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Exact and approximate methods for analyzing the common odds ratio in the combination of 2 x 2 tables are programmed. The approximate methods are improved by incorporating bias and skewness corrections in testing and estimation. Exact methods are done more efficiently by employing network theory. This makes more feasible the exact analyses of sparse data, large numbers of 2 x 2 tables each based on small numbers. Additional tests of interaction, particularly for sparse data, are added. Another feature is the point and interval estimation of the attributable risk. The various aspects of the program are illustrated in three numerical examples. An executable version of this program, for IBM compatible PCs, requires about 314K for up to 500 tables. It is available from the authors upon submission of a PC diskette formatted with MS-DOS.

Consider J independent pairs of mutually independent binomial variates which can be considered as J2×2 tables. We assume a common measure of association, such as the odds ratio, between the tables. This paper studies the asymptotic moments of the «pooled» estimators, i.e. those constructed from the data summed over all the tables. For balanced designs, or proportional subclass numbers, it is shown that the pooled estimator of the common odds ratio is asymptotically biased towards 1, while those of the common ratio and difference are asymptotically unbiased. The asymptotic variances are also compared. A simple test for complete homogeneity is suggested

Consider matched case-control studies with a pair of distinguishable controls, say neighbourhood and hospital, wherein one of the controls may be missing. Liang and Stewart derive an efficient test for the possible difference in controls for the complete data case. They suggest inefficient tests for the situation where some triplets are incomplete. Levin, and Risch and Tibshirani, derive efficient tests for the incomplete triplet case by the methods of maximum likelihood estimator (Wald) tests and likelihood ratio tests, respectively. All these tests require the use of compute algorithms. Using score theory, we derive a simple efficient test for complete triplets which only requires solving a quadratic equation. Furthermore, we show that the usual normal approximation to McNemar's test is equivalent to the score test and is thus fully efficient as well. For incomplete triplets, score theory leads to a fully efficient test which only requires the solution of a cubic equation. We propose a nearly efficient test which is even more simply computed. The example of Liang and Stewart illustrates application of the results.

Recently, Beal (1987, Biometrics 43, 941-950) found Mee's modification of Anbar's approximate interval estimation for the difference in binomial parameters to be a good choice in small sample sizes. As this method can be derived from the score theory of Bartlett, it is easily corrected for skewness. Exact numerical evaluation shows that this correction is not as important for this case as for the ratio of binomial parameters (Gart and Nam, 1988, Biometrics 44, 323-338). The score theory is also used to extend this method to the stratified or multiple-table case. Thus, good approximate interval estimates for differences, ratios, and odds ratios of binomial parameters can all be derived from the same general theory.

Let P1 be the probability of a disease in one population and P2 be the probability of a disease in a second population. The ratio of these quantities, R = P1/P2, is termed the relative risk. We consider first the analyses of the relative risk from retrospective studies. The relation between the relative risk and the odds ratio (or cross-product ratio) is developed. The odds ratio can be considered a parameter of an exponential model possessing sufficient statistics. This permits the development of exact significance tests and confidence intervals in the conditional space. Unconditional tests and intervals are also considered briefly. The consequences of misclassification errors and ignoring matching or stratifying are also considered. The various methods are extended to combination of results over the strata. Examples of case-control studies testing the association between HL-A frequencies and cancer illustrate the techniques. The parallel analyses of prospective studies are given. If P1 and P2 are small with large samples sizes the appropriate model is a Poisson distribution. This yields a exponential model with sufficient statistics. Exact conditional tests and confidence intervals can then be developed. Here we consider the case where two populations are compared adjusting for sex differences as well as for the strata (or covariate) differences such as age. The methods are applied to two examples: (1) testing in the two sexes the ratio of relative risks of skin cancer in people living in different latitudes, and (2) testing over time the ratio of the relative risks of cancer in two cities, one of which fluoridated its drinking water and one which did not.

A previously reported negative association between a high index of dietary vitamin A and lung cancer incidence was confirmed in an extended follow-up, covering 11 1/2 years, of 13,785 men and 2,928 women, Responses to a postal questionnaire provided the dietary information. Relationships between the major dietary items and lung cancer were explored for various diagnostic subsets of the 168 lung cancer cases diagnosed among the study subjects. Analyses were stratified for sex, age, residence characteristics, cigarette smoking and, at times, socioeconomic group. Although the data do not permit a firm interpretation in terms of risk enhancement by a marginal retinoid deficiency, we found that the apparent protection afforded by higher intakes of vitamin A or its provitamins was particularly strong for lung cancer appearing as squamous-cell carcinoma and among those with higher alcohol intakes. The individual food items which showed the strongest negative association with lung cancer were carrots and milk. These two items made a major contribution to the vitamin A index and its variation among the respondents.

