A simple hybrid variance estimator for the Kaplan-Meier survival function

In this paper, we propose a hybrid variance estimator for the Kaplan-Meier survival function. This new estimator approximates the true variance by a Binomial variance formula, where the proportion parameter is a piecewise non-increasing function of the Kaplan-Meier survival function and its upper bound, as described below. Also, the effective sample size equals the number of subjects not censored prior to that time. In addition, we consider an adjusted hybrid variance estimator that modifies the regular estimator for small sample sizes. We present a simulation study to compare the performance of the regular and adjusted hybrid variance estimators to the Greenwood and Peto variance estimators for small sample sizes. We show that on average these hybrid variance estimators give closer variance estimates to the true values than the traditional variance estimators, and hence confidence intervals constructed with these hybrid variance estimators have more nominal coverage rates. Indeed, the Greenwood and Peto variance estimators can substantially underestimate the true variance in the left and right tails of the survival distribution, even with moderately censored data. Finally, we illustrate the use of these hybrid and traditional variance estimators on a data set from a leukaemia clinical trial.     

Finite sample pointwise confidence intervals for a survival distribution with right‐censored data

We review and develop pointwise confidence intervals for a survival distribution with right‐censored data for small samples, assuming only independence of censoring and survival. When there is no censoring, at each fixed time point, the problem reduces to making inferences about a binomial parameter. In this case, the recently developed beta product confidence procedure (BPCP) gives the standard exact central binomial confidence intervals of Clopper and Pearson. Additionally, the BPCP has been shown to be exact (gives guaranteed coverage at the nominal level) for progressive type II censoring and has been shown by simulation to be exact for general independent right censoring. In this paper, we modify the BPCP to create a ‘mid‐p’ version, which reduces to the mid‐p confidence interval for a binomial parameter when there is no censoring. We perform extensive simulations on both the standard and mid‐p BPCP using a method of moments implementation that enforces monotonicity over time. All simulated scenarios suggest that the standard BPCP is exact. The mid‐p BPCP, like other mid‐p confidence intervals, has simulated coverage closer to the nominal level but may not be exact for all survival times, especially in very low censoring scenarios. In contrast, the two asymptotically‐based approximations have lower than nominal coverage in many scenarios. This poor coverage is due to the extreme inflation of the lower error rates, although the upper limits are very conservative. Both the standard and the mid‐p BPCP methods are available in our bpcp R package. Published 2016. This article is US Government work and is in the public domain in the USA.     

Variance estimation of a survival function for interval-censored survival data

Interval-censored survival data often occur in medical studies, especially in clinical trials. In this case, many authors have considered estimation of a survival function. There is, however, relatively little discussion on estimating the variance of estimated survival functions. For right-censored data, a special case of interval-censored data, the most commonly used method for variance estimation is to use the Greenwood formula. In this paper we propose a generalization of the Greenwood formula for variance estimation of a survival function based on interval-censored data. Also a simple bootstrap approach is presented. The two methods are evaluated and compared using simulation studies and a real data set. The simulation results suggest that the methods work well.     

Variance estimation and confidence intervals for the standardized mortality ratio with application to the assessment of a cancer screening program

The effect of a cancer screening program can be measured through the standardized mortality ratio (SMR) statistic. The numerator of the SMR is the observed number of deaths from the screened disease among participants in the screening program, whereas the denominator of the SMR is an estimate of the expected number of deaths in these participants under the assumption that the screening program has no effect. In this article, we propose a variance estimator for the denominator of the SMR when this expected number of deaths is estimated with Sasieni's method. We give both a general formula for this variance as well as formulas for specific disease incidence and survival estimators. We show how this new variance estimator can be used to build confidence intervals for the SMR. We investigate the coverage properties of various types of confidence intervals by simulation and find that intervals that make use of the proposed variance estimator perform well. We illustrate the method by applying it to the Québec Breast Cancer Screening program.     

