A semiparametric approach to analysing dose-response data

In the analysis of a quantal dose-response experiment with grouped data, the most commonly used parametric procedure is logistic regression, commonly referred to as 'logit analysis'. The adequacy of the fit by the logistic regression curve is tested using the chi-square lack-of-fit test. If the lack-of-fit test is not significant, then the logistic model is assumed to be adequate and estimation of effective doses and confidence intervals on the effective doses can be made. When the tolerance distribution of the dose-response data is not known and cannot be assumed by the user, one can use non-parametric methods, such as kernel regression or local linear regression, to estimate the dose-response curve, effective doses and confidence intervals. This research proposes another alternative based on semi-parametric regression to analysing quantal dose-response data called model-robust quantal regression (MRQR). MRQR linearly combines the parametric and non-parametric predictions with the use of a mixing parameter. MRQR uses logistic regression as the parametric portion of the model and local linear regression as the non-parametric portion of the model. Our research has shown that the MRQR procedure can improve the fit of the dose-response curve by producing narrower confidence intervals for predictions while providing improved precision of estimates of the effective doses with respect to either logistic or local linear regression results.     

Inference for cumulative incidence quantiles via parametric and nonparametric approaches

In survival analysis, a point estimate and confidence interval for median survival time have been frequently used to summarize the survival curve. However, such quantile analyses on competing risks data have not been widely investigated. In this paper, we propose parametric inferences for quantiles from the cumulative incidence function and develop parametric confidence intervals for quantiles. In addition, we study a simplified method of inference for the nonparametric approach. We compare the parametric and nonparametric inferences in empirical studies. Simulation studies show that the procedures perform well, with parametric analyses yielding smaller mean square error when the model is not too badly misspecified. We illustrate the methods with data from a breast cancer clinical trial.     

Bootstrap confidence intervals for the sensitivity of a quantitative diagnostic test

We examine bootstrap approaches to the analysis of the sensitivity of quantitative diagnostic test data. Methods exist for inference concerning the sensitivity of one or more tests for fixed levels of specificity, taking into account the variability in the sensitivity due to variability in the test values for normal subjects. However, parametric methods do not adequately account for error, particularly when the data are non-normally distributed, and non-parametric methods have low power. We implement bootstrap methods for confidence limits for the sensitivity of a test for a fixed specificity and demonstrate that under certain circumstances the bootstrap method gives more accurate confidence intervals than do other methods, while it performs at least as well as other methods in many standard situations.     

Intraclass correlation coefficients and bootstrap methods of hierarchical binary outcomes

Intraclass correlation coefficients are designed to assess consistency or conformity between two or more quantitative measurements. When multistage cluster sampling is implemented, no methods are readily available to estimate intraclass correlations of binomial-distributed outcomes within a cluster. Because statistical distribution of the intraclass correlation coefficients could be complicated or unspecified, we propose using a bootstrap method to estimate the standard error and confidence interval within the framework of a multilevel generalized linear model. We compared the results derived from a parametric bootstrap method with those from a non-parametric bootstrap method and found that the non-parametric method is more robust. For non-parametric bootstrap sampling, we showed that the effectiveness of sampling on the highest level is greater than that on lower levels; to illustrate the effectiveness, we analyse survey data in China and do simulation studies.     

CONFIDENCE INTERVALS FOR MEDIAN SURVIVAL TIMES UNDER A PIECEWISE EXPONENTIAL MODEL WITH PROPORTIONAL HAZARDS COVARIATE EFFECTS

Brookmeyer and Crowley derived a non-parametric confidence interval for the median survival time of a homogeneous population by inverting a generalization of the sign test for censored data. The 1−α confidence interval for the median is essentially the set of all values t such that the Kaplan–Meier estimate of the survival function at time t does not differ significantly from one-half at significance level α. Here I extend the method to incorporate covariates into the analysis by assuming an underlying piecewise exponential model with proportional hazards covariate effects. Maximum likelihood estimates of the model parameters are obtained via iterative techniques, from which the estimated (log) survival curve is easily constructed. The delta method provides asymptotic standard errors. Following Brookmeyer and Crowley, I find the confidence interval for the median survival time at a specified value of the covariate vector by inverting the sign test. I illustrate the methods using data from a clinical trial conducted by the Radiation Therapy Oncology Group in cancer of the mouth and throat. It is seen that the piecewise exponential model provides considerable flexibility in accommodating to the shape of the underlying survival curve and thus offers advantages to other, more restrictive, parametric models. Simulation studies indicate that the method provides reasonably accurate coverage probabilities.     