SUMMARY The bias and first four cumulants of the distribution of the logarithm of a binomial variate are studied by means of asymptotic expansions and exact computation. A new estimator of the variance is derived and evaluated. The asymptotic skewness is found to differ from the result of Walter (1975). Applications to point estimation of the one-hit curve and the interval estimation and testing of the ratio of binomial parameters are considered. Because of the bias and nonnormality of such statistics, methods based on likelihood methods or Pearson chi-squared statistics are preferred.

The incidence of most diseases is low enough that in. large populations the number of new cases may be considered a Poisson variate. This paper explores models and methods for analyzing such data Specific cases are the estimation and testing of ratios and the cross-product ratios, both simple and stratified* We assume the Poisson means are exponential functions of the relevant parameters. The resulting sets of sufficient statistics are partitioned into a test statistic and a vector of statistics related to the nuisance parameters . The methods derived are based on the conditional distribution of the test statistic given the other sufficient statistics. The analyses of stratified cross-product ratios are seen to be analogues of the noncentral distribution associated with theanalysis of the common odds ratio in several 2×2 tables. The various methods are illustrated in numerical examples involving incidence rates of cancer in two metropolitan areas adjusting for both age and sex.

Consider I pairs of mutually independent binomial variates xoi and x1i, with corresponding parameters poi and p1i and sample sizes, noi and n1,, where i = 1, ..., I. We assume that the relative risk is constant, that is, b = Pt /izpoi (i = 1, ..., I). Further let qji = 1-pji, and we assume throughout that 0 < pji, qji < 1 for i = 1, ..., I and j = 0,1. We consider asymptotic results as noi's and n1j's become large, but I is fixed. Extending the results of Cochran (1954) for the common odds ratio, Radhakrishna (1965) derived an asymptotically efficient test of the hypothesis b = 1. Recently, for I = 1, Koopman (1984) heuristically derived a useful approximate method for finding confidence limits for 0. In the present paper we use the methods of score statistics, particularly as given by Bartlett (1953a, 1953b, 1955) to derive, for the general case, the asymptotically efficient tests of b = 1, and of its homogeneity over the I tables, as well as approximate confidence limits for the common 0. The test of 0 = 1 is identical with that of Radhakrishna, and Koopman's limits are a special case of these more general results. Thus this rather simple theory gives a coherent structured methodology for analysing ratios of binomial probabilities parallel to that usually employed for common odds ratios; see, for example, Gart (1970, 1971).

A table of exact 95 percent confidence limits for differences and ratios of two proportions and their odds ratio, including one-tailed P values for the Fisher-Irwin exact test, is given for a wide variety of 2 × 2 tables with sample sizes of 20(20)100. The calculations are based on Fisher's [3] conditional theory, and the computer algorithm of Thomas [10 or 11] is used to produce the tables. It is shown how the tables may be used for sample size determination without fixing the power for a given alternative hypothesis.

The performance of three approximate confidence limit methods for the odds ratio, R, is studied at the 95% level in the unconditional sample space. There are: the method proposed by Cornfield (Proceedings of the 3rd Berkeley Symposium 1956;4:135-48), the logit method with 1/2 corrections first considered by Woolf (Ann Hum Genet 1955;19:251-3), and the test-based method proposed by Miettinen (Am J Epidemiol 1976;103:226-35). Cornfield's method comes closest to attaining the normal confidence coefficient. The logit method typically has actual confidence coefficients somewhat too large with disparate tail areas. The latter is ascribed in part to the enhanced skewness induced by the logit transformation itself. The test-based method has actual coefficients uniformly less than nominal when R not equal to 1. This underestimation is worse in finite samples than Halperin found it to be asymptotically. Although the Cornfield and test-based methods have the same confidence coefficients for R = 1, the test-based method is more likely to cover distant values of R not equal to 1 when in fact R = 1. It is concluded that Cornfield's method without the continuity correction is the preferred approximate method in the unconditional space as, with the continuity correction, it was previously found to be in the conditional space.

