Testing homogeneity of difference of two proportions for stratified correlated paired binary data

In ophthalmologic or otolaryngologic study, each subject may contribute paired organs measurements to the analysis. A number of statistical methods have been proposed on bilateral correlated data. In practice, it is important to detect confounding effect by treatment interaction, since ignoring confounding effect may lead to unreliable conclusion. Therefore, stratified data analysis can be considered to adjust the effect of confounder on statistical inference. In this article, we investigate and derive three test procedures for testing homogeneity of difference of two proportions for stratified correlated paired binary data in the basis of equal correlation model assumption. The performance of proposed test procedures is examined through Monte Carlo simulation. The simulation results show that the Score test is usually robust on type I error control with high power, and therefore is recommended among the three methods. One example from otolaryngologic study is given to illustrate the three test procedures.     

Testing the homogeneity of risk differences with sparse count data

ABSTRACT

In this paper, we consider testing the homogeneity of risk differences in independent binomial distributions especially when data are sparse. We point out some drawback of existing tests in either controlling a nominal size or obtaining powers through theoretical and numerical studies. The proposed test is designed to avoid the drawbacks of existing tests. We present the asymptotic null distribution and asymptotic power function for the proposed test. We also provide numerical studies including simulations and real data examples showing the proposed test has reliable results compared to existing testing procedures.     

Tests of odds ratio homogeneity with improved power in sparse fourfold tables

Some new tests of odds ratio homogeneity for fourfold tables are compared with the mixture model score test in the sparse-data case (many tables, small margins per table). Based on general empirical Bayes inequalities, the new tests have competitive power for 1:R matched designs, and superior power for more balanced designs.     

A multivariate zero-inflated binomial model for the analysis of correlated proportional data

In this paper, a new multivariate zero-inflated binomial (MZIB) distribution is proposed to analyse the correlated proportional data with excessive zeros. The distributional properties of purposed model are studied. The Fisher scoring algorithm and EM algorithm are given for the computation of estimates of parameters in the proposed MZIB model with/without covariates. The score tests and the likelihood ratio tests are derived for assessing both the zero-inflation and the equality of multiple binomial probabilities in correlated proportional data. A limited simulation study is performed to evaluate the performance of derived EM algorithms for the estimation of parameters in the model with/without covariates and to compare the nominal levels and powers of both score tests and likelihood ratio tests. The whitefly data is used to illustrate the proposed methodologies.     

Score test of homogeneity for survival data

If follow-up is made for subjects which are grouped into units, such as familial or spatial units then it may be interesting to test whether the groups are homogeneous (or independent for given explanatory variables). The effect of the groups is modelled as random and we consider a frailty proportional hazards model which allows to adjust for explanatory variables. We derive the score test of homogeneity from the marginal partial likelihood and it turns out to be the sum of a pairwise correlation term of martingale residuals and an overdispersion term. In the particular case where the sizes of the groups are equal to one, this statistic can be used for testing overdispersion. The asymptotic variance of this statistic is derived using counting process arguments. An extension to the case of several strata is given. The resulting test is computationally simple; its use is illustrated using both simulated and real data. In addition a decomposition of the score statistic is proposed as a sum of a pairwise correlation term and an overdispersion term. The pairwise correlation term can be used for constructing a statistic more robust to departure from the proportional hazard model, and the overdispesion term for constructing a test of fit of the proportional hazard model.     

