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.     

Comparison of correlated proportions based on paired binary data from clustered samples

In this article, we explore hypothesis testing problems related to correlated proportions from clustered matched-pair binary data. Null hypotheses of equality in proportions, homogeneity, and non-inferiority of one to another are similar testing problems of linear contrasts of correlated proportions with suitable transformation. The covariance estimators of the test statistics are based on moment estimation under the null hypotheses. We present a general framework for testing linear contrasts of the correlated proportions from clustered matched-pair data based upon a class of unbiased estimators of the proportions. The corresponding testing procedures do not impose structure assumptions on the correlation matrix and are easy to use. Simulation results suggest that the proposed method is more likely to maintain the proper significance level and to improve power than the test proposed by Obuchowski.     

Estimation of proportions by group testing with retesting of positive groups

Abstract

When estimating a proportion p by group testing, and it is desired to increase precision, it is sometimes impractical to obtain additional individuals but it is possible to retest groups formed from the individuals within the groups that test positive at the first stage. Hepworth and Watson assessed four methods of retesting, and recommended a random regrouping of individuals from the first stage. They developed an estimator of p for their proposed method, and, because of the analytic complexity, used simulation to examine its variance properties. We now provide an analytical solution to the variance of the estimator, and compare its performance with the earlier simulated results. We show that our solution gives an acceptable approximation in a reasonable range of circumstances.     

A bias-corrected histogram estimator for line transect sampling

The classical histogram method has already been applied in line transect sampling to estimate the parameter f(0), which in turns is used to estimate the population abundance D or the population size N. It is well know that the bias convergence rate for histogram estimator of f(0) is o(h²) as h → 0, under the shoulder condition assumption. If the shoulder condition is not true, then the bias convergence rate is only o(h). This paper proposed two new estimators for f(0), which can be considered as modifications of the classical histogram estimator. The first estimator is derived when the shoulder condition is assumed to be valid and it reduces the bias convergence rate from o(h²) to o(h³). The other one is constructed without using the shoulder condition assumption and it reduces the bias convergence rate from o(h) to o(h²). The asymptotic properties of the proposed estimators are derived and formulas for bin width are also given. The finite properties based on a real data set and an extensive simulation study demonstrated the potential practical use of the proposed estimators.     

One-way analysis of proportions for misclassified binomial data

We consider data with a nominal grouping variable and a binary response variable. The grouping variable is measured without error, but the response variable is measured using a fallible device subject to misclassification. To achieve model identifiability, we use the double-sampling scheme which requires obtaining a subsample of the original data or another independent sample. This sample is then classified by both the fallible device and another infallible device regarding the response variable. We propose two Wald tests for testing the association between the two variables and illustrate the test using traffic data. The Type-I error rate and power of the tests are examined using simulations and a modified Wald test is recommended.     

An empirical comparison of several methods for testing the equality of dependent proportions

Three methods for testing the equality of nonindependent proportions were compared with, the use of Monte Carlo techniques. The three methods included Cochran's test, an ANOVA F test, and Hotelling's T² test. With respect to empirical significance levels, the ANOVA F test is recommended as the preferred method of analysis.

Oftentimes an experimenter is interested in testing the equality of several proportions. When the proportions are independent Kemp and Butcher (1972) and Butcher and Kemp (1974) compared several methods for analysing large sample binomial data for the case of a 3 x 3 factorial design without replication. In addition, Levy and Narula (1977) compared many of the same methods for analyzing binomial data; however, Levy and Narula investigated the relative utility of the methods for small sample sizes.     

Estimation of proportions for multinomial contingency tables subject to marginal constraints

For a two-way table of observed proportions we consider four approaches to the problem of fitting a new set of proportions which-subject to satisfying marginal constraints on row and column totals-corresponds as closely as possible to the original table, The approaches are known to be asymptotically equivalent; but our chief interest is in establishing, at least for the 2x2 table, a hierarchy of preferences under multinomial sampling for moderate sample sizes. Implementation of minimum chi-square is described and recommended.     

