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SUMMARY Tests which are specific for location are developed in the k-sample situation with right-censored data. The tests are based on general estimating functions, and do not require full parametric assumptions for large samples, but in general they are not based only on ranks. One class of tests allows shapes to differ in the k populations, and includes as a special case the median test of Brookmeyer & Crowley (1982). If the distributions are assumed to differ at most in location modified tests are suggested in which survival function estimates are obtained by pooling the data from all samples. Prentice's (1978) median test, with minor modifications, is included in this class.

SUMMARY The use of much of the distribution theory developed for order restricted inference has been limited by the lack of algorithms for the 'level probabilities'. An approximation for these, which accounts for the pattern of large and small 'weights', is developed. This approximation and the equal weights approximation are examined. Both approximations appear to be reasonable for weight sets having a moderate amount of variability. The quality of the equal weights approximation, as a function of the amount of variability in the weights, deteriorates more quickly for certain patterns of large and small weights than for others. Thus, the approximation based upon the pattern of large and small weights is a significant improvement over the equal weights approximation. Finally, Siskind's (1976) approximation, which can be applied if the number of parameters is not too large, is discussed. The chi-bar-squared, %2, and E-bar-squared, P2, distributions are fundamental to the theory of order restricted hypothesis tests. These distributions have tail probabilities which are linear combinations of the tail probabilities of standard distributions and they depend upon the order,restriction through the coefficients in these linear combinations. The values of these coefficients are the probabilities that the order restricted maximum likelihood estimates of normal means assume specified numbers of distinct values, which are called levels. The estimates are based upon independent samples from each of the populations and depend upon the vector of relative precisions, called weights, of the sample means as estimates of the corresponding population means. The use of these tests has been limited because these level probabilities can be virtually impossible to compute if the weights are not all equal. In this paper we describe a technique for approximating the level probabilities for a linear order restriction and for unequal precisions. This approximation is based upon an idea of Chase (1974) and uses the pattern of large and small relative precisions. It seems to be particularly good when the relative precisions have two distinct values. However, it seems to provide a satisfactory approximation so long as the variation in the relative precisions is not too large. This is probably due to the lack of sensitivity of the level probabilities to changes in the weights. The robustness was

SUMMARY Simes (1986) has proposed a modified Bonferroni procedure for the test of an overall hypothesis which is the combination of n individual hypotheses. In contrast to the classical Bonferroni procedure, it is not obvious how statements about individual hypotheses are to be made for this procedure. In the present paper a multiple test procedure allowing statements on individual hypotheses is proposed. It is based on the principle of closed test procedures (Marcus, Peritz & Gabriel, 1976) and controls the multiple level a.

SUMMARY We propose a single unifying approach to bootstrap resampling, applicable to a very wide range of statistical problems. It enables attention to be focused sharply on one or more characteristics which are of major importance in any particular problem, such as coverage error or length for confidence intervals, or bias for point estimation. Our approach leads easily and directly to a very general form of bootstrap iteration, unifying and generalizing present disparate accounts of this subject. It also provides simple solutions to relatively complex problems, such as a suggestion by Lehmann (1986) for 'conditionally' short confidence intervals. We set out a single unifying principle guiding the operation of bootstrap resampling, applicable to a very wide range of statistical problems including bias reduction, shrinkage, hypothesis testing and confidence interval construction. Our principle differs from other approaches in that it focuses attention directly on a measure of quality or accuracy, expressed in the form of an equation whose solution is sought. A very general form of bootstrap iteration is an immediate consequence of iterating the empirical solution to this equation so as to improve accuracy. When employed for bias reduction, iteration of the resampling principle yields a competitor to the generalized jackknife, enabling bias to be reduced to arbitrarily low levels. When applied to confidence intervals it produces the techniques of Hall (1986) and Beran (1987). The resampling principle leads easily to solutions of new, complex problems, such as empirical versions of confidence intervals proposed by Lehmann (1986). Lehmann argued that an 'ideal' confidence interval is one which is short when it covers the true parameter value but not necessarily otherwise. The resampling principle suggests a simple empirical means of constructing such intervals. Section 2 describes the general principle, and ? 3 shows how it leads naturally to bootstrap iteration. There we show that in many problems of practical interest, such as bias reduction and coverage-error reduction in two-sided confidence intervals, each iteration reduces error by the factor n-1, where n is sample size. In the case of confidence intervals our result sharpens one of Beran (1987), who showed that coverage error is reduced by the factor n-2 in two-sided intervals. The main exception to our n-1 rule is coverage error of one-sided intervals, where error is reduced by the factor n-A at each iteration. Our approach to bootstrap iteration serves to unify not just the philosophy of iteration for different statistical problems, but also different techniques of iteration for the same