AN algorithm for robust non-linear analysis of radioimmunoassays and other bioassays

The four-parameter logistic function is an appropriate model for many types of bioassays that have continuous response variables, such as radioimmunoassays. By modelling the variance of replicates in an assay, one can modify the usual parameter estimation techniques )for example, Gauss–Newton or Marquardt–Levenberg( to produce parameter estimates for the standard curve that are robust against outlying observations. This article describes the computation of robust )M-( estimates for the parameters of the four-parameter logistic function. It describes techniques for modelling the variance structure of the replicates, modifications to the usual iterative algorithms for parameter estimation in non-linear models, and a formula for inverse confidence intervals. To demonstrate the algorithm, the article presents examples where the robustly estimated four-parameter logistic model is compared with the logit-log and four-parameter logistic models with least-squares estimates.     

Exposure variability in the workplace: its implications for the assessment of compliance

Day-to-day variations of occupational exposures have important implications for the industrial hygienist trying to assess compliance with an occupational exposure limit. As only a limited number of samples are taken during an observation period, extrapolations are required to estimate exposures over the unsampled period. Compliance may be evaluated using estimates of the geometric mean (GM) and the geometric standard deviation (GSD) to calculate a confidence interval around the mean exposure and compare this interval to a limit value, assuming a lognormal distribution of exposures over time. These confidence intervals are very sensitive to the estimate of GSD. Hence, the questions of when to sample and how many samples to take for a reliable assessment of exposure variability (GSD) are the focus of this paper. Analyses of simulated exposure-time series and 420 data sets of personal exposures with three or more measurements obtained from actual workplaces demonstrate that the small number of samples usually collected during surveys leads to biased estimates of the variance of the exposure distribution. There is a high likelihood of an underestimate of variance, which rapidly increases if 8-hr time-weighted average samples are collected on consecutive days or within a week. The results indicate that in 80% of the within-week exposure-time series, the estimated GSD may be too low, even up to a factor of 2. Evidence is presented that autocorrelation is a likely explanation for the bias observed.(ABSTRACT TRUNCATED AT 250 WORDS)     

Bayesian latent variable modelling in studies of air pollution and health

This paper describes the use of Bayesian latent variable models in the context of studies investigating the short‐term effects of air pollution on health. Traditional Poisson or quasi‐likelihood regression models used in this area assume that consecutive outcomes are independent (although the latter allows for overdispersion), which in many studies may be an untenable assumption as temporal correlation is to be expected. We compare this traditional approach with two Bayesian latent process models, which acknowledge the possibility of short‐term autocorrelation. These include an autoregressive model that has previously been used in air pollution studies and an alternative based on a moving average structure that we describe here. A simulation study assesses the performance of these models when there are different forms of autocorrelation in the data. Although estimated risks are largely unbiased, the results show that assuming independence can produce confidence intervals that are too narrow. Failing to account for the additional uncertainty which may be associated with (positive) correlation can result in confidence/credible intervals being too narrow and thus lead to incorrect conclusions being made about the significance of estimated risks. The methods are illustrated within a case study of the effects of short‐term exposure to air pollution on respiratory mortality in the elderly in London, between 1997 and 2003. Copyright © 2010 John Wiley & Sons, Ltd.     

An application of the LFP survival model to smoking cessation data

We use a limited failure population (LFP) model based on the Weibull distribution to model the times from initial abstinence to return to smoking for subjects enrolled in programmes to help them stop smoking. The model contains a third parameter that corresponds to the proportion of subjects who permanently abstain from smoking. The data are subject to both right and interval censoring. Furthermore, subjects receive treatment in groups, and individuals in the same group may provide correlated outcomes. Use of a maximum likelihood estimation procedure which assumes independent outcomes provides reasonable parameter estimates, but the corresponding standard errors tend to be too small, which results in tests with inflated type I error levels and confidence intervals that tend to be too narrow. We use a bootstrap procedure to obtain more reasonable values for the standard errors and to construct confidence intervals that more nearly achieve the stated coverage probabilities.     

