Families of smooth confidence bands for the survival function under the general random censorship model

Randomly right censored data often arise in industrial life testing and clinical trials. Several authors have proposed asymptotic confidence bands for the survival function when data are randomly censored on the right. All of these bands are based on the empirical estimator of the survival function. In this paper, families of asymptotic (1-alpha) 100% level confidence bands are developed from the smoothed estimate of the survival function under the general random censorship model. The new bands are compared to empirical bands, and it is shown that for small sample sizes, the smooth bands have a higher coverage probability than the empirical counterparts.     

Increasing efficiency from censored survival data by using random effects to model longitudinal covariates

When estimating a survival time distribution, the loss of information due to right censoring results in a loss of efficiency in the estimator. In many circumstances, however, repeated measurements on a longitudinal process which is associated with survival time are made throughout the observation time, and these measurements may be used to recover information lost to censoring. For example, patients in an AIDS clinical trial may be measured at regular intervals on CD4 count and viral load. We describe a model for the joint distribution of a survival time and a repeated measures process. The joint distribution is specified by linking the survival time to subject-specific random effects characterizing the repeated measures, and is similar in form to the pattern mixture model for multivariate data with nonignorable nonresponse. We also describe an estimator of survival derived from this model. We apply the methods to a long-term AIDS clinical trial, and study properties of the survival estimator. Monte Carlo simulation is used to estimate gains in efficiency when the survival time is related to the location and scale of the random effects distribution. Under relatively light censoring (20%), the methods yield a modest gain in efficiency for estimating three-year survival in the AIDS clinical trial. Our simulation study, which mimics characteristics of the clinical trial, indicates that much larger gains in efficiency can be realized under heavier censoring or with studies designed for long term follow up on survival.     

Product-limit survival functions with correlated survival times

A simple variance estimator for product-limit survival functions is demonstrated for survival times with nested errors. Such data arise whenever survival times are observed within clusters of related observations. Greenwood's formula, which assumes independent observations, is not appropriate in this situation. A robust variance estimator is developed using Taylor series linearized values and the between-cluster variance estimator commonly used in multi-stage sample surveys. A simulation study shows that the between-cluster variance estimator is approximately unbiased and yields confidence intervals that maintain the nominal level for several patterns of correlated survival times. The simulation study also shows that Greenwood's formula underestimates the variance when the survival times are positively correlated within a cluster and yields confidence intervals that are too narrow. Extension to life table methods is also discussed.     

Weighing the evidence for hypotheses with small samples of right-censored exponential data

The p-value evidence for an alternative to a null hypothesis regarding the mean lifetime can be unreliable if based on asymptotic approximations when there is only a small sample of right-censored exponential data. However, a guarded weight of evidence for the alternative can always be obtained without approximation, no matter how small the sample, and has some other advantages over p-values. Weights of evidence are defined as estimators of 0 when the null hypothesis is true and 1 when the alternative is true, and they are judged on the basis of the ensuing risks, where risk is mean squared error of estimation. The evidence is guarded in that a pre-assigned bound is placed on the risk under the hypothesis. Practical suggestions are given for choosing the bound and for interpreting the magnitude of the weight of evidence. Acceptability profiles are obtained by inversion of a family of guarded weights of evidence for two-sided alternatives to point hypotheses, just as confidence intervals are obtained from tests; these profiles are arguably more informative than confidence intervals, and are easily determined for any level and any sample size, however small. They can help understand the effects of different amounts of censoring. They are found for several small size data sets, including a sample of size 12 for post-operative cancer patients. Both singly Type I and Type II censored examples are included. An examination of the risk functions of these guarded weights of evidence suggests that if the censoring time is of the same magnitude as the mean lifetime, or larger, then the risks in using a guarded weight of evidence based on a likelihood ratio are not much larger than they would be if the parameter were known.     

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.     