SUMMARY The smooth tests of Neyman (1937) are modified to permit composite alternatives and a choice of orthonormal functions in their construction. These orthonormal functions may be chosen to improve detection of particular alternatives. The tests are easy to apply and have useful orthogonal components. A test for exponentiality is examined, and shown to perform well against its competitors.

SUMMARY A class of linear rank statistics is proposed for the k-sample problem with rightcensored survival data. The class contains as special cases the log rank test (Mantel, 1966; Cox, 1972) and a test essentially equivalent to Peto & Peto's (1972) generalization of the Wilcoxon test. Martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered, and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses. A class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution. Some Monte Carlo results are displayed which present small sample behaviour.

SUMMARY Methods are outlined for testing for serial correlation in large numbers of small samples, emphasis being placed on samples of size three. Some data on pulse rates are analysed in illustration.

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 When data can be presented as a series of k 2 x 2 tables with cell counts (xi, ni-xi; yi, mi - yi), it is often assumed that xi and yi are binomially distributed. This paper deals with inference for the common odds ratio i/l when the binomial assumption is invalid. When k increases, the consistency and asymptotic normality of the Mantel & Haenszel (1959) estimator is derived. The conditional maximum likelihood estimator is shown to be inconsistent and the asymptotic bias is computed when either the first-order Markov chain or beta-binomial model is assumed. The Mantel-Haenszel test for testing f = 1 is also shown to be inappropriate through some simulation studies. Two consistent test statistics are proposed and shown to be comparable to each other in terms of size and efficiency. Some possible further work is described.

SUMMARY The invariant sequential probability ratio test used in testing for a difference between the means of two Gaussian populations is set up. The error probabilities for this test are effectively constant over a rich class of data-dependent allocation rules. The additional risk, average sample number plus (y - 1) times the expected number of observations to the inferior population, for y > 1, is introduced and the optimal allocation rule is found for the continuous-time analogue to this problem. Analytical results show this rule to be asymptotically optimal in discrete time, and simulations indicate its near optimal per- formance for the finite case. The problem of two-population hypothesis testing with data-dependent allocation of observations has been treated by several authors. Also, the applications of this decision model, especially to clinical testing, have been well documented. Recent results show that when the test is sequential and the termination rule is of the sequential probability ratio test type, the probability of correct hypothesis selection is constant, ignoring overshoot, for a rich class of data-dependent allocation rules; see Flehinger, Louis, Robbins & Singer (1972) and an as yet unpublished paper of mine. This constancy permits one to search the class for a rule which performs well with respect to some additional cost structure, one usually based on the number of observations taken on the superior and inferior populations. Flehinger & Louis (1972) and Robbins & Siegmund (1974) give simulations showing that a substantial reduction in the expected number of observations on the inferior popu- lation is possible using data-dependent allocation rules, as opposed to equal assignment, for the case of comparing two Gaussian populations with known variances. The simulation results of Flehinger & Louis (1971) show the same reduction for the exponential distri- bution. In the present paper the risk function formed from the average sample number plus (y - 1) x the expected number of observations allocated to the inferior population, for y > 1, is introduced into the Gaussian testing model. Here y is the relative cost of taking an observation from the inferior as opposed to the superior population, and varying y allows one to balance the two components of risk. In ? 2 first the Gaussian allocation and testing problem is set up and previous results are summarized. Using the above risk function, in Appendix A the optimal allocation and its risk are obtained for the continuous-time idealization, that of comparing the drifts of two Brownian motions. Back in ? 2 this optimal rule is related to the discrete testing situation.

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 two nonlinearity tests for stationary time series. The idea of Tukey's one degree of freedom for nonadditivity test is generalized to the time series setting. The case of concurrent nonlinearity is discussed in detail. Simulation results show that the proposed tests are more powerful than that of Keenan (1985).

