Notes on interval estimation of the gamma correlation under stratified random sampling

We have developed four asymptotic interval estimators in closed forms for the gamma correlation under stratified random sampling, including the confidence interval based on the most commonly used weighted‐least‐squares (WLS) approach (CIWLS), the confidence interval calculated from the Mantel‐Haenszel (MH) type estimator with the Fisher‐type transformation (CIMHT), the confidence interval using the fundamental idea of Fieller's Theorem (CIFT) and the confidence interval derived from a monotonic function of the WLS estimator of Agresti's α with the logarithmic transformation (MWLSLR). To evaluate the finite‐sample performance of these four interval estimators and note the possible loss of accuracy in application of both Wald's confidence interval and MWLSLR using pooled data without accounting for stratification, we employ Monte Carlo simulation. We use the data taken from a general social survey studying the association between the income level and job satisfaction with strata formed by genders in black Americans published elsewhere to illustrate the practical use of these interval estimators.     

Notes on Estimation of the General Odds Ratio and the General Risk Difference for Paired‐Sample Data

Under the matched‐pair design, this paper discusses estimation of the general odds ratio OR_G for ordinal exposure in case‐control studies and the general risk difference RD_G for ordinal outcomes in cross‐sectional or cohort studies. To illustrate the practical usefulness of interval estimators of OR_G and RD_G developed here, this paper uses the data from a case‐control study investigating the effect of the number of beverages drunk at “burning hot” temperature on the risk of possessing esophageal cancer, and the data from a cross‐sectional study comparing the grade distributions of unaided distance vision between two eyes. Finally, this paper notes that using the commonly‐used statistics related to odds ratio for dichotomous data by collapsing the ordinal exposure into two categories: the exposure versus the non‐exposure, tends to be less efficient than using the statistics related to OR_G proposed herein.     

The analysis of stratified 2 x 2 contingency tables

We consider the problem of testing for independence against the consistent superiority of one treatment over another when the response variable is binary and is compared across two treatments in each of several strata. Specifically, we consider the randomized clinical trial setting. A number of issues arise in this context. First, should tables be combined if there are small or zero margins? Second, should one assume a common odds ratio across strata? Third, if the odds ratios differ across strata, then how does the standard test (based on a common odds ratio) perform? Fourth, are there other analyzes that are more appropriate for handling a situation in which the odds ratios may differ across strata? In addressing these issues we find that the frequently used Cochran–Mantel–Haenszel test may have a poor power profile, despite being optimal when the odds ratios are common. We develop novel tests that are analogous to the Smirnov, modified Smirnov, convex hull, and adaptive tests that have been proposed for ordered categorical data. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Monte Carlo evaluation of five interval estimators for the relative risk in sparse data

The relative risk (RR) is one of the most frequently used indices to measure the strength of association between a disease and a risk factor in etiological studies or the efficacy of an experimental treatment in clinical trials. In this paper, we concentrate attention on interval estimation of RR for sparse data, in which we have only a few patients per stratum, but a moderate or large number of strata. We consider five asymptotic interval estimators for RR, including a weighted least-squares (WLS) interval estimator with an ad hoc adjustment procedure for sparse data, an interval estimator proposed elsewhere for rare events, an interval estimator based on the Mantel-Haenszel (MH) estimator with a logarithmic transformation, an interval estimator calculated from a quadratic equation, and an interval estimator derived from the ratio estimator with a logarithmic transformation. On the basis of Monte Carlo simulations, we evaluate and compare the performance of these five interval estimators in a variety of situations. We note that, except for the cases in which the underlying common RR across strata is around 1, using the WLS interval estimator with the adjustment procedure for sparse data can be misleading. We note further that using the interval estimator suggested elsewhere for rare events tends to be conservative and hence leads to loss of efficiency. We find that the other three interval estimators can consistently perform well even when the mean number of patients for a given treatment is approximately 3 patients per stratum and the number of strata is as small as 20. Finally, we use a mortality data set comparing two chemotherapy treatments in patients with multiple myeloma to illustrate the use of the estimators discussed in this paper.     