Variance estimation for stratified propensity score estimators

Propensity score methods are increasingly used to estimate the effect of a treatment or exposure on an outcome in non-randomised studies. We focus on one such method, stratification on the propensity score, comparing it with the method of inverse-probability weighting by the propensity score. The propensity score--the conditional probability of receiving the treatment given observed covariates--is usually an unknown probability estimated from the data. Estimators for the variance of treatment effect estimates typically used in practice, however, do not take into account that the propensity score itself has been estimated from the data. By deriving the asymptotic marginal variance of the stratified estimate of treatment effect, correctly taking into account the estimation of the propensity score, we show that routinely used variance estimators are likely to produce confidence intervals that are too conservative when the propensity score model includes variables that predict (cause) the outcome, but only weakly predict the treatment. In contrast, a comparison with the analogous marginal variance for the inverse probability weighted (IPW) estimator shows that routinely used variance estimators for the IPW estimator are likely to produce confidence intervals that are almost always too conservative. Because exact calculation of the asymptotic marginal variance is likely to be complex, particularly for the stratified estimator, we suggest that bootstrap estimates of variance should be used in practice.     

ESTIMATING CONFIDENCE INTERVALS FOR COST-EFFECTIVENESS RATIOS: AN EXAMPLE FROM A RANDOMIZED TRIAL

Cost-effectiveness ratios usually appear as point estimates without confidence intervals, since the numerator and denominator are both stochastic and one cannot estimate the variance of the estimator exactly. The recent literature, however, stresses the importance of presenting confidence intervals for cost-effectiveness ratios in the analysis of health care programmes. This paper compares the use of several methods to obtain confidence intervals for the cost-effectiveness of a randomized intervention to increase the use of Medicaid's Early and Periodic Screening, Diagnosis and Treatment (EPSDT) programme. Comparisons of the intervals show that methods that account for skewness in the distribution of the ratio estimator may be substantially preferable in practice to methods that assume the cost-effectiveness ratio estimator is normally distributed. We show that non-parametric bootstrap methods that are mathematically less complex but computationally more rigorous result in confidence intervals that are similar to the intervals from a parametric method that adjusts for skewness in the distribution of the ratio. The analyses also show that the modest sample sizes needed to detect statistically significant effects in a randomized trial may result in confidence intervals for estimates of cost-effectiveness that are much wider than the boundaries obtained from deterministic sensitivity analyses.     

Recommendations for the use of Taylor series confidence intervals for estimates of vaccine efficacy

A simple formula for calculating confidence intervals by means of a Taylor series variance approximation has been recommended for gauging the precision of estimates of vaccine efficacy. To evaluate the performance of Taylor series 95% confidence intervals for vaccine efficacy, we conducted a simulation study for commonly expected values of vaccine efficacy, risk of disease in the unvaccinated population, and sample sizes of the vaccinated and unvaccinated groups. In the first simulation, the sample size in the vaccinated group was 500 or 1000, whereas that in the unvaccinated group ranged from 10 to 1000. The confidence intervals were accurate when the sample size in the unvaccinated group was ≥50 and the risk of disease was 0.3-0.9. In contrast, the intervals were too narrow when all three of the following situations occurred: the number of unvaccinated was small (10 or 20), the true vaccine efficacy was relatively low (60% or 80%), and the risk of disease was 0.5-0.9. Furthermore, when the true vaccine efficacy was high (90% or 95%) and the disease risk in the unvaccinated was low (0.1 and 0.2), the confidence intervals were too broad, especially when the unvaccinated sample size was <50. Additional simulations with a sample size in the vaccinated group of 200 gave broad intervals for 95% vaccine efficacy (for all values of disease risk) and for 90% vaccine efficacy when the disease risk was ≤0.3.     

The performance of the two-stage analysis of two-treatment, two-period crossover trials

In the two-treatment, two-period crossover trial, patients are randomly allocated either to one group that receives treatment A followed by treatment B, or to another group that receives the treatments in the reverse order. Grizzle first proposed a two-stage procedure for analysing the data from such a trial. This paper examines the long-run sampling properties of this procedure, in terms of mean square error of point estimates, coverage probability of confidence intervals and actual significance level of hypothesis tests for the differences between the effects of the two treatments. The advantages of incorporating baseline observations into the analysis are also explored. Because the preliminary test for carryover is highly correlated with the analysis of data from the first period only, actual significance levels are higher than nominal levels even when there is no differential carryover. When carryover is present, the nominal level very seriously understates the actual level, and this becomes even worse when baseline observations are ignored. Increasing sample size only exacerbates the problem since this adverse behaviour then occurs at smaller values of the carryover effect. It is concluded that the two-stage analysis is too potentially misleading to be of practical use.     

