Volume 6, Issue 1

The measurement of biallelic pair-wise association called linkage disequilibrium (LD) is an important issue in order to understand the genomic architecture. A plethora of measures of association in two by two tables have been proposed in the literature. Beside the problem of choosing an appropriate measure, the problem of their estimation has been neglected in the literature. It needs to be emphasized that the definition of a measure and the choice of an estimator function for it are conceptually unrelated tasks.In this paper, we compare the performance of various estimators for the three popular LD measures D', r and Y in a simulation study for small to moderate samples sizes (N<=500). The usual frequency-plug-in estimators can lead to unreliable or undefined estimates. Estimators based on the computationally expensive volume measures have been proposed recently as a remedy to this well-known problem. We confirm that volume estimators have better expected mean square error than the naive plug-in estimators. But they are outperformed by estimators plugging-in easy to calculate non-informative Bayesian probability estimates into the theoretical formulae for the measures. Fully Bayesian estimators with non-informative Dirichlet priors have comparable accuracy but are computationally more expensive.We recommend the use of non-informative Bayesian plug-in estimators based on Jeffreys' prior, in particular when dealing with SNP array data where the occurrence of small table entries and table margins is likely.

When comparing the causal effect of peritoneal dialysis (PD) and hemodialysis (HD) treatment on lowering mortality in renal patients, using observational data, it is necessary to adjust for different forms of confounding and informative censoring. Both the type of dialysis treatment that is started with and mortality are affected by baseline covariates. Longitudinal and baseline variables can affect both the probability of switching from one type of dialysis to the other, and mortality. Longitudinal and baseline variables can also affect the probability of receiving a kidney transplant, possibly causing informative censoring. Adjusting for longitudinal variables by including them as covariates in a regression model potentially causes bias, for instance by losing a possible indirect effect of dialysis on mortality via these longitudinal variables. Instead, we fitted a marginal structural model (MSM) to estimate the causal effect of dialysis type, adjusted for confounding and informative censoring. We used the MSM to compare the hazard of death as well as cumulative survival between the potential treatment trajectories "always PD" and "always HD" over time, conditional on age and diabetes mellitus status. We used inverse probability weighting (IPW) to fit the MSM.

Family studies are widely used for research into genetic and environmental influences on human traits. In this paper, we establish statistical methodology for the estimation of a new measure of sib similarity with respect to dichotomous traits measured on each member of within family sib-pair. We call this parameter "excess risk." For inference problems involving a single sample, we construct a large sample confidence interval on the concerned parameter. It has long been suspected that consanguinity is a risk factor for many genetic defects. Therefore, we establish a procedure to test the significance of the difference between excess risk parameters in a sample of consanguineous marriages and another sample of non-consanguineous marriages. We apply the methodology to data from a hospital-based congenital heart defects registry in Saudi Arabia, a population in which consanguinity is quite common.

Self modeling regression (SEMOR) is an approach for modeling sets of observed curves that have a common shape (or sequence of features) but have variability in the amplitude (y-axis) and/or timing (x-axis) of the features across curves. SEMOR assumes the x and y axes for each observed curve can be separately transformed in a parametric manner so that the features across curves are aligned with the common shape, usually represented by non-parametric function. We show that when the common shape is modeled with a regression spline and the transformational parameters are modeled as random with the traditional distribution (normal with mean zero), the SEMOR model may surprisingly suffer from lack of fit and the variance components may be over-estimated. A random effects distribution that restricts the predicted random transformational parameters to have mean zero or the inclusion of a fixed transformational parameter improves estimation. Our work is motivated by arterial pulse pressure waveform data where one of the variance components is a novel measure of short-term variability in blood pressure.