Confidence Intervals of the Attributable Risk under Cross‐Sectional Sampling with Confounders

Since it can account for both the strength of the association between exposure to a risk factor and the underlying disease of interest and the prevalence of the risk factor, the attributable risk (AR) is probably the most commonly used epidemiologic measure for public health administrators to locate important risk factors. This paper discusses interval estimation of the AR in the presence of confounders under cross‐sectional sampling. This paper considers four asymptotic interval estimators which are direct generalizations of those originally proposed for the case of no confounders, and employs Monte Carlo simulation to evaluate the finite‐sample performance of these estimators in a variety of situations. This paper finds that interval estimators using Wald's test statistic and a quadratic equation suggested here can consistently perform reasonably well with respect to the coverage probability in all the situations considered here. This paper notes that the interval estimator using the logarithmic transformation, that is previously found to consistently perform well for the case of no confounders, may have the coverage probability less than the desired confidence level when the underlying common prevalence rate ratio (RR) across strata between the exposure and the non‐exposure is large (≥4). This paper further notes that the interval estimator using the logit transformation is inappropriate for use when the underlying common RR ≐ 1. On the other hand, when the underlying common RR is large (≥4), this interval estimator is probably preferable to all the other three estimators. When the sample size is large (≥400) and the RR ≥ 2 in the situations considered here, this paper finds that all the four interval estimators developed here are essentially equivalent with respect to both the coverage probability and the average length.     

The large-sample relative efficiency of the Mantel-Haenszel estimator in the fixed-strata case

The large-sample relative efficiency of the Mantel-Haenszel estimator psi MH of an odds ratio is investigated for the case in which the number of strata is fixed and the sample sizes within each stratum increase indefinitely. The results show that psi MH is very efficient over a wide range of designs likely to occur in practice. However, conditions are identified under which the relative efficiency of this widely used estimator can be unusually low.     

Estimation of a common effect parameter from sparse follow-up data

Breslow (1981, Biometrika 68, 73-84) has shown that the Mantel-Haenszel odds ratio is a consistent estimator of a common odds ratio in sparse stratifications. For cohort studies, however, estimation of a common risk ratio or risk difference can be of greater interest. Under a binomial sparse-data model, the Mantel-Haenszel risk ratio and risk difference estimators are consistent in sparse stratifications, while the maximum likelihood and weighted least squares estimators are biased. Under Poisson sparse-data models, the Mantel-Haenszel and maximum likelihood rate ratio estimators have equal asymptotic variances under the null hypothesis and are consistent, while the weighted least squares estimators are again biased; similarly, of the common rate difference estimators the weighted least squares estimators are biased, while the estimator employing "Mantel-Haenszel" weights is consistent in sparse data. Variance estimators that are consistent in both sparse data and large strata can be derived for all the Mantel-Haenszel estimators.     

On a Class of Almost Unbiased Ratio Estimators

A class of almost unbiased ratio estimators for population mean σ is derived by weighting sample σ = (1/n) σ y_i, ratio estimators σ and an estimator, σ (y_i/x_i). It is shown that NIETO DE PASCUAL (1961) estimator is a particular member of the class and an optimum estimator in the class (in the minimum variance sense) is identified. The results are illustrated through two numerical examples.     

Estimation of Odds Ratios from Matched Case-Control Studies with Incomplete Data

Matched case-control studies often include pairs with incomplete exposure information. This work presents and compares two estimators for the odds ratio that can be used when the exposures of some of the cases and controls are missing. A simulation study shows that the estimator that uses the marginal exposure frequencies is usually more efficient than the estimator based on discordant pairs.     

On Combination of Ratio and PPS Estimators

AGARWAL and KUMAR (1980) proposed an estimator, combining ratio and pps estimators of population mean and proved that the proposed estimator would always be better (in minimum mean square error sense) than the pps estimator or the ratio estimator under pps sampling scheme for optimum value of constant k (parameter). The optimum value of k is rarely known in practice, hence the alternative is to replace k from the sample-values. In this paper, an estimator depending on estimated optimum value of k based on sample-values, under pps sampling scheme is proposed and studied.     

