A simple approximation for calculating sample sizes for detecting linear trend in proportions

A simple approximate formula for sample sizes for detecting a linear trend in proportions is derived. The formulas for both the uncorrected and corrected Cochran-Armitage test are given. For two binomial proportions these reduce to those given by Casagrande, Pike, and Smith (1978, Biometrics 34, 483-486). Some numerical results of a power study for small sample sizes show that the nominal power corresponding to the approximate sample size is a reasonably good approximation to the actual power.     

Simultaneous confidence intervals for comparing binomial parameters

SUMMARY: To compare proportions with several independent binomial samples, we recommend a method of constructing simultaneous confidence intervals that uses the studentized range distribution with a score statistic. It applies to a variety of measures, including the difference of proportions, odds ratio, and relative risk. For the odds ratio, a simulation study suggests that the method has coverage probability closer to the nominal value than ad hoc approaches such as the Bonferroni implementation of Wald or "exact" small-sample pairwise intervals. It performs well even for the problematic but practically common case in which the binomial parameters are relatively small. For the difference of proportions, the proposed method has performance comparable to a method proposed by Piegorsch (1991, Biometrics 47, 45-52).     

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.     

Approximate interval estimation of the ratio of binomial parameters: a review and corrections for skewness

Various methods for finding confidence intervals for the ratio of binomial parameters are reviewed and evaluated numerically. It is found that the method based on likelihood scores (Koopman, 1984, Biometrics 40, 513-517; Miettinen and Nurminen, 1985, Statistics in Medicine 4, 213-226) performs best in achieving the nominal confidence coefficient, but it may distribute the tail probabilities quite disparately. Using general theory of Bartlett (1953, Biometrika 40, 306-317; 1955, Biometrika 42, 201-203), we correct this method for asymptotic skewness. Following Gart (1985, Biometrika 72, 673-677), we extend this correction to the case of estimating the common ratio in a series of two-by-two tables. Computing algorithms are given and applied to numerical examples. Parallel methods for the odds ratio and the ratio of Poisson parameters are noted.     

Homogeneity score test for the intraclass version of the kappa statistics and sample-size determination in multiple or stratified studies

When the intraclass correlation coefficient or the equivalent version of the kappa agreement coefficient have been estimated from several independent studies or from a stratified study, we have the problem of comparing the kappa statistics and combining the information regarding the kappa statistics in a common kappa when the assumption of homogeneity of kappa coefficients holds. In this article, using the likelihood score theory extended to nuisance parameters (Tarone, 1988, Communications in Statistics-Theory and Methods 17(5), 1549-1556) we present an efficient homogeneity test for comparing several independent kappa statistics and, also, give a modified homogeneity score method using a noniterative and consistent estimator as an alternative. We provide the sample size using the modified homogeneity score method and compare it with that using the goodness-of-fit method (GOF) (Donner, Eliasziw, and Klar, 1996, Biometrics 52, 176-183). A simulation study for small and moderate sample sizes showed that the actual level of the homogeneity score test using the maximum likelihood estimators (MLEs) of parameters is satisfactorily close to the nominal and it is smaller than those of the modified homogeneity score and the goodness-of-fit tests. We investigated statistical properties of several noniterative estimators of a common kappa. The estimator (Donner et al., 1996) is essentially efficient and can be used as an alternative to the iterative MLE. An efficient interval estimation of a common kappa using the likelihood score method is presented.     

Interval estimation of the kappa coefficient with binary classification and an equal marginal probability model

We derive a likelihood score method for interval estimation of the intraclass version of the kappa coefficient of agreement with binary classification using a general theory of Bartlett (1953, Biometrika 40, 306-317). By exact evaluation, we investigate statistical properties of the score method, the chi-square goodness-of-fit procedure (Donner and Eliasziw, 1992, Statistics in Medicine 11, 1511-1519; Hale and Fleiss, 1993, Biometrics 49, 523-534), and a crude confidence interval for small and medium sample sizes. Actual coverage percentages of the score and chi-square methods are satisfactorily close to the nominal confidence coefficient, while that of the crude method is quite unsatisfactory. The expected length of the score method is shorter than that of the chi-square procedure when the response rate is very small or very large.     

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.     

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.     

On the use of historical control data to estimate dose response trends in quantal bioassay.

