Biometrika - current issue

Studentization and deriving accurate p-values

Fraser, D.A.S., Rousseau, J. — 2008-02-28

We have a statistic for assessing an observed data point relative to a statistical model but find that its distribution function depends on the parameter. To obtain the corresponding p-value, we require the minimally modified statistic that is ancillary; this process is called Studentization. We use recent likelihood theory to develop a maximal third-order ancillary; this gives immediately a candidate Studentized statistic. We show that the corresponding p-value is higher-order Un(0, 1), is equivalent to a repeated bootstrap version of the initial statistic and agrees with a special Bayesian modification of the original statistic. More importantly, the modified statistic and p-value are available by Markov chain Monte Carlo simulations and, in some cases, by higher-order approximation methods. Examples, including the Behrens–Fisher problem, are given to indicate the ease and flexibility of the approach.

Distortion of effects caused by indirect confounding

Wermuth, N., Cox, D. R. — 2008-02-28

Undetected confounding may severely distort the effect of an explanatory variable on a response variable, as defined by a stepwise data-generating process. The best known type of distortion, which we call direct confounding, arises from an unobserved explanatory variable common to a response and its main explanatory variable of interest. It is relevant mainly for observational studies, since it is avoided by successful randomization. By contrast, indirect confounding, which we identify in this paper, is an issue also for intervention studies. For general stepwise-generating processes, we provide matrix and graphical criteria to decide which types of distortion may be present, when they are absent and how they are avoided. We then turn to linear systems without other types of distortion, but with indirect confounding. For such systems, the magnitude of distortion in a least-squares regression coefficient is derived and shown to be estimable, so that it becomes possible to recover the effect of the generating process from the distorted coefficient.

Population intervention models in causal inference

Hubbard, A. E., van der Laan, M. J. — 2008-02-28

We propose a new causal parameter, which is a natural extension of existing approaches to causal inference such as marginal structural models. Modelling approaches are proposed for the difference between a treatment-specific counterfactual population distribution and the actual population distribution of an outcome in the target population of interest. Relevant parameters describe the effect of a hypothetical intervention on such a population and therefore we refer to these models as population intervention models. We focus on intervention models estimating the effect of an intervention in terms of a difference and ratio of means, called risk difference and relative risk if the outcome is binary. We provide a class of inverse-probability-of-treatment-weighted and doubly-robust estimators of the causal parameters in these models. The finite-sample performance of these new estimators is explored in a simulation study.

Empirical and counterfactual conditions for sufficient cause interactions

Vanderweele, T. J., Robins, J. M. — 2008-02-28

Sufficient-component causes are discussed within the deterministic potential outcomes framework so as to formalize notions of sufficient causes, synergism and sufficient cause interactions. Doing so allows for the derivation of counterfactual and empirical conditions for detecting the presence of sufficient cause interactions. The conditions are novel in that, unlike other conditions in the literature, they make no assumption about monotonicity. The conditions can also be generalized and the conditions for three-way sufficient cause interactions are given explicitly. The statistical tests derived for sufficient cause interactions are compared with and contrasted to interaction terms in standard statistical models.

Shared parameter models under random effects misspecification

Rizopoulos, D., Verbeke, G., Molenberghs, G. — 2008-02-28

A common objective in longitudinal studies is the investigation of the association structure between a longitudinal response process and the time to an event of interest. An attractive paradigm for the joint modelling of longitudinal and survival processes is the shared parameter framework, where a set of random effects is assumed to induce their interdependence. In this work, we propose an alternative parameterization for shared parameter models and investigate the effect of misspecifying the random effects distribution in the parameter estimates and their standard errors.

Predicting future responses based on possibly mis-specified working models

Cai, T., Tian, L., Solomon, S. D., Wei, L.J. — 2008-02-28

Under a general regression setting, we propose an optimal unconditional prediction procedure for future responses. The resulting prediction intervals or regions have a desirable average coverage level over a set of covariate vectors of interest. When the working model is not correctly specified, the traditional conditional prediction method is generally invalid. On the other hand, one can empirically calibrate the above unconditional procedure and also obtain its crossvalidated counterpart. Various large and small sample properties of these unconditional methods are examined analytically and numerically. We find that the K-fold crossvalidated procedure performs exceptionally well even for cases with rather small sample sizes. The new proposals are illustrated with two real examples, one with a continuous response and the other with a binary outcome.

Flexible generalized t-link models for binary response data

Kim, S., Chen, M.-H., Dey, D. K. — 2008-02-28

A critical issue in modelling binary response data is the choice of the links. We introduce a new link based on the generalized t-distribution. There are two parameters in the generalized t-link: one parameter purely controls the heaviness of the tails of the link and the second parameter controls the scale of the link. Two major advantages are offered by the generalized t-links. First, a symmetric generalized t-link with an unknown shape parameter is much more identifiable than a Student t-link with unknown degrees of freedom and a known scale parameter. Secondly, skewed generalized t-links with both unknown shape and scale parameters provide much more flexible and improved skewed link regression models than the existing skewed links. Various theoretical properties and attractive features of the proposed links are examined and explored in detail. An efficient Markov chain Monte Carlo algorithm is developed for sampling from the posterior distribution. The deviance information criterion measure is used for guiding the choice of links. The proposed methodology is motivated and illustrated by prostate cancer data.