In estimating optimal adaptive treatment strategies, the tailor treatment variables used for patient profiles are typically hand-picked by experts. However these variables may not yield an estimated optimal dynamic regime that is close to the optimal regime which uses all variables. The question of selecting tailoring variables has not yet been answered satisfactorily, though promising new approaches have been proposed. We compare the use of reducts—a variable selection tool from computer sciences—to the S-score criterion proposed by Gunter and colleagues in 2007 for suggesting collections of useful variables for treatment regime tailoring. Although the reducts-based approach promised several advantages such as the ability to account for correlation among tailoring variables, it proved to have several undesirable properties. The S-score performed better, though it too exhibited some disappointing qualities.

Multistate modeling methods are well-suited for analysis of some chronic diseases that move through distinct stages. The memoryless or Markov assumptions typically made, however, may be suspect for some diseases, such as hepatitis C, where there is interest in whether prognosis depends on history. This paper describes methods for multistate modeling where transition risk can depend on any property of past progression history, including time spent in the current stage and the time taken to reach the current stage. Analysis of 901 measurements of fibrosis in 401 patients following liver transplantation found decreasing risk of progression as time in the current stage increased, even when controlled for several fixed covariates. Longer time to reach the current stage did not appear associated with lower progression risk. Analysis of simulation scenarios based on the transplant study showed that greater misclassification of fibrosis produced more technical difficulties in fitting the models and poorer estimation of covariate effects than did less misclassification or error-free fibrosis measurement. The higher risk of progression when less time has been spent in the current stage could be due to varying disease activity over time, with recent progression indicating an "active" period and consequent higher risk of further progression.

A method for clustering incomplete longitudinal data, and gene expression time course data in particular, is presented. Specifically, an existing method that utilizes mixtures of multivariate Gaussian distributions with modified Cholesky-decomposed covariance structure is extended to accommodate incomplete data. Parameter estimation is carried out in a fashion that is similar to an expectation-maximization algorithm. We focus on the particular application of clustering incomplete gene expression time course data. In this application, our approach gives good clustering performance when compared to the results when there is no missing data. Possible extensions of this work are also suggested.

A fast and efficient estimation method is proposed that compensates the distortion in nonlinear transformation models. A likelihood-based estimator is developed that can be computed by an EM-type algorithm. The consistency of the estimator is shown and its limit distribution is provided. The new estimator is particularly well suited for fluorescence lifetime measurements, where only the shortest arrival time of a random number of emitted fluorescence photons can be detected and where arrival times are often modeled by a mixture of exponential distributions. The method is evaluated on real and synthetic data. Compared to currently used methods in fluorescence, the new estimator should allow a reduction of the acquisition time of an order of magnitude.

We consider the problem of constructing confidence intervals for the mean of a Negative Binomial random variable based upon sampled data. When the sample size is large, it is a common practice to rely upon a Normal distribution approximation to construct these intervals. However, we demonstrate that the sample mean of highly dispersed Negative Binomials exhibits a slow convergence in distribution to the Normal as a function of the sample size. As a result, standard techniques (such as the Normal approximation and bootstrap) will construct confidence intervals for the mean that are typically too narrow and significantly undercover at small sample sizes or high dispersions. To address this problem, we propose techniques based upon Bernstein's inequality or the Gamma and Chi Square distributions as alternatives to the standard methods. We investigate the impact of imposing a heuristic assumption of boundedness on the data as a means of improving the Bernstein method. Furthermore, we propose a ratio statistic relating the Negative Binomial's parameters that can be used to ascertain the applicability of the Chi Square method and to provide guidelines on evaluating the length of all proposed methods. We compare the proposed methods to the standard techniques in a variety of simulation experiments and consider data arising in the serial analysis of gene expression and traffic flow in a communications network.

Repeated cross-sectional cluster randomization trials are cluster randomization trials in which the response variable is measured on a sample of subjects from each cluster at baseline and on a different sample of subjects from each cluster at follow-up. One can estimate the effect of the intervention on the follow-up response alone, on the follow-up responses after adjusting for baseline responses, or on the change in the follow-up response from the baseline response. We used Monte Carlo simulations to determine the relative statistical power of different methods of analysis. We examined methods of analysis based on generalized estimating equations (GEE) and a random effects model to account for within-cluster homogeneity. We also examined cluster-level analyses that treated the cluster as the unit of analysis. We found that the use of random effects models to estimate the effect of the intervention on the change in the follow-up response from the baseline response had lower statistical power compared to the other competing methods across a wide range of scenarios. The other methods tended to have similar statistical power in many settings. However, in some scenarios, those analyses that adjusted for the baseline response tended to have marginally greater power than did methods that did not account for the baseline response.