Approximate standard errors and confidence intervals for indices of positive and negative agreement

Indices of positive and negative agreement for observer reliability studies, in which neither observer can be regarded as the standard, have been proposed. In this article, it is demonstrated by means of an example and a small simulation study that a recently published method for constructing confidence intervals for these indices leads to intervals that are too wide. Appropriate asymptotic (i.e., large sample) variance estimates and confidence intervals for the positive and negative agreement indices are presented and compared with bootstrap confidence intervals. We also discuss an alternative method of interval estimation motivated from a Bayesian viewpoint. The asymptotic intervals performed adequately for sample sizes of 200 or more. For smaller samples, alternative confidence intervals such as bootstrap intervals or Bayesian intervals should be considered.     

Asymptotically distribution-free (ADF) interval estimation of coefficient alpha.

[Correction Notice: An erratum for this article was reported in Vol 12(4) of Psychological Methods (see record 2007-18729-004). The sentence describing Equation 1 is incorrect. The corrected sentence is presented in the erratum.] The point estimate of sample coefficient alpha may provide a misleading impression of the reliability of the test score. Because sample coefficient alpha is consistently biased downward, it is more likely to yield a misleading impression of poor reliability. The magnitude of the bias is greatest precisely when the variability of sample alpha is greatest (small population reliability and small sample size). Taking into account the variability of sample alpha with an interval estimator may lead to retaining reliable tests that would be otherwise rejected. Here, the authors performed simulation studies to investigate the behavior of asymptotically distribution-free (ADF) versus normal-theory interval estimators of coefficient alpha under varied conditions. Normal-theory intervals were found to be less accurate when item skewness >1 or excess kurtosis >1. For sample sizes over 100 observations, ADF intervals are preferable, regardless of item skewness and kurtosis. A formula for computing ADF confidence intervals for coefficient alpha for tests of any size is provided, along with its implementation as an SAS macro. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

"Asymptotically distribution-free (ADF) interval estimation of coefficient alpha": Correction.

Reports an error in "Asymptotically distribution-free (ADF) interval estimation of coefficient alpha" by Alberto Maydeu-Olivares, Donna L. Coffman and Wolfgang M. Hartmann (Psychological Methods, 2007[Jun], Vol 12[2], 157-176). The sentence describing Equation 1 is incorrect. The corrected sentence is presented in the erratum. (The following abstract of the original article appeared in record 2007-07830-003.) The point estimate of sample coefficient alpha may provide a misleading impression of the reliability of the test score. Because sample coefficient alpha is consistently biased downward, it is more likely to yield a misleading impression of poor reliability. The magnitude of the bias is greatest precisely when the variability of sample alpha is greatest (small population reliability and small sample size). Taking into account the variability of sample alpha with an interval estimator may lead to retaining reliable tests that would be otherwise rejected. Here, the authors performed simulation studies to investigate the behavior of asymptotically distribution-free (ADF) versus normal-theory interval estimators of coefficient alpha under varied conditions. Normal-theory intervals were found to be less accurate when item skewness >1 or excess kurtosis >1. For sample sizes over 100 observations, ADF intervals are preferable, regardless of item skewness and kurtosis. A formula for computing ADF confidence intervals for coefficient alpha for tests of any size is provided, along with its implementation as an SAS macro. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Sampling variance in the correlation coefficient under range restriction: A Monte Carlo study.

Validity generalization methods require accurate estimates of the sampling variance in the correlation coefficient when the range of variation in the data is restricted. This article presents the results of computer simulations examining the accuracy of the sampling variance estimator under sample range restrictions. Range restriction is assumed to occur by direct selection on the predictor. Sample sizes of 25, 60, and 100 are used, with the selection ratio ranging from .10 to 1.0 and the population correlation ranging from .10 to .90. The estimator is found to have a slight negative bias in unrestricted data. In restricted data, the bias is substantial in sample sizes of 60 or less. In all sample sizes, the negative bias increases as the selection ratio becomes smaller. Implications of the results for studies of validity generalization are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Analyzing bivariate continuous data grouped into categories defined by empirical quantiles of marginal distributions