Analysis of least absolute deviation

Chen, K., Ying, Z., Zhang, H., Zhao, L. — 2008-02-28

We develop a unified L₁-based analysis-of-variance-type method for testing linear hypotheses. Like the classical L₂-based analysis of variance, the method is coordinate-free in the sense that it is invariant under any linear transformation of the covariates or regression parameters. Moreover, it allows singular design matrices and heterogeneous error terms. A simple approximation using stochastic perturbation is proposed to obtain cut-off values for the resulting test statistics. Both test statistics and distributional approximations can be computed using standard linear programming. An asymptotic theory is derived for the method. Special cases of one- and multi-way analysis of variance and analysis of covariance models are worked out in detail. The main results of this paper can be extended to general quantile regression. Extensive simulations show that the method works well in practical settings. The method is also applied to a dataset from General Social Surveys.

Nonparametric regression using local kernel estimating equations for correlated failure time data

Yu, Z., Lin, X. — 2008-02-28

We study nonparametric regression for correlated failure time data. Kernel estimating equations are used to estimate nonparametric covariate effects. Independent and weighted-kernel estimating equations are studied. The derivative of the nonparametric function is first estimated and the nonparametric function is then estimated by integrating the derivative estimator. We show that the nonparametric kernel estimator is consistent for any arbitrary working correlation matrix and that its asymptotic variance is minimized by assuming working independence. We evaluate the performance of the proposed kernel estimator using simulation studies, and apply the proposed method to the western Kenya parasitaemia data.

Bayesian and frequentist confidence intervals arising from empirical-type likelihoods

Chang, I. H., Mukerjee, R. — 2008-02-28

For a general class of empirical-type likelihoods for the population mean, higher-order asymptotics are developed with a view to characterizing its members which allow, for any given prior, the existence of a confidence interval that has approximately correct posterior as well as frequentist coverage. In particular, it is seen that the usual empirical likelihood always allows such a confidence interval, while many of its variants proposed in the literature do not enjoy this property. An explicit form of the confidence interval is also given.

Probability estimation for large-margin classifiers

Wang, J., Shen, X., Liu, Y. — 2008-02-28

Large margin classifiers have proven to be effective in delivering high predictive accuracy, particularly those focusing on the decision boundaries and bypassing the requirement of estimating the class probability given input for discrimination. As a result, these classifiers may not directly yield an estimated class probability, which is of interest itself. To overcome this difficulty, this article proposes a novel method for estimating the class probability through sequential classifications, by using features of interval estimation of large-margin classifiers. The method uses sequential classifications to bracket the class probability to yield an estimate up to the desired level of accuracy. The method is implemented for support vector machines and -learning, in addition to an estimated Kullback–Leibler loss for tuning. A solution path of the method is derived for support vector machines to reduce further its computational cost. Theoretical and numerical analyses indicate that the method is highly competitive against alternatives, especially when the dimension of the input greatly exceeds the sample size. Finally, an application to leukaemia data is described.

Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models

Papaspiliopoulos, O., Roberts, G. O. — 2008-02-28

Inference for Dirichlet process hierarchical models is typically performed using Markov chain Monte Carlo methods, which can be roughly categorized into marginal and conditional methods. The former integrate out analytically the infinite-dimensional component of the hierarchical model and sample from the marginal distribution of the remaining variables using the Gibbs sampler. Conditional methods impute the Dirichlet process and update it as a component of the Gibbs sampler. Since this requires imputation of an infinite-dimensional process, implementation of the conditional method has relied on finite approximations. In this paper, we show how to avoid such approximations by designing two novel Markov chain Monte Carlo algorithms which sample from the exact posterior distribution of quantities of interest. The approximations are avoided by the new technique of retrospective sampling. We also show how the algorithms can obtain samples from functionals of the Dirichlet process. The marginal and the conditional methods are compared and a careful simulation study is included, which involves a non-conjugate model, different datasets and prior specifications.

Two-stage sampling from a prediction point of view when the cluster sizes are unknown

Bjornstad, J. F., Ytterstad, E. — 2008-02-28

We consider the problem of estimating the population total in two-stage cluster sampling when cluster sizes are known only for the sampled clusters, making use of a population model arising from a variance component model. The problem can be considered as one of predicting the unobserved part Z of the total, and the concept of predictive likelihood is studied. Prediction intervals and a predictor for the population total are derived for the normal case, based on predictive likelihood. For a more general distribution-free model, by application of an analysis of variance approach instead of maximum likelihood for parameter estimation, the predictor obtained from the predictive likelihood is shown to be approximately uniformly optimal for large sample size and large number of clusters, in the sense of uniformly minimizing the mean-squared error in a partially linear class of model-unbiased predictors. Three prediction intervals for Z based on three similar predictive likelihoods are studied. For a small number n₀ of sampled clusters, they differ significantly, but for large n₀, the three intervals are practically identical. Model-based and design-based coverage properties of the prediction intervals are studied based on a comprehensive simulation study. The simulation study indicates that for large sample sizes, the coverage measures achieve approximately the nominal level 1 – and are slightly less than 1 – for moderately large sample sizes. For small sample sizes, the coverage measures are about 1 – 2, being raised to 1 – for a modified interval based on the distribution.