Several models with monotone trend in proportions are considered for 2 × I × J contingency tables. For one stratum (I = 1) and broad regularity conditions, the C(α) test statistic for testing for trend is shown to be fully efficient for all choices of the model with monotone trend. For several strata, the efficiency of each C(α) test is compared with the efficiency of the C(α) test appropriate when each of the other models holds. These relative efficiencies are used for choosing the most robust tests in a variety of sampling situations.

The cell cycle of Saccharomyces cerevisiae in populations growing at different rates was analyzed by flow microflourometry. The results demonstrate that (i) variation in growth rate of the population depends mainly on variation in the G1 phase of the cell cycle, and (ii) atall growth rates initiation of DNA synthesis occurs concomitantly with initiation of budding.

Nurminen (1981) gives conditions for the asymptotic efficiency of general weighted mean estimators. His application is a noniterative estimator of a common relative risk for binomial variates. This summary estimator and, by implication, the Mantel-Haenszel summary odds ratio estimator are asserted to be asymptotically efficient. We examine conditions for the asymptotic efficiency in the unconditional sample space of the MantelHaenszel estimator for a common odds ratio, the estimator of a common relative risk considered by Nurminen, and Tarone's (1981) estimator of a common relative risk for binomial variables. Contrary to Nurminen, we find that, except for a few degenerate cases, these estimators are inefficient whenever the common odds ratio or common relative risk is not equal to 1.

SUMMARY The bias and first four cumulants of the empirical logit transformation of a binomial variate are studied by means of asymptotic expansions and exact computation. The covariance of the empirical logit and its estimated variance is derived. Neither the + 2 correction of Haldane (1955) and Anscombe (1956) nor the -2 suggested by Cox (1970) for weighted logit regression is universally effective in reducing bias. Other corrections are considered. The distribution of the empirical logit is shown to be considerably more skewed and to have a much larger kurtosis than the comparable binomial distribution. Methods based directly on sufficient statistics are preferred to those based on the empirical logit.

SUMMARY An exact test for comparing proportions in matched samples is derived for the situation where order within pairs may be important. The analysis is based on an extension of the logistic model of Cox (1958 c). The test is formally equivalent to Fisher's exact test for 2 x 2 tables. The McNemar test (1947) has long been used in comparing matched or paired responses each of which take only two values, 0 or 1. The test consists of counting only those pairs which are unlike, (0, 1) or (1, 0) and testing whether the distribution of these two types is consistent with a binomial distribution with a probability of 2. Cox (1958c) showed that this test may be derived from a logistic model, and found a confidence interval for the relevant parameter of non-centrality. Cochran (1950) extended the test to more than two proportions. In many experimental situations, matching entails an ordering within the pairs. For instance, in testing certain drugs the matching involves a patient receiving the two drugs in two different time periods. Sometimes the order of the drug administration has appreciable effect on the patient's response; see, for instance, Meier, Free & Jackson (1958). In such instances, the McNemar test may ignore pertinent and important information on order within pairs. To fix ideas, we recall an example cited by Mosteller (1952). Two drugs, A and B, are used on each of 100 subjects, the response being either 'nausea' or 'not nausea'. Of these subjects, 81 never had nausea, denoted by (0, 0); 9 subjects had nausea with A but not with B, (1, 0); 1 had nausea with B but not with A, (0, 1); and 9 had nausea with both drugs, (1, 1). Mosteller considered only the unlike pairs, and calculated from the binomial distribution the onetailed significance level

The method of maximum likelihood was proposed as a general method of estimation by Fisher (1912). However, rigorous proofs of the asymptotic properties of the maximum likelihood estimators (hereafter called ML estimators) were not developed until the work of Dugue (1946), Wald (1943, 1949), Cramer (1946), Huzurbazar (1948) and Chanda (1954), all of whom published within the last 25 years. Each of these authors assumed observations from a single population. Neyman & Scott (1948) and Kraft & LeCam (1956) have pointed out situations wherein the ML estimator is inefficient and not consistent. Their examples involved sampling from a number of subpopulations with different distributions but with only a finite number of observations from each of the subpopulations. Some situations arise where observations do not come from a single population but from distinct but related populations, related in the sense that some populations reasonably may be assumed to have some parameters in common. We shall call such populations associated. We consider two cases of associated populations. In the first, a finite number of populations are considered and limiting results regarding ML estimators are obtained as numbers of observations from the distinct populations become large in constant ratios. In the second, one observation is obtained from each population and the number of populations sampled becomes large. The second case may include the first as some populations sampled are identical but for applications to investigations of specific ML estimators it seems better to consider the two cases separately. In this paper we extend the basic theory on ML estimation to associated populations. Under regularity conditions developed, the ML estimators for parameters in associated populations are shown to be consistent and asymptotically normal with a variance-covariance matrix as derived. The asymptotic X2-distribution of -2 ln A, A being the likelihood ratio, holds and, for the non-null distributions, the parameter of non-centrality is set forth. Our objective is to set forth results as simply as possible and in usable forms; more generalized theorems could be obtained. We limit consideration to ML estimators that are roots of the normal equations for the maximization process. The consistency of the more generally defined estimator can be proved for associated populations following the approach of Wald (1943). Our demonstrations follow those of Chanda closely; most of the results for associated populations are easy generalizations of results for a single population and we state results without complete proofs when this is so. Two simple examples involving associated populations are included. One of these applies to the work of Dykstra (1960) which is dependent upon results given here.

