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Summary Models for analysing multivariate datasets with missing values require strong, often unassessable, assumptions. The most common of these is that the mechanism that created the missing data is ignorable, which is a two-fold assumption dependent on the mode of inference. The first part, which is the focus here, under the Bayesian and direct-likelihood paradigms requires that the missing data be missing at random; in contrast, the frequentist-likelihood paradigm demands that the missing data mechanism always produce missing at random data, a condition known as missing always at random. Under certain regularity conditions, assuming missing always at random leads to a condition that can be tested using the observed data alone, namely that the missing data indicators depend only on fully observed variables. In this note we propose three different diagnostic tests that not only indicate when this assumption is incorrect but also suggest which variables are the most likely culprits. Although missing always at random is not a necessary condition to ensure validity under the Bayesian and direct-likelihood paradigms, it is sufficient, and evidence of its violation should encourage the careful statistician to conduct targeted sensitivity analyses.

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.

We consider the testing of mutual independence among all entries in a $d$-dimensional random vector based on $n$ independent observations. We study two families of distribution-free test statistics, which include Kendall's tau and Spearman's rho as important examples. We show that under the null hypothesis the test statistics of these two families converge weakly to Gumbel distributions, and propose tests that control the type I error in the high-dimensional setting where $d>n$. We further show that the two tests are rate-optimal in terms of power against sparse alternatives, and outperform competitors in simulations, especially when $d$ is large., Comment: to appear in Biometrika

We consider inference on a scalar parameter of interest in the presence of a nuisance parameter, using a likelihood-based statistic which is asymptotically normally distributed under the null hypothesis. Higher-order expansions are used to compare the repeated sampling distribution, under a general contiguous alternative hypothesis, of $p$-values calculated from the asymptotic normal approximation to the null sampling distribution of the statistic with the distribution of $p$-values calculated by bootstrap approximations. The results of comparisons in terms of power of different testing procedures under an alternative hypothesis are closely related to differences under the null hypothesis, specifically the extent to which testing procedures are conservative or liberal under the null. Empirical examples are given which demonstrate that higher-order asymptotic effects may be seen clearly in small-sample contexts.

We describe a class of statistical tests for the comparison of two or more survival curves, typically estimated using the Kaplan–Meier method. The class is based on the construction of O’Quigley (2003), and some special cases are of particular interest. Underlying the inferential development are the arguments of Efron & Hinkley (1978), leading to a theoretical sampling model that is in some sense closer to the observed data. The log-rank and weighted log-rank tests arise as special members of the class. In practice the log-rank test will often be a suboptimal, even poor, test due to the presence of non-proportional hazards. The proposed test maintains good power and, in all the cases considered, has greater power than the log-rank test under non-proportional hazards. The power will depend on the alternatives being considered, and under reasonable assumptions on the alternatives, we conclude that the proposed test is more powerful than the log-rank test. Simulations support these conclusions. An example is given as an illustration.

We develop gatekeeping procedures that focus on comparing multiple treatments with a control when there are multiple endpoints. Our procedures utilize estimated correlations among individual test statistics without parametric assumptions. We make comparisons with other gatekeeping procedures with respect to properties of the trade-off in statistical power between families of hypotheses. We introduce a reward function to facilitate these comparisons. We illustrate our methods by simulation and an analysis of data from a randomized, multi-armed clinical trial. Copyright 2013, Oxford University Press.

Statistical analyses often involve testing multiple hypotheses that are naturally grouped into a fixed sequence of families. An effective approach to control the familywise error rate is to prioritize the importance of prespecification in the testing order. A gatekeeping testing procedure examines the first family with no multiple adjustment and then examines the subsequent family depending on the decision made with respect to the previous one. In this paper, we describe the union closure method that can be used to design gatekeeping procedures. A bipolar disorder trial with three primary and two secondary outcomes is presented as an example. Power comparisons based on the bipolar disorder trial show that the proposed gatekeeping procedures under the union closure framework are more powerful than competing methods. Copyright 2011, 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.