Sample size determination for testing equality in Poisson frequency data under an AB/BA crossover trial

Assuming that the frequency of occurrence follows the Poisson distribution, we develop sample size calculation procedures for testing equality based on an exact test procedure and an asymptotic test procedure under an AB/BA crossover design. We employ Monte Carlo simulation to demonstrate the use of these sample size formulae and evaluate the accuracy of sample size calculation formula derived from the asymptotic test procedure with respect to power in a variety of situations. We note that when both the relative treatment effect of interest and the underlying intraclass correlation between frequencies within patients are large, the sample size calculation based on the asymptotic test procedure can lose accuracy. In this case, the sample size calculation procedure based on the exact test is recommended. On the other hand, if the relative treatment effect of interest is small, the minimum required number of patients per group will be large, and the asymptotic test procedure will be valid for use. In this case, we may consider use of the sample size calculation formula derived from the asymptotic test procedure to reduce the number of patients needed for the exact test procedure. We include an example regarding a double‐blind randomized crossover trial comparing salmeterol with a placebo in exacerbations of asthma to illustrate the practical use of these sample size formulae. Copyright © 2013 John Wiley & Sons, Ltd.     

Performance of and software for a modified mantel-haenszel statistic for correlated data

This paper presents the results of a small sample simulation study designed to evaluate the performance of a recently proposed test statistic for the analysis of correlated binary data. The new statistic is an adjusted Mantel-Haenszel test, which may be used in testing for association between a binary exposure and a binary outcome of interest across several fourfold tables when the data have been collected under a cluster sampling design. Al- though originally developed for the analysis of periodontal data, the proposed method may be applied to clustered binary data arising in a variety of settings, including longitu- dinal studies, family studies, and school-based research. The features of the simulation are intended to mimic those of a research study of periodontal health, in which a large number of observations is made on each of a relatively small number of patients. The simulation reveals that the adjusted test statistic performs well in finite samples, having empirical type I error rates close to nominal and empirical power similar to that of more complicated marginal regression methods. Software for computing the adjusted statistic is also provided.     

Testing equality of proportions with incomplete correlated data

Let (ψ_i,φ_i) be independent, identically distributed pairs of zero-one random variables with (possible) dependence of ψ_i and φ_i within the pair. For n pairs, both variables are observed, but for m₁ additional pairs only ψ_i is observed and for m₂ others φ_i is observed. If π_1· = P{ψ_i = 1} and π_·1=P{φ_i, the problem is to test π_1·=π_·1. Maximum likelihood estimates of π_1· and π_·1 are obtained via the EM algorithm. A test statistic is developed whose null distribution is asymptotically chi-square with one degree of freedom (as n and either m₁ or m₂ tend to infinity). If m₁ = m₂ = 0 the statistic reduces to that of McNemar's test; if n = 0, it is equivalent to the statistic for testing equality of two independent proportions. This test is compared with other tests by means of Pitman efficiency. Examples are presented.     

Design and inference for 3‐stage bioequivalence testing with serial sampling data

A bioequivalence test is to compare bioavailability parameters, such as the maximum observed concentration (C_max) or the area under the concentration‐time curve, for a test drug and a reference drug. During the planning of a bioequivalence test, it requires an assumption about the variance of C_max or area under the concentration‐time curve for the estimation of sample size. Since the variance is unknown, current 2‐stage designs use variance estimated from stage 1 data to determine the sample size for stage 2. However, the estimation of variance with the stage 1 data is unstable and may result in too large or too small sample size for stage 2. This problem is magnified in bioequivalence tests with a serial sampling schedule, by which only one sample is collected from each individual and thus the correct assumption of variance becomes even more difficult. To solve this problem, we propose 3‐stage designs. Our designs increase sample sizes over stages gradually, so that extremely large sample sizes will not happen. With one more stage of data, the power is increased. Moreover, the variance estimated using data from both stages 1 and 2 is more stable than that using data from stage 1 only in a 2‐stage design. These features of the proposed designs are demonstrated by simulations. Testing significance levels are adjusted to control the overall type I errors at the same level for all the multistage designs.     

Exact methods of testing the homogeneity of prevalences for correlated binary data

Correlated binary data arise in many ophthalmological and otolaryngological clinical trials. To test the homogeneity of prevalences among different groups is an important issue when conducting these trials. The equal correlation coefficients model proposed by Donner in 1989 is a popular model handling correlated binary data. The asymptotic chi-square test works well when the sample size is large. However, it would fail to maintain the type I error rate when the sample size is relatively small. In this paper, we propose several exact methods to deal with small sample scenarios. Their performances are compared with respect to type I error rate and power. The ‘M approach’ and the ‘E + M approach’ seem to outperform the others. A real work example is given to further explain how these approaches work. Finally, the computational efficiency of the exact methods is discussed as a pressing issue of future work.     