Epidemiologists sometimes study the association between two measurements of exposure on the same subjects by grouping the original bivariate continuous data into categories that are defined by the empirical quantiles of the two marginal distributions. Although such grouped data are presented in a two-way contingency table, the cell counts in this table do not have a multinomial distribution. We describe the joint distribution of counts in such a table by the term empirical bivariate quantile-partitioned (EBQP) distribution. Blomqvist (1950, Annals of Mathematical Statistics 21, 539-600) gave an asymptotic EBQP theory for bivariate data partitioned by the sample medians. We demonstrate that his asymptotic theory is not correct, however, except in special cases. We present a general asymptotic theory for tables of arbitrary dimensions and apply this theory to construct confidence intervals for the kappa statistic. We show by simulations that the confidence interval procedures we propose have near nominal coverage for sample sizes exceeding 60 for both 2 x 2 and 3 x 3 tables. These simulations also illustrate that the asymptotic theory of Blomqvist (1950) and the methods that Fleiss, Cohen, and Everitt (1969, Psychological Bulletin 72, 323-327) give for multinomial tables can yield subnominal coverage for kappa calculated from EBQP tables, although in some cases the coverage for these procedures is near nominal levels.     

The importance of local control in the conservative treatment of breast cancer

The purpose of this study was to examine the meaning of local control, especially on survival, in breast cancer patients treated by lumpectomy with or without radiotherapy. We analyzed the survival results of four major published randomized trials that compare conservation surgery with or without radiation using three different statistical approaches: p-values, confidence intervals, and Bayesian techniques. All four trials report statistically significant increased local control and improved survival for the irradiated patients. Survival based on p-values and confidence intervals shows statistical significance for long-term follow-up of the NSABP-B06 trial, but not for the other trials, probably because of small sample sizes and short follow-up. At 10 years, the overall survival rates for the NSABP-B06 were 65% and 71% for lumpectomy alone or with radiation respectively. Interpreted in a Bayesian framework, the expected advantage in 10-year survival was 6% (the mean of NSABP-B06 10-year survival) with an 83% probability that the 10-year survival difference may lie between 2% and 10%. An 85% probability that 3% of patients will survive at 10 years because of irradiation translates into a 30% reduction in annual odds of death several years after treatment in stage I good prognosis patients and 15% in stage I poor prognosis patients. Analysis of the randomized trials comparing lumpectomy with or without radiation indicate a clear improvement in survival for the irradiated patients associated with increased local control. Combination of improved survival with the reduced psychological and economic costs associated with local recurrence argues well for the inclusion of radiation for many breast cancer patients.     

Upward bias in estimates of pacemaker reliability: effect of unreported patient mortality

OBJECTIVES: This study attempted to determine the effect of unreported patient deaths on estimates of pacemaker reliability. BACKGROUND: The reliability of pacemakers is usually reported with reference to implant registration data and returned product analysis without censoring when follow-up data are missing. METHODS: We studied 73 patients (mean [+/- SD] age 77 +/- 8 years) undergoing implantation of a ventricular-inhibited (VVI) pacemaker who were subsequently found to be at increased risk of experiencing premature pacemaker failure. Survival curves for patients and pacemakers were constructed by the Kaplan-Meier method with appropriate censoring at the time of unrelated death or elective explantation of a normal device. To examine the effect of unreported loss of follow-up data, patient mortality was then ignored, and follow-up for pacemakers without known failure was assumed to continue to the date of analysis. RESULTS: There were 13 device failures, with a median pacemaker survival time of 37 months. Twenty-three patients died, all of causes unrelated to the pacemaker system; median patient survival time was only 44 months. Ignoring this attrition inflated follow-up time from 122 to 188 patient-years and reduced the apparent pacemaker failures at 30 months by almost half, from 37% to only 20%. Modeling the process shows that when the patient mortality rate is more than half the pacemaker failure rate, ignoring censoring inflates the device survival estimate by > or = 10% from the median survival onward. CONCLUSIONS: When medical device survival curves are generated by implant registration data and returned product analysis, they should be adjusted for unreported loss of follow-up.     