A Product-Type Estimator in Double Sampling

The use of ratio and product estimators, using auxiliary information, for estimating the mean of a finite population is well known. The efficiency of ratio estimator or product estimator is high depending on whether the auxiliary character is highly positively or negatively coorelated with the main character of interest. This paper proposes a product-type estimator which is more efficient than the usual ratio and product estimators in practical situations. We consider the case of double sampling from which the single sampling results may easily be derived.     

Performance of the Peto odds ratio compared to the usual odds ratio estimator in the case of rare events

For the calculation of relative measures such as risk ratio (RR) and odds ratio (OR) in a single study, additional approaches are required for the case of zero events. In the case of zero events in one treatment arm, the Peto odds ratio (POR) can be calculated without continuity correction, and is currently the relative effect estimation method of choice for binary data with rare events. The aim of this simulation study is a variegated comparison of the estimated OR and estimated POR with the true OR in a single study with two parallel groups without confounders in data situations where the POR is currently recommended. This comparison was performed by means of several performance measures, that is the coverage, confidence interval (CI) width, mean squared error (MSE), and mean percentage error (MPE). We demonstrated that the estimator for the POR does not outperform the estimator for the OR for all the performance measures investigated. In the case of rare events, small treatment effects and similar group sizes, we demonstrated that the estimator for the POR performed better than the estimator for the OR only regarding the coverage and MPE, but not the CI width and MSE. For larger effects and unbalanced group size ratios, the coverage and MPE of the estimator for the POR were inappropriate. As in practice the true effect is unknown, the POR method should be applied only with the utmost caution.     

Estimation of Finite Population Variance Using Ratio and Product Methods of Estimation

For estimating finite population variance σ_y² of a character y under our study, estimators using auxiliary information on a character x in the form of ratio, product, ratio-type or product-type estimators have been suggested, and their comparative study with the conventional unbiased estimator s_y² of σ_y² has been made in simple random sampling with replacement. A generalized estimator representing a class of estimators for the finite populations variance, has also been studied.     

Least Squares and Minimum MSE Estimators of Variance Components in Mixed Linear Models

The paper deals with the quadratic invariant estimators of the linear functions of variance components in mixed linear model. The estimator with locally minimal mean square error with respect to a parameter ? is derived. Under the condition of normality of the vector Y the theoretical values of MSE of several types of estimators are compared in two different mixed models; under a different types of distributions a simulation study is carried out for the behaviour of derived estimators.     

Five interval estimators for proportion ratio under a stratified randomized clinical trial with noncompliance

It is not uncommon that we may encounter a randomized clinical trial (RCT) in which there are confounders which are needed to control and patients who do not comply with their assigned treatments. In this paper, we concentrate our attention on interval estimation of the proportion ratio (PR) of probabilities of response between two treatments in a stratified noncompliance RCT. We have developed and considered five asymptotic interval estimators for the PR, including the interval estimator using the weighted-least squares (WLS) estimator, the interval estimator using the Mantel-Haenszel type of weight, the interval estimator derived from Fieller's Theorem with the corresponding WLS optimal weight, the interval estimator derived from Fieller's Theorem with the randomization-based optimal weight, and the interval estimator based on a stratified two-sample proportion test with the optimal weight suggested elsewhere. To evaluate and compare the finite sample performance of these estimators, we apply Monte Carlo simulation to calculate the coverage probability and average length in a variety of situations. We discuss the limitation and usefulness for each of these interval estimators, as well as include a general guideline about which estimators may be used for given various situations.     

Optimal approximate conversions of odds ratios and hazard ratios to risk ratios

Odds ratios approximate risk ratios when the outcome under consideration is rare but can diverge substantially from risk ratios when the outcome is common. In this paper, we derive optimal analytic conversions of odds ratios and hazard ratios to risk ratios that are minimax for the bias ratio when outcome probabilities are specified to fall in any fixed interval. The results for hazard ratios are derived under a proportional hazard assumption for the exposure. For outcome probabilities specified to lie in symmetric intervals centered around 0.5, it is shown that the square-root transformation of the odds ratio is the optimal minimax conversion for the risk ratio. General results for any nonsymmetric interval are given both for odds ratio and for hazard ratio conversions. The results are principally useful when odds ratios or hazard ratios are reported in papers, and the reader does not have access to the data or to information about the overall outcome prevalence.     