We consider the problem of score testing for certain low dimensional parameters of interest in a model that could include finite but high dimensional secondary covariates and associated nuisance parameters. We investigate the possibility of the potential gain in power by reducing the dimensionality of the secondary variables via oracle estimators such as the Adaptive Lasso. As an application, we use a recently developed framework for score tests of association of a disease outcome with an exposure of interest in the presence of a possible interaction of the exposure with other co-factors of the model. We derive the local power of such tests and show that if the primary and secondary predictors are independent, then having an oracle estimator does not improve the local power of the score test. Conversely, if they are dependent, there is the potential for power gain. Simulations are used to validate the theoretical results and explore the extent of correlation needed between the primary and secondary covariates to observe an improvement of the power of the test by using the oracle estimator. Our conclusions are likely to hold more generally beyond the model of interactions considered here.

Models, such as logistic regression and Poisson regression models, are often used to estimate treatment effects in randomized trials. These models leverage information in variables collected before randomization, in order to obtain more precise estimates of treatment effects. However, there is the danger that model misspecification will lead to bias. We show that certain easy to compute, model-based estimators are asymptotically unbiased even when the working model used is arbitrarily misspecified. Furthermore, these estimators are locally efficient. As a special case of our main result, we consider a simple Poisson working model containing only main terms; in this case, we prove the maximum likelihood estimate of the coefficient corresponding to the treatment variable is an asymptotically unbiased estimator of the marginal log rate ratio, even when the working model is arbitrarily misspecified. This is the log-linear analog of ANCOVA for linear models. Our results demonstrate one application of targeted maximum likelihood estimation.

Ordinal regression analysis is a convenient tool for analyzing ordinal response variables in the presence of covariates. In this paper we extend this methodology to the case of continuous self-rating scales such as the Visual Analog Scale (VAS) used in pain assessment, or the Linear Analog Self-Assessment (LASA) scales in quality of life studies. These scales measure subjects' perception of an intangible quantity, and cannot be handled as ratio variables because of their inherent nonlinearity. We express the likelihood in terms of a function connecting the scale with an underlying continuous latent variable and approximate this function either parametrically or non-parametrically. Then a general semi-parametric regression framework for continuous scales is developed. Two data sets have been analyzed to compare our method to the standard discrete ordinal regression model, and the parametric to the non-parametric versions of the model. The first data set uses VAS data from a study on the efficacy of low-level laser therapy in the treatment of chronic neck pain; the second comes from a study on chemotherapy treatments in advanced breast cancer and looks at the impact of different drugs on patients' quality of life. The continuous formulation of the ordinal regression model has the advantage of no loss of precision due to categorization of the scores and no arbitrary choice of the number and boundaries of categories. The semi-parametric form of the model makes it a flexible method for analysis of continuous ordinal scales.

The analysis of dose-response relationships is a common problem in pre-clinical studies. For example, proportions such as mortality rates and histopathological findings are of particular interest in repeated toxicity studies. Commonly applied designs consist of an untreated control group and several, possibly unequally spaced, dosage groups. The Williams test can be formulated as a multiple contrast test and is a powerful option to evaluate such data. In this paper, we consider simultaneous inference for Williams-type multiple contrasts when the response variable is binomial and sample sizes are only moderate. Approximate simultaneous confidence limits can be constructed using the quantiles of a multivariate normal distribution taking the correlation into account. Alternatively, multiplicity-adjusted p-values can be calculated as well. A simulation study shows that a simple correction based on adding pseudo observations leads to acceptable performance for moderate sample sizes, such as 40 per group. In addition, the calculation of adjusted p-values and approximate power is presented. Finally, the proposed methods are applied to example data from two toxicological studies; the methods are available in an R-package.

