We describe an algorithm for the sequential sampling of entries in multiway contingency tables with given constraints. The algorithm can be used for computations in exact conditional inference. To justify the algorithm, a theory relates sampling values at each step to properties of the associated toric ideal using computational commutative algebra. In particular, the property of interval cell counts at each step is related to exponents on lead indeterminates of a lexicographic Gröbner basis. Also, the approximation of integer programming by linear programming for sampling is related to initial terms of a toric ideal. We apply the algorithm to examples of contingency tables which appear in the social and medical sciences. The numerical results demonstrate that the theory is applicable and that the algorithm performs well. 1. Introduction. Sampling

Existing methods for setting confidence intervals for the difference � between binomial proportions based on paired data perform inadequately. The asymptotic method can produce limits outside the range of validity. The ‘exact ’ conditional method can yield an interval which is effectively only one-sided. Both these methods also have poor coverage properties. Better methods are described, based on the profile likelihood obtained by conditionally maximizing the proportion of discordant pairs. A refinement (methods 5 and 6) which aligns 1! � with an aggregate of tail areas produces appropriate coverage properties. A computationally simpler method based on the score interval for the single proportion also performs well (method 10). � 1998 John Wiley & Sons, Ltd. 1.

Summary We analysed data from a case-control investigation conducted in Milan, Northern Italy, to evaluate the relation between the use of combination oral contraceptives and the risk of cancers of the breast, ovary, endometrium and cervix uteri. For the present analysis, 776 cases of histologically confirmed breast cancer, 406 of epithelial overian cancer and 170 of endometrial cancer aged under 60 were compared with a group of 1,282 subjects below age 60 admitted for a spectrum of acute conditions apparently unrelated to oral contraceptive use or to any of the known or potential risk factors for the diseases under study. Likewise, 225 cases of invasive cervical cancer were compared with 225 age-matched inpatient controls, and 202 cases of cervical intra-epithelial neoplasia with 202 outpatient controls identified in the same screening clinics. The age-adjusted relative risk estimates for ever vs. never use of combination oral contraceptives were 1.04 (95% confidence interval (CI) 0.73-1.37) for breast cancer, 0.68 (95 % CI=0.48-0.97) for epithelial ovarian cancer, 0.50 (95 % CI=0.23-1.12) for endometrial cancer, 1.49 (95 % CI=0.88-2.55) for cervical cancer and 0.77 (95 % CI=0.50-1.18) for cervical intra-epithelial neoplasia. The risk of ovarian cancer decreased and that of invasive cervical cancer increased with longer duration of use. Neither duration of oral contraceptive use nor time since first or last use significantly altered a user's risk of other neoplasms considered. Likewise, analysis of sub-groups of age, parity or other potentially important covariates did not show any important interaction,

In order to control for confounding variables, epidemiologists often obtain data in the form of a 2 x 2 table. One variable is usually the disease status, while the other variable represents a dichotomous exposure variable that is suspected of being a risk factor. If a confounding variable is present, the data are often stratified into several 2 x 2 tables. The objectives of the analysis are to test for the association between the suspected risk factor and the disease and to estimate the strength of this relationship. Before estimating a common odds ratio, it is important to check whether the odds ratios are homogeneous. This paper presents the results of a Monte Carlo study that was performed to determine the size and power of a number of tests of association and homogeneity when the data are sparse. We also evaluated the performance of three estimators of the common odds ratio. For the Monte Carlo studies, equal numbers of cases and controls were used in a wide variety of sparse data situations. On the basis of these studies, we recommend the Breslow-Day test for nonsparse data, and the T4 and T5 statistics for sparse data to test for homogeneity. The Mantel-Haenszel test of association is recommended for sparse and nonsparse data sets. With sparse data, none of the odds ratio estimators are entirely satisfactory.