Confidence bands for low-dose risk estimation with quantal response data

We study the use of simultaneous confidence bands for low-dose risk estimation with quantal response data, and derive methods for estimating simultaneous upper confidence limits on predicted extra risk under a multistage model. By inverting the upper bands on extra risk, we obtain simultaneous lower bounds on the benchmark dose (BMD). Monte Carlo evaluations explore characteristics of the simultaneous limits under this setting, and a suite of actual data sets are used to compare existing methods for placing lower limits on the BMD.     

Multiplicity-adjusted inferences in risk assessment: benchmark analysis with quantal response data

A primary objective in quantitative risk or safety assessment is characterization of the severity and likelihood of an adverse effect caused by a chemical toxin or pharmaceutical agent. In many cases data are not available at low doses or low exposures to the agent, and inferences at those doses must be based on the high-dose data. A modern method for making low-dose inferences is known as benchmark analysis, where attention centers on the dose at which a fixed benchmark level of risk is achieved. Both upper confidence limits on the risk and lower confidence limits on the "benchmark dose" are of interest. In practice, a number of possible benchmark risks may be under study; if so, corrections must be applied to adjust the limits for multiplicity. In this short note, we discuss approaches for doing so with quantal response data.     

Benchmark Dose Profiles for Joint‐Action Quantal Data in Quantitative Risk Assessment

Summary Benchmark analysis is a widely used tool in public health risk analysis. Therein, estimation of minimum exposure levels, called Benchmark Doses (BMDs), that induce a prespecified Benchmark Response (BMR) is well understood for the case of an adverse response to a single stimulus. For cases where two agents are studied in tandem, however, the benchmark approach is far less developed. This article demonstrates how the benchmark modeling paradigm can be expanded from the single‐dose setting to joint‐action, two‐agent studies. Focus is on response outcomes expressed as proportions. Extending the single‐exposure setting, representations of risk are based on a joint‐action dose–response model involving both agents. Based on such a model, the concept of a benchmark profile (BMP) – a two‐dimensional analog of the single‐dose BMD at which both agents achieve the specified BMR – is defined for use in quantitative risk characterization and assessment. The resulting, joint, low‐dose guidelines can improve public health planning and risk regulation when dealing with low‐level exposures to combinations of hazardous agents.     

Functional Generalized Linear Models with Images as Predictors

Summary .  Functional principal component regression (FPCR) is a promising new method for regressing scalar outcomes on functional predictors. In this article, we present a theoretical justification for the use of principal components in functional regression. FPCR is then extended in two directions: from linear to the generalized linear modeling, and from univariate signal predictors to high-resolution image predictors. We show how to implement the method efficiently by adapting generalized additive model technology to the functional regression context. A technique is proposed for estimating simultaneous confidence bands for the coefficient function; in the neuroimaging setting, this yields a novel means to identify brain regions that are associated with a clinical outcome. A new application of likelihood ratio testing is described for assessing the null hypothesis of a constant coefficient function. The performance of the methodology is illustrated via simulations and real data analyses with positron emission tomography images as predictors.     

Simultaneous confidence intervals for a success probability and intraclass correlation,with an application to screening mammography

This paper provides asymptotic simultaneous confidence intervals for a success probability and intraclass correlation of the beta‐binomial model, based on the maximum likelihood estimator approach. The coverage probabilities of those intervals are evaluated. An application to screening mammography is presented as an example. The individual and simultaneous confidence intervals for sensitivity and specificity and the corresponding intraclass correlations are investigated. Two additional examples using influenza data and sex ratio data among sibships are also considered, where the individual and simultaneous confidence intervals are provided.     

Risk assessment for quantitative responses using a mixture model

A problem that frequently occurs in biological experiments with laboratory animals is that some subjects are less susceptible to the treatment than others. A mixture model has traditionally been proposed to describe the distribution of responses in treatment groups for such experiments. Using a mixture dose-response model, we derive an upper confidence limit on additional risk, defined as the excess risk over the background risk due to an added dose. Our focus will be on experiments with continuous responses for which risk is the probability of an adverse effect defined as an event that is extremely rare in controls. The asymptotic distribution of the likelihood ratio statistic is used to obtain the upper confidence limit on additional risk. The method can also be used to derive a benchmark dose corresponding to a specified level of increased risk. The EM algorithm is utilized to find the maximum likelihood estimates of model parameters and an extension of the algorithm is proposed to derive the estimates when the model is subject to a specified level of added risk. An example is used to demonstrate the results, and it is shown that by using the mixture model a more accurate measure of added risk is obtained.     