The covariate-adaptive randomization method was proposed for clinical trials long ago but little theoretical work has been done for statistical inference associated with it. Practitioners often apply test procedures available for simple randomization, which is controversial since procedures valid under simple randomization may not be valid under other randomization schemes. In this paper, we provide some theoretical results for testing hypotheses after covariate-adaptive randomization. We show that one way to obtain a valid test procedure is to use a correct model between outcomes and covariates, including those used in randomization. We also show that the simple two sample t-test, without using any covariate, is conservative under covariate-adaptive biased coin randomization in terms of its Type I error, and that a valid bootstrap t-test can be constructed. The powers of several tests are examined theoretically and empirically. Our study provides guidance for applications and sheds light on further research in this area. Copyright 2010, Oxford University Press.

This paper presents a default model-selection procedure for Gaussian graphical models that involves two new developments. First, we develop a default version of the hyper-inverse Wishart prior for restricted covariance matrices, called the hyper-inverse Wishart g-prior, and show how it corresponds to the implied fractional prior for selecting a graph using fractional Bayes factors. Second, we apply a class of priors that automatically handles the problem of multiple hypothesis testing. We demonstrate our methods on a variety of simulated examples, concluding with a real example analyzing covariation in mutual-fund returns. These studies reveal that the combined use of a multiplicity-correction prior on graphs and fractional Bayes factors for computing marginal likelihoods yields better performance than existing Bayesian methods. Copyright 2009, Oxford University Press.

We present a theoretical basis for testing related endpoints. Typically, it is known how to construct tests of the individual hypotheses, but not how to combine them into a multiple test procedure that controls the familywise error rate. Using the closure method, we emphasize the role of consonant procedures, from an interpretive as well as a theoretical viewpoint. Surprisingly, even if each intersection test has an optimality property, the overall procedure obtained by applying closure to these tests may be inadmissible. We introduce a new procedure, which is consonant and has a maximin property under the normal model. The results are then applied to PROactive, a clinical trial designed to investigate the effectiveness of a glucose-lowering drug on macrovascular outcomes among patients with type 2 diabetes. Copyright 2009, Oxford University Press.

We study tools for checking the validity of a parametric regression model. When the dimension of the regressors is large, many of the existing tests face the curse of dimensionality or require some ordering of the data. Our tests are based on the residual empirical process marked by proper functions of the regressors. They are able to detect local alternatives converging to the null at parametric rates. Parametric and nonparametric alternatives are considered. In the latter case, through a proper principal component decomposition, we are able to derive smooth directional tests which are asymptotically distribution-free under the null model. The new tests take into account precisely the 'geometry of the model'. A simulation study is carried through and an application to a real dataset is illustrated. Copyright 2008, Oxford University Press.

A frequently encountered challenge in high-dimensional regression is the detection of relevant variables. Variable selection suffers from instability and the power to detect relevant variables is typically low if predictor variables are highly correlated. When taking the multiplicity of the testing problem into account, the power diminishes even further. To gain power and insight, it can be advantageous to look for influence not at the level of individual variables but rather at the level of clusters of highly correlated variables. We propose a hierarchical approach. Variable importance is first tested at the coarsest level, corresponding to the global null hypothesis. The method then tries to attribute any effect to smaller subclusters or even individual variables. The smallest possible clusters, which still exhibit a significant influence on the response variable, are retained. It is shown that the proposed testing procedure controls the familywise error rate at a prespecified level, simultaneously over all resolution levels. The method has power comparable to the Bonferroni--Holm procedure on the level of individual variables and dramatically larger power for coarser resolution levels. The best resolution level is selected adaptively. Copyright 2008, Oxford University Press.

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. Copyright 2008, Oxford University Press.

We develop diagnostic measures for assessing the influence of individual observations when using empirical likelihood with general estimating equations, and we use these measures to construct goodness-of-fit statistics for testing possible misspecification in the estimating equations. Our diagnostics include case-deletion measures, local influence measures and pseudo-residuals. Our goodness-of-fit statistics include the sum of local influence measures and the processes of pseudo-residuals. Simulation studies are conducted to evaluate our methods, and real datasets are analyzed to illustrate the use of our diagnostic measures and goodness-of-fit statistics. Copyright 2008, Oxford University Press.