Predicting cumulative incidence probability by direct binomial regression

Scheike, T. H., Zhang, M.-J., Gerds, T. A. — 2008-02-28

We suggest a new simple approach for estimation and assessment of covariate effects for the cumulative incidence curve in the competing risks model. We consider a semiparametric regression model where some effects may be time-varying and some may be constant over time. Our estimator can be implemented by standard software. Our simulation study shows that the estimator works well and has finite-sample properties comparable with the subdistribution approach. We apply the method to bone marrow transplant data and estimate the cumulative incidence of death in complete remission following a bone marrow transplantation. Here death in complete remission and relapse are two competing events.

Nonparametric estimation of bivariate failure time associations in the presence of a competing risk

Bandeen-Roche, K., Ning, J. — 2008-02-28

Most research on the study of associations among paired failure times has either assumed time invariance or been based on complex measures or estimators. Little has accommodated competing risks. This paper targets the conditional cause-specific hazard ratio, henceforth called the cause-specific cross ratio, a recent modification of the conditional hazard ratio designed to accommodate competing risks data. Estimation is accomplished by an intuitive, nonparametric method that localizes Kendall's tau. Time variance is accommodated through a partitioning of space into ‘bins’ between which the strength of association may differ. Inferential procedures are developed, small-sample performance is evaluated, and the methods are applied to the investigation of familial association in dementia onset.

Nonparametric estimation of cause-specific cross hazard ratio with bivariate competing risks data

Cheng, Y., Fine, J. P. — 2008-02-28

We propose an alternative representation of the cause-specific cross hazard ratio for bivariate competing risks data. The representation leads to a simple plug-in estimator, unlike an existing ad hoc procedure. The large sample properties of the resulting inferences are established. Simulations and a real data example demonstrate that the proposed methodology may substantially reduce the computational burden of the existing procedure, while maintaining similar efficiency properties.

A note on path-based variable selection in the penalized proportional hazards model

Zou, H. — 2008-02-28

We propose an efficient and adaptive shrinkage method for variable selection in the Cox model. The method constructs a piecewise-linear regularization path connecting the maximum partial likelihood estimator and the origin. Then a model is selected along the path. We show that the constructed path is adaptive in the sense that, with a proper choice of regularization parameter, the fitted model works as well as if the true underlying submodel were given in advance. A modified algorithm of the least-angle-regression type efficiently computes the entire regularization path of the new estimator. Furthermore, we show that, with a proper choice of shrinkage parameter, the method is consistent in variable selection and efficient in estimation. Simulation shows that the new method tends to outperform the lasso and the smoothly-clipped-absolute-deviation estimators with moderate samples. We apply the methodology to data concerning nursing homes.

Testing hypotheses in order

Rosenbaum, P. R. — 2008-02-28

In certain circumstances, one wishes to test one hypothesis only if certain other hypotheses have been rejected. This ordering of hypotheses simplifies the task of controlling the probability of rejecting any true hypothesis. In an example from an observational study, a treated group is shown to be further from both of two control groups than the two control groups are from each other.

A Note on repeated p-values for group sequential designs

Posch, M., Wassmer, G., Brannath, W. — 2008-02-28

One-sided confidence intervals and overall p-values for group-sequential designs are typically based on a sample space ordering which determines both the overall p-value and the corresponding confidence bound. Accordingly, the strength of evidence against the null hypothesis is consistently measured by both quantities such that the order of the p-values of two distinct sample points is consistent with the order of the respective confidence bounds. An exception is the commonly used repeated p-values and repeated confidence intervals. We show that they are not ordering-consistent in the above sense and propose an alternative repeated p-value which is ordering-consistent and has the monitoring property of the classical repeated p-value in being valid even when deviating from the prefixed stopping rule.

Asymptotic inference for a nonstationary double AR(1) model

Ling, S., Li, D. — 2008-02-28

We investigate the nonstationary double ar(1) model, where > 0, > 0, the _t are independent standard normal random variables and Elog | + _t| >= 0. We show that the maximum likelihood estimator of (, ) is consistent and asymptotically normal. Combination of this result with that in Ling ([11]) for the stationary case gives the asymptotic normality of the maximum likelihood estimator of for any in the real line, with a root-n rate of convergence. This is in contrast to the results for the classical ar(1) model, corresponding to = 0.