A Note on the Log‐Rank Test in Life Table Analysis with Correlated Observations

Survival data consisting of independent sets of correlated failure times may arise in many situations. For example, we may take repeated observations of the failure time of interest from each patient or observations of the failure time on siblings, or consider the failure times on littermates in toxicological experiments. Because the failure times taken on the same patient or related family members or from the same litter are likely correlated, use of the classical log‐rank test in these situations can be quite misleading with respect to type I error. To avoid this concern, this paper develops two closed‐form asymptotic summary tests, that account for the intraclass correlation between the failure times within patients or units. In fact, one of these two test includes the classical log‐rank test as a special case when the intraclass correlation equals 0. Furthermore, to evaluate the finite‐sample performance of the two tests developed here, this paper applies Monte Carlo simulation and notes that they can actually perform quite well in a variety of situations considered here.     

Interval Estimation of Simple Difference in Dichotomous Data with Repeated Measurements

When comparing two treatments, we often use the simple difference between the probabilities of response to measure the efficacy of one treatment over the other. When the measurement of outcome is unreliable or the cost of obtaining additional subjects is high relative to that of additional measurements from the obtained subjects, we may often consider taking more than one measurement per subject to increase the precision of an interval estimator. This paper focuses discussion on interval estimation of simple difference when we take repeated measurements per subject. This paper develops four asymptotic interval estimators of simple difference for any finite number of measurements per subject. This paper further applies Monte Carlo simulation to evaluate the finite‐sample performance of these estimators in a variety of situations. Finally, this paper includes a discussion on sample size determination on the basis of both the average length and the probability of controlling the length of the resulting interval estimate proposed elsewhere.     

Notes in Case-Control Studies with Matched Pairs under Inverse Sampling

In case-control studies with matched pairs, the traditional point estimator of odds ratio (OR) is well-known to be biased with no exact finite variance under binomial sampling. In this paper, we consider use of inverse sampling in which we continue to sample subjects to form matched pairs until we obtain a pre-determined number (>0) of index pairs with the case unexposed but the control exposed. In contrast to use of binomial sampling, we show that the uniformly minimum variance unbiased estimator (UMVUE) of OR does exist under inverse sampling. We further derive an exact confidence interval of OR in closed form. Finally, we develop an exact test and an asymptotic test for testing the null hypothesis H₀: OR = 1, as well as discuss sample size determination on the minimum required number of index pairs for a desired power at α-level.     

Notes on Testing Equality in Dichotomous Data with Matched Pairs

In attempting to improve the efficiency of McNemar's test statistic, we develop two test procedures that account for the information on both the discordant and concordant pairs for testing equality between two comparison groups in dichotomous data with matched pairs. Furthermore, we derive a test procedure derived from one of the most commonly‐used interval estimators for odds ratio. We compare these procedures with those using McNemar's test, McNemar's test with the continuity correction, and the exact test with respect to type I error and power in a variety of situations. We note that the test procedures using McNemar's test with the continuity correction and the exact test can be quite conservative and hence lose much efficiency, while the test procedure using McNemar's test can actually perform well even when the expected number of discordant pairs is small. We also find that the two test procedures, which incorporate the information on all matched pairs into hypothesis testing, may slightly improve the power of using McNemar's test without essentially losing the precision of type I error. On the other hand, the test procedure derived from an interval estimator of adds ratio with use of the logarithmic transformation may have type I error much larger than the nominal α‐level when the expected number of discordant pairs is not large and therefore, is not recommended for general use.     

Interval Estimation of Simple Difference under Independent Negative Binomial Sampling

When the sample size is not large or when the underlying disease is rare, to assure collection of an appropriate number of cases and to control the relative error of estimation, one may employ inverse sampling, in which one continues sampling subjects until one obtains exactly the desired number of cases. This paper focuses discussion on interval estimation of the simple difference between two proportions under independent inverse sampling. This paper develops three asymptotic interval estimators on the basis of the maximum likelihood estimator (MLE), the uniformly minimum variance unbiased estimator (UMVUE), and the asymptotic likelihood ratio test (ALRT). To compare the performance of these three estimators, this paper calculates the coverage probability and the expected length of the resulting confidence intervals on the basis of the exact distribution. This paper finds that when the underlying proportions of cases in both two comparison populations are small or moderate (≤0.20), all three asymptotic interval estimators developed here perform reasonably well even for the pre-determined number of cases as small as 5. When the pre-determined number of cases is moderate or large (≥50), all three estimators are essentially equivalent in all the situations considered here. Because application of the two interval estimators derived from the MLE and the UMVUE does not involve any numerical iterative procedure needed in the ALRT, for simplicity we may use these two estimators without losing efficiency.     

