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SUMMARY The bias of several logit estimators and their corresponding variance estimators is investigated in small samples. Their use in quantal bioassay is similarly explored. The logit transformation has been suggested in the analysis of higher dimensional contingency tables, by Woolf (1954) and many others more recently, and also in estimating the parameters of the logistic function in the quantal bioassay problem (Berkson, 1944, 1953). Various modifications of the logit have been suggested by Berkson (1953), Haldane (1955), Anscombe (1956), Tukey, mentioned by Anscombe (1956), and Hitchcock (1962). Modifications of its usual variance estimator have been proposed by Haldane, Goodman (1964) and Gart (1966). More recently Goodman, in an unpublished paper, has derived several further modifications of his estimator. In this paper we present a numerical comparison of the bias of these estimators and give conditions under which one or the other may be preferred. The use of logits in the quantal bioassay problem is briefly explored with particular reference to the asymptotic results of Hitchcock regarding the bias of the estimators.

This paper deals with the design of the likelihood continual reassessment method, which is an increasingly widely used model-based method for dose-finding studies. It is common to implement the method in a two-stage approach, whereby the model-based stage is activated after an initial sequence of patients has been treated. While this two-stage approach is practically appealing, it lacks a theoretical framework, and it is often unclear how the design components should be specified. This paper develops a general framework based on the coherence principle, from which we derive a design calibration process. A real clinical-trial example is used to demonstrate that the proposed process can be implemented in a timely and reproducible manner, while offering competitive operating characteristics. We explore the operating characteristics of different models within this framework and show the performance to be insensitive to the choice of dose-toxicity model.

This paper considers benchmarking issues in the context of small area estimation. We find optimal estimators within the class of benchmarked linear estimators under linear constraints. This extends existing results for external and internal benchmarking, and also links the two. Necessary and sufficient conditions for self-benchmarking are found for an augmented model. Most results of this paper are found using ideas of orthogonal projection Copyright 2013, Oxford University Press.

This paper derives the asymptotic null distribution of a quasilikelihood ratio test statistic for an autoregressive moving average model against its threshold extension. The null hypothesis is that of no threshold, and the error term could be dependent. The asymptotic distribution is rather complicated, and all existing methods for approximating a distribution in the related literature fail to work. Hence, a novel bootstrap approximation based on stochastic permutation is proposed in this paper. Besides being robust to the assumptions on the error term, our method enjoys more flexibility and needs less computation when compared with methods currently used in the literature. Monte Carlo experiments give further support to the new approach, and an illustration is reported. Copyright 2011, Oxford University Press.

A Gaussian copula regression model gives a tractable way of handling a multivariate regression when some of the marginal distributions are non-Gaussian. Our paper presents a general Bayesian approach for estimating a Gaussian copula model that can handle any combination of discrete and continuous marginals, and generalises Gaussian graphical models to the Gaussian copula framework. Posterior inference is carried out using a novel and efficient simulation method. The methods in the paper are applied to simulated and real data.

This paper considers the problem of using observational data in the presence of selection bias to identify causal effects in the framework of linear structural equation models. We propose a criterion for testing whether or not observed statistical dependencies among variables are generated by conditioning on a common response variable. When the answer is affirmative, we further provide formulations for recovering the covariance matrix of the whole population from that of the selected population. The results of this paper provide guidance for reliable causal inference, based on the recovered covariance matrix obtained from the statistical information with selection bias.

SUMMARY Properties of the periodogram are seldom studied in the setting of nonparametric regression, although that is the context in which the periodogram is widely applied in astronomy. There it is a competitor with more recent least-squares methods. The periodogram has the advantage of providing significant graphical insight into statistical and numerical aspects of the problem. However, as we show in the present paper, it also has drawbacks. The estimator that it produces has somewhat higher variance than its least-squares counterpart, and a periodogram-based approach is more prone to suffer difficulties caused by periodicity of the observation schedule. While the periodogram remains a very attractive tool, the information provided in this paper allows users to assess more readily the extent to which it can be relied upon in a nonparametric setting. This aspect of our contributions is discussed theoretically and illustrated by numerical studies involving a real dataset.

SUMMARY This paper is concerned with a paradox associated with parameter estimation in the presence of nuisance parameters. In a statistical model with unknown nuisance parameters, the efficiency of an estimator of a parameter usually increases when the nuisance para meters are known. However the opposite phenomenon can sometimes occur. In this paper, we elucidate the occurrence of this paradox by examining estimating functions. In particular, we focus on the projected estimating function, which is defined by the projection of the score function on to a given estimating function. A sufficient condition for the paradox to occur is the orthogonality of the two components of the projected estimating functions corresponding to parameters of interest and nuisance parameters. In addition, a numerical assessment is conducted in the context of a simple model to investigate the improvement of the asymptotic efficiency of estimators.

This paper addresses the task of locating and identifying an unknown number of objects of different types in an image. Baddeley & Van Lieshout (1993) advocate marked point processes as object priors, whereas Grenander & Miller (1994) use deformable template models. In this paper elements of both approaches are combined to handle scenes containing variable numbers of objects of different types, using reversible jump Markov chain Monte Carlo methods for inference (Green, 1995). The naive application of these methods here leads to slow mixing and we adapt the model and algorithm in tandem in proposing three strategies to deal with this. The first two expand the model space by introducing an additional 'unknown' object type and the idea of a variable resolution template. The third strategy, utilising the first two, augments the algorithm with classes of updates which provide intuitive transitions between realisations containing different numbers of cells by splitting or merging nearby objects.

SUMMARY We put forward a state space model where the unobservable state variable can be any Gaussian stochastic process. We discuss both maximum likelihood estimation and Bayesian inference for this generalised model. The methodology developed in this paper is particularly important for the class of long memory plus noise models. Armed with the simulation smoother introduced in this paper, we can estimate a class of non-Gaussian measurement time series models with long memory in the state equation.

SUMMARY This paper presents a class of Mantel & Haenszel (1959) type statistics for testing independence between pairs of failure times. This class is extended to include a weight function in order to enhance test power. Large sample distributions of these statistics are given under the null hypothesis of independence. The paper also presents a simplified test, the covariance test, which is asymptotically equivalent to the Mantel-Haenszel-type test. In order to account for heterogeneity among subjects, both tests are generalised to include subject-specific covariates. In a simulation study, the weighted Mantel-Haenszel and covariance tests are shown to lead to an increase in power for detecting dependence, compared to unweighted tests and to a score test (Commenges & Andersen, 1995) that is based on a random effects model. An example is used to illustrate the methodology.