SUMMARY A locally optimal test for a null variance ratio is considered in the context of the one-way random effects model, when normality is assumed. Using an asymptotic design sequence with an increasing number of groups with bounded but unequal sizes, this test has the correct asymptotic level of significance for nonnormal data, a property which is not shared by the competing Wald test. Asymptotic power calculations demonstrate that the locally optimal test may be more powerful than the Wald test even in situations where the actual significance level of the Wald test overstates the nominal level.

SUMMARY Nonparametric tests are useful when definite parametric alternatives are not available. The behaviour under censoring of extensions of the tests is used to delineate three distinct classes of censored data rank tests: asymptotically efficient, approximately distribution-free and approximately unbiased. Section 2 gives qualitative comparisons; ? 3 gives some representative numerical comparisons. The Pitman efficiencies approximated by Lee, Desu & Gehan (1975) from simulations with 50 observations in each group are shown to agree with asymptotic calculations. 1. WEIGHTED LOG RANK TESTS It is assumed that two samples of randomly censored observations are available, that is, that the true lifetimes Yij have cumulative distribution functions F(y I Oj) for j = 1, ..., n(i); i = 1,2, that the censoring times Cij have cumulative distribution functions H(y I Of), and that the observable quantities are Xij = min (Yij, C) and Dij = [Xij = Yij], where [A] denotes the indicator function of the set A. It is assumed that 01 = 0, that F(t 1 0) is continuous in 0 and that the null hypothesis is 61 = 02 = 0. The 'true' value of 02 will be denoted by 0 and the functions F(t I 0) and H(t I 0) will be abbreviated to F and H. Survival distributions will be denoted by a bar, so that F(t) = 1- F(t). The hazard rates of F and F(t 10) will be denoted by A and AO, respectively. The test statistics studied are functions of: Di(t), the number of deaths observed before time t in the ith sample; Ri(t), the number at risk at time t in the ith sample; and Aj, an estimator of the cumulative hazard function for the ith sample. These functions are defined formally by n(i) n(i)

Many tests have been proposed for detecting a preassigned number, k 1, of outliers in a sample of size n assumed to be from a normal population. Barnett & Lewis (1984, ? 6.3) list most of these and also discuss their rationale. Pearson & Chandrasekar (1936), Dixon (1950) and Grubbs (1950) had already observed that, if we are testing for a single outlier and if in fact there is more than one outlier, say two, present, then the test for a single outlier would fail to detect even a single outlier. This phenomenon was termed a masking effect by R. B. Murphy in his Princeton Ph.D. thesis. The masking effect has been discussed widely in the literature although mostly by examples. Bendre & Kale (1985) introduced a simple measure of the masking effect as the loss in power due to the presence of more than the number of outliers under test, and studied this for the test for a single outlier in samples from an exponential distribution. The object of the present paper is to use the same measure on some of the commonly applied tests for detecting outliers in samples from a normal distribution. We illustrate the technique by first considering Grubbs's test for a single outlier, which is equivalent to the test based on the maximum studentized deviation first proposed by Pearson & Chandrasekar (1936), and then briefly indicating how the technique can be applied to tests based on skewness proposed by Ferguson (1961) as a locally most powerful invariant test and to Dixon's (1950) test. We conclude that Ferguson's test is much less vulnerable to a masking effect, whereas Dixon's test is highly susceptible, with Grubbs's test falling between the two. As pointed out by a referee, this conclusion is based on a single measure of masking effect and the Dixon test, even though suffering from the masking effect, could be recommended on other grounds. While only three tests are considered, the techniques developed here enable study of the masking effect on most of the tests used for detecting outliers in normal samples.

SUMMARY In the one-way analysis of variance, a researcher may believe that the ordering of the means is known and would like to incorporate this information into the test of hypotheses. Bartholomew developed likelihood ratio tests for such situations and proposed a twomoment chi-squared approximation for the minimum power of the test for the case of known variances. The accuracy of a closely related approximation is studied and an approximation based on mixtures of noncentral F variables is developed for the case of unknown variances. The simple ordering, the simple tree ordering and a class of slippage alternatives are considered, and an example concerning the choice of sample size needed to provide a specified power at a particular alternative is treated. For powers in the range typically of interest the two-moment approximation seems adequate.

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 The problem considered is that of testing a sequence of independent normal random variables with constant, known or unknown, variance for no change in mean versus alternatives with a single change-point. Various tests, such as those based on the likelihood ratio and recursive residuals, are studied. Power approximations are developed by integrating approximations for conditional boundary crossing probabilities. A comparison of several tests is made.