Dynamic inference in general nested case‐control designs

Nested case‐control designs are attractive in studies with a time‐to‐event endpoint if the outcome is rare or if interest lies in evaluating expensive covariates. The appeal is that these designs restrict to small subsets of all patients at risk just prior to the observed event times. Only these small subsets need to be evaluated. Typically, the controls are selected at random and methods for time‐simultaneous inference have been proposed in the literature. However, the martingale structure behind nested case‐control designs allows for more powerful and flexible non‐standard sampling designs. We exploit that structure to find simultaneous confidence bands based on wild bootstrap resampling procedures within this general class of designs. We show in a simulation study that the intended coverage probability is obtained for confidence bands for cumulative baseline hazard functions. We apply our methods to observational data about hospital‐acquired infections.     

Risk‐adjusted monitoring of time to event in the presence of long‐term survivors

The use of control charts for monitoring schemes in medical context should consider adjustments to incorporate the specific risk for each individual. Some authors propose the use of a risk‐adjusted survival time cumulative sum (RAST CUSUM) control chart to monitor a time‐to‐event outcome, possibly right censored, using conventional survival models, which do not contemplate the possibility of cure of a patient. We propose to extend this approach considering a risk‐adjusted CUSUM chart, based on a cure rate model. We consider a regression model in which the covariates affect the cure fraction. The CUSUM scores are obtained for Weibull and log‐logistic promotion time model to monitor a scale parameter for nonimmune individuals. A simulation study was conducted to evaluate and compare the performance of the proposed chart (RACUF CUSUM) with RAST CUSUM, based on optimal control limits and average run length in different situations. As a result, we note that the RAST CUSUM chart is inappropriate when applied to data with a cure rate, while the proposed RACUF CUSUM chart seems to have similar performance if applied to data without a cure rate. The proposed chart is illustrated with simulated data and with a real data set of patients with heart failure treated at the Heart Institute (InCor), at the University of São Paulo, Brazil.     

Empirical likelihood for cumulative hazard ratio estimation with covariate adjustment

In medical studies, it is often of scientific interest to evaluate the treatment effect via the ratio of cumulative hazards, especially when those hazards may be nonproportional. To deal with nonproportionality in the Cox regression model, investigators usually assume that the treatment effect has some functional form. However, to do so may create a model misspecification problem because it is generally difficult to justify the specific parametric form chosen for the treatment effect. In this article, we employ empirical likelihood (EL) to develop a nonparametric estimator of the cumulative hazard ratio with covariate adjustment under two nonproportional hazard models, one that is stratified, as well as a less restrictive framework involving group-specific treatment adjustment. The asymptotic properties of the EL ratio statistic are derived in each situation and the finite-sample properties of EL-based estimators are assessed via simulation studies. Simultaneous confidence bands for all values of the adjusted cumulative hazard ratio in a fixed interval of interest are also developed. The proposed methods are illustrated using two different datasets concerning the survival experience of patients with non-Hodgkin's lymphoma or ovarian cancer.     

Obtaining Critical Values for Simultaneous Confidence Intervals and Multiple Testing

There are many situations where it is desired to make simultaneous tests or give simultaneous confidence intervals for linear combinations (contrasts) of population or treatment means. Somerville (1997, 1999) developed algorithms for calculating the critical values for a large class of simultaneous tests and simultaneous confidence intervals. Fortran 90 and SAS‐IML batch programs and interactive programs were developed. These programs calculate the critical values for 15 different simultaneous confidence interval procedures (and the corresponding simultaneous tests) and for arbitrary procedures where the user specifies a combination of one and two sided contrasts. The programs can also be used to obtain the constants for “step‐down” testing of multiple hypotheses. This paper gives examples of the use of the algorithms and programs and illustrates their versatility and generality. The designs need not be balanced, multiple covariates may be present and there may be many missing values. The use of multiple regression and dummy variables to obtain the required variance covariance matrix is illustrated. Under weak normality assumptions the methods are “exact” and make the use of approximate methods or “simulation” unnecessary.     