SUMMARY This paper advocates likelihood analysis for regression models with measurement errors in explanatory variables, for data problems in which the relevant distributions can be adequately modelled. Although computationally difficult, maximum likelihood estimates are more efficient than those based on first and second moment assumptions, and likelihood ratio inferences can be substantially better than those based on asymptotic normality of estimates. The EM algorithm is presented as a straightforward approach for likelihood analysis of normal linear regression with normal explanatory variables, and normal replicate measurements. Likelihood methods for regression analysis with explanatory variable measurement error have received little attention relative to methods based on first and second moment assumptions, largely because of computational difficulties, uncertainty about the robustness of likelihood methods, and the belief that the simpler methods perform just as well in practice. With improved computational tools for likelihood analysis, as discussed, for example, by Tanner (1993), it is worthwhile to reinvestigate these issues. In particular, the common belief that the simpler methods perform just as well as likelihood methods may be naive. First, the comparison is usually cast in terms of efficiency, without regard to test and confidence interval validity. Likelihood ratio tests and confidence intervals can be substantially better than tests and confidence intervals based on estimates and standard errors of the commonly-used method of moments approaches. The sampling distributions of the estimators are very often skewed, especially if the measurement errors are large. Furthermore, knowledge about the relative efficiency of the moment methods comes from a small number of efficiency comparisons from simulations of isolated situations, in papers promoting these simpler methods (Carroll, Rupert & Stefanski, 1995, ? 7.1). An important and often overlooked issue is that the moment methods may be especially inappropriate when the distribution of the true covariates is skewed (Pierce et al., 1992). Section 2 discusses computational approaches and difficulties associated with likelihood analysis in the measurement error problem generally. Section 3 illustrates the EM algorithm (Dempster, Laird & Rubin, 1977) for likelihood analysis of normal linear regression with a normally-distributed explanatory variable measured with normally-distributed error, when there are replicate measurements for at least some of the cases. Sections 5 and 6

SUMMARY The primary aim of this paper is to show how graphical models can be used as a mathematical language for integrating statistical and subject-matter information. In particular, the paper develops a principled, nonparametric framework for causal inference, in which diagrams are queried to determine if the assumptions available are sufficient for identifying causal effects from nonexperimental data. If so the diagrams can be queried to produce mathematical expressions for causal effects in terms of observed distributions; otherwise, the diagrams can be queried to suggest additional observations or auxiliary experiments from which the desired inferences can be obtained.

SUMMARY This paper offers a study in the application of Bayes linear methods. We outline a general approach to the analysis of mean effects for grouped multivariate repeated measurement studies, based upon partial belief specification. We suggest a method for coherent partial prior specification for such structures, based on moment evaluations for exchangeable data. We describe the general collection of interpretive and diagnostic tools termed 'Bayes linear methods', and suggest a simple criterion for trial design. The theory is illustrated by analysis of a crossover trial concerned with side effects of kidney dialysis. Bayes linear methodology is an alternative to full Bayes analysis which may be used when we wish to introduce our prior information explicitly into the analysis but, due to the complexity of the problem, we are unwilling to make a full prior specification. The methodology incorporates general approaches to modelling and fitting supported by a systematic collection of interpretive and diagnostic tools. This paper offers a study in the application of this methodology, by developing the Bayes linear analysis of mean effects in grouped multivariate repeated measurement studies, with particular emphasis on systematic methods for building meaningful partial prior specifications. We apply our approach to the study of crossover trials and analyse a particular trial for kidney dialysis. In ? 2, we discuss the role of partial prior specification and describe a general approach for coherent partial prior specification for grouped multivariate repeated measurement studies, based on moment evaluations for exchangeable data. This approach is applied, with discussion, in ? 3 to the specification of beliefs for crossover trials. In ? 4, we describe and specify beliefs for a particular crossover trial concerned with side effects of kidney dialysis. In ? 5, we summarize basic interpretive and diagnostic features of Bayes linear methodology. In ? 6, we apply the methodology to analyse the data from the crossover trial for kidney dialysis. In ? 7 we suggest and apply a simple criterion for trial design. Section 8 contains concluding comments on our approach.

SUMMARY In a sequence of papers, McLeish (1984), Hutton & Nelson (1986) and Godambe & Heyde (1987) have motivated the use of the quasi-score function as an estimating function for semiparametric models by showing that it is the projection of the true score into a class of unbiased estimating functions. Thus the quasi-score approximates the score within this class of functions. In this paper, we introduce a class of inference functions for independent observations that contain analogues of likelihood ratios under semiparametric assumptions. This class of functions, which contains the semiparametric estimating functions as a subspace, is built through the use of tensor products for models with independent observations. We consider the projection of the likelihood ratios into this class of inference functions. It is shown that this projected likelihood is first order equivalent to the quasi-likelihood.

SUMMARY New methods are proposed which allow Bayes factors to be computed for hypotheses which impose nonlinear restrictions on the parameters. Projection methods are used to induce the prior distribution over the restricted parameter space which is required for computation of the Bayes factor. Various distance metrics are introduced to define the projection, including a utility-based metric which gives Kullback-Leibler divergence as a special case. Draws from the restricted and unrestricted prior distributions are used to construct marginal distributions of the likelihood which is shown to have additional diagnostic value over and above the Bayes factor. These methods are applied to hypotheses Statisticians are frequently asked to entertain hypotheses which are nonlinear in the model parameters. Nonlinear hypotheses arise both from subject matter theory as well as from the process of model selection. To address nonlinear hypotheses, researchers use frequentist methods which are usually implemented using asymptotic approximations. Berger (1987), Zellner (1971, 1978, 1984) and others have discussed the weaknesses of the classical significance testing approach and suggest Bayesian alternatives such as odds ratios. The odds ratio is the ratio of posterior probabilities of two competing hypotheses and is the product of the prior odds ratio and the Bayes factor. The odds ratio has the conceptual advantages of providing a natural probability metric for assessing the strength of data and prior beliefs. Because of the difficulties of assessing priors under the nonlinear restrictions and the computation of the required integrals, few Bayes factors have been computed for nonlinear hypotheses; see Lindley (1988) for an exception. In the present paper, we outline a general method for computing Bayes factors which is flexible, conceptually appealing and computationally tractable. The literature on Bayes factors starts with the work of Jeffreys (1961). Other references include Berger & Delampady (1987), Dickey (1971), Good (1985), Kass & Vaidyanathan (1992), Smith & Spiegelhalter (1980) and Zellner (1971, 1978, 1984). These papers deal with the testing of linear restrictions on the parameter space, often in the context of linear models. Even the widely-used Schwarz (1978) model selection criterion considers only linear restrictions in models of the exponential family form. The key difficulty which has kept researchers from developing Bayes factors for nonlinear hypotheses is the