Analysis of proportions for clustered binary data in the presence of unequal dispersions

This paper investigates the test procedures for testing the homogeneity of the proportions in the analysis of clustered binary data in the context of unequal dispersions across the treatment groups. We introduce a simple test procedure based on adjusted proportions using a sandwich estimator of the variance of the proportion estimators obtained by the generalized estimating equations approach of Zeger and Liang (1986) [Biometrics 42, 121-130]. We also extend the exiting test procedures of testing the hypothesis of proportions in this context. These test procedures are then compared, by simulations, in terms of size and power. Moreover, we derive the score test for testing the homogeneity of the dispersion parameters among several groups of clustered binary data. An illustrative application of the recommended test procedures is also presented.     

Simultaneous estimation of the hardy-weinberg proportions

We consider the problem of simultaneously estimating k + 1 related proportions, with a special emphasis on the estimation of Hardy-Weinberg (HW) proportions. We prove that the uniformly minimum-variance unbiased estimator (UMVUE) of two proportions which are individually admissible under squared-error loss are inadmissible in estimating the proportions jointly. Furthermore, rules that dominate the UMVUE are given. A Bayesian analysis is then presented to provide insight into this inadmissibility issue: The UMVUE is undesirable because the two estimators are Bayes rules corresponding to different priors. It is also shown that there does not exist a prior which yields the maximum-likelihood estimators simultaneously. When the risks of several estimators for the HW proportions are compared, it is seen that some Bayesian estimates yield significantly smaller risks over a large portion of the parameter space for small samples. However, the differences in risks become less significant as the sample size gets larger.     

Estimating the proportions of two populations in a mixture using linear maps

We address the problem of estimating the proportions of two statistical populations in a given mixture on the basis of an unlabeled sample of n–dimensional observations on the mixture. Assuming that the expected values of observations on the two populations are known, we show that almost any linear map from Rⁿ to R¹ yields an unbiased consistent estimate of the proportion of one population in a very easy way. We then find that linear map for which the resulting proportion estimate has minimum variance among all estimates so obtained. After deriving a simple expression for the minimum-variance estimate, we discuss practical aspects of obtaining this and related estimates.     

Comparison of proportions with asymmetric prior information

In this paper a Bayesian model is developed for comparing two binomial proportions. A two stage hierarchical prior distribution is used to represent prior dependence. Prior exchangeability and independence are shown to be but special cases. The relevant distributions have to be computed numerically and some examples are presented.     

New asymptotic inferences about the difference,ratio, and linear combination of two independent proportions

In statistics it is customary to realize asymptotic inferences about the difference d, the ratio R or a linear combination L of two independent proportions. In this article the authors evaluate ten inference methods, and conclude that for α = 1% some of the new procedures behave better than the classical. In cases d, R, or L, the optimal method consists in adding 0.5, 0.5 or 1 to all the data, respectively, and then applying a modification of the arc-sine transformation (d or R) or the likelihood ratio test (L). A free program may be obtained at ULR http://www.ugr.es/local/bioest/Z_LINEAR_K.EXE.     

Bayesian analysis of the difference of two proportions

In this article we give the expression of the prior distribution of p₁-P₂, where P₁ and P₂ and independent proportions with a beta prior each. The expression derived for the posterior distribution of P₁-P₂ then shows the closure of the beta-difference family for independent dual Bernoulli samples. Other bayesian results are also presented.     

A unified sequential test procedure for simultaneous testing the equality of several binomial proportions to a specified standard

In this article, we propose a unified sequentially rejective test procedure for testing simultaneously the equality of several independent binomial proportions to a specified standard. The proposed test procedure is general enough to include some well-known multiple testing procedures such as the Ordinary Bonferroni procedure, Hochberg procedure and Rom procedure. It involves multiple tests of significance based on the simple binomial tests (exact or approximate) which can be easily found in many elementary standard statistics textbooks. Unlike the traditional Chi-square test of the overall hypothesis, the procedure can identify the subset of the binomial proportions, which are different from the prespecified standard with the control of the familywise type I error rate. Moreover, the power computation of the procedure is provided and the procedure is illustrated by two real examples from an ecological study and a carcinogenicity study.     