New tests for trend in proportions, in the presence of historical control data, are proposed. One such test is a simple score statistic based on a binomial likelihood for the "current" study and beta-binomial likelihoods for each historical control series. A closely related trend statistic based on estimating equations is also proposed. Trend statistics that allow overdispersed proportions in the current study are also developed, including a version of Tarone's (1982, Biometrics 38, 215-220) test that acknowledges sampling variation in the beta distribution parameters, and a trend statistic based on estimating equations. Each such trend test is evaluated with respect to size and power under both binomial and beta-binomial sampling conditions for the current study, and illustrations are provided.     

Bias-corrected maximum likelihood estimator of the negative binomial dispersion parameter

We derive a first-order bias-corrected maximum likelihood estimator for the negative binomial dispersion parameter. This estimator is compared, in terms of bias and efficiency, with the maximum likelihood estimator investigated by Piegorsch (1990, Biometrics46, 863-867), the moment and the maximum extended quasi-likelihood estimators investigated by Clark and Perry (1989, Biometrics45, 309-316), and a double-extended quasi-likelihood estimator. The bias-corrected maximum likelihood estimator has superior bias and efficiency properties in most instances. For ease of comparison we give results for the two-parameter negative binomial model. However, an example involving negative binomial regression is given.     

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.     

Exact unconditional inference for risk ratio in a correlated 2 x 2 table with structural zero

In this article, we consider small-sample statistical inference for rate ratio (RR) in a correlated 2 x 2 table with a structural zero in one of the off-diagonal cells. Existing Wald's test statistic and logarithmic transformation test statistic will be adopted for this purpose. Hypothesis testing and confidence interval construction based on large-sample theory will be reviewed first. We then propose reliable small-sample exact unconditional procedures for hypothesis testing and confidence interval construction. We present empirical results to evince the better confidence interval performance of our proposed exact unconditional procedures over the traditional large-sample procedures in small-sample designs. Unlike the findings given in Lui (1998, Biometrics 54, 706-711), our empirical studies show that the existing asymptotic procedures may not attain a prespecified confidence level even in moderate sample-size designs (e.g., n = 50). Our exact unconditional procedures on the other hand do not suffer from this problem. Hence, the asymptotic procedures should be applied with caution. We propose two approximate unconditional confidence interval construction methods that outperform the existing asymptotic ones in terms of coverage probability and expected interval width. Also, we empirically demonstrate that the approximate unconditional tests are more powerful than their associated exact unconditional tests. A real data set from a two-step tuberculosis testing study is used to illustrate the methodologies.     

Tests for Trends in Incidence Rate Ratios

Breslow (1984) described an efficient score test for trend in incidence density rate ratios for cohort studies under a conditional Poisson or binomial model employing maximum likelihood estimation of the rate parameters. In this communication, an alternative derivation of this statistic that is based on an unconditional approach is provided, along with an examination of associated goodness-of-fit tests and methods of confidence interval estimation. The procedures are illustrated by a cohort study of ischemic heart disease mortality following industrial exposure to carbon disulfide.     

Regression analysis of doubly censored failure time data using the additive hazards model

Doubly censored failure time data arise when the survival time of interest is the elapsed time between two related events and observations on occurrences of both events could be censored. Regression analysis of doubly censored data has recently attracted considerable attention and for this a few methods have been proposed (Kim et al., 1993, Biometrics 49, 13-22; Sun et al., 1999, Biometrics 55, 909-914; Pan, 2001, Biometrics 57, 1245-1250). However, all of the methods are based on the proportional hazards model and it is well known that the proportional hazards model may not fit failure time data well sometimes. This article investigates regression analysis of such data using the additive hazards model and an estimating equation approach is proposed for inference about regression parameters of interest. The proposed method can be easily implemented and the properties of the proposed estimates of regression parameters are established. The method is applied to a set of doubly censored data from an AIDS cohort study.     

