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.     

A Note on Interval Estimation of the Simple Difference in Data with Correlated Matched Pairs

When we employ cluster sampling to collect data with matched pairs, the assumption of independence between all matched pairs is not likely true. This paper notes that applying interval estimators, that do not account for the intraclass correlation between matched pairs, to estimate the simple difference between two proportions of response can be quite misleading, especially when both the number of matched pairs per cluster and the intraclass correlation between matched pairs within clusters are large. This paper develops two asymptotic interval estimators of the simple difference, that accommodate the data of cluster sampling with correlated matched pairs. This paper further applies Monte Carlo simulation to compare the finite sample performance of these estimators and demonstrates that the interval estimator, derived from a quadratic equation proposed here, can actually perform quite well in a variety of situations.     

Sample Size Determination for Grouped Exponential Observations: A Cost Function Approach

Calculating the required sample size for a desired power at a given type I error level, we often assume that we know the exact time of all subject responses whenever they occur during our study period. It is very common, however, in practice that we only monitor subjects periodically and, therefore, we know only whether responses occur or not during an interval. This paper includes a quantitative discussion of the effect resulting from data grouping or interval censoring on the required sample size when we have two treatment groups. Furthermore, with the goal of exploring the optimum in the number of subjects, the number of examinations per subject for test responses, and the total length of a study time period, this paper also provides a general guideline about how to determine these to minimize the total cost of a study for a desired power at a given α-level. A specified linear cost function that incorporates the costs of obtaining subjects, periodic examinations for test responses of subjects, and the total length of a study period, is assumed, primarily for illustrative purpose.     

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.     

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)     

Robust best linear estimator for Cox regression with instrumental variables in whole cohort and surrogates with additive measurement error in calibration sample

Biomedical researchers are often interested in estimating the effect of an environmental exposure in relation to a chronic disease endpoint. However, the exposure variable of interest may be measured with errors. In a subset of the whole cohort, a surrogate variable is available for the true unobserved exposure variable. The surrogate variable satisfies an additive measurement error model, but it may not have repeated measurements. The subset in which the surrogate variables are available is called a calibration sample. In addition to the surrogate variables that are available among the subjects in the calibration sample, we consider the situation when there is an instrumental variable available for all study subjects. An instrumental variable is correlated with the unobserved true exposure variable, and hence can be useful in the estimation of the regression coefficients. In this paper, we propose a nonparametric method for Cox regression using the observed data from the whole cohort. The nonparametric estimator is the best linear combination of a nonparametric correction estimator from the calibration sample and the difference of the naive estimators from the calibration sample and the whole cohort. The asymptotic distribution is derived, and the finite sample performance of the proposed estimator is examined via intensive simulation studies. The methods are applied to the Nutritional Biomarkers Study of the Women's Health Initiative.     

Comparing Accuracy in an Unpaired Post‐market Device Study with Incomplete Disease Assessment

The sensitivity and specificity of a new medical device are often compared relative to that of an existing device by calculating ratios of sensitivities and specificities. Although it would be ideal for all study subjects to receive the gold standard so true disease status was known for all subjects, it is often not feasible or ethical to obtain disease status for everyone. This paper proposes two unpaired designs where each subject is only administered one of the devices and device results dictate which subjects are to receive disease verification. Estimators of the ratio of accuracy and corresponding confidence intervals are proposed for these designs as well as sample size formulae. Simulation studies are performed to investigate the small sample bias of the estimators and the performance of the variance estimators and sample size formulae. The sample size formulae are applied to the design of a cervical cancer study to compare the accuracy of a new device with the conventional Pap smear.     

Use of baseline data for estimation of treatment effects in the presence of regression to the mean.

We consider a regression to the mean problem with a very large sample for the first measurement and relatively small subsample for the second measurement, selected on the basis of the initial measurement. This is a situation that often occurs in screening trials. We propose to estimate the unselected population mean and variance from the first measurement in the larger sample. Using these estimates, the correlation between the two measurements, as well as an effect of treatment, can be estimated in simple and explicit form. Under the condition that the size of the subsample is of a smaller order, the new estimators for all the four parameters are as asymptotically efficient as the usual maximum likelihood estimators. Tests based on this new approach are also discussed. An illustration from a cholesterol screening study is included.     

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.     

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.     