Benchmark dose profiles for joint‐action continuous data in quantitative risk assessment

Benchmark analysis is a widely used tool in biomedical and environmental risk assessment. Therein, estimation of minimum exposure levels, called benchmark doses (BMDs), that induce a prespecified benchmark response (BMR) is well understood for the case of an adverse response to a single stimulus. For cases where two agents are studied in tandem, however, the benchmark approach is far less developed. This paper demonstrates how the benchmark modeling paradigm can be expanded from the single‐agent setting to joint‐action, two‐agent studies. Focus is on continuous response outcomes. Extending the single‐exposure setting, representations of risk are based on a joint‐action dose–response model involving both agents. Based on such a model, the concept of a benchmark profile—a two‐dimensional analog of the single‐dose BMD at which both agents achieve the specified BMR—is defined for use in quantitative risk characterization and assessment.     

Estimation and confidence regions for multi-dimensional effective dose

The problem of finding confidence regions for multiple predictor variables corresponding to given expected values of a response variable has not been adequately resolved. Motivated by an example from a study on hyperbaric exposure using a logistic regression model, we develop a conceptual framework for the estimation of the multi-dimensional effective dose for binary outcomes. The k -dimensional effective dose can be determined by conditioning on k - 1 components and solving for the last component as a conditional univariate effective dose. We consider various approaches for calculating confidence regions for the multi-dimensional effective dose and compare them via a simulation study for a range of possible designs. We analyze data related to decompression sickness to illustrate our procedure. Our results provide a practical approach to finding confidence regions for predictor variables for a given response value.     

Simultaneous confidence regions for closed tests,including Holm‐, Hochberg‐, and Hommel‐related procedures

The derivation of simultaneous confidence regions for some multiple‐testing procedures (MTPs) of practical interest has remained an unsolved problem. This is the case, for example, for Hochberg's step‐up MTP and Hommel's more powerful MTP that is neither a step‐up nor a step‐down procedure. It is shown in this article how the direct approach used previously by the author to construct confidence regions for certain closed‐testing procedures (CTPs) can be extended to a rather general setup. The general results are then applied to a situation with one‐sided inferences and CTPs belonging to a class studied by Wei Liu. This class consists of CTPs based on ordered marginal p‐values. It includes Holm's, Hochberg's, and Hommel's MTPs. A property of the confidence regions derived for these three MTPs is that no confidence assertions sharper than rejection assertions can be made unless all null hypotheses are rejected. Briefly, this is related to the fact that these MTPs are quite powerful. The class of CTPs considered includes, however, also MTPs related to Holm's, Hochberg's, and Hommel's MTPs that are less powerful but are such that confidence assertions sharper than rejection assertions are possible even if not all null hypotheses are rejected. One may thus choose and prespecify such an MTP, though this is at the cost of less rejection power.     

Failure time analysis of HIV vaccine effects on viral load and antiretroviral therapy initiation

The world's first efficacy trial of a preventive HIV vaccine was completed in 2003. Study participants who became HIV infected were followed for 2 years and monitored for HIV viral load and initiation of antiretroviral therapy (ART). In order to determine if vaccination may have altered HIV progression in persons who acquired HIV, a pre-specified objective was to compare the time until a composite endpoint between the vaccine and placebo arms, where the composite endpoint is the first event of ART initiation or viral failure (HIV viral load exceeds a threshold x(vl) copies/ml). Specifically, with vaccine efficacy, VE(tau, x(vl)), defined as one minus the ratio (vaccine/placebo) of the cumulative probability of the composite endpoint (with failure threshold x(vl)) occurring by tau months, the aim was to estimate the four parameters {VE(tau, x(vl)): x(vl) is an element of {1500, 10,000, 20,000, 55,000} copies/ml} with simultaneous 95% confidence bands. A Gaussian multipliers simulation method is devised for constructing confidence bands for VE(tau, x(vl)) with x(vl) spanning multiple discrete values or a continuous range. The new method is evaluated in simulations and is applied to the vaccine trial data set.     

Combining One‐sided and Two‐sided Confidence Interval Procedures for Successive Comparisons of Ordered Treatment Effects

Lee and Spurrier (1995) present one‐sided and two‐sided confidence interval procedures for making successive comparisons between ordered treatments. Their procedures have important applications for problems where the treatments can be assumed to satisfy a simple ordering, such as for a sequence of increasing dose‐levels of a drug. The two‐sided procedure provides both upper and lower bounds on the differences between successive treatments, whereas the one‐sided procedure only provides lower bounds on these differences. However, the one‐sided procedure allows sharper inferences regarding which treatments can be declared to be better than their previous ones. In this paper we apply the results obtained in Hayter , Miwa , and Liu (2000) to develop a new procedure which combines the good aspects of both the one‐sided and the two‐sided procedures. This new procedure maintains the inferential sensitivity of the one‐sided procedure while also providing both upper and lower bounds on the differences between successive treatments. Some new critical points are needed which are tabulated for the balanced case where the sample sizes are all equal. The application of the new procedure is illustrated with an example.     