Small area estimation of proportions in business surveys

Binary data are often of interest in business surveys, particularly when the aim is to characterize grouping in the businesses making up the survey population. When small area estimates are required for such binary data, use of standard estimation methods based on linear mixed models (LMMs) becomes problematic. We explore two model-based techniques of small area estimation for small area proportions, the empirical best predictor (EBP) under a generalized linear mixed model and the model-based direct estimator (MBDE) under a population-level LMM. Our empirical results show that both the MBDE and the EBP perform well. The EBP is a computationally intensive method, whereas the MBDE is easy to implement. In case of model misspecification, the MBDE also appears to be more robust. The mean-squared error (MSE) estimation of MBDE is simple and straightforward, which is in contrast to the complicated MSE estimation for the EBP.     

Bias correction of estimated proportions using inverse binomial group testing

Group testing, in which individuals are pooled together and tested as a group, can be combined with inverse sampling to estimate the prevalence of a disease. Alternatives to the MLE are desirable because of its severe bias. We propose an estimator based on the bias correction method of Firth (1993), which is almost unbiased across the range of prevalences consistent with the group testing design. For equal group sizes, this estimator is shown to be equivalent to that derived by applying the correction method of Burrows (1987), and better than existing methods. For unequal group sizes, the problem has some intractable elements, but under some circumstances our proposed estimator can be found, and we show it to be almost unbiased. Calculation of the bias requires computer‐intensive approximation because of the infinite number of possible outcomes.     

Revisiting retesting in the estimation of proportions by group testing

In the estimation of a proportion p by group testing (pooled testing), retesting of units within positive groups has received little attention due to the minimal gain in precision compared to testing additional units. If acquisition of additional units is impractical or too expensive, and testing is not destructive, we show that retesting can be a useful option. We propose the retesting of a random grouping of units from positive groups, and compare it with nested halving procedures suggested by others. We develop an estimator of p for our proposed method, and examine its variance properties. Using simulation we compare retesting methods across a range of group testing situations, and show that for most realistic scenarios, our method is more efficient.     

Approximate percentiles for ratios of f-variates

The need for percentiles of ratios of independent F-variates arises in many types of applications. Some tabulated values are available mostly for equal or nearly equal degrees-of-freedom cases, but it is not possible to provide comprehensive tables for the numerous combinations of possible degrees-of-freedom and probability levels. A simple approximation for the percentiles based on the Cornish-Fisher expansion technique is provided which is adequate for larger degrees-of-freedom and a more complicated approximation is provided which is suitable for all cases. The accuracy of the approximation is shown over a wide range of typical cases     

Best linear corrector of classification estimates of proportions of objects in several unknown classes

It has been recognized that counting the objects allocated by a rule of classification to several unknown classes often does not provide good estimates of the true class proportions of the objects to be classified. We propose a linear transformation of these classification estimates, which minimizes the mean squared error of the transformed estimates over all possible sets of true proportions. This so-called best-linear-corrector (BLC) transformation is a function of the confusion (classification-error) matrix and of the first and second moments of the prior distribution of the vector of proportions. When the number of objects to be classified increases, the BLC tends to the inverse of the confusion matrix. The estimates that are obtained directly by this inverse-confusion corrector (ICC) are also the maximum-likelihood unbiased estimates of the probabilities that the objects originate from one or the other class, had the objects been preselected with those probabilities. But for estimating the actual proportions, the ICC estimates behave less well than the raw classification estimates for some collections. In that situation, the BLC is substantially superior to the ICC even for some large collections of objects and is always substantially superior to the raw estimates. The statistical model is applied concretely to the measure of forest covers in remote sensing.