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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 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 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.

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.

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.

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.

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.

SUMMARY A probability sampling plan is suggested in this paper for drawing the sample so that the linear least-squares predictor of the total of a finite population is made asymptotically design-unbiased. Conditions are given for the sampling plan to reduce to the more familiar sample designs discussed in the literature. The linear least-squares predictor of the total of a finite population (Royall, 1970, 1976) derived under a superpopulation model with explanatory variables omitted is still best under the correct model provided that it is unbiased under the correct model (Tam, 1986). The present paper specifies probability sampling schemes which render the linear least-squares predictor asymptoti- cally design-unbiased. Such probability schemes enable some general form of balancing in the sample for the explanatory variables omitted from the assumed superpopulation model, and provide protection of the predictor against this type of model failure. Conditions are also presented in this paper for the specified sample design to become the equal probability selection sample design or the optimal sample design considered by Godambe (1955, 1982).

The Akaike information criterion for model selection presupposes that the parameter space is not subject to order restrictions or inequality constraints. Anraku (1999) proposed a modified version of this criterion, called the order-restricted information criterion, for model selection in the one-way analysis of variance model when the population means are monotonic. We propose a generalization of this to the case when the population means may be restricted by a mixture of linear equality and inequality constraints. If the model has no inequality constraints, then the generalized order-restricted information criterion coincides with the Akaike information criterion. Thus, the former extends the applicability of the latter to model selection in multi-way analysis of variance models when some models may have inequality constraints while others may not. Simulation shows that the information criterion proposed in this paper performs well in selecting the correct model.

Random imputation methods are often used in practice because they tend to preserve the distribution of the variable being imputed, which is an important property when the goal is to estimate population quantiles. However, this type of imputation method introduces additional variability, the imputation variance, due to the random selection of residuals. In this paper, we propose a class of random balanced imputation methods under which the imputation variance is eliminated while the distribution of the variable being imputed is preserved. The rationale behind balanced imputation is to select residuals at random so that appropriate constraints are satisfied. We describe an algorithm for selecting the random residuals that can be viewed as an adaptation of the cube algorithm proposed in the context of balanced sampling (Deville & Tille, 2004). Results of a simulation study support our findings. Copyright 2011, Oxford University Press.

SUMMARY This paper focuses on the Akaike information criterion, AIC, for linear mixed-effects models in the analysis of clustered data. We make the distinction between questions regarding the population and questions regarding the particular clusters in the data. We show that the AIC in current use is not appropriate for the focus on clusters, and we propose instead the conditional Akaike information and its corresponding criterion, the conditional AIC, CAIC. The penalty term in CAIC is related to the effective degrees of freedom p for a linear mixed model proposed by Hodges & Sargent (2001); p reflects an intermediate level of complexity between a fixed-effects model with no cluster effect and a corresponding model with fixed cluster effects. The CAIC is defined for both maximum likelihood and residual maximum likelihood estimation. A pharmacokinetics data appli cation is used to illuminate the distinction between the two inference settings, and to illustrate the use of the conditional AIC in model selection.

SUMMARY This paper considers estimation of the unknown size N of a population based on multiple capture-recapture samples. We extend the Bayesian multiple recapture model to accommodate possible heterogeneity and dependence among the samples and possible heterogeneity within the samples. In the dependent model, we show that posterior inference for N is independent of almost all the nuisance parameters. We develop a flexible Bayesian model for heterogeneity within samples and demonstrate how Gibbs sampling can be used to calculate the Bayesian estimator for N and other quantities of interest. The performance of the proposed estimators is evaluated by simulation under both correct and incorrect model specifications, and we illustrate our methods in two examples about software review and estimation of a cottontail rabbit population.

Recently, new concepts have been proposed for assessing bioequivalence of two drug formulations, namely the so-called population and individual bioequivalence. Using moment-based and probability-based measures for evaluating the proposed bioequivalence concepts, criteria have been formulated to decide whether two formulations should be regarded as bioequivalent or not. This decision has of course to be based on an adequate statistical method where the Food and Drug Administration (FDA) guidance (1997) recommends the use of a bootstrap percentile interval. In this paper, we discuss theoretical properties such as consistency and accuracy of the recommended bootstrap intervals. We focus our investigations on the concept of individual bioequivalence and here especially on the scaled versions of the moment-based as well as the probability-based measures as recommended by the FDA. As estimates for the former, we consider those obtained from an according analysis of variance and restricted maximum likelihood estimators under mixed effect models, where an unbiased estimator of the latter can be derived from the corresponding relative frequencies.