A Comparison of Nonparametric Error Rate Estimation Methods in Classification Problems

Assessment of the misclassification error rate is of high practical relevance in many biomedical applications. As it is a complex problem, theoretical results on estimator performance are few. The origin of most findings are Monte Carlo simulations, which take place in the “normal setting”: The covariables of two groups have a multivariate normal distribution; The groups differ in location, but have the same covariance matrix and the linear discriminant function LDF is used for prediction. We perform a new simulation to compare existing nonparametric estimators in a more complex situation. The underlying distribution is based on a logistic model with six binary as well as continuous covariables. To study estimator performance for varying true error rates, three prediction rules including nonparametric classification trees and parametric logistic regression and sample sizes ranging from 100‐1,000 are considered. In contrast to most published papers we turn our attention to estimator performance based on simple, even inappropriate prediction rules and relatively large training sets. For the major part, results are in agreement with usual findings. The most strikingly behavior was seen in applying (simple) classification trees for prediction: Since the apparent error rate Êrr.app is biased, linear combinations incorporating Êrr.app underestimate the true error rate even for large sample sizes. The .632+ estimator, which was designed to correct for the overoptimism of Efron's .632 estimator for nonparametric prediction rules, performs best of all such linear combinations. The bootstrap estimator Êrr.B0 and the crossvalidation estimator Êrr.cv, which do not depend on Êrr.app, seem to track the true error rate. Although the disadvantages of both estimators – pessimism of Êrr.B0 and high variability of Êrr.cv – shrink with increased sample sizes, they are still visible. We conclude that for the choice of a particular estimator the asymptotic behavior of the apparent error rate is important. For the assessment of estimator performance the variance of the true error rate is crucial, where in general the stability of prediction procedures is essential for the application of estimators based on resampling methods. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison of non-iterative and iterative estimators of heterogeneity variance for the standardized mortality ratio

This paper continues work presented in B?hning et al. (2002b, Annals of the Institute of Statistical Mathematics 54, 827-839, henceforth BMSRB) where a class of non-iterative estimators of the variance of the heterogeneity distribution for the standardized mortality ratio was discussed. Here, these estimators are further investigated by means of a simulation study. In addition, iterative estimators including the Clayton-Kaldor procedure as well as the pseudo-maximum-likelihood (PML) approach are added in the comparison. Among all candidates, the PML estimator often has the smallest mean square error, followed by the non-iterative estimator where the weights are proportional to the external expected counts. This confirms the theoretical result in BMSRB in which an asymptotic efficiency could be proved for this estimator (in the class of non-iterative estimators considered). Surprisingly, the Clayton-Kaldor iterative estimator (often recommended and used by practitioners) performed poorly with respect to the MSE. Given the widespread use of these estimators in disease mapping, medical surveillance, meta-analysis and other areas of public health, the results of this study might be of considerable interest.     

Asymptotic Relative Efficiency of Some Jackknife Estimators of a Common Odds Ratio

In the situation of several 2 × 2 tables the asymptotic relative efficiencies of certain jackknife estimators of a common odds ratio are investigated in the case that the number of tables is fixed while the sample sizes within each table tend to infinity. The estimators show very good results over a wide range of parameters. Some situations in which the estimators have low asymptotic relative efficiency are pointed out:.     

A semiparametric odds ratio model for measuring association

We propose a semiparametric odds ratio model to measure the association between two variables taking discrete values, continuous values, or a mixture of both. Methods for estimation and inference with varying degrees of robustness to model assumptions are studied. Semiparametric efficient estimation and inference procedures are also considered. The estimation methods are compared in a simulation study and applied to the study of associations among genital tract bacterial counts in HIV infected women.