Implementation and applications of bootstrap methods for the National Immunization Survey

In complex probability sample surveys, numerous adjustments are customarily made to the survey weights to reduce potential bias in survey estimates. These adjustments include sampling design (SD) weight adjustments, which account for features of the sampling plan, and non-sampling design (NSD) weight adjustments, which account for non-sampling errors and other effects. Variance estimates prepared from complex survey data customarily account for SD weight adjustments, but rarely account for all NSD weight adjustments. As a result, variance estimates may be biased and standard confidence intervals may not achieve their nominal coverage levels. We describe the implementation of the bootstrap method to account for the SD and NSD weight adjustments for complex survey data. Using data from the National Immunization Survey (NIS), we illustrate the use of the bootstrap (i). for evaluating the use of standard confidence intervals that use Taylor series approximations to variance estimators that do not account for NSD weight adjustments, (ii). for obtaining confidence intervals for ranks estimated from weighted survey data, and (iii). for evaluating the predictive power of logistic regressions using receiver operating characteristic curve analyses that account for the SD and NSD adjustments made to the survey weights.     

Marginal modeling of nonnested multilevel data using standard software

Epidemiologic data are often clustered within multiple levels that may not be nested within each other. Generalized estimating equations are commonly used to adjust for correlation among observations within clusters when fitting regression models; however, standard software does not currently accommodate nonnested clusters. This paper introduces a simple generalized estimating equation strategy that uses available commercial or public software for the regression analysis of nonnested multilevel data. The authors describe how to obtain empirical standard error estimates for constructing valid confidence intervals and conducting statistical hypothesis tests. The method is evaluated using simulations and illustrated with an analysis of data from the Breast Cancer Surveillance Consortium that estimates the influence of woman, radiologist, and facility characteristics on the positive predictive value of screening mammography. Performance with a small number of clusters is discussed. Both the simulations and the example demonstrate the importance of accounting for the correlation within all levels of clustering for proper inference.     

Accounting for error due to misclassification of exposures in case-control studies of gene-environment interaction

We consider analysis of data from an unmatched case-control study design with a binary genetic factor and a binary environmental exposure when both genetic and environmental exposures could be potentially misclassified. We devise an estimation strategy that corrects for misclassification errors and also exploits the gene-environment independence assumption. The proposed corrected point estimates and confidence intervals for misclassified data reduce back to standard analytical forms as the misclassification error rates go to zero. We illustrate the methods by simulating unmatched case-control data sets under varying levels of disease-exposure association and with different degrees of misclassification. A real data set on a case-control study of colorectal cancer where a validation subsample is available for assessing genotyping error is used to illustrate our methods.     

Helping doctors to draw appropriate inferences from the analysis of medical studies

Most clinicians and many medical statisticians interpret standard frequentist confidence intervals by invoking the Bayesian concept of subjective probability. Fortunately, the assumptions that render this interpretation acceptable are often quite reasonable in the setting of the practical day-to-day analysis of medical data. This article takes the subjective interpretation of confidence intervals to its logical conclusion and argues that the inferential understanding of clinicians and public health physicians could potentially be improved if, where it was appropriate, standard inferential statements - point estimates, 95 per cent confidence intervals and P-values - were supplemented by estimates of the subjective posterior probability, assuming a uniform prior density, that the true value of a parameter to be estimated exceeds one or a series of thresholds that are clinically critical or easily interpretable. Many decision makers in the health care arena draw totally inappropriate inferences from analyses where the point estimate indicates a clinically valuable effect but the null hypothesis cannot formally be rejected, and, although the proposed approach could be of potential value in a range of settings, it is argued that it could be of particular use in the rational interpretation of underpowered studies that must inform critical clinical or public health decisions.     