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 The asymptotic expansions of the distributions of the likelihood ratio criterion and Wald's statistic are derived for a composite hypothesis under a sequence of local alternative hypotheses converging to the null hypothesis when the sample size tends to infinity. Comparisons between the two statistics are made.

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 Previous work on the use of gaps or spacings to test for uniformity of a sample has been in terms of distances between successive order statistics of the sample. This paper generalizes this notion of first-order gaps to mth-order gaps, and considers the sum of the logarithms of the mth-order gaps as a test statistic of uniformity. Asymptotic normality of this test statistic is shown under the null hypothesis of -uniformity, even when m grows at a moderate rate with the sample size. The test is compared with the most powerful test symmetric in the first-order gaps, and it is shown that the Pitman asymptotic relative efficiency increases for large m approximately linearly in m.

SUMMARY A test is given to detect clustering in disease incidence or mortality data. The test statistic is the mean distance between all pairs of disease cases. Its null mean and variance and its asymptotic normality are derived under assumptions that accommodate differences in population distribution among demographic subgroups at different disease risk. The test is illustrated on 63 cases of anal and rectal squamous cell carcinoma in San Francisco during 1973-1981. Patterns of disease incidence and mortality over time, space, or occupational categories can provide clues to the cause of the disease. An aetiologic agent can produce a spatial, temporal or occupational cluster of disease cases. It is seldom difficult to detect clusters of rare diseases like angiosarcoma of the liver among men occupationally exposed to polyvinyl chloride. However more common diseases present two problems. First, clusters may be obscured by the scattered occurrence of cases unrelated to the cause of the clusters. Secondly, clusters may be produced by factors unrelated to the disease process, such as variations in the overall population distribution, or variations in the distributions of demographic subgroups at high disease risk. It is therefore useful to have a method for detecting clusters that will adjust for such factors. Previous tests for clustering, for example, Pinkel & Nefzger (1959), Knox (1964), Mantel (1967), are not satisfactory for the study of chronic disease such as cancer. Those tests are designed to determine whether cases are clustered both in space and in time simultaneously. However, cases of chronic disease caused by a spatially localized agent may be close in space, but they are unlikely to be close in time, because of long and variable time periods between exposure and diagnosis. Thus there is need for alternative methods to detect spatial clusters in regional disease incidence or mortality data. Consider the set X = {xl,... , XK } of points called census tracts, with Nk cases of disease, called cancer, in tract Xk. Under the hypothesis Ho of no clustering, the Nk are independent Poisson variables, with the mean of Nk proportional to the population size ek in tract xk

SUMMARY This paper investigates the distribution of the locally best invariant test for coefficient constancy, against the random walk alternative, in a regression model which may contain nonvarying coefficients. Unfortunately, the distribution of the locally best invariant test depends critically on the exogenous variables of the model, even asymptotically. A modification is suggested which has a known asymptotic distribution regardless of the regression effect.

SUMMARY Many statistical procedures are, strictly speaking, only appropriate when a simplifying assumption about the shape of an error distribution is made, and such an assumption may need to be checked against departures in specific parametric directions. A number of 'diagnostic' test procedures are summarized in which location and scale may be unknown, and examples are given in the context of testing for an exponential or Gaussian shape. Approximate efficient scores are emphasized which lead to both well known and novel test statistics. In particular, we show that the standard measures of skewness and kurtosis are asymptotically locally optimal test statistics for normality against departures in the log gamma and Student t family respectively. The properties of two new test statistics are briefly investigated.

Sensitivity surfaces and contours for test statistics are described for comparing several test procedures for normality, using complete samples. The contours are deterministic for a given sample size and illustrate the behaviour of the test statistics over a wide range of nonnorinal data configurations. Five test statistics are considered in detail showing that plotting of sensitivity surfaces provides a compact means of comparing tests for normality without the need for large-scale empirical investigations.

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 Distribution theory for the likelihood ratio statistics for the comparison of several treatments with a control is discussed. These test statistics account for prior information that the treatments are at least as effective as the control. Their null distributions are mixtures of chi-squared or beta distributions and the mixing coefficients, which are 'level probabilities', are intractable for even a moderate number of treatments and unequal weights, which are typically the sample sizes. The distribution corresponding to equal weights is considered as an approximation and an approximation based on the pattern of large and small weights is developed. Both are adequate for moderate variations in the weights, but the approximation based on patterns in the weight set is considerably more accurate for larger variations.