Summary When modelling survival data, it is common to assume that the survival time $T$ is conditionally independent of the censoring time $C$ given a set of covariates. However, there are numerous situations in which this assumption is not realistic. The goal of this paper is therefore to develop a semiparametric normal transformation model which assumes that, after a proper nonparametric monotone transformation, the vector $(T, C)$ follows a linear model, and the vector of errors in this bivariate linear model follows a standard bivariate normal distribution with a possibly nondiagonal covariance matrix. We show that this semiparametric model is identifiable, and propose estimators of the nonparametric transformation, the regression coefficients and the correlation between the error terms. It is shown that the estimators of the model parameters and the transformation are consistent and asymptotically normal. We also assess the finite-sample performance of the proposed method by comparing it with an estimation method under a fully parametric model. Finally, our method is illustrated using data from the AIDS Clinical Trial Group 175 study.

Summary Cross-sectional sampling is often used when investigating inter-event times, resulting in left-truncated and right-censored data. In this paper, we consider a semiparametric truncation model in which the truncating variable is assumed to belong to a certain parametric family. We examine two methods of estimating both the truncation and the lifetime distributions. We obtain asymptotic representations of the estimators for the lifetime distribution and establish their weak convergence. Both of the proposed estimators perform better than Wang’s (1991) nonparametric maximum likelihood estimator in terms of the integrated mean squared error, when the parametric family for the truncation is sufficiently close to its true distribution. The full likelihood approach is preferable to the conditional likelihood approach in estimating the lifetime distribution, though not necessarily the truncation distribution. In an application to Alzheimer’s disease data, hypothesis tests reject the uniform truncation distribution, but several other parametric models lead to similar behaviour of the truncation and lifetime distributions after disease onset.

Summary This paper proposes a new set of testable implications for the instrumental variable independence assumption for discrete treatment, but unrestricted outcome and instruments: generalized instrumental inequalities. When outcome and treatment are both binary, but instruments are unrestricted, we show that the generalized instrumental inequalities are necessary and sufficient to detect all observable violations of the instrumental variable independence assumption. To test the generalized instrumental inequalities, we propose an approach combining a sample splitting procedure and an inference method for intersection bounds. This idea allows one to easily implement the test using existing Stata packages. We apply our proposed strategy to assess the validity of the instrumental variable independence assumption for various instruments used in the returns to college literature.

Summary Consider a high-dimensional linear regression problem, where the number of covariates is larger than the number of observations and the interest is in estimating the conditional variance of the response variable given the covariates. A conditional and an unconditional framework are considered, where conditioning is with respect to the covariates, which are ancillary to the parameter of interest. In recent papers, a consistent estimator was developed in the unconditional framework when the marginal distribution of the covariates is normal with known mean and variance. In the present work, a certain Bayesian hypothesis test is formulated under the conditional framework, and it is shown that the Bayes risk is a constant. This implies that no consistent estimator exists in the conditional framework. However, when the marginal distribution of the covariates is normal, the conditional error of the above consistent estimator converges to zero, with probability converging to one. It follows that even in the conditional setting, information about the marginal distribution of an ancillary statistic may have a significant impact on statistical inference. The practical implication in the context of high-dimensional regression models is that additional observations where only the covariates are given are potentially very useful and should not be ignored. This finding is most relevant to semi-supervised learning problems where covariate information is easy to obtain.

SUMMARY An important and often discussed research problem in statistics is how to compare several populations; examples arise in medical science, engineering, finance and other fields. Often population means or medians are compared. However, one population may have a higher mean income, for example, because of a small number of super-rich individuals; the mean therefore may not reflect the wealth of the general population. Instead, an index of the degree of stochastic dominance of one population over another would better reflect their relative wealth. Currently, we can estimate such an index under restrictive conditions, but there is no generic estimator with a known asymptotic distribution. In this paper, we suggest linking the populations via the density ratio model. Under this model, we develop an empirical likelihood estimator and establish its asymptotic normality. In addition, we improve the estimation efficiency by examining the similarities between the populations. Furthermore, we provide a valid bootstrap method for hypothesis testing and the construction of confidence intervals. Simulation experiments show that the proposed estimator substantially improves the estimation efficiency and power of the test, and leads to confidence intervals with satisfactorily precise coverage probabilities. It is also robust with respect to mild model misspecification. Two examples are given to demonstrate the usefulness of both the method and the concept.

SUMMARY Consider a semiparametric model indexed by a Euclidean parameter of interest and an infinite-dimensional nuisance parameter. In many applications, pseudolikelihood provides a convenient way to infer the parameter of interest, where the nuisance parameter is replaced by a consistent estimator. The purpose of this paper is to establish the asymptotic behaviour of the pseudolikelihood ratio statistic under semiparametric models. In particular, we consider testing the hypothesis that the parameter of interest lies on the boundary of its parameter space. Under regularity conditions, we establish the equivalence between the asymptotic distributions of the pseudolikelihood ratio statistic and a likelihood ratio statistic for a normal mean problem with a misspecified covariance matrix. This result holds when the nuisance parameter is estimated at a rate slower than the usual rate in parametric models. We study three examples in which the asymptotic distributions are shown to be mixtures of chi-squared variables. We conduct simulation studies to examine the finite-sample performance of the pseudolikelihood ratio test.

SUMMARY This paper compares the use of two posterior probability methods to deal with outliers in linear models. We show that putting together diagnostics that come from the mean-shift and variance-shift models yields a procedure that seems to be more effective than the use of probabilities computed from the posterior distributions of actual realized residuals. The relation of the suggested procedure to the use of a certain predictive distribution for diagnostics is derived. Two main procedures for outlier detection have emerged when using the Bayesian approach. The first of these confines itself to postulating a null model for the generation of the data and then seeks identification methods for outliers with no alternative model to the null entertained. Examples of work in the category are (i) use of the predictive distribution for detection, (ii) using the posterior probabilities of various unobserved perturbations, and (iii) looking at the change in a posterior of interest when some observations are deleted. These methods will be discussed in ? 3 of this paper, after the basic model is set out in ? 2. The second procedure takes into account an alternative model for the generation of a subset of the sample. As examples, various authors have proposed and utilized the mean-shift model and the variance-inflation model. These methods are discussed in ? 4 of this paper. We show in this paper that these procedures can be classified as those which identify outliers by looking at some functions of the predictive density ordinate, and those that look at the posterior probabilities of the unobserved residuals. The first seems to be effective in overcoming the problem of high leverage outliers, in contrast to the latter, where outliers are masked by high leverage. These results are illustrated by examples in ? 5.