Reference curves based on non-parametric quantile regression

Reference curves which take time into account, such as those for age, are often required in medicine, but simple systematic and efficient statistical methods for constructing them are lacking. Classical methods are based on parametric fitting (polynomial curves). Semi-parametric methods are also widely used especially in Europe. Here, we propose a new methodology for the estimation of reference intervals for data sets, based on non-parametric estimation of conditional quantiles. The derived methods should be applicable to all clinical (or more generally biological) variables that are measured on a continuous quantitative scale. As an example, we analyse a data set collected to establish reference curves for biophysical properties of the skin of healthy French women. The results are compared to those obtained with Royston's polynomial parametric method and the semi-parametric LMS approach.     

Applications of global statistics in analysing quality of life data

Quality of life (QOL) instruments usually consist of a number of components, each of which deals specifically with a particular functionally related dysfunction. In a clinical trial whose primary aim is the evaluation of the treatment by means of QOL instruments, analysis of each of the components usually consists of either univariate analysis of variance (ANOVA) or some non-parametric methods. This multiple testing approach can produce an increase in false positive findings. One attempt to correct for this is the Bonferroni adjustment. Another approach is to apply global statistics (parametric or non-parametric) for the null hypothesis of no treatment difference versus the alternative hypothesis that one treatment is uniformly better than the other for QOL instruments as a whole. Data from a randomized double-blind trial of 111 congestive heart failure patients, which involved four QOL instruments, were analysed with univariate ANOVA, Bonferroni adjustment, parametric and non-parametric global statistics. The global statistics complemented the univariate methods and made the presentation of QOL data very effective. I recommend the general use of global statistics in analysis of QOL data.     

Definition, interpretation and calculation of cost-effectiveness acceptability curves

This paper discusses the definition, interpretation and computation of cost-effectiveness (CE) acceptability curves. A formal definition of the CE acceptability curve based on the net benefit approach is provided. The curve can be computed using parametric or non-parametric techniques and for both computational approaches we establish a formal relation between the CE acceptability curve and statistical inference based on confidence intervals and P values in CE analysis.     

Using the bootstrap to improve estimation and confidence intervals for regression coefficients selected using backwards variable elimination

Applied researchers frequently use automated model selection methods, such as backwards variable elimination, to develop parsimonious regression models. Statisticians have criticized the use of these methods for several reasons, amongst them are the facts that the estimated regression coefficients are biased and that the derived confidence intervals do not have the advertised coverage rates. We developed a method to improve estimation of regression coefficients and confidence intervals which employs backwards variable elimination in multiple bootstrap samples. In a given bootstrap sample, predictor variables that are not selected for inclusion in the final regression model have their regression coefficient set to zero. Regression coefficients are averaged across the bootstrap samples, and non-parametric percentile bootstrap confidence intervals are then constructed for each regression coefficient. We conducted a series of Monte Carlo simulations to examine the performance of this method for estimating regression coefficients and constructing confidence intervals for variables selected using backwards variable elimination. We demonstrated that this method results in confidence intervals with superior coverage compared with those developed from conventional backwards variable elimination. We illustrate the utility of our method by applying it to a large sample of subjects hospitalized with a heart attack.     

Generalized P-values and confidence intervals: a novel approach for analyzing lognormally distributed exposure data

The problem of assessing occupational exposure using the mean of a lognormal distribution is addressed. The novel concepts of generalized p-values and generalized confidence intervals are applied for testing hypotheses and computing confidence intervals for a lognormal mean. The proposed methods perform well, they are applicable to small sample sizes, and they are easy to implement. Power studies and sample size calculation are also discussed. Computational details and a source for the computer program are given. The procedures are also extended to compare two lognormal means and to make inference about a lognormal variance. In fact, our approach based on generalized p-values and generalized confidence intervals is easily adapted to deal with any parametric function involving one or two lognormal distributions. Several examples involving industrial exposure data are used to illustrate the methods. An added advantage of the generalized variables approach is the ease of computation and implementation. In fact, the procedures can be easily coded in a programming language for implementation. Furthermore, extensive numerical computations by the authors show that the results based on the generalized p-value approach are essentially equivalent to those based on the Land's method. We want to draw the attention of the industrial hygiene community to this accurate and unified methodology to deal with any parameter associated with the lognormal distribution.     