SUMMARY A Bayesian approach to estimation and hypothesis testing for a Poisson process with a change-point is developed, and an example given.

SUMMARY Five procedures for testing the equality of two correlated means with incomplete data on both responses are compared by using Monte Carlo studies; three procedures were suggested by Lin & Stivers (1974) and two are suggested in this paper. The results are that the two tests based on a modified maximum likelihood estimator are to be preferred, the one due to Lin & Stivers when the number of complete pairs is large and the one proposed in this paper otherwise, provided the variances of the two responses do not differ by much. When the correlation between the two responses is small two other tests may be used; a test proposed in this paper when the homoscedasticity assumption is not strongly violated, and a Welch type statistic suggested by Lin & Stivers otherwise. Some key word8: Equality of means; Missing data; Monte Carlo study; Student's t test; Welch approximate test. 1. INTRODIUCTION The problem of testing the equality of two correlated means with incomplete data has been treated by several authors. Ekbohm (1976) has carried out some simulation studies for the case with missing data on one response and recommends Morrison's (1973) statistic when the variances are not too different. A similar comparison is given by Lin & Stivers (1975). Naik (1975) has given a test which may be appropriate in the heteroscedastic case with negative or small positive values of the correlation coefficient. For the case with missing values on both responses Lin & Stivers ( 1974) have suggested some test statistics and it is the purpose of this paper to suggest two more and to give the results of some simulation studies. In constructing test statistics two different approaches are commonly used. We may consider some appropriate linear model, regard all effects in the model as fixed in deriving the sums of squares, and then obtain the expected mean squares and a test ratio under the assumed mixed model. In the present case this procedure leads to the complete paired test, using only the complete pairs and discarding the extra observations. Another approach is to start by obtaining an appropriate estimator, find an estimator of its variance and then forming a statistic which, at least for large samples, is approximately normally distributed. If the maximum likelihood estimator is easily obtainable it will as a rule be a good starting-point. In the present case there is no explicit expression for the maximum likelihood estimator (Ratkowsky, 1974), and so some modifications have to be done in order to construct a noniterative test statistic. Among the five statistics compared, two, one due to Lin & Stivers (1974) and one proposed in this paper, are based on a modified maximum likelihood estimator; the other three are based on the simple mean difference. The main results of the simulation studies

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.

SUMMARY Sequential trimmed t tests are proposed using the restricted scheme and wedge scheme. The formulations of these plans are given and their performances are compared with the usual restricted and wedge t tests under normality and long-tailed distributions. 1. INTRODUCT10N Let xi be random observations taken sequentially from a normal population with mean uz and unknown variance cr2. In testing the hypothesis Ho: ,u/a = 0 against H1: I Ha o = A, one could use sequential t tests such as Wald's open plan (1947, pp. 83, 46), Armitage's restricted plan (1957), Schneiderman & Armitage's wedge plan (1962b) and others. These test statistics, as shown by Yuen (1977), are sensitive to long-tailed distributions and this led us to consider other alternatives which are more robust. The formulation of this new test statistic is a modification of the usual restricted sequential t test which consists of a portion of the same upper rejection boundary as in the open sequential t test (National Bureau of Standards, 1951), and an acceptance boundary providing for termination at a fixed sample size N1. This terminal sample size is chosen such that the test will have specified risks of 2a and /3.

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 The problem of comparing two regression planes without the assumption of equal variances is considered. A natural analogue of the Behrens-Fisher statistic is proposed and its distribution bounded in terms of the studentized maximum and minimum chi-squared distributions. The results are used to construct conservative tests and confidence regions for comparing the regression parameters. Numerical comparisons with more specialized bounds due to Hsu (1938), Haijek (1962) and others are made.

SUMMARY We consider graphs, confidence procedures and tests that can be used with censored survival data to compare the hazard experience of a treatment group with that of a control group. In particular, we consider the relative change A(t) in a cumulative rate function which is used to define a generalized proportional hazard model. This model is equivalent to the model obtained by introducing a gamma distributed frailty term in the proportional hazard model. We introduce an estimate A(t) or A(t) based on the Kaplan-Meier and Nelson-Aalen estimates, and obtain the asymptotic distribution of the process A(t) - A(t) appropriately scaled. This result is used to construct asymptotic pointwise confidence intervals for A(t). Under the proportional hazard model assumption, Doob's transformation of Gaussian processes is used to derive asymptotic simultaneous confidence bands for A(t) from a general class of test statistics defined as sup norms of weighted differences between A(t) and its estimate A(t). We also consider procedures based on estimates r(u) of the relative trend function r(u) between cumulative hazards. The methods are illustrated on a data set from a study of the effect of radiation on mice, and on a data set from a study of the effect of tumour size on the survival of cancer patients.

