disease outcomes. With such datasets, it is often of interest to assess the overall effect of a set of genetic markers, assembled based on biological knowledge. Genetic marker-set analyses have been advocated as more reliable and powerful approaches compared with the traditional marginal approaches (Curtis and others, 2005. Pathways to the analysis of microarray data. TRENDS in Biotechnology 23, 429-435; Efroni and others, 2007. Identification of key processes underlying cancer phenotypes using biologic pathway analysis. PLoS One 2, 425). Procedures for testing the overall effect of a marker-set have been actively studied in recent years. For example, score tests derived under an Empirical Bayes (EB) framework (Liu and others, 2007. Semiparametric regression of multidimensional genetic pathway data: least-squares kernel machines and linear mixed models. Biometrics 63, 1079-1088; Liu and others, 2008. Estimation and testing for the effect of a genetic pathway on a disease outcome using logistic kernel machine regression via logistic mixed models. BMC bioinformatics 9, 292-2; Wu and others, 2010. Powerful SNP-set analysis for case-control genome-wide association studies. American Journal of Human Genetics 86, 929) have been proposed as powerful alternatives to the standard Rao score test (Rao, 1948. Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation. Mathematical Proceedings of the Cambridge Philosophical Society, 44, 50-57). The advantages of these EB-based tests are most apparent when the markers are correlated, due to the reduction in the degrees of freedom. In this paper, we propose an adaptive score test which up- or down-weights the contributions from each member of the marker-set based on the Z-scores of their effects. Such an adaptive procedure gains power over the existing procedures when the signal is sparse and the correlation among the markers is weak. By combining evidence from both the EB-based score test and the adaptive test, we further construct an omnibus test that attains good power in most settings. The null distributions of the proposed test statistics can be approximated well either via simple perturbation procedures or via distributional approximations. Through extensive simulation studies, we demonstrate that the proposed procedures perform well in finite samples. We apply the tests to a breast cancer genetic study to assess the overall effect of the FGFR2 gene on breast cancer risk.     

Using Weighted Kaplan-Meier Statistics in Nonparametric Comparisons of Paired Censored Survival Outcomes

This research introduces methods for nonparametric testing of weighted integrated survival differences in the context of paired censored survival designs. The current work extends work done by Pepe and Fleming (1989, Biometrics 45, 497-507), which considered similar test statistics directed toward independent treatment group comparisons. An asymptotic closed-form distribution of the proposed family of tests is presented, along with variance estimates constructed under null and alternative hypotheses using nonparametric maximum likelihood estimates of the closed-form quantities. The described method allows for additional information from individuals with no corresponding matched pair member to be incorporated into the test statistic in sampling scenarios where singletons are not prone to selection bias. Simulations presented over a range of potential dependence in the paired censored survival data demonstrate substantial power gains associated with taking into account the dependence structure. Consequences of ignoring the paired nature of the data include overly conservative tests in terms of power and size. In fact, simulation results using tests for independent samples in the presence of positive correlation consistently undershot both size and power targets that would have been attained in the absence of correlation. This additional worrisome effect on operating characteristics highlights the need for accounting for dependence in this popular family of tests.     

Estimation of microbial densities from dilution count experiments

Although dilution counts have been widely used in quantitative microbiology, their interpretation has always been widely discussed both in microbiology and in applied statistics. Maximum-likelihood (most-probable-number) methods hae generally been used to estimate densities from dilution experiments. It has not been widely recognized that these methods are intrinsically and statistically biased at the sample sizes used in microbiology. This paper presents an analysis of proposed method for correction of such biases, and the method was found to be robust for moderate deviations from Poisson behavior. For analyses at greater variance with the Poisson assumptions, the use of the Spearman-Karber method is analyzed and shown to yield an estimate of density of lesser bias than that produced by the most-probable-number method. Revised methods of constructing confidence limits proposed by Loyer and Hamilton (M.W. Loyer and M.A. Hamilton, Biometrics 40:907-916, 1984) are also discussed, and charts for the three- and four-decimal dilution series with five tubes per dilution are presented.