Analysis of cost data in randomized trials: an application of the non-parametric bootstrap

Health economic evaluations are now more commonly being included in pragmatic randomized trials. However a variety of methods are being used for the presentation and analysis of the resulting cost data, and in many cases the approaches taken are inappropriate. In order to inform health care policy decisions, analysis needs to focus on arithmetic mean costs, since these will reflect the total cost of treating all patients with the disease. Thus, despite the often highly skewed distribution of cost data, standard non-parametric methods or use of normalizing transformations are not appropriate. Although standard parametric methods of comparing arithmetic means may be robust to non-normality for some data sets, this is not guaranteed. While the randomization test can be used to overcome assumptions of normality, its use for comparing means is still restricted by the need for similarly shaped distributions in the two groups. In this paper we show how the non-parametric bootstrap provides a more flexible alternative for comparing arithmetic mean costs between randomized groups, avoiding the assumptions which limit other methods. Details of several bootstrap methods for hypothesis tests and confidence intervals are described and applied to cost data from two randomized trials. The preferred bootstrap approaches are the bootstrap-t or variance stabilized bootstrap-t and the bias corrected and accelerated percentile methods. We conclude that such bootstrap techniques can be recommended either as a check on the robustness of standard parametric methods, or to provide the primary statistical analysis when making inferences about arithmetic means for moderately sized samples of highly skewed data such as costs.     

Empirical likelihood‐based confidence intervals for length‐biased data

Logistic or other constraints often preclude the possibility of conducting incident cohort studies. A feasible alternative in such cases is to conduct a cross‐sectional prevalent cohort study for which we recruit prevalent cases, that is, subjects who have already experienced the initiating event, say the onset of a disease. When the interest lies in estimating the lifespan between the initiating event and a terminating event, say death for instance, such subjects may be followed prospectively until the terminating event or loss to follow‐up, whichever happens first. It is well known that prevalent cases have, on average, longer lifespans. As such, they do not constitute a representative random sample from the target population; they comprise a biased sample. If the initiating events are generated from a stationary Poisson process, the so‐called stationarity assumption, this bias is called length bias. The current literature on length‐biased sampling lacks a simple method for estimating the margin of errors of commonly used summary statistics. We fill this gap by using the empirical likelihood‐based confidence intervals by adapting this method to right‐censored length‐biased survival data. Both large and small sample behaviors of these confidence intervals are studied. We illustrate our method by using a set of data on survival with dementia, collected as part of the Canadian Study of Health and Aging. Copyright © 2012 John Wiley & Sons, Ltd.     

Variation over time of the effects of prognostic factors in a population-based study of colon cancer: comparison of statistical models

The authors compare the performance of different regression models for censored survival data in modeling the impact of prognostic factors on all-cause mortality in colon cancer. The data were for 1,951 patients, who were diagnosed in 1977-1991, recorded by the Registry of Digestive Tumors of C?te d'Or, France, and followed for up to 15 years. Models include the Cox proportional hazards model and its three generalizations that allow for hazard ratio to change over time: 1) the piecewise model where hazard ratio is a step function; 2) the model with interaction between a predictor and a parametric function of time; and 3) the non-parametric regression spline model. Results illustrate the importance of accounting for non-proportionality of hazards, and some advantages of flexible non-parametric modeling of time-dependent effects. The authors provide empirical evidence for the dependence of the results of piecewise and parametric models on arbitrary a priori choices, regarding the number of time intervals and specific parametric function, which may lead to biased estimates and low statistical power. The authors demonstrate that a single, a priori selected spline model recovers a variety of patterns of changes in hazard ratio and fits better than other models, especially when the changes are non-monotonic, as in the case of cancer stages.     

Confidence intervals for the risk ratio under inverse sampling

In this paper, we investigate various confidence intervals for the risk ratio under inverse sampling (also known as negative binomial sampling). Three existing confidence intervals (namely, the confidence intervals that are based on Fieller's theorem, the delta method and the F-statistic) are reviewed and three new confidence intervals (namely, the score, likelihood ratio and saddlepoint approximation (SA)-based confidence intervals) are developed. Comparative studies among these confidence intervals through Monte Carlo simulations are evaluated in terms of their coverage probabilities and expected interval widths under different settings. Our simulation results suggest that the SA-based confidence interval is generally more appealing. We illustrate these confidence interval construction methods with real data sets from a drug comparison study and a congenital heart disease study.     