Confidence Limits on the Ratio of the Mean Failure Times Based on Paired Exponential Observations

This paper discusses interval estimation for the ratio of the mean failure times on the basis of paired exponential observations. This paper considers five interval estimators: the confidence interval using an idea similar to Fieller's theorem (CIFT), the confidence interval using an exact parametric test (CIEP), the confidence interval using the marginal likelihood ratio test (CILR), the confidence interval assuming no matching effect (CINM), and the confidence interval using a locally most powerful test (CIMP). To evaluate and compare the performance of these five interval estimators, this paper applies Monte Carlo simulation. This paper notes that with respect to the coverage probability, use of the CIFT, CILR, or CIMP, although which are all derived based on large sample theory, can perform well even when the number of pairs n is as small as 10. As compared with use of the CILR, this paper finds that use of the CIEP with equal tail probabilities is likely to lose efficiency. However, this loss can be reduced by using the optimal tail probabilities to minimize the average length when n is small (<20). This paper further notes that use of the CIMP is preferable to the CIEP in a variety of situations considered here. In fact, the average length of the CIMP with use of the optimal tail probabilities can even be shorter than that of the CILR. When the intraclass correlation between failure times within pairs is 0 (i.e., the failure times within the same pair are independent), the CINM, which is derived for two independent samples, is certainly the best one among the five interval estimators considered here. When there is an intraclass correlation but which is small (<0.10), the CIFT is recommended for obtaining a relatively short interval estimate without sacrificing the loss of the coverage probability. When the intraclass correlation is moderate or large, either the CILR or the CIMP with the optimal tail probabilities is preferable to the others. This paper also notes that if the intraclass correlation between failure times within pairs is large, use of the CINM can be misleading, especially when the number of pairs is large.     

Curve fitting for repeated measurements made at irregular time-points

In longitudinal studies of human populations, it is often not feasible to measure all subjects at the same time-points. This precludes the use of classical methods of curve fitting for repeated measurements. When the total interval of follow-up is short for all subjects in the study, an intraclass correlation matrix is assumed for the measurements on each subject. An estimation procedure based on iteratively reweighted least squares is described. The model is then generalized to incorporate covariables, with little modification in the estimation procedure. The proposed method is applied to data from a longitudinal study of bone mass in postmenopausal women.     

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.     

On Method of Moments Estimation in Linear Mixed Effects Models with Measurement Error on Covariates and Response with Application to a Longitudinal Study of Gene-Environment Interaction

We study a linear mixed effects model for longitudinal data, where the response variable and covariates with fixed effects are subject to measurement error. We propose a method of moment estimation that does not require any assumption on the functional forms of the distributions of random effects and other random errors in the model. For a classical measurement error model we apply the instrumental variable approach to ensure identifiability of the parameters. Our methodology, without instrumental variables, can be applied to Berkson measurement errors. Using simulation studies, we investigate the finite sample performances of the estimators and show the impact of measurement error on the covariates and the response on the estimation procedure. The results show that our method performs quite satisfactory, especially for the fixed effects with measurement error (even under misspecification of measurement error model). This method is applied to a real data example of a large birth and child cohort study.     

Computing for incomplete repeated measures

Repeated-measures experiments involve two or more intended measurements per subject. If the within-subjects design is the same for each subject and no data are missing, then the analysis is relatively simple and there are readily available programs that do the analysis automatically. However, if the data are incomplete, and do not have the same arrangement for each subject, then the analysis becomes much more difficult. Beginning with procedures that are not optimal but are comparatively simple, we discuss unbalanced linear model analysis and then normal maximum likelihood (ML) procedures. Included are ML and REML (restricted maximum likelihood) estimators for the mixed model and also estimators for a model that allows arbitrary within-subject covariance matrices. The objective is to give procedures that can be implemented with available software.     

Estimation of Box's ε for low‐ and high‐dimensional repeated measures designs with unequal covariance matrices

We present new inference methods for the analysis of low‐ and high‐dimensional repeated measures data from two‐sample designs that may be unbalanced, the number of repeated measures per subject may be larger than the number of subjects, covariance matrices are not assumed to be spherical, and they can differ between the two samples. In comparison, we demonstrate how crucial it is for the popular Huynh‐Feldt (HF) method to make the restrictive and often unrealistic or unjustifiable assumption of equal covariance matrices. The new method is shown to maintain desired α‐levels better than the well‐known HF correction, as demonstrated in several simulation studies. The proposed test gains power when the number of repeated measures is increased in a manner that is consistent with the alternative. Thus, even increasing the number of measurements on the same subject may lead to an increase in power. Application of the new method is illustrated in detail, using two different real data sets. In one of them, the number of repeated measures per subject is smaller than the sample size, while in the other one, it is larger.     

