Six criteria suitable for measures of ecological coexistence are proposed. For twenty such measures are examined whether they satisfy these criteria or not. Four of them satisfy all six criteria. Three of them, suggested by Ochiai, Dice and Jaccard are recommended. For them asymptotic standard errors are given. An example is given with asymptotic confidence intervals for the three measures recommended.

SUMMARY The inter-and intra-observer reproducibility of three grading systems for breast cancer has been analysed: those of the World Health Organisation, of Black, and of Hartveit. In addition, the components forming the basis of these grading systems were analysed separately with regard to reproducibility and interrelationship. In this analysis, degree of differentiation was also included as a parameter.The grading systems of WHO and of Black gave a stratification of the material into three categories of tumours, but that of Hartveit did not. All three systems had a low inter-and intra-observer reproducibility.Truncated component analysis indicated that the grading systems of WHO and of Black are closely related to each other and to the 'nuclear lobulation' component of Hartveit's system. The components 'pleomorphism', 'mitotic frequency', and 'hyperchromasia' of the WHO system were closely interrelated. Descriptors such as 'differentiation' and 'tubule formation' were interrelated but acted in a different direction from the others.It is surprising that few of the many suggestions for the classification and grading of breast carcinoma have been analysed with regard to intra-and interobserver reproducibility. This paper analyses the subjective grading of breast cancer suggested by the World Health Organisation (WHO) (Bloom and Richardson, 1957;Scarff and Torloni, 1968), Speer (1957), andHartveit (1971) in order to determine: (a) intra-and inter-observer reproducibility; and (b) interrelationships between the different parameters forming the basis of these grading systems.Any valid grading of breast cancer must use uniform diagnostic criteria which can be applied consistently by individual observers. Moreover, the grading should correlate with survival.Some authors who have analysed the significance of nuclear atypia have not confirmed that it is important in regard to the prognosis (Kister et al., 1969). Others have expressed scepticism concerning the value of prognosis based on histological features Received for publication 21 February 1978 Revised version received 22 March 1979 in connection with breast cancer (Evans, 1933;Warren et al., 1933;Richards, 1948;Lewison et al., 1953;Ackerman, 1954; Sternberg, 1958). Most grading systems have considered the nuclear morphology of the tumour cell population as important, but in some, other factors have also been evaluated such as histological structure, manner of growth, and extent of lymphocytic infiltration.The concept that the nuclear morphology of tumour cells could have implications ior their biological behaviour is due to von Hansemann (1893), and his studies have been the starting point for many grading systems of carcinoma. He considered the mitotic rate in the tumour and the occurrence of abnormal mitoses as important characteristics (Hansemann, 1892) and concluded (Hansemann, 1902) that the higher the degree of anaplasia, the greater the tendency of the tumour to metastasise. Broders (1940) introduced a grading system for squamous carcinoma of the lip a...

If the same categories are used for two nominal scale variables, this information should be used in similarity measures between those variables. Two such similarity measures, one proposed by Goodman & Kruskal and one (kappa) by Cohen, are examined. Two alternative coefficients, called C and S are further proposed. They are found to be generalizations of the G index and the phi coefficient respectively. Both of them seem to have many desirable characteristics, e.g., they are both E-coefficients. They may also be used as measures of similarity between persons classifying into categories defined beforehand.

The possibility of using component analysis for nominal data is discussed. Particularly, two nominal scale correlation coefficients are applicable, namely, Tschuprow's coefficient and the J index. The reason is that they are E-correlation coefficients; that is, they satisfy the requirements of a scalar product between normalized vectors in a Eu-clidean space. Some characteristics of these coefficients are described. The contingency coefficient and Cramér's V are shown not to be applicable in a component analysis. An example of a truncated component analysis on artifical nominal data is included with both the J index and Tschuprow's coefficient. One frequently used vector-algebra-based method in psychological research is truncated component analysis (see Gorsuch, 1974). By using some kind of similarity measure between test items as the scalar product between the normalized vectors corresponding to the items, a Euclidean space is created. Certain choices of base vectors of a linear subspace will then be interpreted as the main factors of the item structure. Not all similarity measures are applicable. In order to clarify this problem, Vegelius (1973, 1976a, in press) introduced the concept E-(correlation) coefficient, where E stands for Euclidean. An E-coefficient is a similarity measure which satisfies the requirements of a scalar product between normalized vectors in a Euclidean space. From the definition of a scalar product, it is possible to deduce that the following conditions must be fulfilled for all variables u, v, and w in order for a similarity measure r to be an E-coefficient (Shilov, 1961):

Summary.-A number of recent studies have shown an association between breastcancer risk and height, weight and dietary habits, especially fat consumption. In the present study, height and weight were determined for 179 consecutive, unselected, breast-cancer patients and age-matched controls selected from a computerized population register. Height and weight for these two groups were compared, including two different indices for overweight (Quetelet's index and Broca's index). Comparisons were repeated after subdivision into pre-and postmenopausal women. In all calculations, the mean values of patients and controls were very similar and without significant difference. It therefore seems improbable that increased height and weight or obesity constitute risk factors for breast cancer. Earlier studies may have shown differences as the result of selection mechanisms not present in this study.

The concentrations of estrone (E1), androstenedione (A), testosterone (T) and sex-hormone-binding globulin (SHBG) in the serum were determined in 122 postmenopausal women, unselected with respect to age and stage of disease and with a newly diagnosed breast cancer. The results were compared with those in 122 age-matched women without breast cancer, selected from the population register. The patients were found to have a significantly higher mean level than the controls of E1 (132 and 108 pmol/l), A (2.5 and 1.6 nmol/l) and T (1.54 and 1.38 nmol/l) and a lower level of SHBG (40.2 and 47.3 nmol/l) in the serum. A multiple regression analysis revealed in the control group that the serum level of E1 was significantly correlated to A (r = 0.48, p less than 0.001) and T (r = 0.45, p less than 0.001). In the patient group E1 was only slightly correlated to T (r = 0.25, p less than 0.01) and not to A (r = 0.10, p greater than 0.05). A significant negative correlation was found between SHBG and weight in both groups. Otherwise no significant correlations were found between any of the hormone levels and age, stage of disease of weight. It was concluded that an increased availability of A and T, leading to an increased androgenic stimulation--and therefore decreased SHBG--and an increased E1 level, is the most reasonable explanation for the findings. The lack of correlation between E1 and A in the patient group is however difficult to explain and the results do not seem to fit into a definite hypothesis.

Some characteristics of Hubert's Г, as a measure of nominal scale response agreement, are shown, including the characteristic that in a contingency table with equal frequencies its value will normally not be zero. By making a slight modification of its definition, some of these characteristics can be eliminated. As another alternative, the J-index is suggested. It is closely related to Г but does not have the same problematic characteristics. Some asymptotic variance formulas for the J-index are given, together with a numerical example.

Eleven criteria are suggested as suitable for measures of similarity between distributions, For seven measures it is discussed whether they satisfy these measures or not. Two measures, the proportional similarity and the Hellinger coefficient satisfy all the eleven criteria.