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. Copyright 2008, Oxford University Press.

We consider that observations come from a general normal linear model and that it is desirable to test a simplifying null hypothesis about the parameters. We approach this problem from an objective Bayesian, model-selection perspective. Crucial ingredients for this approach are 'proper objective priors' to be used for deriving the Bayes factors. Jeffreys-Zellner-Siow priors have good properties for testing null hypotheses defined by specific values of the parameters in full-rank linear models. We extend these priors to deal with general hypotheses in general linear models, not necessarily of full rank. The resulting priors, which we call 'conventional priors', are expressed as a generalization of recently introduced 'partially informative distributions'. The corresponding Bayes factors are fully automatic, easily computed and very reasonable. The methodology is illustrated for the change-point problem and the equality of treatments effects problem. We compare the conventional priors derived for these problems with other objective Bayesian proposals like the intrinsic priors. It is concluded that both priors behave similarly although interesting subtle differences arise. We adapt the conventional priors to deal with nonnested model selection as well as multiple-model comparison. Finally, we briefly address a generalization of conventional priors to nonnormal scenarios. Copyright 2007, Oxford University Press.

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 In this note we give an alternative formulation of the nonparametric estimators of Wang (2003) with the identity link. This results in a closed form of the estimator that has computational advantages and gives insight into the rationale behind the estimator.

We give several definitions of residual autocorrelations and derive their joint asymptotic distribution for the panel time series model of Hjellvik & Tjostheim (1999a). A portmanteau goodness-of-fit test arises naturally from the asymptotic distribution. Simulation results show that the asymptotic standard errors compared satisfactorily with the empirical standard errors, that the goodness-of-fit test has reasonable empirical size, and that it is powerful enough to be useful with a modest sample size. The results of this paper are illustrated with a real-data example. © 2002 Biometrika Trust.

SUMMARY We derive the limiting distribution of the partial likelihood ratio under general conditions. The multiplicative hazards models being fitted may be nonnested and misspecified. The true model is not assumed to contain either model under consideration. The null hypothesis is that the models are equidistant in Kullback-Leibler metric applied to the rank likelihood. The statistic is consistent for the model which is closer to the truth. Its distribution depends on the unknown data-generating mechanism. A sequential testing procedure is proposed for nonnested comparisons which is valid regardless of the true model. This involves a novel statistic for the equality of the fitted models which is separate from the partial likelihood. The methodology has important applications in model assessment. Simulations and a real example demonstrate its utility in selecting the functional forms of covariates and relative risks.

The results of Davies (1977, 1987) are extended to a linear model situation with unknown residual variance.

SUMMARY We propose a method for comparing survival distributions when cause-of-failure information is missing for some individuals. We use multiple imputation to impute missing causes of failure, where the probability that a missing cause is that of interest may depend on auxiliary covariates, and combine log-rank statistics computed from several 'completed' datasets into a test statistic that achieves asymptotically the nominal level. Simulations demonstrate the relevance of the theory in finite samples.

SUMMARY We extend the optimal symmetric group sequential tests of Eales & Jennison (1992) to the broader class of asymmetric designs. Two forms of asymmetry are considered, involving unequal type I and type II error rates and different emphases on expected sample sizes at the null and alternative hypotheses. We discuss the properties of our optimal designs and use them to assess the efficiency of the family of tests proposed by Pampallona & Tsiatis (1994) and two families of one-sided tests defined through error spending functions. We show that the error spending designs are highly efficient, while the easily implemented tests of Pampallona & Tsiatis are a little less efficient but still not far from optimal. Our results demonstrate that asymmetric designs can decrease the expected sample size under one hypothesis, but only at the expense of a significantly larger expected sample size under the other hypothesis.