Multilevel logistic regression models are increasingly being used to analyze clustered data in medical, public health, epidemiological, and educational research. Procedures for estimating the parameters of such models are available in many statistical software packages. There is currently little evidence on the minimum number of clusters necessary to reliably fit multilevel regression models. We conducted a Monte Carlo study to compare the performance of different statistical software procedures for estimating multilevel logistic regression models when the number of clusters was low. We examined procedures available in BUGS, HLM, R, SAS, and Stata. We found that there were qualitative differences in the performance of different software procedures for estimating multilevel logistic models when the number of clusters was low. Among the likelihood-based procedures, estimation methods based on adaptive Gauss-Hermite approximations to the likelihood (glmer in R and xtlogit in Stata) or adaptive Gaussian quadrature (Proc NLMIXED in SAS) tended to have superior performance for estimating variance components when the number of clusters was small, compared to software procedures based on penalized quasi-likelihood. However, only Bayesian estimation with BUGS allowed for accurate estimation of variance components when there were fewer than 10 clusters. For all statistical software procedures, estimation of variance components tended to be poor when there were only five subjects per cluster, regardless of the number of clusters.

Collaborative double robust targeted maximum likelihood estimators represent a fundamental further advance over standard targeted maximum likelihood estimators of a pathwise differentiable parameter of a data generating distribution in a semiparametric model, introduced in van der Laan, Rubin (2006). The targeted maximum likelihood approach involves fluctuating an initial estimate of a relevant factor (Q) of the density of the observed data, in order to make a bias/variance tradeoff targeted towards the parameter of interest. The fluctuation involves estimation of a nuisance parameter portion of the likelihood, g. TMLE has been shown to be consistent and asymptotically normally distributed (CAN) under regularity conditions, when either one of these two factors of the likelihood of the data is correctly specified, and it is semiparametric efficient if both are correctly specified.In this article we provide a template for applying collaborative targeted maximum likelihood estimation (C-TMLE) to the estimation of pathwise differentiable parameters in semi-parametric models. The procedure creates a sequence of candidate targeted maximum likelihood estimators based on an initial estimate for Q coupled with a succession of increasingly non-parametric estimates for g. In a departure from current state of the art nuisance parameter estimation, C-TMLE estimates of g are constructed based on a loss function for the targeted maximum likelihood estimator of the relevant factor Q that uses the nuisance parameter to carry out the fluctuation, instead of a loss function for the nuisance parameter itself. Likelihood-based cross-validation is used to select the best estimator among all candidate TMLE estimators of Q0 in this sequence. A penalized-likelihood loss function for Q is suggested when the parameter of interest is borderline-identifiable.We present theoretical results for "collaborative double robustness," demonstrating that the collaborative targeted maximum likelihood estimator is CAN even when Q and g are both mis-specified, providing that g solves a specified score equation implied by the difference between the Q and the true Q0. This marks an improvement over the current definition of double robustness in the estimating equation literature.We also establish an asymptotic linearity theorem for the C-DR-TMLE of the target parameter, showing that the C-DR-TMLE is more adaptive to the truth, and, as a consequence, can even be super efficient if the first stage density estimator does an excellent job itself with respect to the target parameter.This research provides a template for targeted efficient and robust loss-based learning of a particular target feature of the probability distribution of the data within large (infinite dimensional) semi-parametric models, while still providing statistical inference in terms of confidence intervals and p-values. This research also breaks with a taboo (e.g., in the propensity score literature in the field of causal inference) on using the relevant part of likelihood to fine-tune the fitting of the nuisance parameter/censoring mechanism/treatment mechanism.