SUMMARY We consider fitting categorical regression models to data obtained by either stratified or nonstratified case-control, or response selective, sampling from a finite population with known population totals in each response category. With certain models, such as the logistic with appropriate constant terms, a method variously known as conditional maximum likelihood (Breslow & Cain, 1988) or pseudo-conditional likelihood (Wild, 1991), which involves the prospective fitting of a pseudo-model, results in maximum likelihood estimates of case-control data. We extend these results by showing the maximum likelihood estimates for any model can be found by iterating this process with a simple updating of offset parameters. Attention is also paid to estimation of the asymptotic covariance matrix. One benefit of the results of this paper is the ability to obtain maximum likelihood estimates of the parameters of logistic models for stratified case-control studies, compare Breslow & Cain (1988), Scott & Wild (1991), using an ordinary logistic regression program, even when the stratum constants are modelled.

Attributable fractions are commonly used to measure the impact of risk factors on disease incidence in the population. These static measures can be extended to functions of time when the time to disease occurrence or event time is of interest. The present paper deals with nonparametric and semiparametric estimation of attributable fraction functions for cohort studies with potentially censored event time data. The semiparametric models include the familiar proportional hazards model and a broad class of transformation models. The proposed estimators are shown to be consistent, asymptotically normal and asymptotically efficient. Extensive simulation studies demonstrate that the proposed methods perform well in practical situations. A cardiovascular health study is provided. Connections to causal inference are discussed. Copyright 2010, Oxford University Press.

SUMMARY We develop several methods for estimating the treatment effect difference defined as the overall log-odds ratio of favourable response in a multicentre clinical trial comparing two treatments with binary response. A simulation study compares the bias and mean squared error of the point estimates and the exact coverage probabilities of confidence intervals obtained distributions. Multicentre randomized clinical trials are frequently used to test the efficacy of new medical regimens. These trials are attractive due to the ease of getting the desired sample size in a short period of time and the broad coverage of the patient population and medical practitioners. The analysis of data from such trials requires combining information from the centres in a way that properly accounts for the variation due to the centres and the differential efficacy of treatments between the centres. In this paper we investigate several approaches to pooling the data across centres and for handling the effects associated with the centres. Chakravarthi & Grizzle (1975) and Beitler & Landis (1985) argue for treating these effects as random, and Fleiss (1986) argues against because the centres are seldom chosen at random from a well-defined population of clinics and therefore the effects cannot be associated with any population. We prefer the random effect model approach for the following reasons. The goal of a trial is to extend the results of the study to the general population, duly accounting for variation in the patient population and the medical skills of practitioners. Hence, if we believe that the variability between centres as estimated by the data is typical of the variability in the population of centres, then the inference based on the random effects model is more appropriate than the one based on the fixed effects model. Furthermore, the variation among the centres may be more than that predicted from samples from a binomial distribution. Treating the centre effects as random introduces this intra-class correlation into our model assumptions. Let the outcome variable from each centre be a pair of binomial random variables XAj

SUMMARY In finite population inference problems, auxiliary population information is often available. We show in this paper that the empirical likelihood method can be naturally applied to such problems to make effective use of the auxiliary information. We prove that the resulting estimates have smaller asymptotic variances than the usual estimates which do not use auxiliary information. A Bahadur-type representation for empirical likelihood sample quantiles is given. Simulation results show that the empirical likelihood estimates perform well among a number of competitors and are model robust.

SUMMARY Poststratification is often used to increase the precision of survey estimators. When applied to regression estimators, it assumes that the regression relationships in the various strata are different. This gives rise to the use of the separate regression estimator. In this paper, we consider situations where the strata affiliation of the population units is not specified in the frame so that the separate regression estimator is not available, the strata means of the regressor variable and the strata sizes being unknown. We propose a new regression type estimator which accounts for the different regression relationships in the various strata, but no longer depends on the unknown strata means and sizes. We show that this estimator is more precise than estimators that ignore the differences in the regression relationships. The theoretical comparisons are illustrated by simulation results.