In causal inference the effect of confounding may be controlled using regression adjustment in an outcome model, propensity score adjustment, inverse probability of treatment weighting or a combination of these. Approaches based on modelling the treatment assignment mechanism, along with their doubly robust extensions, have been difficult to motivate using formal likelihood-based or Bayesian arguments, as the treatment assignment model plays no part in inferences concerning the expected outcomes. On the other hand, forcing dependency between the outcome and treatment assignment models by allowing the former to be misspecified results in loss of the balancing property of the propensity scores and the loss of any double robustness. In this paper, we explain in the framework of misspecified models why doubly robust inferences cannot arise from purely likelihood-based arguments. As an alternative to Bayesian propensity score analysis, we propose a Bayesian posterior predictive method for constructing doubly robust estimation procedures by incorporating the inverse treatment assignment probabilities as importance sampling weights in Monte Carlo integration.

Interval censoring arises frequently in clinical, epidemiological, financial, and sociological studies, where the event or failure of interest is known only to occur within an interval induced by periodic monitoring. We formulate the effects of potentially time-dependent covariates on the interval-censored failure time through a broad class of semiparametric transformation models that encompasses proportional hazards and proportional odds models. We consider nonparametric maximum likelihood estimation for this class of models with an arbitrary number of monitoring times for each subject. We devise an EM-type algorithm that converges stably, even in the presence of time-dependent covariates, and show that the estimators for the regression parameters are consistent, asymptotically normal, and asymptotically efficient with an easily estimated covariance matrix. Finally, we demonstrate the performance of our procedures through extensive simulation studies and application to an HIV/AIDS study conducted in Thailand., Comment: This paper has been withdrawn by the author due to some errors in the proofs

SUMMARY The standard method for the comparison of two or more sample covariance matrices is the likelihood ratio test. The purpose of the present paper is to show how this test can be made more informative by hierarchically partitioning the test statistic into three components. Confronted with the comparison of several sample covariance matrices, most statisticians will probably consider using the likelihood ratio test (Seber, 1984, p. 449). The only reservation that needs to be made about the use of the test is its known sensitivity to nonnormality in the data (Layard, 1974; Olson, 1974; Manly, 1986). In the present paper we assume that the data are consistent with multivariate normality. The purpose of the present paper is to show that the test statistic for the standard likelihood ratio test can be partitioned into three components. These reflect differences between the covariance matrices due to: (a) matrices being proportional, (b) changing variances, and (c) changing correlations. The components can be tested for significance

SUMMARY This paper considers from a Bayesian viewpoint inferences about the size of a closed animal population from data obtained by a multiple-recapture sampling scheme. The method developed enables prior information about the population size and the catch probabilities to be utilized to produce considerable improvements in certain cases on ordinary maximum likelihood methods. Several ways of expressing such prior information are explored and a practical example of the uses of these ways is given. The main result of the paper is an approximation to the posterior distribution of sample size that exhibits the contributions made by the likelihood and the prior ideas. The multiple-recapture sampling scheme involves taking samples from a population of animals, at each stage counting the number of marked animals in the sample, marking the previously unmarked animals and returning the sample to the population. The literature is reviewed by Cormack (1968) and the papers most relevant to our problem are those of Chapman (1952) and Darroch (1958). In these papers, maximum likelihood estimates for the population parameters and variances for these estimates are given. Our approach differs from previous ones by introducing prior information about the population size, and about the propensities of the animals to be captured, called 'catch probabilities' by some previous authors. The incorporation of these propensities is a feature not shared by many previous approaches. Darroch (1958) found their introduction into his model does not change the maximum likelihood estimates of the population size. Our approach will show that strong prior knowledge about the catch probabilities can greatly affect inference about the population size, and it is in this respect that the greatest difference from previous approaches lies. Although it is recognized that models of an open population incorporating death and immigration such as those of Jolly (1965) and Seber (1965) are more practically realistic, we feel that our method, which deals only with a closed population, is worth investigating as a step towards providing a Bayesian treatment of the open population problem.

SUMMARY Bootstrap methods are not inherently conditional, but they can be made so by appropri- ate stratification of the simulated samples which bootstrap produces. We show how stratification can work in a bootstrap analysis of mean-shift in Nile river flow data. The results are compared with both parametric and semiparametric likelihood analyses. The paper ends with some general remarks on conditional bootstraps. Bootstrap methods (Efron, 1982) are simulation methods for assessing frequency properties of statistical analyses, the simulation model being that which is fitted to the actual data. Often the simulation involves Monte Carlo, rather than theoretical, evaluation. Unfortunately the bootstrap as usually described is inherently unconditional; Rubin's (1981) discussion bears on this point. However, if the bootstrap simulation results can be properly stratified, then it is possible to estimate relevant conditional probabilities. Stratification can sometimes be effected in the model itself, as when fixing the values of sample sizes or explanatory variables. Otherwise one must run the unconditional bootstrap and stratify its output. In the present paper we illustrate the latter situation in the analysis of a mean-shift model. Section 2 describes the application and its analysis by parametric methods. Sections 3 and 4 produce alternative bootstrap analyses, the first of which uses the stratification of an unconditional Monte Carlo simulation; the second bootstrap analysis involves estimation of a likelihood function. The final section contains some brief general remarks about conditional bootstraps.