A concrete example of the collaborative double-robust targeted likelihood estimator (C-TMLE) introduced in a companion article in this issue is presented, and applied to the estimation of causal effects and variable importance parameters in genomic data. The focus is on non-parametric estimation in a point treatment data structure. Simulations illustrate the performance of C-TMLE relative to current competitors such as the augmented inverse probability of treatment weighted estimator that relies on an external non-collaborative estimator of the treatment mechanism, and inefficient estimation procedures including propensity score matching and standard inverse probability of treatment weighting. C-TMLE is also applied to the estimation of the covariate-adjusted marginal effect of individual HIV mutations on resistance to the anti-retroviral drug lopinavir. The influence curve of the C-TMLE is used to establish asymptotically valid statistical inference. The list of mutations found to have a statistically significant association with resistance is in excellent agreement with mutation scores provided by the Stanford HIVdb mutation scores database.

We present a nonparametric methodology for evaluation of agreement between multiple methods of measurement of a continuous variable. Our approach is unified in that it can deal with any scalar measure of agreement currently available in the literature, and can incorporate repeated and unreplicated measurements, and balanced as well as unbalanced designs. Our key idea is to treat an agreement measure as a functional of the joint cumulative distribution function of the measurements from multiple methods. This measure is estimated nonparametrically by plugging-in a weighted empirical counterpart of the joint distribution function. The resulting estimator is shown to be asymptotically normal under some specified assumptions. A closed-form expression is provided for the asymptotic standard error of the estimator. This asymptotic normality is used to derive a large-sample distribution-free methodology for simultaneously comparing the multiple measurement methods. The small-sample performance of this methodology is investigated via simulation. The asymptotic efficiency of the proposed nonparametric estimator relative to the normality-based maximum likelihood estimator is also examined. The methodology is illustrated by applying it to a blood pressure data set involving repeated measurements from three measurement methods.

Several recent studies have reported that women who have used hormone replacement therapy (HRT), and developed breast cancer, tend to have a better prognosis than women with breast cancer who have not used HRT. One possible explanation is that tumors caused by HRT are more benign than tumors caused by other factors. Although it is relevant to quantify differences in prognostic factors across subtypes of breast cancer, it is not obvious how to do this correctly. This is because the tumors which occur among women who are treated with HRT are a mixture of HRT-induced and other tumors. We propose a framework based on principal stratification to distinguish women with HRT-induced tumors from women with tumors caused by other factors. To estimate the difference in prognosis for these two groups, we propose two estimation methods, which can be used under both cohort and case-control sampling schemes.

Current methods used to analyze time to event data either rely on highly parametric assumptions which result in biased estimates of parameters which are purely chosen out of convenience, or are highly unstable because they ignore the global constraints of the true model. By using Targeted Maximum Likelihood Estimation (TMLE) one may consistently estimate parameters which directly answer the statistical question of interest. Targeted Maximum Likelihood Estimators are substitution estimators, which rely on estimating the underlying distribution. However, unlike other substitution estimators, the underlying distribution is estimated specifically to reduce bias in the estimate of the parameter of interest. We will present here an extension of TMLE for observational time to event data, the Collaborative Targeted Maximum Likelihood Estimator (C-TMLE) for the treatment specific survival curve. Through the use of a simulation study we will show that this method improves on commonly used methods in both robustness and efficiency. In fact, we will show that in certain situations the C-TMLE produces estimates whose mean square error is lower than the semi-parametric efficiency bound. We will also demonstrate that a semi-parametric efficient substitution estimator (TMLE) outperforms a semi-parametric efficient non-substitution estimator (the Augmented Inverse Probability Weighted estimator) in sparse data situations. Lastly, we will show that the bootstrap is able to produce valid 95 percent confidence intervals in sparse data situations, while influence curve based inference breaks down.

Outcome-dependent sampling probabilities can be used to increase efficiency in observational studies. For continuous outcomes, appropriate consideration of sampling design in estimating parameters of interest is often computationally cumbersome. In this article, we suggest a Stochastic EM type algorithm for estimation when ascertainment probabilities are known or estimable. The computational complexity of the likelihood is avoided by filling in missing data so that an approximation of the full data likelihood can be used. The method is not restricted to any specific distribution of the data and can be used for a broad range of statistical models.