生存率置信区间的五种估计方法

生存率是医学随访研究资料分析中常用的指标，例如适用于小样本资料的Ｋａｐｌａｎ－Ｍｅｉｅｒ乘积限估计和大样本资料的寿命表法生存率估计。本文对生存率置信区间的估计方法进行了讨论。主要介绍了五种置信区间的估计方法：经典法（基于Ｇｒｅｅｎｗｏｏｄ方差公式）、校正法、反正旋转换法、ｌｏｇ（－ｌｏｇ）转换法及ｌｏｇｉｔ转换法。文中给出了两个实例，并就生存率９５％置信区间的估计做了详细介绍，还进一步讨论了它们在     

Variance estimation when using inverse probability of treatment weighting (IPTW) with survival analysis

Propensity score methods are used to reduce the effects of observed confounding when using observational data to estimate the effects of treatments or exposures. A popular method of using the propensity score is inverse probability of treatment weighting (IPTW). When using this method, a weight is calculated for each subject that is equal to the inverse of the probability of receiving the treatment that was actually received. These weights are then incorporated into the analyses to minimize the effects of observed confounding. Previous research has found that these methods result in unbiased estimation when estimating the effect of treatment on survival outcomes. However, conventional methods of variance estimation were shown to result in biased estimates of standard error. In this study, we conducted an extensive set of Monte Carlo simulations to examine different methods of variance estimation when using a weighted Cox proportional hazards model to estimate the effect of treatment. We considered three variance estimation methods: (i) a naïve model‐based variance estimator; (ii) a robust sandwich‐type variance estimator; and (iii) a bootstrap variance estimator. We considered estimation of both the average treatment effect and the average treatment effect in the treated. We found that the use of a bootstrap estimator resulted in approximately correct estimates of standard errors and confidence intervals with the correct coverage rates. The other estimators resulted in biased estimates of standard errors and confidence intervals with incorrect coverage rates. Our simulations were informed by a case study examining the effect of statin prescribing on mortality. © 2016 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.     

A comparison of confidence interval methods for the intraclass correlation coefficient in cluster randomized trials

This study compared different methods for assigning confidence intervals to the analysis of variance estimator of the intraclass correlation coefficient (rho). The context of the comparison was the use of rho to estimate the variance inflation factor when planning cluster randomized trials. The methods were compared using Monte Carlo simulations of unbalanced clustered data and data from a cluster randomized trial of an intervention to improve the management of asthma in a general practice setting. The coverage and precision of the intervals were compared for data with different numbers of clusters, mean numbers of subjects per cluster and underlying values of rho. The performance of the methods was also compared for data with Normal and non-Normally distributed cluster specific effects. Results of the simulations showed that methods based upon the variance ratio statistic provided greater coverage levels than those based upon large sample approximations to the standard error of rho. Searle's method provided close to nominal coverage for data with Normally distributed random effects. Adjusted versions of Searle's method to allow for lack of balance in the data generally did not improve upon it either in terms of coverage or precision. Analyses of the trial data, however, showed that limits provided by Thomas and Hultquist's method may differ from those of the other variance ratio statistic methods when the arithmetic mean differs markedly from the harmonic mean cluster size. The simulation results demonstrated that marked non-Normality in the cluster level random effects compromised the performance of all methods. Confidence intervals for the methods were generally wide relative to the underlying size of rho suggesting that there may be great uncertainty associated with sample size calculations for cluster trials where large clusters are randomized. Data from cluster based studies with sample sizes much larger than those typical of cluster randomized trials are required to estimate rho with a reasonable degree of precision.     

Bayesian methods of confidence interval construction for the population attributable risk from cross‐sectional studies

Population attributable risk measures the public health impact of the removal of a risk factor. To apply this concept to epidemiological data, the calculation of a confidence interval to quantify the uncertainty in the estimate is desirable. However, because perhaps of the confusion surrounding the attributable risk measures, there is no standard confidence interval or variance formula given in the literature. In this paper, we implement a fully Bayesian approach to confidence interval construction of the population attributable risk for cross‐sectional studies. We show that, in comparison with a number of standard Frequentist methods for constructing confidence intervals (i.e. delta, jackknife and bootstrap methods), the Bayesian approach is superior in terms of percent coverage in all except a few cases. This paper also explores the effect of the chosen prior on the coverage and provides alternatives for particular situations. Copyright © 2016 John Wiley & Sons, Ltd.