A nonparametric measure of intraclass correlation based on the probability of certain types of concordances among the observations and which can be estimated from ranked data is proposed. One estimator whose variance can be estimated in an unbiased way is shown to be asymptotically normally distributed. Properties of the measure and its estimator are studied for a normal model; in particular the method is shown to provide a relatively powerful test of the null hypothesis of zero intraclass correlation in a normal population. The intraclass correlation coefficient is often used as a measure of the degree to which individuals from the same family resemble one another in some variable such as height or weight. From a random sample of families and a random selection of individuals from each family, both the between and within family components of variance can be estimated and an estimate of the intraclass correlation obtained. When the observations are ranks this approach is not feasible since direct estimates of the components of variance are not available. An example occurs in a study of aggressiveness in male red grouse (Moss, Watson & Parr, 1974). One measure of aggressiveness of an individual bird is a dominance rank, the position of that bird in the dominance hierarchy of a given group of birds. Comparison of the degree of familial resemblance in different populations is complicated by the fact that the data are ranks and that dominance hierarchies can be variable in size with different family sizes. The development of standard test statistics, such as the Kruskal-Wallis statistic (Kruskal, 1952), as measures of intraclass correlation is unsatisfactory because of the complicated dependence on the structure of the sample; see ? 6. In this paper a measure of intraclass correlation is proposed which overcomes this problem. It is independent of the structure of the sample and it can be estimated in an unbiased way from ranked data. The interclass rank correlation coefficients, Kendall's Xr and Spearman's p (Kendall, 1955), can be interpreted in terms of probabilities of concordances among the observations from a bivariate distribution. In response to Hill (1974), Kerridge (1975) has used this probabilistic interpretation of r in an analysis of Football League results. The measure presented in the present paper is based on concordances among observations from a population in which individuals occur in families. Its properties for a general class of populations are discussed and some particular results for a one-way classification model with normally distributed effects are presented. The power of the associated test is assessed for a normal model for a range of sample structures.

The purpose of the present paper is to make a quantitative comparison of the Shiryayev-Roberts and Page procedures. We do this in the context of continuous time in order to use the machinery of diffusion processes to perform explicitly certain calculations, which seem impossible in discrete time. Although the continuous time results are not especially good approximations to the corresponding quantities in discrete time, they provide very useful comparative information on which to base selection of a stopping rule. This paper is organized as follows. The Shiryayev-Roberts process is defined and shown to be a novel diffusion process with some surprising properties. We also specify more precisely the basis for our comparison of the two procedures and give the results of some elementary calculations. These developments contain an asymptotic evaluation. We define a modification of our basic procedure and give an asymptotic evaluation of its average run length. Numerical comparisons and a discussion of their significance are contained next. Our conclusions are roughly these. In simple situations where the two procedures can be directly compared, neither seems dramatically superior to the other. However, the Shiryayev-Roberts procedure is more easily adapted to complex circumstances and consequently warrants additional study.

SUMMARY This paper is devoted to a study of the unmatched cross-product estimator of relative risks in a 1 :, matched case-control study. It is shown that this estimator is consistent for the 'case-adjusted relative risk' in rare disease settings. The cross-product estimator has a much wider range of applicability than the classical matched methods, which require a conditionally independent binomial model with a common odds ratio. In general, the methods estimate different parameters. However, in the special situation where the risk factor classification of cases and matched controls are unconditionally independent, the two definitions of relative risk do agree. When this does occur, the proposed method is asymptotically superior to the classical methods. This paper also considers the optimal number of matched controls per case and sample size selection.

SUMMARY Lancaster's additive approach to absence of interaction in contingency tables has been used quite frequently in agriculture in recent years, sometimes incorrectly. In this paper, we wish to compare the frequency distributions of response for a number of treatment groups which are factorially structured. An additive model is proposed which enables us to examine the role of the individual factors in explaining any differences among responses. Test statistics are developed, and the theory is illustrated with a number of agricultural experiments. The analysis of multidimensional contingency tables has received a good deal of attention in recent years. The most commonly used analyses are based on log linear models and the multiplicative definition of zero higher-order interactions. However, there has also been some interest in the additive model developed by Lancaster (1949, 1951, 1971). The two approaches have been compared by Darroch (1974), Murphy (1978) and O'Neill (1982). Although they often lead to the same conclusions there may be instances when large differences arise. This paper considers the partition of x2 in three-way factorial experiments involving frequency data, in which one of the factors is a response variable and the other two factors have bivariate marginal totals fixed by the experimental design. Such experiments are common in agriculture, where for example a number of animals are allocated randomly to treatment groups factorially structured and then classified according to their response. Lancaster's method is based on an expansion of the theoretical probabilities using orthonormal functions which are simply related to the columns of an orthogonal matrix. The partitioning produces main effects and interactions in a similar way to the traditional analysis of variance, with a further decomposition into single degrees of freedom corresponding to linear and nonlinear trends, or to comparisons within and between treatment groups. However the method has a few complications. First, even under the null hypothesis of no second-order interaction, the test statistic is not asymptotically x2 if first-order interactions are present. This problem was pointed out by many authors including Plackett (1962), Darroch (1962), Goodman (1964), Lancaster

SUMMARY This paper considers unimodality properties of certain likelihoods derived from grouped data. Three main types of model are considered: exponential families, linear models and empirical Bayes models. Partial answers are offered to questions raised in recent papers and areas needing further clarification are indicated. The paper shows that a link between the models is a theorem given by Prekopa on strongly unimodal probability densities and indicates ways of using the theorem to formulate models for analysing grouped multimodal univariate data.

SUMMARY We discuss the use of normal order statistics plots, based on deviance residuals, to check distributional assumptions in regression models. Continuous and discrete error distributions are considered, as are censored data. Misspecified error distributions and discrimination between competing models are discussed, with an example. Residual plots to detect inadequacies in normal linear regression models have a long history. An example is the normal scores or rankit plot: a plot of ordered residuals against normal order statistics, which is used to detect outliers and to check distributional assumptions. Under- or overdispersion in such a plot may also indicate a misspecified systematic component of the model. The purpose of this paper is to discuss such plots for regressions with nonnormal errors, such as generalized linear models, for which deviance residuals are commonly used. Deviance residuals are known to be approximately normal in many cases. In this paper we briefly describe the properties of rankit plots based on them for continuous distributions, outline analogous results for discrete and censored data, and describe a method for discriminating between competing models with different error distributions. Most roads lead to Rome for the normal distribution in the sense that many definitions of residuals are functions of (y - jt)/ o. Not so for other distributions, for which various