The evaluation of the ability of a diagnostic test to separate diseased subjects from non-diseased subjects is a crucial issue in modern medicine. The accuracy of a continuous-scale test at a chosen cut-off level can be measured by its sensitivity and specificity, i.e. by the probabilities that the test correctly identifies the diseased and non-diseased subjects, respectively.In practice, sensitivity and specificity of the test are unknown. Moreover, which cut-off level to use is also generally unknown in that no preliminary indications driving its choice could be available.In this paper, we address the problem of making joint inference on pairs of quantities defining accuracy of a diagnostic test, in particular, when one of the two quantities is the cut-off level. We propose a technique based on an empirical likelihood statistic that allows, within a unified framework, to build bivariate confidence regions for the pair (sensitivity, cut-off level) at a fixed value of specificity as well as for the pair (specificity, cut-off level) at a fixed value of sensitivity or the pair (sensitivity, specificity) at a fixed cut-off value.A simulation study is carried out to assess the finite-sample accuracy of the method. Moreover, we apply the method to two real examples.

Targeted maximum likelihood estimation of a parameter of a data generating distribution, known to be an element of a semi-parametric model, involves constructing a parametric model through an initial density estimator with parameter ? representing an amount of fluctuation of the initial density estimator, where the score of this fluctuation model at ? = 0 equals the efficient influence curve/canonical gradient. The latter constraint can be satisfied by many parametric fluctuation models since it represents only a local constraint of its behavior at zero fluctuation. However, it is very important that the fluctuations stay within the semi-parametric model for the observed data distribution, even if the parameter can be defined on fluctuations that fall outside the assumed observed data model. In particular, in the context of sparse data, by which we mean situations where the Fisher information is low, a violation of this property can heavily affect the performance of the estimator. This paper presents a fluctuation approach that guarantees the fluctuated density estimator remains inside the bounds of the data model. We demonstrate this in the context of estimation of a causal effect of a binary treatment on a continuous outcome that is bounded. It results in a targeted maximum likelihood estimator that inherently respects known bounds, and consequently is more robust in sparse data situations than the targeted MLE using a naive fluctuation model.When an estimation procedure incorporates weights, observations having large weights relative to the rest heavily influence the point estimate and inflate the variance. Truncating these weights is a common approach to reducing the variance, but it can also introduce bias into the estimate. We present an alternative targeted maximum likelihood estimation (TMLE) approach that dampens the effect of these heavily weighted observations. As a substitution estimator, TMLE respects the global constraints of the observed data model. For example, when outcomes are binary, a fluctuation of an initial density estimate on the logit scale constrains predicted probabilities to be between 0 and 1. This inherent enforcement of bounds has been extended to continuous outcomes. Simulation study results indicate that this approach is on a par with, and many times superior to, fluctuating on the linear scale, and in particular is more robust when there is sparsity in the data.