Exact one-sided confidence bounds for the risk ratio in 2 x 2 tables with structural zero

This paper examines exact one-sided confidence limits for the risk ratio in a 2 x 2 table with structural zero. Starting with four approximate lower and upper limits, we adjust each using the algorithm of Buehler (1957) to arrive at lower (upper) limits that have exact coverage properties and are as large (small) as possible subject to coverage, as well as an ordering, constraint. Different Buehler limits are compared by their mean size, since all are exact in their coverage. Buehler limits based on the signed root likelihood ratio statistic are found to have the best performance and recommended for practical use.     

A two-stage stepwise estimation procedure

Summary .   This article proposes a two-stage simultaneous confidence procedure for the comparisons of k pairs of population means, without using multiplicity adjustment of more than two populations. The proposed procedure can be broadly applied to parametric or nonparametric models. It is robust and versatile because its derivation only utilizes a partitioning approach in conjunction with a bivariate adjustment, without any assumption on the underlying distribution. To elucidate the application, the proposed procedure is intertwined with the estimation of the therapeutic window of a drug. It provides confidence limits for the efficacy and the toxicity of the effective doses, highest ineffective dose, safe doses, and lowest unsafe dose, simultaneously. Such estimation information facilitates follow-up studies in clinical trials. As an illustrative example, the new procedure is applied to analyze a data set on molecular cancer therapeutics regarding the apoptotic killing effects of different chemical compounds on two leukemia cell lines.     

Point and interval estimation of baseline risk and treatment effect based on logistic model for observational studies

In observational studies with dichotomous outcome of a population, researchers usually report treatment effect alone, although both baseline risk and treatment effect are needed to evaluate the significance of the treatment effect to the population. In this article, we study point and interval estimates including confidence region of baseline risk and treatment effect based on logistic model, where baseline risk is the risk of outcome of the population under control treatment while treatment effect is measured by the risk difference between outcomes of the population under active versus control treatments. Using approximate normal distribution of the maximum‐likelihood (ML) estimate of the model parameters, we obtain an approximate joint distribution of the ML estimate of the baseline risk and the treatment effect. Using the approximate joint distribution, we obtain point estimate and confidence region of the baseline risk and the treatment effect as well as point estimate and confidence interval of the treatment effect when the ML estimate of the baseline risk falls into specified range. These interval estimates reflect nonnormality of the joint distribution of the ML estimate of the baseline risk and the treatment effect. The method can be easily implemented by using any software that generates normal distribution. The method can also be used to obtain point and interval estimates of baseline risk and any other measure of treatment effect such as risk ratio and the number needed to treat. The method can also be extended from logistic model to other models such as log‐linear model.     

Exact Multiple Comparisons of Three or More Regression Lines: Pairwise Comparisons and Comparisons with a Control

The problem of finding exact simultaneous confidence bounds for differences in regression models for k groups via the union‐intersection method is considered. The error terms are taken to be iid normal random variables. Under an assumption slightly more general than having identical design matrices for each of the k groups, it is shown that an existing probability point for the multivariate studentized range can be used to find the necessary probability point for pairwise comparisons of regression models. The resulting methods can be used with simple or multiple regression. Under a weaker assumption on the k design matrices that allows more observations to be taken from the control group than from the k‐1 treatment groups, a method is developed for computing exact probability points for comparing the simple linear regression models of the k‐1 groups to that of the control. Within a class of designs, the optimal design for comparisons with a control takes the square root of (k‐1) times as many observations from the control than from each treatment group. The simultaneous confidence bounds for all pairwise differences and for comparisons with a control are much narrower than Spurrier's intervals for all contrasts of k regression lines.     

Operability region equivalence: simultaneous confidence bands for the equivalence of two regression models over restricted regions

In many scientific problems the purpose of comparing two linear regression models is to demonstrate that they have only negligible differences and so can be regarded as being practically equivalent. The frequently used statistical approach of testing the homogeneity null hypothesis of the two models by using a partial F test is not appropriate for this purpose. In this paper, a simultaneous confidence band is proposed which provides an upper bound on the largest possible difference between the two models, in units of the standard error of the observations, over a given region of the covariates. This is demonstrated to be a more practical method for assessing the equivalence of the two regression models.