The relationship between chickenpox and yaws epidemics occurring among children in a village in Netherlands New Guinea is analysed using the mathematical model for a simple deterministic epidemic. It is shown that the yaws epidemic accelerated significantly in the month following the chickenpox epidemic, but that it reverted to its previous rate in the succeeding months. The number of yaws cases attributable to the influence of the chickenpox is estimated from the projected course of the yaws epidemic. It is statistically verified that those children contracting chickenpox were more likely to become yaws cases in the subsequent month.These results point to the danger of yaws's spread being much more rapid among a population which has recently been subject to an epidemic of chickenpox.

Cramer ([6], p. 474 ff.), Darmois [8], Frechet [10], and Rao [14] derived independently a lower bound for the mean square error of an estimate $t$ of a parameter which appears in a frequency function of a specified form. This expression, alternately termed the Cramer-Rao inequality or the information limit, is \begin{equation*}\tag{1.1}]E(t - \alpha)^2 \geqq \lbrack E(t) - \alpha\rbrack^2 + \frac{\big\lbrack\frac{\partial E(t)}{\partial\alpha}\big\rbrack^2}{E\big(\frac{\partial\ln\phi} {\partial\alpha}\big)^2}\end{equation*} where $\phi$ is the likelihood of the sample. The expression $E(\partial \ln \phi/\partial\alpha)^2$ is called the information on $\alpha$ and is sometimes denoted by $I(\alpha)$. Under rather general conditions it can be shown equal to $E( - \partial^2 \ln \phi/\partial\alpha^2)$. The equality in (1.1) is reached if and only if, \begin{equation*}\tag{1.2}\phi = \phi_1e^{tV(\alpha) + W(\alpha)}\end{equation*} where $t$ and $\phi_1$ are functions of the observations alone and $V(\alpha)$ and $W(\alpha)$ are functions of $\alpha$ alone. By the results of Pitman [13] and Koopman [12], the form of (1.2) implies that $t$ must be a sufficient statistic. The fact that this form of the likelihood yields a minimum variance estimate was first pointed out by Aitken and Silverstone [1]. If we have $n$ observations which are independently and identically distributed, the frequency function of the underlying population must be of the so-called Pitman-Koopman form, \begin{equation*}\tag{1.3}f(x; \alpha) = (\alpha)h(x)e^{P(\alpha)g(x)}\end{equation*} and $t$ must be a function of $\sum^n_{i = 1} g(x_i)$ for the equality in (1.2) to hold. Several extensions of the basic inequality have been derived. Bhattacharyya [4] and Chapman and Robbins [5] have derived results which yield more stringent inequalities in certain instances. Wolfowitz [21] has extended the result to sequential sampling situations. Cramer [7], Darmois [8], and Barankin [2] have considered joint bounds on sets of estimates of parameters and Hammersley [11] has derived a lower bound of the mean square error of an estimate for the situation in which the parameter to be estimated can only assume discrete values. Barankin [3] has also considered lower bounds on the general absolute central moments of the estimate. All these results assume that the parameters involved are constants. Here we shall consider the case where the parameters are random variables. Thus the lower bound of the mean square error of an estimate will take into account the variability due to both the observations and the parameters involved. Necessary and sufficient conditions for equality of the extended inequality are derived. Most unfavorable distributions, i.e., distributions which maximize the lower bound, are defined, and several examples are given. Extensions analogous to those of Bhattacharyya [4] and Wolfowitz [21] are also considered. Finally, bounds on the variance of linear estimates of the mean of the parameter are derived.