Lately, bivariate zero-inflated (BZI) regression models have been used in many instances in the medical sciences to model excess zeros. Examples include the BZI Poisson (BZIP), BZI negative binomial (BZINB) models, etc. Such formulations vary in the basic modeling aspect and use the EM algorithm (Dempster, Laird and Rubin, 1977) for parameter estimation. A different modeling formulation in the Bayesian context is given by Dagne (2004). We extend the modeling to a more general setting for multivariate ZIP models for count data with excess zeros as proposed by Li, Lu, Park, Kim, Brinkley and Peterson (1999), focusing on a particular bivariate regression formulation. For the basic formulation in the case of bivariate data, we assume that Xi are (latent) independent Poisson random variables with parameters ? i, i = 0, 1, 2. A bi-variate count vector (Y1, Y2) response follows a mixture of four distributions; p0 stands for the mixing probability of a point mass distribution at (0, 0); p1, the mixing probability that Y2 = 0, while Y1 = X0 + X1; p2, the mixing probability that Y1 = 0 while Y2 = X0 + X2; and finally (1 - p0 - p1 - p2), the mixing probability that Yi = Xi + X0, i = 1, 2. The choice of the parameters {pi, ? i, i = 0, 1, 2} ensures that the marginal distributions of Yi are zero inflated Poisson (? 0 + ? i). All the parameters thus introduced are allowed to depend on co-variates through canonical link generalized linear models (McCullagh and Nelder, 1989). This flexibility allows for a range of real-life applications, especially in the medical and biological fields, where the counts are bivariate in nature (with strong association between the processes) and where there are excess of zeros in one or both processes. Our contribution in this paper is to employ a fully Bayesian approach consolidating the work of Dagne (2004) and Li et al. (1999) generalizing the modeling and sampling-based methods described by Ghosh, Mukhopadhyay and Lu (2006) to estimate the parameters and obtain posterior credible intervals both in the case where co-variates are not available as well as in the case where they are. In this context, we provide explicit data augmentation techniques that lend themselves to easier implementation of the Gibbs sampler by giving rise to well-known and closed-form posterior distributions in the bivariate ZIP case. We then use simulations to explore the effectiveness of this estimation using the Bayesian BZIP procedure, comparing the performance to the Bayesian and classical ZIP approaches. Finally, we demonstrate the methodology based on bivariate plant count data with excess zeros that was collected on plots in the Phoenix metropolitan area and compare the results with independent ZIP regression models fitted to both processes.

Regression models for functional responses and scalar predictors are often fitted by means of basis functions, with quadratic roughness penalties applied to avoid overfitting. The fitting approach described by Ramsay and Silverman in the 1990s amounts to a penalized ordinary least squares (P-OLS) estimator of the coefficient functions. We recast this estimator as a generalized ridge regression estimator, and present a penalized generalized least squares (P-GLS) alternative. We describe algorithms by which both estimators can be implemented, with automatic selection of optimal smoothing parameters, in a more computationally efficient manner than has heretofore been available. We discuss pointwise confidence intervals for the coefficient functions, simultaneous inference by permutation tests, and model selection, including a novel notion of pointwise model selection. P-OLS and P-GLS are compared in a simulation study. Our methods are illustrated with an analysis of age effects in a functional magnetic resonance imaging data set, as well as a reanalysis of a now-classic Canadian weather data set. An R package implementing the methods is publicly available.

We propose statistical methods for comparing phenomics data generated by the Biolog Phenotype Microarray (PM) platform for high-throughput phenotyping. Instead of the routinely used visual inspection of data with no sound inferential basis, we develop two approaches. The first approach is based on quantifying the distance between mean or median curves from two treatments and then applying a permutation test; we also consider a permutation test applied to areas under mean curves. The second approach employs functional principal component analysis. Properties of the proposed methods are investigated on both simulated data and data sets from the PM platform.

When patients are monitored for potentially recurrent events such as infections or tumor metastases, it is common for clinicians to ask patients to come back sooner for follow-ups based on the results of the most recent exam. This means that subjects' observation times will be irregular and related to subject-specific factors. Previously proposed methods for handling such panel count data assume that the dependence between the events process and the observation time process is governed by time-independent factors. This article considers situations where the observation times are predicted by time-varying factors such as the outcome observed at the last visit or cumulative exposure. Using a joint modelling approach, we propose a class of inverse-intensity-rate-ratio weighted estimators that are root-n consistent and asymptotically normal. The proposed estimators use estimating equations and are fairly simple and easy to compute. We demonstrate the performance of the method using simulated data and illustrate the approach using a cancer study dataset.

Sample size requirements that achieve a prespecified expected lower limit for a confidence interval about the intraclass kappa statistic are supplied for the case of multiple raters and a binary outcome variable. The expected lower confidence limit achievable for a given number of subjects and raters is also presented. These results should be useful in the planning stages of an interobserver agreement study in which the focus is on interval estimation rather than hypothesis-testing.