Synthesis of pharmacokinetic parameters of vancomycin via bootstrap methods

OBJECTIVE: For the adjustment of individual vancomycin dosages, we estimate the important pharmacokinetic quantities half-life, clearance, and volume of distribution. MATERIAL: To obtain reliable information 293 observations from 244 patients were extracted from 23 published studies on vancomycin. Information about vancomycin's pharmacokinetics out of different sources represents an increase in sample size and, therefore, interpretive power. METHODS: Once the whole of the data had been stratified into a small number of homogeneous clusters based on cofactors, different (robust) estimators (mean, median, Winsorized, and trimmed mean) were calculated for the expected value of the pharmacokinetic parameters of vancomycin within the clusters. Measures of the statistical accuracy such as standard error, bias, mean square error, and confidence interval were estimated via bootstrap methods from large bootstrap sample sizes to compare the quality of the estimators. RESULTS: Due to the homogenization of the data all individual estimator functions yield very similar results and the empirical mean works fairly well as an estimate. The most frequently used estimator with the smallest estimated mean square error was the Winsorized mean.     

Decision support issues using a physiology based score

OBJECTIVE: As physiology based assessments of mortality risk become more accurate, their potential utility in clinical decision support and resource rationing decisions increases. Before these prediction models can be used, however, their performance must be statistically evaluated and interpreted in a clinical context. We examine the issues of confidence intervals (as estimates of survival ranges) and confidence levels (as estimates of clinical certainty) by applying Pediatric Risk of Mortality III (PRISM III) in two scenarios: (1) survival prediction for individual patients and (2) resource rationing. DESIGN: A non-concurrent cohort study. SETTING: 32 pediatric intensive care units (PICUs). PATIENTS: 10608 consecutive patients (571 deaths). INTERVENTIONS: None. MEASUREMENTS AND RESULTS: For the individual patient application, we investigated the observed survival rates for patients with low survival predictions and the confidence intervals associated with these predictions. For the resource rationing application, we investigated the maximum error rate of a policy which would limit therapy for patients with scores exceeding a very high threshold. For both applications, we also investigated how the confidence intervals change as the confidence levels change. The observed survival in the PRISM III groups >28, >35, and >42 were 6.3, 5.3, and 0%, with 95% upper confidence interval bounds of 10.5, 13.0, and 13.3%, respectively. Changing the confidence level altered the survival range by more than 300% in the highest risk group, indicating the importance of clinical certainty provisions in prognostic estimates. The maximum error rates for resource allocation decisions were low (e. g., 29 per 100000 at a 95% certainty level), equivalent to many of the risks of daily living. Changes in confidence level had relatively little effect on this result. CONCLUSIONS: Predictions for an individual patient's risk of death with a high PRISM score are statistically not precise by virtue of the small number of patients in these groups and the resulting wide confidence intervals. Clinical certainty (confidence level) issues substantially influence outcome ranges for individual patients, directly affecting the utility of scores for individual patient use. However, sample sizes are sufficient for rationing decisions for many groups with higher certainty levels. Before there can be widespread acceptance of this type of decision support, physicians and families must confront what they believe is adequate certainty.     

Probabilistic Nonroad Mobile Source Emission Factors

Quantitative methods for characterizing both variability and uncertainty are applied to case studies of nitrogen oxides (NOx) and total hydrocarbon air pollutant emission factors for construction, farm and industrial engines. A database was developed and statistical methods were used to identify statistically significant categories, including gasoline engines, two-stroke diesel engines, and four-stroke diesel engines. Emissions were not found to be statistically significantly different for older versus newer nonroad diesel engines nor when comparing naturally aspirated and turbocharged engines. Interengine variability in emissions was quantified using Weibull, gamma, and lognormal distributions. Bootstrap simulation was used to characterize confidence intervals for the mean. The 95% confidence intervals for the mean emission factors were as small as ?10 to +11% and as large as ?48 to +49%, with an average range of ?26 to +27%. Asymmetry in the confidence intervals is attributable to small sample sizes and skewness in the data. Estimates of uncertainty convey information regarding the quality of the emission factors and serve as a basis for calculation of uncertainty in emission inventories.     