We consider the test of homogeneity of relative risks or odds ratios for multiplymatched case-control studies suggested by Ejigou & McHugh (1984). It is shown that this test may be simply derived as a special case of Pearsonian chi-squared tests with an estimated parameter (Cramer, 1946, Ch. 30). Consider J independent pairs of variates whose components are a single zero-one variate for the case and a matched binomial variate of sample size R for the control. Let the associated probabilities of a one, say exposure, be Plj and P2j respectively. We wish to test the hypothesis of a constant odds ratio, that is

SUMMARY Stevens (1951) gives approximate expressions for the conditional mean and variance of an entry in a contingency table with fixed marginals. Cornfield (1956) shows these to be the moments of the asymptotic normal approximation to the noncentral hypergeometric distribution. McCullagh (1984) and Levin (1984) suggest corrections to the expression for the mean. We show that the bias correction preferred by Levin for the conditional case is virtually identical to that derived by Gart (1985a) for the unconditional case. This correction is based on Bartlett's (1955) general results for score statistics. Related theory (Bartlett, 1953) yields Stevens's approximate variance formula and an expression for the third central moment.

Fourteen chemicals of varied uses were tested for carcinogenicity by oral administration in male and female Charles River CD rats. Under the conditions of the tests, propane sultone, propylene imine, and ethylenethiourea, in addition to the positive control N-2-fluorenylacetamide, were carcinogenic. Avadex, bis(2-chloroethyl) ether, the potassium salt of bis(2-hydroxyethyl) dithiocarbamic acid, ethylene carbonate, and semicarbazide hydrochloride were not carcinogenic under the test conditions. Dithiooxamide, glycerol alpha-monochlorohydrin, and thiosemicarbazide gave somewhat ambiguous results, though administered at high enough dose levels to be toxic. An inadequate number of animals survived treatments with sodium azide, sodium bisulfide, and vinylene carbonate, or the animals may not have received sufficiently high doses of the test chemicals to provide maximum test sensitivity. However, there were no indications that these three chemicals were carcinogenic under the test conditions.

The purpose of this study was to compare the antioxidant activities of a-tocopherol and y-tocopherol in protecting the red cell mem brane against lipid peroxidation. Tocopherols were incorporated into the red cell membrane by incubating cells with solutions of the tocopherols in bovine albumin. The cells were then washed and subjected to the dialuric acid hemolysis test. Analysis of variance of the response curves revealed that y-tocopherol had 38% of the activity of a-tocopherol. No evidence was found for an interaction between the two tocopherols when present in the red cell membranes simultaneously. J. Nutr. 106: 124-127, 1976.

The Poisson distribution enjoys several convenient statistical properties. It is additive and complete; it possesses a sufficient statistic. Its exponential structure permits the construction of “exact” conditional tests. This paper discusses how these properties can be exploited in developing useful statistical methods. Part I contains the derivation of conditional tests of the Poisson based on the variance and the third and fourth sample cumulants. The “variance” test is also extended to case of left-truncated samples. In Part II significance tests for cross-product ratios of Poisson means are derived. All the methods are illustrated with practical examples in biomedical research.

The interpretation of chronic bioassay tests for carcinogenicity requires that the data be appropriately recorded. A "case history" for each animal links the pathology data for each organ of each animal to the length of its life. This information can be used in interpretation of the tumor incidences in light of the survival information. The role of historical controls was discussed, and the use of significance tests in a multidisciplinary approach to the assessment of the pattern of tumor response was suggested. Multiple comparison methods valid for the interpretation of continuous (or measurement) data do not apply to the discrete data analyses used in these studies. The ideas and methods of these studies were applied to an animal study of chloroform.

Tests for linear trend and homogeneity among proportions and analogous time adjusted tests for life table data are programmed for stratified data. For proportions, the maximum likelihood estimator of the common logistic slope and its approximate standard error is given. This parameter is related to the relative risk for the factor under test. The model is a simpler and more relevant subcase of the general linear multiple logistic model. Tests of fit include a chi-square test of departure from trend for both types of data, and, for proportions, a test for the equality of slopes among groups of strata. The program is particularly useful for large prospective studies in epidemiology, but may also prove helpful for case-control studies, clinical trials, and combining animal studies. Several optional analyses are included. The program generalizes most aspects of a previously published program by Thomas, Breslow, and Gart (Trend and homogeneity analyses of proportions and life table data. Comp. Biomed. Res. 10, 373 (1977)). A FORTRAN card deck with documentation is available upon request.