A score test for monotonic trend in hazard rates for grouped survival data in stratified populations

Models for monotone trends in hazard rates for grouped survival data in stratified populations are introduced, and simple closed form score statistics for testing the significance of these trends are presented. The test statistics for some of the models understudy are shown to be independent of the assumed form of the function which relates the hazard rates to the sets of monotone scores assigned to the time intervals. The procedure is applied to test monotone trends in the recovery rates of erythematous response among skin cancer patients and controls that have been irradiated with a ultraviolent challenge.     

Extending conditional likelihood in models for stratified binary data

The conditional likelihood is widely used in logistic regression models with stratified binary data. In particular, it leads to accurate inference for the parameters of interest, which are common to all strata, eliminating stratum-specific nuisance parameters. The modified profile likelihood is an accurate approximation to the conditional likelihood, but has the advantage of being available for general parametric models. Here, we propose the modified profile likelihood as an ideal extension of the conditional likelihood in generalized linear models for binary data, with generic link function. An important feature is that for the implementation we only need standard outputs of routines for generalized linear models. The accuracy of the method is supported by theoretical properties and is confirmed by simulation results.This research was supported by MIUR COFIN 2001-2003.     

Sample size re-estimation for survival data in clinical trials with an adaptive design

In clinical trials with survival data, investigators may wish to re-estimate the sample size based on the observed effect size while the trial is ongoing. Besides the inflation of the type-I error rate due to sample size re-estimation, the method for calculating the sample size in an interim analysis should be carefully considered because the data in each stage are mutually dependent in trials with survival data. Although the interim hazard estimate is commonly used to re-estimate the sample size, the estimate can sometimes be considerably higher or lower than the hypothesized hazard by chance. We propose an interim hazard ratio estimate that can be used to re-estimate the sample size under those circumstances. The proposed method was demonstrated through a simulation study and an actual clinical trial as an example. The effect of the shape parameter for the Weibull survival distribution on the sample size re-estimation is presented.     

Statistical inference of risk difference in K correlated 2×2 tables with structural zero

K correlated 2×2 tables with structural zero are commonly encountered in infectious disease studies. A hypothesis test for risk difference is considered in K independent 2×2 tables with structural zero in this paper. Score statistic, likelihood ratio statistic and Wald‐type statistic are proposed to test the hypothesis on the basis of stratified data and pooled data. Sample size formulae are derived for controlling a pre‐specified power or a pre‐determined confidence interval width. Our empirical results show that score statistic and likelihood ratio statistic behave better than Wald‐type statistic in terms of type I error rate and coverage probability, sample sizes based on stratified test are smaller than those based on the pooled test in the same design. A real example is used to illustrate the proposed methodologies. Copyright © 2009 John Wiley & Sons, Ltd.     

Simultaneous confidence interval construction for many-to-one comparisons of proportion differences based on correlated paired data

In some medical researches such as ophthalmological, orthopaedic and otolaryngologic studies, it is often of interest to compare multiple groups with a control using data collected from paired organs of patients. The major difficulty in performing the data analysis is to adjust the multiplicity between the comparison of multiple groups, and the correlation within the same patient''s paired organs. In this article, we construct asymptotic simultaneous confidence intervals (SCIs) for many-to-one comparisons of proportion differences adjusting for multiplicity and the correlation. The coverage probabilities and widths of the proposed CIs are evaluated by Monte Carlo simulation studies. The methods are illustrated by a real data example.     