A brief review, with references, of the literature on capture-recapture theory is given in Jolly (1963). More recently, Cormack (1964) gives a solution, including asymptotic variances, for a specific situation involving the marking and release of a non-random sample of fulmar petrels. His model is stochastic and will be referred to in ? 4. Seber (1962, 1965) has also produced some interesting solutions, and these together with Darroch (1958, 1959) are most directly relevant to our present problem. Darroch (1958, 1959) shows that in a fully stochastic model with either immigration (often called dilution) or death (or emigration), the population parameters can be easily estimated by maximum likelihood. For the more general case when death and immigration are operating simultaneously he derives estimation equations by equating certain observations to their expectations, but does not give variances or covariances for the estimates. In a later paper, (Darroch, 1961), he considers estimation for a closed population consisting of different strata. Seber (1962) establishes a stochastic model for what he calls the multi-sample single recapture census in which an individual cannot be recaptured more than once. This situation arises, for example, when the recaptures are made in the course of hunting or fishing. He allows for both death and immigration in the population, provides explicit maximumlikelihood estimates of the parameters with variances, and suggests tests for certain of the assumptions. In a second paper, (Seber, 1965), he considers a multiple-recapture model differing only slightly from that of Darroch (1959), with both death and immigration. Again, he provides explicit maximum-likelihood estimates of the parameters with variances. A test is also given for equi-catchability in a closed population of individuals with different capture histories. The first purpose of the present paper is to derive a general probability distribution designed to fit the majority of capture-recapture problems involving a 'single' population. The word 'single' here denotes a population covering an area within whose boundaries the animals (or, in general, individuals or members) are free to move and to mix with others of their kind, but which is regarded as a single area in respect of which parameters are to be estimated. The type of situation which is thus excluded by this definition is one where the population is split into a number of defined areas, and separate population estimates are required for each area as well as for numbers of animals moving from one area to another. The single population, however, need not be homogeneous but may consist of different classes of animals behaving in different ways. The other assumptions underlying the model are stated with the notation in ? 2, and the generalized probability distribution is derived in ?3.

In a recent paper on inference about the number of defectives in a finite population, Wright (1992) provides a Bayesian view on the sample size required to give high assurance that the probability of unobserved defectives is low. We reconsider Wright's approach and derive alternative formulae appropriate when there is considerable, but not infinite, prior information

SUMMARY Possibilities for moderating effects of a subgrouping variable on strength or direction of an association have been much discussed by social scientists but have not been given satisfactory statistical formulations. The results concern directed measures of associations in linear models containing just three variables. Some key words: Analysis of covariance; Analysis of variance; cG-distribution; Conditional independence; Graphical chain model; Parallel regressions; Yule-Simpson paradox. 1. INTRODUCTION Linear models are commonly used as a framework to estimate and test how a continuous response variable depends on potential influencing variables. This paper is concerned with the situation in which two influences are random variables as well, which may be discrete or continuous. The continuous random variables are to stand for quantitative properties of observational units, while the discrete variables either capture qualitative properties or they represent subgroups of the population to which the observational units belong. To think of all variables as random variables is usually appropriate for data obtained from observational studies, where one cannot control which levels or values of the influencing variables are to be observed. The linear model permits an analysis conditional on fixed levels and values of the influences. But frequently, the association structure among the influences is, in social science applications, of interest itself and, as it will turn out, it is important for a correct interpretation of how the response depends on the influences. Awareness of such problems in interpretation is widespread among social scientists as is documented by extensive discussions of moderating effects; see, for example, Saunders (1956), Zedeck (1971) and Baron & Kenny (1986). There, a variable is called a moderator if its presence changes the strength or direction of an association. However, methods recommended to social scientists for identifying moderating effects given in the above literature or by Cohen & Cohen (1983, pp. 310-4) have been shown to be seriously deficient (Wermuth, 1988). Thus, there is need for clarification. We treat the case of a potential discrete moderator variable and call its categories or levels the subgroups in the linear model. A moderating effect is closely tied to the notion of consistent results on an association (Wermuth, 1987). In the present context results are said to be weakly consistent if the associations within subgroups coincide in direction and strongly consistent if the associ- ations within subgroups coincide in direction and strength. This notion depends typically on the chosen measure of association. In linear models we look at directed measures of