Estimating mutation rate and generation time from longitudinal samples of DNA sequences

We present in this paper a simple method for estimating the mutation rate per site per year which also yields an estimate of the length of a generation when mutation rate per site per generation is known. The estimator, which takes advantage of DNA polymorphisms in longitudinal samples, is unbiased under a number of population models, including population structure and variable population size over time. We apply the new method to a longitudinal sample of DNA sequences of the env gene of human immunodeficiency virus type 1 (HIV-1) from a single patient and obtain 1.62 x 10(-2) as the mutation rate per site per year for HIV-1. Using an independent data set to estimate the mutation rate per generation, we obtain 1.8 days as the length of a generation of HIV-1, which agrees well with recent estimates based on viral load data. Our estimate of generation time differs considerably from a recent estimate by Rodrigo et al. when the same mutation rate per site per generation is used. Some factors that may contribute to the difference among different estimators are discussed.     

Cost-efficient study designs for binary response data with Gaussian covariate measurement error.

When mismeasurement of the exposure variable is anticipated, epidemiologic cohort studies may be augmented to include a validation study, where a small sample of data relating the imperfect exposure measurement method to the better method is collected. Optimal study designs (i.e., least expensive subject to specified power constraints) are developed that give the overall sample size and proportion of the overall sample size allocated to the validation study. If better exposure measurements can be collected on a sample of subjects, an optimal design can be suggested that conforms to realistic budgetary constraints. The properties of three designs--those that include an internal validation study, those where the validated subsample is derived from subjects external to the primary investigation, and those that use the better method of exposure assessment on all subjects--are compared. The proportion of overall study resources allocated to the validation substudy increases with increasing sample disease frequency, decreasing unit cost of the superior exposure measurement relative to the imperfect one, increasing unit cost of outcome ascertainment, increasing distance between two alternative values of the relative risk between which the study is designed to discriminate, and increasing magnitude of hypothesized values. This proportion also depends in a nonlinear fashion on the severity of measurement error, and when the validation study is internal, measurement error reaches a point after which the optimal design is the smaller, fully validated one.     

Efficiency considerations in the analysis of inter-observer agreement

The reliability of binary assessments is often measured by the proportion of agreement above chance, as estimated by the kappa statistic. In this paper, we develop a model to estimate inter-rater and intra-rater reliability when each of the two observers has the opportunity to obtain a pair of replicate measurements on each subject. The model is analogous to the nested beta-binomial model proposed by Rosner (1989, 1992). We show that the gain in precision obtained from increasing the number of measurements per rater from one to two may allow fewer subjects to be included in the study with no net loss in efficiency for estimating the inter-rater reliability.     

Effect of radiation dose on the height of atomic bomb survivors: a longitudinal study

We conducted a longitudinal analysis of height after age 20 for atomic bomb survivors in the Adult Health Study (AHS) cohort. The measurements we used were made from July 1958 to June 1998 (AHS examination cycles 1-20). We analyzed only the subjects with known atomic bomb radiation doses, excluding those who were not in the city at the time of bombing (ATB) and those exposed in utero. We also excluded from the analysis measurements made after the occurrence of vertebral fracture. The total number of subjects was 11,862, and the total number of measurements was 109,770; the mean number of measurements per subject was 9.25. Assuming that stature after age 20 is approximately constant, a simple mixed-effects model was fitted to stature after age 20, and linear dose effects for young ATB subjects were modeled for both sexes. The estimated mean heights for subjects born in 1945 in Hiroshima were 166.0 cm for men and 155.4 cm for women. The sex difference in height was 10.6 cm, with men significantly taller than women (P < 0.001). The difference between the cities was not significant (P = 0.162). The birth cohort effects per decade were -1.7 cm for men (P < 0.001) and -2.1 cm for women (P < 0.001). A reduction of stature due to radiation exposure was observed for individuals of both sexes who were below 19 years of age ATB (95% confidence interval, 17-21 years), and the dose effect was larger for women than for men (P = 0.028). The estimated effects per gray for those who were age 0 ATB were -1.2 cm for men and -2.0 cm for women and for those who were age 10 ATB were-0.57 cm for men and -0.96 cm for women.