Complementary Log–Log Regression for the Estimation of Covariate‐Adjusted Prevalence Ratios in the Analysis of Data from Cross‐Sectional Studies

We assessed complementary log–log (CLL) regression as an alternative statistical model for estimating multivariable‐adjusted prevalence ratios (PR) and their confidence intervals. Using the delta method, we derived an expression for approximating the variance of the PR estimated using CLL regression. Then, using simulated data, we examined the performance of CLL regression in terms of the accuracy of the PR estimates, the width of the confidence intervals, and the empirical coverage probability, and compared it with results obtained from log–binomial regression and stratified Mantel–Haenszel analysis. Within the range of values of our simulated data, CLL regression performed well, with only slight bias of point estimates of the PR and good confidence interval coverage. In addition, and importantly, the computational algorithm did not have the convergence problems occasionally exhibited by log–binomial regression. The technique is easy to implement in SAS (SAS Institute, Cary, NC), and it does not have the theoretical and practical issues associated with competing approaches. CLL regression is an alternative method of binomial regression that warrants further assessment.     

Efficiency of haplotype frequency estimation when nuclear family information is included

In genetic studies the haplotype structure of the regarded population is expected to carry important information. Experimental methods to derive haplotypes, however, are expensive and none of them has yet become standard methodology. On the other hand, maximum likelihood haplotype estimation from unphased individual genotypes may incur inaccuracies. We therefore investigated the relative efficiency of haplotype frequency estimation when nuclear family information is included compared to estimation from experimentally derived haplotypes. Efficiency was measured in terms of variance ratios of the estimates. The variances were derived from the binomial distribution for experimentally derived haplotypes, and from the Fisher information matrix corresponding to the general likelihood function of the haplotype frequency parameters, including family information. We subsequently compared these variance ratios to the variance ratios for the case of estimation from individual genotypes. We found that the information gained from a single child compensates missing phase information to a high degree, resulting in estimates almost as reliable as those derived from observed haplotypes. Thus, if children have already been genotyped for other reasons, it is highly recommendable to include them into the estimation. If child information is not already present, it depends on the number of loci and the haplotype diversity if it is useful to genotype a single child just to reduce phase ambiguity. In general, if the number of loci is less than or equal to three or if the number of haplotypes with a frequency >5% is less than or equal to four, haplotype estimation from individuals is quite good already and the improvement gained from a single child can not compensate the genotyping effort for it. On the other hand, under scenarios with many loci and high haplotype diversity, haplotype frequency estimation from trios can be more efficient than haplotype frequency estimation from individuals also on a per genotype base.     

A simple binomial test for estimating sequencing errors in public repository 16S rRNA sequences

Sequences in public databases may contain a number of sequencing errors. A double binomial model describing the distribution of indel-excluded similarity coefficients (S) among repeatedly sequenced 16S rRNA was previously developed and it produced a confidence interval of S useful for testing sequence identity among sequences of 400-bp length. We characterized patterns in sequencing errors found in nearly complete 16S rRNA sequences of Vibrionaceae as highly variable in reported sequence length and containing a small number of indels. To accommodate these characteristics, a simple binomial model for distribution of the similarity coefficient (H) that included indels was derived from the double binomial model for S. The model showed good fit to empirical data. By using either a pre-determined or bootstrapping estimated standard probability of base matching, we were able to use the exact binomial test to determine the relative level of sequencing error for a given pair of duplicated sequences. A limitation of the method is the requirement that duplicated sequences for the same template sequence be paired, but this can be overcome by using only conserved regions of 16S rRNA sequences and pairing a given sequence with its highest scoring BLAST search hit from the nr database of GenBank.     

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.     

Confidence Limits for the Ratio of Two Binomial Proportions Based on Likelihood Scores: Non-Iterative Method

The interval estimation of the ratio of two binomial proportions based on the score statistic is superior over other methods. Iterative algorithms for calculating the approximate confidence interval have been provided by, e.g., KOOPMAN (1984, Biometrics 40:513–517) and GART and NAM (1988a, Biometrics 44:323–338). This note presents the analytical solutions for upper and lower confidence limits in a closed form and gives examples for numerical illustration. The non-iterative method is generally more desirable than the iterative method.     

Methods of estimation in log odds ratio regression models

McCullagh's (1984, Journal of the Royal Statistical Society, Series B 46, 250-256) approximation to the conditional maximum likelihood estimator in log odds ratio regression models is shown to have negligible asymptotic bias unless the odds ratios are large and the sample sizes in individual 2 X 2 tables are very small. In application to two sets of case-control data, it yields results virtually indistinguishable from those of the conditional analysis. A generalization of the Mantel-Haenszel estimator proposed by Davis (1985, Biometrics 41, 487-495) does not approximate the conditional results nearly as well.