SUMMARY In many tumour growth inhibition studies, tumour sizes are recorded over a period of time, forming a growth curve for each experimental animal. A frequently asked question in these studies is whether or not different treatment agents/inhibitors result in different growth rates. To compare tumour growth rates under two treatments, either the t-test or the Wilcoxon-Mann-Whitney test could be used under suitable assumptions, provided that a one-dimensional variable, such as the area under the curve, is used to summarise a growth curve. Such tests may not be valid in the presence of informative heterogeneous censoring, in that a subject's continued participation in the experiment may depend on the tumour size and/or the level of toxicity of the treatment given. We propose test statistics that naturally correct the bias caused by the censorship and retain high efficiency. They are easy to construct and are nonparametric in nature, making no assumption on the distributions of the growth curves. The method is illustrated with an example from a tumour growth inhibition study on mice. Simulation results are also reported showing that the method performs well with moderate sample sizes.

SUMMARY Generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified. However, under such misspecification, the estimator of the regression parameter can be inefficient. In this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter, and that remains optimal even if the working correlation structure is misspecified. The idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices, a representation that is valid for the working correlations most commonly used. Both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations. This approach also provides a chi-squared inference function for testing nested models and a chi-squared regression misspecification test. Furthermore, the test statistic follows a chi-squared distribution asymptotically whether or not the working correlation structure is correctly specified.

SUMMARY This paper generalises four types of disturbance commonly used in univariate time series analysis to the multivariate case, highlights the differences between univariate and multivariate outliers, and investigates dynamic effects of a multivariate outlier on individual components. The effect of a multivariate outlier depends not only on its size and the underlying model, but also on the interaction between the size and the dynamic structure of the model. The latter factor does not appear in the univariate case. A multivariate outlier can introduce various types of outlier for the marginal component models. By comparing and contrasting results of univariate and multivariate outlier detections, one can gain insights into the characteristics of an outlier. We use real examples to demonstrate the proposed analysis.

Summary A simple, nonparametric two-sample test for equality of a given collection of quantiles is developed which can be applied to a variety of empirical distribution functions, including the Kaplan‐Meier estimator, a self-consistent estimator for doubly-censored data and an estimator for repeated measures data. The null hypothesis tested is that the quantiles are equal but other aspects of the distributions may diVer between the two samples. This procedure can also be applied to quantile testing in group sequential clinical trials with staggered patient entry. A simple simulation study demonstrates that the moderate sample size properties of this procedure are reasonable.

Although the U.S. Food and Drug Administration (FDA, 1992) recommends testing bioequivalence of individual pharmacokinetic parameters one at a time, it seems reasonable and interesting to conduct a test simultaneously for all the parameters. In this paper, we discuss several ways to construct such tests. It is shown that the confidence set approach leads to a test which can be uniformly improved by the intersection of Schuirmann's two one-sided tests procedure. The latter test can further be improved upon noticeably by using the one-dimensional unbiased test of Brown, Hwang & Munk (1997). Numerical calculations of powers are given to support this claim.

SUMMARY Conditional frequentist tests of a precise hypothesis versus a composite alternative have recently been developed, and have been shown to be equivalent to conventional Bayes tests in the very strong sense that the reported frequentist error probabilities equal the posterior probabilities of the hypotheses. These results are herein extended to sequential testing, and yield fully frequentist sequential tests that are considerably easier to use than are conventional sequential tests. Among the interesting properties of these new tests is the lack of dependence of the reported error probabilities on the stopping rule, seeming to lend frequentist support to the stopping rule principle.

SUMMARY The additive risk model assumes that the hazard function associated with a set of covariates is the sum of the baseline hazard function and the regression function of covariates. We propose two different test procedures for checking the adequacy of two-sample additive risk models for randomly censored observations. One is based on the martingale residuals and the other on the difference between weighted estimators of the excess risk. The test statistics are shown to be asymptotically normal under appropriate regularity conditions and consistent under any model misspecifications. Finally, two real examples are provided, along with results of a simulation study.

SUMMARY Bayes/frequentist correspondences between the p-value and the posterior probability of the null hypothesis have been studied in univariate hypothesis testing situations. This paper extends these comparisons to multiple testing and in particular to the Bonferroni multiple testing method, in which p-values are adjusted by multiplying by k, the number of tests considered. In the Bayesian setting, prior assessments may need to be adjusted to account for multiple hypotheses, resulting in corresponding adjustments to the posterior probabilities. Conditions are given for which the adjusted posterior probabilities roughly correspond to Bonferroni adjusted p-values.