Constructing confidence intervals for cost-effectiveness ratios: an evaluation of parametric and non-parametric techniques using Monte Carlo simulation

The statistic of interest in most health economic evaluations is the incremental cost-effectiveness ratio. Since the variance of a ratio estimator is intractable, the health economics literature has suggested a number of alternative approaches to estimating confidence intervals for the cost-effectiveness ratio. In this paper, Monte Carlo simulation techniques are employed to address the question of which of the proposed methods is most appropriate. By repeatedly sampling from a known distribution and applying the different methods of confidence interval estimation, it is possible to calculate the coverage properties of each method to see if these correspond to the chosen confidence level. As the results of a single Monte Carlo experiment would be valid only for that particular set of circumstances, a series of experiments was conducted in order to examine the performance of the different methods under a variety of conditions relating to the sample size, the coefficient of variation of the numerator and denominator of the ratio, and the covariance between costs and effects in the underlying data. Response surface analysis was used to analyse the results and substantial differences between the different methods of confidence interval estimation were identified. The methods, both parametric and non-parametric, which assume a normal sampling distribution performed poorly, as did the approach based on simply combining the separate intervals on costs and effects. The choice of method for confidence interval estimation can lead to large differences in the estimated confidence limits for cost-effectiveness ratios. The importance of such differences is an empirical question and will depend to a large extent on the role of hypothesis testing in economic appraisal. However, where it is suspected that the sampling distribution is skewed, normal approximation methods produce particularly poor results and should be avoided.     

Effect measures in non-parametric regression with interactions between continuous exposures

In many biomedical studies, interest is often attached to calculating effect measures in the presence of interactions between two continuous exposures. Traditional approaches based on parametric regression are limited by the degree of arbitrariness involved in transforming these exposures into categorical variables or imposing a parametric form on the regression function. In this paper, we present: (a) a flexible non-parametric method for estimating effect measures through generalized additive models including interactions; and (b) bootstrap techniques for (i) testing the significance of interaction terms, and (ii) constructing confidence intervals for effect measures. The validity of our methodology is supported by simulations, and illustrated using data from a study of possible risk factors for post-operative infection. This application revealed a hitherto unreported effect: for patients with high plasma glucose levels, increased risk is associated, not only with low, but also with high percentages of lymphocytes.     

Smooth non-parametric receiver operating characteristic (ROC) curves for continuous diagnostic tests

Receiver operating characteristic (ROC) curves have seen increasing use to assess the accuracy of diagnostic tests that yield continuous test results. We can calculate a fully non-parametric ROC curve from the empirical false positive rate (FPR) and true positive rate (TPR) at each possible decision threshold, but it may be quite jagged. We can obtain a smooth ROC curve by directly fitting a parametric model, for example, binormal or bilogistic, to the actual test results, but substantial lack-of-fit may result if the distributional assumptions are not valid. A recently proposed alternative algorithm ‘LABROC4’ is insensitive to such departures, but is not fully flexible in that the form of the resulting ROC curve is restricted to be binormal. We propose a smooth non-parametric ROC curve derived from kernel density estimates of the two test result distributions. We obtain pointwise standard errors for the estimated TPR at each FPR and use them to construct pointwise confidence intervals for the ROC curve, using a logit transformation. We also obtain standard errors for the estimated TPR and FPR at given thresholds and use them to construct confidence rectangles in (FPR, TPR)-space that correspond to a specific threshold. We adapt existing methods for the unsmoothed non-parametric ROC curve to obtain the area under the ROC curve and its standard error; we also give partial areas and corresponding standard errors. We have created a FORTRAN algorithm ‘ROC-&-ROL’, available upon request. We compare ROC curves and areas obtained by applying our methods and its competitors to two data sets, one of which is fit well by parametric methods and LABROC4, the other of which is not. © 1997 John Wiley & Sons, Ltd.     

CHARACTERIZATION OF HIV INCUBATION DISTRIBUTIONS AND SOME COMPARATIVE STUDIES

In this paper we use a general stochastic model to characterize the HIV incubation distributions. We generate some Monte Carlo data under different conditions and compare the fitting of HIV incubation distributions by some well known parametric models and some non-parametric methods. The parametric models include most of those that have appeared in the literature. The non-parametric methods include the Kaplan–Meier method, the EMS method, the spline approximation and the Bacchetti method. The comparison criteria are the chi-square statistic, the residual sum of squares, the AIC and the BIC. We show that the non-parametric methods, especially the EMS method, provide excellent fits in almost all cases; for the parametric models, the generalized log-logistic distributions with three and with four parameters fit better than other parametric models.