The attributable risk, often called the population attributable risk, is in many epidemiological contexts a more relevant measure of exposure-disease association than the excess risk, relative risk, or odds ratio. When estimating attributable risk with case-control data and a rare disease, we present a simple bias correction to the standard approach, which also makes it more stable and less variable. As with analogous corrections given by Jewell (1986) for other measures of association, the adjustment often won't make a substantial difference unless the sample size is very small or point estimates are desired within fine strata, but we discuss the possible utility for applications.

Stochastic search variable selection (SSVS) algorithms provide an appealing and widely used approach for searching for good subsets of predictors while simultaneously estimating posterior model probabilities and model-averaged predictive distributions. This article proposes a two-level generalization of SSVS to account for missing predictors while accommodating uncertainty in the relationships between these predictors. Bayesian approaches for allowing predictors that are missing at random require a model on the joint distribution of the predictors. We show that predictive performance can be improved by allowing uncertainty in the specification of predictor relationships in this model. The methods are illustrated through simulation studies and analysis of an epidemiologic data set.

Health economic decision models compare costs and health effects of different interventions over the long term and usually incorporate survival data. Since survival is often extrapolated beyond the range of the data, inaccurate model specification can result in very different policy decisions. However, in this area, flexible survival models are rarely considered, and model uncertainty is rarely accounted for. In this article, various survival distributions are applied in a decision model for oral cancer screening. Flexible parametric models are compared with Bayesian semiparametric models, in which the baseline hazard can be made arbitrarily complex while still enabling survival to be extrapolated. A fully Bayesian framework is used for all models so that uncertainties can be easily incorporated in estimates of long-term costs and effects. The fit and predictive ability of both parametric and semiparametric models are compared using the deviance information criterion in order to account for model uncertainty in the cost-effectiveness analysis. Under the Bayesian semiparametric models, some smoothing of the hazard function is required to obtain adequate predictive ability and avoid sensitivity to the choice of prior. We determine that one flexible parametric survival model fits substantially better than the others considered in the oral cancer example.

In epidemiological cohort studies, the probability of developing a disease for individuals in a treatment/intervention group is compared with that of a control group. The groups involve varying cluster sizes, and the binary responses within each cluster cannot be assumed independently. Three major measures of association used to report the efficacy of treatments or effectiveness of public health intervention programs in case of prospective studies are Risk Difference (RD), Risk Ratio (RR) and Relative Risk Difference (RED). The preference of one measure of association over the other in drawing statistical inference depends on design of study. Lui (2004) discusses a number of methods of constructing confidence intervals for each of these measures. Specifically, Lui (2004) discusses four methods for RD, four methods for RR and three methods for RED. For the construction of confidence intervals for RD, Paul and Zaihra (2008) compare the four methods discussed by Lui (2004), using extensive simulations with a method based on an estimator of the variance of a ratio estimator by Cochran (1977) and a method based on a sandwich estimator of the variance of the regression estimator using the generalized estimating equations approach of Zeger and Liang (1986). Paul and Zaihra (2008) conclude that the method based on an estimate of the variance of a ratio estimator performs best overall. In this paper, we extend the two new methodologies introduced in Paul and Zaihra (2008) to confidence interval construction of the risk measures RR and RED. Extensive simulations show that the method based on an estimate of the variance of a ratio estimator performs best overall for constructing confidence interval for the other two risk measures RR and RED as well. This method involves a very simple variance expression which can be implemented with a very few computer codes. Therefore, it can be considered as an easily implementable alternative for all the three measures of association.

We offer an explanation to the simulation result of Austin (2009) regarding rate ratios, and argue that unmatched analysis of propensity score matched count data results in conservative statistical inferences on the rate ratios.

In their presentation on measures of predictive capacity Gu and Pepe say little about calibration. This comment distinguishes conditional and unconditional calibration and how these relate to the stated results.

This is a response to Nancy Cook's Readers' Reaction to "Measures to Summarize and Compare the Predictive Capacity of Markers."