The assumption of proportional hazards in different populations leads to elegant analysis of survival data as initiated by Cox (1972). However, this model is not appropriate in all situations. In recent literature there have been several instances of crossing hazards, which may be modelled by an increasing hazard ratio. We present a U-statistic test for checking constancy of the hazard ratio against this alternative. The test is suitable for censored data and is found to have reasonable power in certain parametric families. Next we propose a test for checking proportionality of the hazards of two competing risks, in the presence of a third risk, against the alternative of increasing hazard ratio. Analyses of a few real data sets and some results based on computer simulations are also given.

SUMMARY In an observational study, detecting hidden bias involves checking that treatment effects appear where they should, and not elsewhere. For instance, treated and control groups are often compared with respect to outcomes the treatment should not affect. This paper uses such a test for bias to obtain a confidence set for an unobserved covariate. The impact of this unobserved covariate is indicated by a sensitivity analysis with the covariate confined to the confidence set. In this way, a test for bias may indicate either the presence and magnitude of a hidden bias or else a reduction in the sensitivity to bias.

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 We calculate the asymptotic efficiency of logistic regression relative to linear discriminant analysis for testing hypotheses about the parameters when the explanatory variables are normally distributed. The Pitman asymptotic efficiency for hypothesis tests in this context is the same as the asymptotic relative efficiency calculated using misclassification rates and is independent, of the number of explanatory variables and the number of parameters to be tested.

SUMMARY Consider a binary random variable having outcomes subject to misclassification errors where the probabilities of misclassification errors depend on outcome. The asymptotic relative efficiencies are derived for comparing two binomial populations, comparing stratified binomial populations and for paired comparisons. The probabilities of misclassification can vary with strata or be different for each pair. The criterion for optimum experimental designs is shown to be the ratio of the asymptotic relative efficiency to expected cost per observation. This criterion implies that mixtures of different experiments can never be optimal. A number of experimental strategies is discussed.

SUMMARY We consider exact conditional methods for identifying outliers in logistic regression data. Tests for a single outlier and multiple outliers are developed assuming a logistic slippage model. The p-values for these tests are determined using an explicit enumeration of all possible responses consistent with the observed value of the sufficient statistic. Justifications are given for preferring this computationally intensive approach to standard methods based on asymptotic approximations. The techniques are applied to two examples.

SUMMARY We propose a test of the null hypothesis of no treatment effect in a randomized clinical trial that is based on the randomization distribution of residuals. These residuals result from regressing the response on covariates, but not treatment. In contrast to model-based score tests, this procedure maintains nominal size when the model is misspecified, and, in particular, when relevant covariates are omitted from the regression. The efficiency of the procedure is evaluated for regressions with some, but not all, required covariates. For many generalized linear models and survival models, conventional model-based score tests are shown to have supranominal size when relevant covariates are omitted, but logistic regression and the proportional hazards model are robust.

SUMMARY Standard chi-squared, X2, or likelihood ratio, G2, test statistics for logistic regression analysis, involving a binary response variable, are adjusted to take account of the survey design. These adjustments are based on certain generalized design effects. Logistic regression diagnostics to detect any outlying cell proportions in the table and influential points in the factor space are also developed, taking account of the survey design. Finally, the results are used to analyse some data from the October 1980 Canadian Labour Force Survey.

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 Limit distributions both under the null and a sequence of alternative hypotheses are derived for the likelihood ratio and related statistics for testing the homogeneity of several autoregressive processes. These results are extended to mixed autoregressive and moving average processes. It is often required to test whether two or more sets of data can be considered as coming from a common time series model. If the hypothesis of homogeneity is accepted one can obviously achieve parsimony in model specification and also obtain better estimates of the model parameters by pooling the data sets. On the other hand, if there is a significant difference between the sets one may be interested in alternative models which may vary between sets but are still close to each other for large samples. When dealing with regression models the homogeneity tests usually lead to analysis of variance techniques. In this note we consider the comparison of autoregressive and also of mixed autoregressive-moving average models. A unified method of deriving limit distributions of the likelihood ratio and related test statistics is suggested in ? 2. The method is illustrated in ? 3 for autoregressive processes augmented by regression variables, and for mixed autoregressive-moving average processes.