A FORTRAN program for computing various statistical tests relating proportions to dose is described. Life table curves are computed and printer plots may be obtained. Exact and approximate tests of linear trend for proportions are given. Finally two methods of testing for linear trend while adjusting for the time element are programmed. In both cases, tests of homogeneity and departures from trend are also given. The program is available from authors.

The effect of an epidemic of chickenpox on the course of an epidemic of yaws in a village of New Guinea was studied by Gart and deVries (1966) and Gart (1968). A deterministic continuous time model was used to confirm that the yaws epidemic accelerated dramatically in the month following the chickenpox epidemic. However, by the nature of the model, no statistical tests are possible.

Healy (1969) recently pointed out that various test criteria may rank in differing orders the possible outcomes in a contingency table with fixed marginals. In particular, he noted a 2 X 3 table where the likelihood ratio criterion and the individual frequency criterion lead to somewhat different rankings. For the general r X c contingency table with fixed marginals, Gart (1966) showed that a modified form of the likelihood ratio statistic will very often lead to the same ranking as the individual frequency criterion. In order to review that result we introduce notation for the general case. Let {ai,}, i = 1, 2, ... r; j = 1, 2, * * c; be the entries in an r X c contingency table with fixed marginal totals Ai. = ] a , A. = E] ai , i = 1, 2, * * * r; j = 1, 2, * * c;andA.. = E E ai . The individual frequency of a particular table is

Gart and Nam (1984, Biometrics 40, 887-894) consider the analysis of an ABO-like model based on the Hardy-Weinberg law in the commonly occurring case of human leukocyte antigen (HLA) data where there are no double blanks, that is, no recessive homozygotes. They derive a score test, based on the truncated likelihood, of the hypothesis that the true recessive gene (or allele) frequency is zero. Yasuda (1968, Biometrics 24, 915-935) considers a similar codominant system wherein the true recessive gene frequency is assumed zero, but the Hardy-Weinberg law does not hold. In particular, he considers the possibility of a nonzero inbreeding coefficient. We show that the two models are equivalent; each likelihood can be shown to be a reparameterization of the other. Furthermore, the score test of the zero gene frequency in Gart and Nam is identical to the score test for a zero inbreeding coefficient given by Yasuda. The results are applied to an example wherein it appears that the Hardy-Weinberg model is appropriate. Thus, it is not possible in this population to identify homozygous individuals without error from phenotypic data alone.

Various methods for finding confidence intervals for the ratio of binomial parameters are reviewed and evaluated numerically. It is found that the method based on likelihood scores (Koopman, 1984, Biometrics 40, 513-517; Miettinen and Nurminen, 1985, Statistics in Medicine 4, 213-226) performs best in achieving the nominal confidence coefficient, but it may distribute the tail probabilities quite disparately. Using general theory of Bartlett (1953, Biometrika 40, 306-317; 1955, Biometrika 42, 201-203), we correct this method for asymptotic skewness. Following Gart (1985, Biometrika 72, 673-677), we extend this correction to the case of estimating the common ratio in a series of two-by-two tables. Computing algorithms are given and applied to numerical examples. Parallel methods for the odds ratio and the ratio of Poisson parameters are noted.

In generalized ABO-like genetic systems, particularly for certain human leukocyte antigen (HLA) loci, it is quite likely, even for large sample sizes, that no double recessives (or 'double blanks') may be found. It has been suggested that all such data should be analyzed as if the true recessive gene frequency were 0. We derive an easily computed score [or C(alpha)] statistic which tests for the possibility that, in such instances, this frequency may exceed 0. An associated point estimator is suggested. The methods are applied to the data analyzed by Smouse and Williams (1982, Biometrics 38, 757-768).

Shannon's measure of entropy,-E Pi log Pi, has been widely used to describe the diversity of ecological communities. There are, however, several different sets of proportions (or probabilities) that arise in the specification of a community and the study of samples obtained from it. The purpose of this note is to examine the relationships between the corresponding entropies. We consider a community comprising a finite or countable set of species of similar type. The abundance of Species i, denoted by Ai, is defined to be the expected number of individuals of that species caught by the expenditure of unit trapping effort. In the case of an infinite set of species it will be assumed that 2 Ai =A