Five Confidence Intervals of the Closed Population Size in the Capture‐recapture Problem under Inverse Sampling with Replacement

In the capture‐recapture problem for two independent samples, the traditional estimator, calculated as the product of the two sample sizes divided by the number of sampled subjects appearing commonly in both samples, is well known to be a biased estimator of the population size and have no finite variance under direct or binomial sampling. To alleviate these theoretical limitations, the inverse sampling, in which we continue sampling subjects in the second sample until we obtain a desired number of marked subjects who appeared in the first sample, has been proposed elsewhere. In this paper, we consider five interval estimators of the population size, including the most commonly‐used interval estimator using Wald's statistic, the interval estimator using the logarithmic transformation, the interval estimator derived from a quadratic equation developed here, the interval estimator using the χ²‐approximation, and the interval estimator based on the exact negative binomial distribution. To evaluate and compare the finite sample performance of these estimators, we employ Monte Carlo simulation to calculate the coverage probability and the standardized average length of the resulting confidence intervals in a variety of situations. To study the location of these interval estimators, we calculate the non‐coverage probability in the two tails of the confidence intervals. Finally, we briefly discuss the optimal sample size determination for a given precision to minimize the expected total cost. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

A Test Procedure of Equivalence in Ordinal Data with Matched‐Pairs

There are many epidemiologic studies or clinical trials, in which we may wish to establish an equivalence rather than to detect a difference between the distributions of responses. In this paper, we develop test procedures to detect equivalence with respect to the tail marginal distributions and the marginal proportions when the underlying data are on an ordinal scale with matched pairs. We include a numerical example concerning the unaided distance vision of two eyes over 7477 women to illustrate the practical usefulness of the proposed procedure. Finally, we include a brief discussion on the relation between the test procedures developed here and an asymptotic interval estimator proposed elsewhere for the simple difference in dichotomous data with matched‐pairs.     

Confidence Intervals of the Simple Difference between the Proportions of a Primary Infection and a Secondary Infection,Given the Primary Infection

This paper discusses interval estimation of the simple difference (SD) between the proportions of the primary infection and the secondary infection, given the primary infection, by developing three asymptotic interval estimators using Wald's test statistic, the likelihood‐ratio test, and the basic principle of Fieller's theorem. This paper further evaluates and compares the performance of these interval estimators with respect to the coverage probability and the expected length of the resulting confidence intervals. This paper finds that the asymptotic confidence interval using the likelihood ratio test consistently performs well in all situations considered here. When the underlying SD is within 0.10 and the total number of subjects is not large (say, 50), this paper further finds that the interval estimators using Fieller's theorem would be preferable to the estimator using the Wald's test statistic if the primary infection probability were moderate (say, 0.30), but the latter is preferable to the former if this probability were large (say, 0.80). When the total number of subjects is large (say, ≥200), all the three interval estimators perform well in almost all situations considered in this paper. In these cases, for simplicity, we may apply either of the two interval estimators using Wald's test statistic or Fieller's theorem without losing much accuracy and efficiency as compared with the interval estimator using the asymptotic likelihood ratio test.     

Estimation of Spatial Variation in Risk Using Matched Case‐control Data

A common problem in environmental epidemiology is to estimate spatial variation in disease risk after accounting for known risk factors. In this paper we consider this problem in the context of matched case‐control studies. We extend the generalised additive model approach of Kelsall and Diggle (1998) to studies in which each case has been individually matched to a set of controls. We discuss a method for fitting this model to data, apply the method to a matched study on perinatal death in the North West Thames region of England and explain why, if spatial variation is of particular scientific interest, matching is undesirable.     

Joint Estimation of Diagnostic Accuracy Measures for Paired Organs – Application in Ophthalmology

Diagnostic studies in ophthalmology frequently involve binocular data where pairs of eyes are evaluated, through some diagnostic procedure, for the presence of certain diseases or pathologies. The simplest approach of estimating measures of diagnostic accuracy, such as sensitivity and specificity, treats eyes as independent, consequently yielding incorrect estimates, especially of the standard errors. Approaches that account for the inter‐eye correlation include regression methods using generalized estimating equations and likelihood techniques based on various correlated binomial models. The paper proposes a simple alternative statistical methodology of jointly estimating measures of diagnostic accuracy for binocular tests based on a flexible model for correlated binary data. Moments' estimation of model parameters is outlined and asymptotic inference is discussed. The resulting estimates are straightforward and easy to obtain, requiring no special statistical software but only elementary calculations. Results of simulations indicate that large‐sample and bootstrap confidence intervals based on the estimates have relatively good coverage properties when the model is correctly specified. The computation of the estimates and their standard errors are illustrated with data from a study on diabetic retinopathy.     

A Note on the Estimation of the Population Mean in Stratified Random Sampling Using Two Auxiliary Variables

In this paper, a two‐phase sampling estimator for a stratified population mean using two auxiliary variables x and z is considered when the stratum mean of x is unknown but that of z is known. The suggested estimator under its optimal condition is found to be more efficient than the one using only x.     