SUMMARY Rank procedures are discussed for testing location in matched pairs data. The tests may be applied to samples containing censored as well as uncensored observations. It is assumed that members of a given pair have equal censoring times while members from distinct pairs may have different times of censoring. In addition, it is assumed that censoring is independent of the random variables under study, and that censoring, if it occurs, is on the right. The test statistics are derived under the assumption that within-pair differences are symmetrically and identically distributed about a common median. The application of the techniques to problems in survival data analysis is also discussed.

SUMMARY Maximum likelihood methods are used to test for a change in a sequence of independent exponential family random variables, with particular emphasis on the exponential distribution. The exact null and alternative distributions of the test statistics are found, and the power is compared with a test based on a linear trend statistic. Exact and approximate confidence regions for the change-point are based on the values accepted by a level x likelihood ratio test and a modification of the method proposed by Cox & Spj0tvoll (1982). The methods are applied to a classical data set on the time intervals between coal mine explosions, and the change in variation of stock market returns. In both cases the confidence regions for the change-point cover historical events that may have caused the changes.

SUMMARY A simply parameterized class of models for the estimation of survival probability in a marked population is introduced which allows tests of dependence between sighting probabilities and constancy of survival or sighting probabilities. Estimation and hypothesis testing based on the likelihood function are discussed. By simulation, the asymptotic properties of the test for dependence are shown to hold for quite modest sample sizes. The degree of bias in estimation of survival probabilities when dependence between sightings is not taken into account is shown to increase as the probability of a nonsighting following a nonsighting increases. A comprehensive analysis of the data from a mark-recapture experiment dealing with survival of abalone is given.

SUMMARY Some current nonparametric methods related to the log rank test are extended to cover ordered categories of response, e.g. response, no change, progression, and to incorporate information on duration of response. As an example, data from a protocol for the treatment of advanced breast cancer are analysed. The powers of several tests for this special case of multistate survival data are compared by Monte Carlo simulation for both small and moderate samples from exponential and Weibull distributions.

A simple procedure for consideration of specified treatment contrasts or sets of contrasts in a paired comparisons experiment is developed. General likelihood estimation and likelihood ratio tests are given. Specified treatment comparisons as appropriate in a particular experiment may be made. The procedure may be used for consideration of factor effects and interactions when the treatments in paired comparisons are factorial treatment combinations. An example is given of a taste preference experiment on coffee with factors, brew strength, roast color and brand, each at two levels. Results are summarized in an analysis of chi-square table very analogous to the typical analysis of variance summary.

SUMMARY In observational studies, treatments are not randomly assigned to experimental units, so that randomization tests and their associated interval estimates are not generally applicable. In an effort to compensate for the lack of randomization, treated and control units are often matched on the basis of observed covariates; however, the possibility remains of bias due to residual imbalances in unobserved covariates. A general though simple method is proposed for displaying the sensitivity of permutation inferences to a range of assumptions about unobserved covariates in matched observational studies. The sensitivity analysis is applicable to Wilcoxon's signed rank test, to the McNemar-Cox test for paired binary responses, and to some matching problems with a variable number of controls.

In considering life test experiments from a behavioristic Bayes point of view, we may speculate on the effect of model misspecification on the posterior mean under alternative stopping rules. Of course, under the life distribution model, we expect the mean of the posterior to equal the mean of the prior. However, if we think the model may have been misspecified, we would no longer expect this equality. We show that for Bayes estimators of mean life under the exponential model and general priors on mean life, we expect, based on a preposterior analysis, that the mean of the posterior will increase with sample size when the true model has an increasing failure rate and certain fixed stopping rules are used. Also, we show that for Bayes estimators of mean life based on the natural conjugate prior for the exponential model and complete samples, the Bayes risk is less when the coefficient of variations is less than for the exponential model. Recommendations relative to use of life test sampling plans based on an exponential model are made. (Author)