SUMMARY This paper considers a wide class of latent structure models. These models can serve as possible explanations of the observed relationships among a set of m manifest polytomous variables. The class of models considered here includes both models in which the parameters are identifiable and also models in which the parameters are not. For each of the models considered here, a relatively simple method is presented for calculating the maximum likeli- hood estimate of the frequencies in the m-way contingency table expected under the model, and for determining whether the parameters in the estimated model are identifiable. In addition, methods are presented for testing whether the model fits the observed data, and for replacing unidentifiable models that fit by identifiable models that fit. Some illus- trative applications to data are also included. This paper deals with the relationships among m polytomous variables, i.e. with the analysis of an m-way contingency table. These m variables are manifest variables in that, for each observed individual in a sample, his class with respect to each of the m variables is observed. We also consider here polytomous variables that are latent in that an individ- ual's class with respect to these variables is not observed. The classes of a latent variable will be called latent classes. Consider first a 4-way contingency table which cross-classifies a sample of n individuals with respect to four manifest polytomous variables A, B, C and D. If there is, say, some latent dichotomous variable X, so that each of the n individuals is in one of the two latent classes with respect to this variable, and within the tth latent class the manifest variables (A, B, C, D) are mutually independent, then this two-class latent structure would serve as a simple explanation of the observed relationships among the variables in the 4-way con- tingency table for the n individuals. There is a direct generalization when the latent variable has T classes. We shall present some relatively simple methods for determining whether the observed relationships among the variables in the m-way contingency table can be explained by a T-class structure, or by various modifications and extensions of this latent structure. To illustrate the methods we analyze Table 1, a 24 contingency table presented earlier by Stouffer & Toby (1951, 1962, 1963), which cross-classifies 216 respondents with respect to whether they tend towards universalistic values ( + ) or particularistic values (-) when confronted by each of four different situations of role conflict. The letters A, B, C and D in

SUMMARY Rank tests for association between a time variable and a continuous covariate, and between two time variables are given in the presence of censoring. In the first case, these tests are related to the semiparametric tests of Cox (1972) and Prentice (1978) but differ in that the covariate is replaced by a score computed from its rank. In the second case, the test statistic is again a simple inner product, which in the proportional hazards model reduces to the correlation between the two vectors of log rank scores. The loss of efficiency arising from incorrect score functions is also explored when there is no censoring. In this paper we consider a general model for association between pairs of observations in which either variable or both variables may be right censored. The aim is to develop tests based on the generalized ranks of both variables which have maximal power at alternatives which are near to the null hypothesis of independence. Attention will be concentrated on two special cases of interest in survival analysis: tests for association between a censored survival time and a possibly censored continuous covariate, and tests for association between the times to different events. Beginning with Cox (1972), the first situation has been studied extensively in a semiparametric set-up in which information based on ranks is used for the time variable but exact values are used for the covariate. Procedures which use only the ranks of both variables have been developed by Brown, Hollander & Korwar (1974) and more recently by O'Brien (1978). However, neither of these procedures have been developed with a particular model in mind and one of the achievements of the present paper is to derive rank tests with good local power when a particular model such as the proportional hazards model holds. In the case of a proportional hazards model with a continuous covariate, the present development may be preferable to the semiparametric methods when the functional form of the relation- ship between the hazard and the covariate is not known in advance, or when errors or outliers may exist in the observed covariate values. Thus, significance levels may be obtained which are not inflated by multiple regroupings or transformations of the covariate data, although it would be expected that parametric model building with variables found to be associated with survival would often be carried out in further analyses.

SUMMARY A procedure is proposed for estimating a parameter of an epidemic model from household data. The procedure is essentially a method of moments for martingales. It is computationally convenient and is less restrictive in its model assumptions than the maximum likelihood methods used so far. The approach is illustrated with measles data and is also shown to be useful when households contain susceptibles of different types. This paper is concerned with data generated by the spread of an infectious disease within households. Each of the households initially has one or more infectives, and susceptibles may become infected by adequately close contacts with infectives from the same household. A newly infected individual passes through a stage during which he is latent and then through a stage during which he is infectious, before being removed by isolation, by death or by naturally losing his infectiousness and becoming immune for the remaining duration of the epidemic. Such data are usually analysed by maximum likelihood methods using one of two types of epidemic models to describe the spread of the disease within a household. Either a chain binomial model, like those initially introduced by Greenwood and Reed-Frost, is used or the Kermack-McKendrick model, as formulated in its stochastic form by Bartlett (1949), is used. An extensive discussion of these models and methods is given in the book by Bailey (1975). The method proposed in this paper is based on an approach suggested for counting processes in a Berkeley Ph.D. thesis by 0. 0. Aalen and is essentially a method of moments for martingales constructed from the epidemic model. There are two advantages of the resulting estimator and its asymptotic normality. One advantage is its computational convenience and another is that it applies for an epidemic model with more realistic assump- tions about the durations of the latent and infectious periods than those made in the chain binomial and Kermack-McKendrick models. We take for our epidemic model a continuous time model. The duration of the latent period and the duration of the infectious period are taken to be independent random variables which, apart from in ? 4, are taken to have arbitrary distributions. We let v denote the mean dura- tion of the infectious period. Infection occurs, within households, according to an orderly point process with intensity at any time being jointly proportional to the number of suscep- tibles and the number of infectious individuals in the household at that time. The constant of proportionality ,B is the within-household infection rate.

SUMMARY An extension of the growth curve model of Potthoff & Roy (1964) is presented. The extension arises naturally when parallel profiles are required, or profiles are concurrent at some known time point and may be of use for the analysis of longitudinal data from designed experiments. Linearization of a nonlinear model also results in this model. Under certain conditions, transformation of the model leads to seemingly unrelated multivariate regressions. In this case questions of identifiability and estimation are handled easily. An analysis of data on the growth of lambs is presented to illustrate the approach. In this paper we consider the analysis of repeated measurements on experimental units which may be grouped according to some experimental design. Usually successive measurements are taken over time, but this need not be the case. We restrict ourselves to the case where the same number of observations at identical time points are made on all the experimental units. This is likely to be the case in designed experiments, though not perhaps in observational studies. The approach given here extends that of Potthoff & Roy (1964), and we show below that the potential applications are quite widespread. We emphasize that in this paper that we are attempting to present what we regard as a useful and practical extension to the classical growth curve model. It is likely to be useful if the aim of the analysis is the estimation of an explicit model for interpretative or predictive purposes. This is rather in contrast to the traditional approach to the analysis of such data where the aim is more directed towards testing for effects of treatments or covariates. If testing for effects is the primary purpose of analysis, then the simpler traditional methods, as given initially by Wishart (1938) and subsequently by many authors, such as Rowell & Walters (1976), may well suffice. In this approach selected linear functions of the response over time are calculated and treatment effects are assessed using the analysis of variance. Since this approach concentrates on individual linear functions seriatim and generally, but not always, ignores correlations between such linear functions, we would regard this as essentially a univariate technique. However if testing is the main concern, the loss of efficiency in ignoring the multivariate character of the data may not be severe. Our approach is more complex but the aim is more comprehensive. We aim to provide an explicit model for the mean response over time. If the model is then used to answer questions that could have been considered by separate tests, the answers will, of course, usually be essentially the same as those obtained by the simpler traditional approach.