Confidence intervals for differences in correlated binary proportions

An experiment to assess the efficacy of a particular treatment or process often produces dichotomous responses, either favourable or unfavourable. When we administer the treatment on two occasions to the same subjects, we often use McNemar's test to investigate the hypothesis of no difference in the proportions on the two occasions, that is, the hypothesis of marginal homogeneity. A disadvantage in using McNemar's statistic is that we estimate the variance of the sample difference under the restriction that the marginal proportions are equal. A competitor to McNemar's statistic is a Wald statistic that uses an unrestricted estimator of the variance. Because the Wald statistic tends to reject too often in small samples, we investigate an adjusted form that is useful for constructing confidence intervals. Quesenberry and Hurst and Goodman discussed methods of construction that we adapt for constructing confidence intervals for the differences in correlated proportions. We empirically compare the coverage probabilities and average interval lengths for the competing methods through simulation and give recommendations based on the simulation results.     

Using rare mutations to estimate population divergence times: a maximum likelihood approach

In this paper we propose a method to estimate by maximum likelihood the divergence time between two populations, specifically designed for the analysis of nonrecurrent rare mutations. Given the rapidly growing amount of data, rare disease mutations affecting humans seem the most suitable candidates for this method. The estimator RD, and its conditional version RDc, were derived, assuming that the population dynamics of rare alleles can be described by using a birth-death process approximation and that each mutation arose before the split of a common ancestral population into the two diverging populations. The RD estimator seems more suitable for large sample sizes and few alleles, whose age can be approximated, whereas the RDc estimator appears preferable when this is not the case. When applied to three cystic fibrosis mutations, the estimator RD could not exclude a very recent time of divergence among three Mediterranean populations. On the other hand, the divergence time between these populations and the Danish population was estimated to be, on the average, 4,500 or 15,000 years, assuming or not a selective advantage for cystic fibrosis carriers, respectively. Confidence intervals are large, however, and can probably be reduced only by analyzing more alleles or loci.     

Point estimates and confidence intervals in measures of association.

Examined 6 statistics for estimating the population correlation ratio, ρ–2 (the proportionate reduction in error or variance explained index), for bias. Findings reveal that the expected value of an adjusted version of the sample correlation ratio, η?–2, was slightly less biased and less consistent than ω–2 or ε–2 with small population effects and sample sizes. A simplified method for generating approximate confidence intervals for ρ–2 was developed and found to be efficient relative to computation time. (25 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Environmental influences in hand preference: an African point of view

In the analysis of survival data from clinical trials and other studies, the censoring generally precludes estimation of the mean survival time. To accommodate censoring, Irwin (1949) proposed, as an alternative, estimation of the mean lifetime restricted to a suitably chosen time T. In this article we consider the use of Irwin's restricted mean as an index for comparing survival in different groups, using as an example published data from a randomized clinical trial in patients with primary biliary cirrhosis. Irwin's method, originally based on the actuarial survival estimator, is extended to incorporate covariates into the analysis through the use of piecewise exponential models. For comparing two survival curves, the logrank test is known to be optimal under proportional hazards alternatives. However, comparison of restricted means may outperform the logrank test in situations involving nonproportional hazard rates. We examine the size and power of these two procedures under various proportional and nonproportional hazards alternatives, with and without covariate adjustment. For survival curves that separate early in time the censored data generalization of the Wilcoxon test is known to exhibit high power, and we examine how the comparison of restricted means performs relative to this procedure also.     

Estimating standardized linear contrasts of means with desired precision.

L. Wilkinson and the Task Force on Statistical Inference (1999) recommended reporting confidence intervals for measures of effect sizes. If the sample size is too small, the confidence interval may be too wide to provide meaningful information. Recently, K. Kelley and J. R. Rausch (2006) used an iterative approach to computer-generate tables of sample size requirements for a standardized difference between 2 means in between-subjects designs. Sample size formulas are derived here for general standardized linear contrasts of k ≥ 2 means for both between-subjects designs and within-subjects designs. Special sample size formulas also are derived for the standardizer proposed by G. V. Glass (1976). (PsycINFO Database Record (c) 2010 APA, all rights reserved)