SUMMARY A comparison is made between play-the-winner sampling and vector-at-a-time sampling for selecting the better of two binomial populations when the selection requirement is defined in terms of the ratio of the single trial probabilities of success rather than the difference. Thus the probability of correct selection is to be at least P* when the true ratio of probabilities of success 0 is at least 0*, where P* and 0* are prescribed. Termination rules based on the difference in the number of successes and inverse sampling are considered. It is shown that play-the-winner sampling is uniformly preferable to vector-at-a-time sampling for both termination rules, and that for 0* close to one the success lead termination rule leads to a smaller expected number of trials on the poorer treatment unless 0 is even closer to one. The problem of selecting the better of two binomial populations, i.e. the one with the higher probability of success p on a single trial, has usually been stated using the framework of ranking and selection procedures as follows. If P* and A* are preassigned constants, with < P* < 1 and 0 < A* < 1, the probability of a correct selection is to be at least P* when the true difference in the p-values is at least A*. Sobel & Weiss (1970) consider two sequential procedures for this selection problem in which sampling is terminated when the absolute difference in the number of successes for the two populations first reaches a pre- determined integer. We will refer to this as the success-lead rule for termination. In the first procedure, play-the-winner sampling is used in which sampling continues with the same population after each success and switches to the other population after each failure. In the second procedure, vector-at-a-time sampling is used in which two observations are taken at each stage, one from each population. In many contexts, it is desirable to have a small expected number of trials on the poorer population or treatment but using the success-lead rule it is not true uniformly in the parameter space for the p's that play-the- winner sampling is superior to vector-at-a-time sampling in this respect. In a second paper Sobel & Weiss (1971) consider the use of an inverse sampling rule when sampling is termi- nated when any one population has r successes. They show that in this case play-the-winner sampling is uniformly preferable to vector-at-a-time sampling in the sense that for the same probability requirement, the expected number of trials on the poorer treatment is always smaller. Truncated versions of the inverse sampling rule and the success-lead rule for play- the-winner sampling were proposed by Hoel (1972) and Fushimi (1973). Their stopping rules include the number of failures as well as the number of successes and mean that excessive sample sizes are avoided when the population probabilities of success are small.

Taking account of heterogeneity between the individuals in population based mortality studies is important. A systematic way of describing heterogeneity is by an unobserved quantity called frailty, entering the hazard multiplicatively. Until now most studies have used a gamma distributed frailty, which is mathematically convenient; for example, the distribution among survivors is also gamma. This paper shows that several other distributions have equally simple properties, the main example being the inverse Gaussian distribution. Consequences of the different distributions are examined; the inverse Gaussian makes the population homogeneous with time, whereas for the gamma the relative heterogeneity is constant.

SUMMARY Suppose that we have a sample from a bivariate normal population with variances 0-2, 0-2 and correlation coefficient p, but some observations on one of the two variables are missing. Based on these data, we wish to test the hypothesis of equality of means of the two variables. In this paper, classes of test procedures are given for when the alternative is one-sided and for when it is two-sided. The tests satisfy the property that the probability of type I error does not exceed a preassigned level a when p is negative, and when p is positive but small. The test that is to be recommended in a particular situation depends on what information, if any, is available about the nuisance parameters. Comparisons of the tests with the paired t test and the Mehta & Gurland (1969, 1973) test are also included.

SUMMARY In an earlier paper Sobel & Weiss (1970) examined two sampling schemes for choosing the binomial population with the larger probability of success. The two sequential allocation procedures were discussed in the light of their ability to reach a decision while using as small a number as possible of tests with the poorer population. It was shown that neither sampling scheme was uniformly better than the other with respect to that criterion. However, one sampling procedure, the Play-the-Winner rule (Robbins, 1956; Zelen, 1969), was superior, provided that the probability of success of the better population was greater than a number roughly equal to A. Here we investigate a two-stage procedure in which the first stage is used to estimate the probability of success of the better population. It is shown that a fairly small estimation stage, approximately five tests with each population, allows one to benefit considerably from the choice between the two sampling methods available in the second stage.

A multiple decision approach to the problem of ranking populations according to their population means has been formulated by Bechhofer (1954). A single-sample solution to this problem was presented by him for the case of normal populations with known variances. In the present paper the case of normal populations with unknown equal variances is considered. A two-sample procedure is proposed as a solution to the latter problem; this procedure is of the type used by Stein (1945) for obtaining a test of Student's hypothesis with power independent of the variance. Tables are available which enable the experimenter to apply this procedure with little computational effort to the ranking of two or three populations with unknown but equal variances. The same tables can be used when the variance ratios are arbitrary and known. Graphic comparisons of the expected sample size are made for the single-sample and two-sample procedures.