SUMMARY This paper is concerned with the analysis of contingency tables which are constructed using categorical data from a complex sampling scheme. For the mutual independence hypothesis, we extend Tavare's (1983) results on two-dimensional contingency tables to higher dimensions. In particular, with respect to Pearson's statistic for testing the independence of two variables, we strengthen the robustness result proven by Tavare, thus making it applicable more widely. This paper is concerned with the analysis of contingency tables which are constructed using categorical data from a complex sampling scheme, i.e. from data containing some form of between-observation dependence which renders the multinomial sampling scheme invalid. Throughout, we are interested in how departures from multinomial sampling affect Pearson's statistic, X2, which is appropriate for multinomial data. In the literature to date, two distinct approaches to the problem of categorical data from complex sampling schemes have been discussed. First, Holt, Scott & Ewings (1980) demonstrate that, for survey data, analyses based on x2 can be extremely misleading and propose the one-moment adjustment of X2. Related references are Rao & Scott (1981, 1984) and Bedrick (1983). Secondly, Altham (1976), Cohen (1976) and Brier (1980) model inherent clustering in the sample. They show that to avoid invalid analyses, some adjustment of X2 is then necessary. More recently Tavare & Altham (1983) and Tavare (1983) have considered two-dimensional contingency tables which are constructed from data generated by Markov chains. Analytic results for the null asymptotic distribution of X2 are obtained and, once more, it is shown that some adjustment of X2 will usually be necessary. However, Tavare (1983) also obtains a very interesting robustness result: let Y1 and Y2 be independent variables, where Y1 is an independent trials process and Y2 is an arbitrary, irreducible first-order Markov chain. Then, the null asymptotic distribution of X2, for testing the independence of Y1 and Y2, is x2 on the appropriate degrees of freedom. In other words, no adjustment of X2 is necessary. Of the literature cited above, the present

SUMMARY The log rank test is widely used for comparison of survival curves. This paper examines various estimators for the variance of the log rank test statistic. These include the Mantel-Haenszel variance, the permutation variance of Peto & Peto and several newly proposed estimators. The results generalize to a class of test statistics proposed by Tarone & Ware. The log rank test is widely used for comparison of survival curves. There are two versions of the test which differ in the choice of variance for the same test statistic T. These are the Mantel-Haenszel variance, an estimate of var (T) which is unbiased independently of differences in censoring, and the permutation variance of Peto & Peto (1972), for which the variance is computed under the assumption of equal censoring. This paper examines the choice of variance. It is observed that for equal sample sizes the Mantel-Haenszel variance tends to underestimate the true variance when T is large in absolute value. As this is the situation where rejection of the null hypothesis is to be seriously considered, it appears that the true type I error will tend to be higher than that assumed, and the reported P value will tend to be too low. This phenomena is discussed also by Morton (1978). In ? 4, it is shown that under equal sample sizes the permutation variance tends to overestimate the true variance when censoring is unequal. It thus appears prudent to use the permutation variance rather than the Mantel-Haenszel variance in assessing significance. In ? 5 alternative variance estimators are proposed. The results immediately generalize to a class of test statistics proposed by Tarone & Ware (1977). This point is discussed in ? 6.

The main intent of this paper is to introduce a new statistical procedure for testing a complete sample for normality. The test statistic is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. This ratio is both scale and origin invariant and hence the statistic is appropriate for a test of the composite hypothesis of normality. Testing for distributional assumptions in general and for normality in particular has been a major area of continuing statistical research-both theoretically and practically. A possible cause of such sustained interest is that many statistical procedures have been derived based on particular distributional assumptions-especially that of normality. Although in many cases the techniques are more robust than the assumptions underlying them, still a knowledge that the underlying assumption is incorrect may temper the use and application of the methods. Moreover, the study of a body of data with the stimulus of a distributional test may encourage consideration of, for example, normalizing transformations and the use of alternate methods such as distribution-free techniques, as well as detection of gross peculiarities such as outliers or errors. The test procedure developed in this paper is defined and some of its analytical properties described in ? 2. Operational information and tables useful in employing the test are detailed in ? 3 (which may be read independently of the rest of the paper). Some examples are given in ? 4. Section 5 consists of an extract from an empirical sampling study of the comparison of the effectiveness of various alternative tests. Discussion and concluding remarks are given in ?6. 2. THE W TEST FOR NORMALITY (COMPLETE SAMPLES) 2 1. Motivation and early work This study was initiated, in part, in an attempt to summarize formally certain indications of probability plots. In particular, could one condense departures from statistical linearity of probability plots into one or a few 'degrees of freedom' in the manner of the application of analysis of variance in regression analysis? In a probability plot, one can consider the regression of the ordered observations on the expected values of the order statistics from a standardized version of the hypothesized distribution-the plot tending to be linear if the hypothesis is true. Hence a possible method of testing the distributional assumptionis by means of an analysis of variance type procedure. Using generalized least squares (the ordered variates are correlated) linear and higher-order

SUMMARY The probability integral for the distribution of range of a sample from a population whose distribution can be represented by the first few terms of an Edgeworth series has been obtained in this paper. The numerical values of the corrective functions arising due to nonnormality are tabulated. The new theoretical results are compared with the earlier results, where available. The distribution of range of samples from nonnormal populations was first studied empirically by Pearson & Adyanthdya (1928) and later, among others, by Pearson (1950), Cox (1954) and David (1954). These studies have been limited mainly to the mean range and to the probability integral in some simple nonnormal cases, and from these likely effects of nonnormality on the distribution of range have been conjectured. Singh (1967) obtained some theoretical results regarding the expectation and the variance of range of samples from a population whose distribution can be represented by the first few terms of an Edgeworth series. These results provided some additional information regarding the effects of parental excess and skewness on the mean and variance of the range. In the present paper the probability integral for the distribution of range of samples from the same type of population has been obtained and evaluated for small samples to examine the effects of parental excess and skewness.

A great deal of use has undoubtedly been made of least squares regression methods in circumstances in which they are known to be inapplicable. In particular, they have often been employed for the analysis of time series and similar data in which successive observations are serially correlated. The resulting complications are well known and have recently been studied from the standpoint of the econometrician by Cochrane & Orcutt (1949). A basic assumption underlying the application of the least squares method is that the error terms in the regression model are independent. When this assumption—among others—is satisfied the procedure is valid whether or not the observations themselves are serially correlated. The problem of testing the errors for independence forms the subject of this paper and its successor. The present paper deals mainly with the theory on which the test is based, while the second paper describes the test procedures in detail and gives tables of bounds to the significance points of the test criterion adopted. We shall not be concerned in either paper with the question of what should be done if the test gives an unfavourable result.