A Note on the Estimation of Coverage Errors in Demographic Data

A simple formula for the lower bound of coverage errors in demographic data is obtained. The formula is derived under general assumptions, the validity of which is discussed. Three numerical examples are given.     

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.     

A Revisit on Comparing the Asymptotic Interval Estimators of Odds Ratio in a Single 2 × 2 Table

It is well known that Cornfield 's confidence interval of the odds ratio with the continuity correction can mimic the performance of the exact method. Furthermore, because the calculation procedure of using the former is much simpler than that of using the latter, Cornfield 's confidence interval with the continuity correction is highly recommended by many publications. However, all these papers that draw this conclusion are on the basis of examining the coverage probability exclusively. The efficiency of the resulting confidence intervals is completely ignored. This paper calculates and compares the coverage probability and the average length for Woolf s logit interval estimator, Gart 's logit interval estimator of adding 0.50, Cornfield 's interval estimator with the continuity correction, and Cornfield 's interval estimator without the continuity correction in a variety of situations. This paper notes that Cornfield 's interval estimator with the continuity correction is too conservative, while Cornfield 's method without the continuity correction can improve efficiency without sacrificing the accuracy of the coverage probability. This paper further notes that when the sample size is small (say, 20 or 30 per group) and the probability of exposure in the control group is small (say, 0.10) or large (say, 0.90), using Cornfield 's method without the continuity correction is likely preferable to all the other estimators considered here. When the sample size is large (say, 100 per group) or when the probability of exposure in the control group is moderate (say, 0.50), Gart 's logit interval estimator is probably the best.     

A Note on Penalized Regression Spline Estimation in the Secondary Analysis of Case-Control Data

Primary analysis of case-control studies focuses on the relationship between disease (D) and a set of covariates of interest (Y,X). A secondary application of the case-control study, often invoked in modern genetic epidemiologic association studies, is to investigate the interrelationship between the covariates themselves. The task is complicated due to the case-control sampling, and to avoid the biased sampling that arises from the design, it is typical to use the control data only. In this paper, we develop penalized regression spline methodology that uses all the data, and improves precision of estimation compared to using only the controls. A simulation study and an empirical example are used to illustrate the methodology.     

A Note on Variance Estimation of the Aalen–Johansen Estimator of the Cumulative Incidence Function in Competing Risks,with a View towards Left‐Truncated Data

The Aalen–Johansen estimator is the standard nonparametric estimator of the cumulative incidence function in competing risks. Estimating its variance in small samples has attracted some interest recently, together with a critique of the usual martingale‐based estimators. We show that the preferred estimator equals a Greenwood‐type estimator that has been derived as a recursion formula using counting processes and martingales in a more general multistate framework. We also extend previous simulation studies on estimating the variance of the Aalen–Johansen estimator in small samples to left‐truncated observation schemes, which may conveniently be handled within the counting processes framework. This investigation is motivated by a real data example on spontaneous abortion in pregnancies exposed to coumarin derivatives, where both competing risks and left‐truncation have recently been shown to be crucial methodological issues (Meister and Schaefer (2008), Reproductive Toxicology 26 , 31–35). Multistate‐type software and data are available online to perform the analyses. The Greenwood‐type estimator is recommended for use in practice.     

A Note on Estimation of Several Missing Values in Two Way Classified Data with Many Observations Per Cell

In this paper an attempt has been made to reduce the computational complexities involved in estimation of several missing values. As a result it has been shown that one can estimate m missing values by developing only k (≤m) linear equations, where m and k are respectively the number of missing values and missing cells. The procedure is also illustrated with the help of a numerical example.     

A novel method for estimating the strength of positive mating preference by similarity in the wild

Mating preference can be a driver of sexual selection and assortative mating and is, therefore, a key element in evolutionary dynamics. Positive mating preference by similarity is the tendency for the choosy individual to select a mate which possesses a similar variant of a trait. Such preference can be modelled using Gaussian‐like mathematical functions that describe the strength of preference, but such functions cannot be applied to empirical data collected from the field. As a result, traditionally, mating preference is indirectly estimated by the degree of assortative mating (using Pearson's correlation coefficient, r) in wild captured mating pairs. Unfortunately, r and similar coefficients are often biased due to the fact that different variants of a given trait are nonrandomly distributed in the wild, and pooling of mating pairs from such heterogeneous samples may lead to “false–positive” results, termed “the scale‐of‐choice effect” (SCE). Here we provide two new estimators of mating preference (C_rough and C_scaled) derived from Gaussian‐like functions which can be applied to empirical data. Computer simulations demonstrated that r coefficient showed robust estimations properties of mating preference but it was severely affected by SCE, C_rough showed reasonable estimation properties and it was little affected by SCE, while C_scaled showed the best properties at infinite sample sizes and it was not affected by SCE but failed at biological sample sizes. We recommend using C_rough combined with the r coefficient to infer mating preference in future empirical studies.