Optimum allocation in multivariate stratified sampling design in the presence of nonresponse with Gamma cost function

In multivariate stratified sample survey with L strata, let p-characteristics are defined on each unit of the population. To estimate the unknown p-population means of each characteristic, a random sample is taken out from the population. In multivariate stratified sample survey, the optimum allocation of any characteristic may not be optimum for others. Thus the problem arises to find out an allocation which may be optimum for all characteristics in some sense. Therefore a compromise criterion is needed to workout such allocation. In this paper, the procedure of estimation of p-population means is discussed in the presence of nonresponse when the use of linear cost function is not advisable. A solution procedure is suggested by using lexicographic goal programming problem. The numerical illustrations are given for its practical utility.     

Comparison of disease prevalence in two populations under double-sampling scheme with two fallible classifiers

A disease prevalence can be estimated by classifying subjects according to whether they have the disease. When gold-standard tests are too expensive to be applied to all subjects, partially validated data can be obtained by double-sampling in which all individuals are classified by a fallible classifier, and some of individuals are validated by the gold-standard classifier. However, it could happen in practice that such infallible classifier does not available. In this article, we consider two models in which both classifiers are fallible and propose four asymptotic test procedures for comparing disease prevalence in two groups. Corresponding sample size formulae and validated ratio given the total sample sizes are also derived and evaluated. Simulation results show that (i) Score test performs well and the corresponding sample size formula is also accurate in terms of the empirical power and size in two models; (ii) the Wald test based on the variance estimator with parameters estimated under the null hypothesis outperforms the others even under small sample sizes in Model II, and the sample size estimated by this test is also accurate; (iii) the estimated validated ratios based on all tests are accurate. The malarial data are used to illustrate the proposed methodologies.     

Confidence intervals for assessing equivalence of two treatments with combined unilateral and bilateral data

Responses from the paired organs are generally highly correlated in bilateral studies, statistical procedures ignoring the correlation could lead to incorrect results. Note the intraclass correlation in the study of combined unilateral and bilateral outcomes; 11 confidence intervals (CIs) including 7 asymptotic CIs and 4 Bootstrap-resampling CIs for assessing the equivalence of 2 treatments are derived under Rosner''s correlated binary data model. Performance is evaluated with respect to the empirical coverage probability (ECP), the empirical coverage width (ECW) and the ratio of the mesial non-coverage probability to the non-coverage probability (RMNCP) via simulation studies. Simulation results show that (i) all CIs except for the Wald CI and the bias-corrected Bootstrap percentile CI generally produce satisfactory ECPs and hence are recommended; (ii) all CIs except for the bias-corrected Bootstrap percentile CI provide preferred RMNCPs and are more symmetrical; (iii) as the measurement of the dependence increases, the ECWs of all CIs except for the score CI and the profile likelihood CI show increasing patterns that look like linear, while there is no obvious pattern on the ECPs of all CIs except for the profile likelihood CI. A data set from an otolaryngologic study is used to illustrate the proposed methods.     

Incorporating historical two‐arm data in clinical trials with binary outcome: A practical approach

The feasibility of a new clinical trial may be increased by incorporating historical data of previous trials. In the particular case where only data from a single historical trial are available, there exists no clear recommendation in the literature regarding the most favorable approach. A main problem of the incorporation of historical data is the possible inflation of the type I error rate. A way to control this type of error is the so‐called power prior approach. This Bayesian method does not “borrow” the full historical information but uses a parameter 0 ≤ δ ≤ 1 to determine the amount of borrowed data. Based on the methodology of the power prior, we propose a frequentist framework that allows incorporation of historical data from both arms of two‐armed trials with binary outcome, while simultaneously controlling the type I error rate. It is shown that for any specific trial scenario a value δ > 0 can be determined such that the type I error rate falls below the prespecified significance level. The magnitude of this value of δ depends on the characteristics of the data observed in the historical trial. Conditionally on these characteristics, an increase in power as compared to a trial without borrowing may result. Similarly, we propose methods how the required sample size can be reduced. The results are discussed and compared to those obtained in a Bayesian framework. Application is illustrated by a clinical trial example.