In this paper we consider the following problem. Given a multivariate normal population with known variance matrix, what test is appropriate to determine whether the means are slipped to the right? In the case when the population is univariate normal, this problem can be solved by the ordinary one-sided test using either the normal or the t-distribution functions. It is the purpose of this paper to develop what may be termed a multivariate analogue of the one-sided test of significance.

In an earlier paper* we have endeavoured to emphasise the importance of placing in a logical sequence the stages of reasoning adopted in the solution of certain statistical problems, which may be termed problems of inference. In testing whether a given sample, Z, is likely to have been drawn from a population H, we have started from the simple principle that appears to be used in the judgments of ordinary life-that the degree of confidence placed in an hypothesis depends upon the relative probability or improbability of alternative hypotheses. From this point of view any criterion which is to assist in scaling the degree of confidence with which we accept or reject the hypothesis that X has been randomly drawn from HI should be one which decreases as the probability (defined in some definite manner) of alternative hypotheses becomes relatively greater. Now it is of course impossible in practice to scale the confidence with which we form a judgment with any single numerical criterion, partly because there will nearly always be present certain a priori conditions and limitations which cannot be expressed in exact terms. Yet though it may be irmpossible to bring the ideal situation into agreement with the real, some form of numerical measure is essential as a guide and control. In our previous paper we have made use of the criterion of likelihood. That there may be other forms of criteria or that this one can be interpreted in a different manner is very possible, but our object has been to find a single principle connecting the various sampling tests already in use, and one which could be extended to new problems.

It is clear that a p-variate body of data arranged in a q-way classification will formally look llke a (p+q)-dimensional contingency table, but a distinction can be made between a 'variate' and a ‘way of classification’ in that along the direction of a 'variate' the marginal frequencies are supposed to be stochastic variates while along a 'way of classification' the marginal frequencies are supposed to be fixed. When, along certain directions, the marginals are fixed, an approach based on a conditional probability argument has been used. In the present paper (i) the conditional probability approach is abandoned and we start either from a single multinomial distribution or a product of an appropriate number of different multinomial distributions according as, with multi-way frequency data, all ways are 'variates' or some are 'variates' and some are 'ways of classification'. (ii) Also the hypotheses that are posed are of different kinds altogether according as we have a 'multivariate analysis' situation or an 'analysis of variance' situation. The hypotheses that are meaningful for one situation would not be too meaningful for the other and vice versa. Since the conditional probability approach is altogether abandoned, the mathematical theorems to which appeal is made are the two theorems as stated and proved by Cramer (1946, chapter 30) and a number of other such theorems which have been proved the same way and which, between them, take care of all the hypotheses discussed in this paper. When all ways are 'variates' the hypotheses are analogous to those in the usual multivariate analysis, and when some ways are 'variates' and some are 'ways of classification' the hypotheses are analogous to those in the analysis of variance. The general methods discussed in this paper arose out of an attempt to analyse a large mass of categorical data. The analysis has been carried out, and a few typical cases (together with the numerical analysis) illustrating different parts of the theoretical development will be presented in a subsequent paper. In this paper only the large-sample tests are considered. How 'large' the sample size has to be for the validity of the use of these asymptotic techniques, or in other words, some results on the nature of the approximation involved will also be discussed in a later paper.

In this paper we assume that each person has his individual age of onset. There will be no restriction for the distribution of age of onset. We are going to develop a method to estimate the gene frequencies and the penetrance and to test simple hypotheses such as the hypothesis of dominance or of recessiveness. In ? 2 we formulate the basic assumptions concerning the trait, the genes, and the age of onset. We try to make the assumptions as (realistic' as possible so that we can apply the model to a concrete case. The reader may be troubled because we use many notations, but this seems to be inevitable if we want to avoid misunderstanding. In ? 3 we derive the distribution law for the frequencies of the characteristic under the assumption that there is no selection of families whatsoever. In order to connect the theory with the observations of a sample we study two methods of selection or ascertainment in ? 4. The following section contains the theoretical distributions under the assumption of ascertainment. To estimate parameters and to test genetical hypotheses we have to compare the theoretical frequencies with the observed frequencies. The procedure will be explained briefly at the end of ? 5. A special application concerning the atopic diseases of the respiratory tract is explained in ? 6. In the past, several papers have been published dealing with genetical problems of those diseases. A critical survey is given in ? 7.

SUMMARY This paper gives a two-sample procedure for selecting the population with the largest mean from k normal populations with unknown variances. The method is based on a twosample procedure proposed by Stein (1945). Tables necessary for the application of the procedure are given for selected values of k. Comparisons of the minimum values of the expected sample sizes using the proposed procedure are made with the corresponding single-sample sizes for known variances (Bechhofer, 1954). Comparisons are also made of the expected total sample sizes for the single-sample procedure, the two-sample procedure given in this paper and the two-sample procedure proposed by Bechhofer, Dunnett & Sobel (1954) which assumes that the populations have known variance ratios. It is shown that the expected total sample sizes are not much increased by ignorance of the variance ratios.

An effect modifier is a pretreatment covariate that affects the magnitude of the treatment effect or its stability. When there is effect modification, an overall test that ignores an effect modifier may be more sensitive to unmeasured bias than a test that combines results from subgroups defined by the effect modifier. If there is effect modification, one would like to identify specific subgroups for which there is evidence of effect that is insensitive to small or moderate biases. In this paper, we propose an exploratory method for discovering effect modification, and combine it with a confirmatory method of simultaneous inference that strongly controls the familywise error rate in a sensitivity analysis, despite the fact that the groups being compared are defined empirically. A new form of matching, strength-$k$ matching, permits a search through more than $k$ covariates for effect modifiers, in such a way that no pairs are lost, provided that at most $k$ covariates are selected to group the pairs. In a strength-$k$ match, each set of $k$ covariates is exactly balanced, although a set of more than $k$ covariates may exhibit imbalance. We apply the proposed method to study the effects of the earthquake that struck Chile in 2010.

The use of high-throughput data to study the changing behavior of biological pathways has focused mainly on examining the changes in the means of pathway genes. In this paper, we propose instead to test for changes in the co-regulated and unregulated variability of pathway genes. We assume that the eigenvalues of previously defined pathways capture biologically relevant quantities, and we develop a test for biologically meaningful changes in the eigenvalues between classes. This test reflects important and often ignored aspects of pathway behavior and provides a useful complement to traditional pathway analyses.