SUMMARY The problem of testing the adequacy of an autoregressive-moving average time series model is considered. A new approach is proposed, and some Monte Carlo results on the finite sample size and power of the test procedures of this paper are reported. The test statistics discussed by Box & Pierce (1970) and Ljung & Box (1978) are also examined.

SUMMARY The electric response of the brain related to some event, i.e. stimulus, is usually estimated by repeated stimulation and subsequent averaging of the activity recorded time-locked to the stimulus. The present paper deals with the model that the single responses may have varying latency, i.e. arrival time, after stimulus onset. For estimating the single latencies, a maximum likelihood approach employing iterative Fisher-scoring is developed. Further a statistic for testing on the presence of latency variation is derived. Simulations and real data applications are discussed.

SUMMARY Robust tests of general linear hypotheses in linear models are developed. These are likeli- hood ratio type tests in the same sense that M-estimates are maximum likelihood type estimates. Construction of the tests suggests a decomposition of the data into terms analogous to classical sums of squares, providing a robust analysis of variance. Asymptotic efficiency and robustness properties of the tests are the same as those of the M-estimates upon which they are based. Parameter estimation is usually only a first step in the analysis of data arising fromn a linear model. A classical least squares analysis often focuses upon the analysis of variance, which tests simultaneous hypotheses on large subsets of the parameters. Since the terms in a classical analysis of variance are quadratic forms in least squares estimates, one would expect that the sensitivity of the estimates to departures from normality should be inherited by the tests. In fact, for moderate to heavy tailed error distributions or in the presence of outliers, it appears that the classical F test does lose power. Calculations of relative efficiency for proce- dures proposed in this paper substantiate on theoretical grounds the possible inefficiency and lack of power of classical F tests. In this paper, Huber's M-estimates are adapted to hypothesis tests which can be termed likelihood ratio type tests. These procedures naturally generalize and bear a striking re- semblance to classical F tests. Robustness and efficiency properties of M-estimates apply directly to the proposed tests. Hence the case to be made for using likelihood ratio type tests rather than classical F tests is the same as that for using M-estimates in favour of least squares estimates: possible poor performance of the classical methods may be overcome with methods which perform well both when classical assumptions are met and when they are not. The proposed methods are natural, intuitive and as easily computed as M-estimates.

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)

Cox (1972) proposed a regression model for life tables and derived certain hypothesis testing procedures. If there is no censoring and we wish to test a simple null hypothesis, his test statistic is similar to a permutation test. This relation is explored and it is shown how to modify any rank test to allow for censored data. Cox's test and the log rank test are compared and it is concluded that if no censoring occurs the nonparametric test is to be preferred if the sample size is very small. When censoring occurs tests like Cox's are recommended.

SUMMARY Wolfe (1977a) introduced the concept of a partially sequential two-sample hypothesis test and studied some of the properties for a general class of such procedures. In this paper we deal with criteria for selecting an indicator set for use in a partially sequential test. In particular, we show how the Neyman-Pearson lemma can be used to generate an asymptotically optimal indicator set, in the sense that it corresponds to an asymptotically most powerful partially sequential test. In addition, we provide an accurate upper bound for the asymptotic expected sample size of the sequentially obtained observations, and obtain a closed form approximation for the maximum number of these observations necessary in order to obtain prespecified asymptotic power bounds against alternatives of interest. Some key word8: Asymptotically most powerful partially sequential test; Asymptotic power bound; Expected sample size; Negative binomial distribution; Optimal indicator set; Partially sequential procedure; Two-sample test.

The detection of two outliers in a linear model is considered. The case of two outliers in the same direction is studied in detail. The first Bonferroni inequality is used to obtain nominal upper percentage points. For a random sample from a normal distribution, the proposed statistic reduces to the well known Murphy's test statistic. Similarly, for two outliers on either direction, our statistic reduces to the studentized range statistic. An approximate relationship between the critical points of Murphy's test and the studentized range statistic is investigated.