All the methods which have been presented in recent years for the analysis of capturerecapture data have aimed at estimating total population size, survival rates and dilution rates. These methods, whether deterministic or stochastic, all assume that the animals captured on each occasion are a random sample from the whole population. The proportion of marked animals in each sample then contains information about the size of the population at that instant. Darroch (1958, 1959) has shown for a stochastic model that, in the cases where the population is subject to either death or to immigration but not both, the parameters of the population together with their variances and covariances can be very simply estimated. In the more general cases with both death and immigration, he obtains estimation equations which can readily be solved numerically, and suggests that the variances of the estimates can be obtained by standard asymptotic techniques. For a deterministic survival rate Jolly (1963) obtains explicit formulae for population size and survival and dilution rates, and asymptotic formulae for their variance and covariances. This theory has been derived to meet the case where the animals are sampled completely at random. However, for some species, the markings which are applied to render an animal individually identifiable may be visible without capture. For the past thirteen years zoologists from the University of Aberdeen have been carrying out a study of the fulmar petrel (Fulmarus glacialis) on a small island in Orkney, the details of which have been describedby Carrick & Dunnet (1954) andDunnet, Anderson & Cormack (1963). Throughout this study birds have been caught individually on their nests and have been marked by an identifying set of coloured rings on the leg. These rings are clearly visible in flight, so that in years subsequent to marking the continued presence of a bird in the colony may be established without any capture being effected. New individuals may be marked each year by further capture. The aim of this paper is to examine a model applicable to this biological situation. Clearly no useful inference about the population size is possible, but valid estimates of survival rate can be obtained. In ?? 2 and 3 the notation used and the general formulation of the theory are set out. Maximum-likelihood estimates of survival rates are obtained in ? 4, and their asymptotic variances and covariances in ? 5. These are applied to the over-all survival rate and the life span in ? 6. In ? 7, the theory is illustrated for records from the study of fulmars. 2. NOTATION

SUMMARY The problem considered here is that of testing equality of means in the presence of correlation when a sample from a bivariate normal population is available. The procedure studied for this purpose involves a preliminary test of a hypothesis concerning the population correlation coefficient. When some observations on one of the variables are missing a statistic used to test the hypothesis of equality of means is suggested. The power of this test is compared with that of the paired t test. It is well known that in testing for equality of means of two independent normal populations the t statistic based on two independent samples is appropriate. On the other hand, in testing for equality of means of two correlated normal populations the t statistic based on the differences of the observations in the two samples is appropriate. It may happen, however, that one is uncertain whether or not the populations from which the samples are taken are correlated. In such a situation the paired t test could be employed but such a solution to the problem would leave something to be desired. To be more specific, if the populations are actually independent but the experimenter does not know this and uses a paired t test, this could result in a loss in sensitivity because a test based on independent samples has twice the number of degrees of freedom as the paired t test based on correlated samples. In the presence of uncertainty of correlation, therefore, it is important to utilize the data to the best advantage. One purpose of this paper is to develop a procedure for testing equality of means in the presence of possible correlation or lack of it. A further problem which will also be considered is how to test equality of means when some observations from one of the populations are missing.

This paper considers the problem of selecting the best one of k populations, a decision that must frequently be taken in industrial practice where the populations may correspond to alternative varieties or treatments. For definiteness the best population will be taken to be that with the least mean, though the situation is basically unaltered if least is replaced by greatest or mean replaced by any other parameter of the populations. Apart from their mean values, the populations are supposed equally desirable, thus implying equal costs and no side benefits. In particular we suppose that differences in the variances of the populations

THE theoretical distribution of many statistics calculated from small samples is known when the population is normal, but when it is not normal we know very little about the distribution of such statistics. Such work as has been done has generally assumed population forms of standard types, but we may occasionally come up against samples from populations which do not appear to fit into any known type. This has led to many attempts being made to build up, by experimental sampling from non-normal data, partial populations of samples from which can be inferred in an empirical way the laws of distribution followed by derived statistics. A list of papers dealing with this subject which have come to the author's notice is given on pp. 79, 80 below. In many cases it has been found that in sampling from curves with one mode not at the end of the range, the distributions of statistics such as "Student's " "t", the correlation coefficient and, in certain cases, Fisher's "z ", differ very slightly from one population to another. On the whole these investigations have suggested that in such cases we can neglect the departure of the population from normality without introducing serious error into our tests of significance. The possibility of further theoretical work must not be overlooked, but unless our results are independent of population form (as, for instance, in recent work by Pitman and Welch) it is unlikely that we shall be able to make much practical use of the results. It is customary to designate a non-normal population by the values of 1 and f2; but in the case of samples of 100 or less from a normal population the range of values of ,81 and /2 excluding 5 % of the total at each end is comparable with the range of 1 and /2 in the non-normal populations which have been used for sampling experiments. Further, this range of populations is considered by E. S. Pearson to cover most cases which will be found to occur in practice. On these grounds I think that conclusions of practical value are most likely to be reached by further sampling. No attempt appears to have been made to carry out an experimental sampling from a bivariate population in which the distribution surface is not normal and in which the correlation coefficient is high, or to take sets of samples from a univariate non-normal population and to assign the samples to blocks and treatments in a randomized block experiment, taking a completely fresh