SUMMARY Simple Monte Carlo significance testing has many applications, particularly in the preliminary analysis of spatial data. The method requires the value of the test statistic to be ranked among a random sample of values generated according to the null hypothesis. However, there are situations in which a sample of values can only be conveniently generated using a Markov chain, initiated by the observed data, so that independence is violated. This paper describes two methods that overcome the problem of dependence and allow exact tests to be carried out. The methods are applied to the Rasch model, to the finite lattice Ising model and to the testing of association between spatial processes. Power is discussed in a simple case.

SUMMARY The cumulative chi-squared method has been shown by Hirotsu (1979) and by Takeuchi & Hirotsu (1982) to have good power for testing against ordered alternatives in various statistical models. In this paper a related but simpler statistic is proposed. It is shown to have comparable power against ordered alternatives in analysis of variance models. The components of the statistic are optimal for testing deviations from the null hypothesis expressible in terms of Fourier cosine series. Both the new statistic and the cumulative chi-squared statistic are also shown to be Bayes tests.

Cumulant methods for analysisng the independence of r-way contingency tables and goodness-of-fit frequency data are presented. These methods are based on the exact first three cumulants of both the classical Pearson chi-squared statistic and a recent advantageous modification of this statistic. These approximate methods provide major distributional improvements over existing asymptotic methods

SUMMARY An asymptotic theory for canonical correlation analysis is given for multivariate populations with finite fourth moments. The asymptotic distributions of the sample canonical correlation coefficients and of statistics used for testing hypotheses about the population coefficients involve the fourth order cumulants of the parent population and are sensitive to departures from normality. These asymptotic distributions have surprisingly simple forms in the case of elliptical populations; here a modified test statistic with a chi-squared approximation can be used for testing the hypothesis that some of the population coefficients are zero. Finally we note that, when sampling from elliptical populations, the asymptotic distributions of test statistics used in some other multivariate procedures are similarly simple.

On montre que la re-randomisation de la mediane a la place de la moyenne transforme un gros probleme de calcul en un probleme convenable pour une calculatrice

SUMMARY The standard statistical test for assessing adverse effects in rodent tumourigenicity experiments is based on a time-adjusted analysis of lethal and incidental tumours. This test is shown to arise as the sum of two score statistics for fatal and incidental lesions respectively, the former optimal against extreme value hazard alternatives and the latter optimal against logistic prevalence alternatives in terms of asymptotic relative efficiency in the Pitman sense. An efficiency robust test is developed which retains near full efficiency in the extreme value/logistic situation as well as in other situations where the standard test may be only 70% efficient. Rodent tumourigenicity experiments are used to determine whether or not a particular agent results in an increase in tumour occurrence in animals exposed to this substance throughout their lifetimes as compared to unexposed controls. The tumour status of an individual animal is normally determined at the time of death. Since many animals will die either from natural mortality or as a result of tumour development prior to the end of the study period, information on tumourigenicity will accrue throughout the course of the experiment, typically lasting about two years. Animals surviving to the end of the experiment are sacrificed and subjected to the same pathological evaluation as those dying prior to the termination time. Peto et al. (1980) have proposed life-table analysis of the lethal or fatal tumours and a prevalence analysis of the nonlethal or incidental tumours. In the present paper, we show that, when the substance under study affects the occurrence of fatal and incidental lesions in a similar fashion, their test can be formally derived as the sum of two score statistics for fatal and incidental tumours respectively. The former statistic is the score test under extreme value hazard alternatives whereas the latter statistic is optimal against logistic prevalence alternatives. Because the true form of the underlying hazard and prevalence functions is unknown, we develop in ? 3 an efficiency robust version of this test. This is done by maximizing the minimum asymptotic relative efficiency within a finite class of hazard and prevalence functions. Applications of the maximin efficiency robust test are discussed in ? 4.

SUMMARY Three tests for zero restrictions on regression coefficients that are known to be nonnegative are considered: the classical F test, the likelihood ratio test, and a one-sided t test in a particular direction. Critical values for the likelihood ratio test are given for the cases of two and three restrictions, and the power function is calculated for the case of two restrictions. The analysis is conducted in terms of a characterization of the clas all similar tests for the problem, of which each of the above tests is a member. The likelihood ratio test emerges as the preferred test.