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1. The accurate form of the probability distribution of an estimated correlation coefficient in a simple, the true correlation being taken as a datum, has been given by Fisher in two well-known papers (1915, 1921). Pearson and others (1917) have studied the adaptation of the formula, which is rather complicated, for numerical work, but a transformation suggested by Fisher in his earlier paper and applied in his late one offers great advantages. The present paper deals with the inverse problem: given the estimate from a sample, what is the probability distribution of the true value, the latter being supposed initially unknown? Much of the mathematics of correlation depends on the hypothesis of the normal correlation surface. As this is a generalization of the normal law of error for one variable, it is subject to similar criticisms. It can be derived from the hypothesis that the two variables considered are subject to a number of independent component disturbances, m of which effect both in the same sense and n in opposite senses (Jeffreys 1935, pp. 213-17). If they are all equal in amount, then even if m and n are fairly small they lead to a good approximation to the normal correlation surface with correlation ( m - n )/( m + n ). We could regard m /( m + n ) as a frequency to be estimated from a sample, and such frequencies are taken, in the absence of special knowledge, as having their prior probabilities uniformly distributed. The idea of the normal correlation surface therefore suggests that we should take the prior probability of the correlation coefficient as uniformly distributed between - 1 and + 1.

In testing a hypothesis concerning the correlation coefficient in a bivariate normal distribution where all the parameters are unknown, the Pearson product moment statistic is appropriate. It may happen, however, that there are relations among the parameters in the distribution, in which case the Pearson statistic would not utilize this information. In particular if the variances of the two marginal distributions are equal, it is possible to test the correlation coefficient by means of a simpler statistic which makes use of this information. In this paper we explain how this statistic arises and present some properties of its distribution. This statistic as well as its properties developed here are utilized in the latter part of this paper where we consider the problem of estimating the difference of the means when some of the observations corresponding to one of the variables are missing.

This paper describes a general method for deriving optimal procedures for problems where the covariance matrices are patterned under both null and alternative hypotheses. The pattern considered in this paper was first suggested by Olkin (1970) and is a generalization of the intraclass correlation model of Wilks (1946) and arises in the study of interchangeable random variables. We prove a theorem showing how we can transform most such problems to products of problems where the covariance matrices are unpatterned. This theorem is applied to two problems, the multivariate analysis of variance problem and the multivariate classification problem where in both cases the covariance matrix is assumed patterned. We use theorems about products to derive optimal procedures for these problems. We then look at Olkin's pattern for the mean vector, and show that most problems where both the mean vector and covariance matrix are patterned can be transformed to a product of problems, one of which is trivial. The same two examples are studied where now both mean vectors and covariance matrices are assumed patterned. We also consider the problem of testing that the mean vector is patterned when we know the covariance matrix is.

In practical applications it is frequently assumed that the values obtained by a sampling process are independently drawn from the same normal population. Then confidence intervals and significance tests which were derived under the assumption of independence are applied using these values. Often the assumption of independence between the values may be at best only approximately valid. For some cases, however, it may be permissible to assume that the correlation between each two values is the same (intraclass correlation). The purpose of this paper is to investigate the effect of this intraclass correlation on the confidence coefficients and significance levels of several well known confidence intervals and significance tests which were derived under the assumption of independence, and to extend these considerations to the case of two sets of values. In the first part of the paper the relations given in Table I are used to compute tables which show the effect of intraclass correlation on the confidence coefficients and significance levels of the confidence intervals and significance tests listed in Table II. The second part of the paper consists of the proofs of the relations given in Table I.

The same one-hour essay question was included in the final examinations in the Contracts course at 16 law schools. The professor (P) at each school sent to ETS five answers from his class. A Xerox copy of the set of 80 answers was then sent to each P who graded them on a five-point scale. An analysis of these grades indicated that: (1) the students were consistent in their performance on different essays since grades on the common question (CQ) correlated .71 with their grades in the course, (2) the professors agreed with each other highly in how the papers should be graded (the intraclass correlation was .58 and the reliability of the mean grades across all Ps was .96), and (3) the average grade on the CQ was significantly related to a number of other variables such as LSAT scores (.45), Handwriting quality (.25), the Length of the answer (.56), and first-year GPA in law school (.62). The average grades assigned to the papers by two laymen and the combination of LSAT and Length correlated .63 and .68, respectively, with the Ps' average grade on the CQ. The implications of these findings are discussed.

The problem considered in this paper is that of developing a mode of inference about the population mean from a sample (from a normal distribution) which are correlated, using structural methods of statistical inference. The result for the general correlation model has been applied to the intraclass correlation model and the first order auto-regressive model.

The sample mean and sample covariance matrix are unbiased and consistent estimates of population mean and covariance matrix only if the samples are independent. In practical applications of Bayes' procedure these estimates are used in place of population means and covariance matrices on the assumption of independence among the training samples. This practice has often given, especially in remote sensing data analysis, misclassification probabilities much higher than that can be accounted for theoretically. The reason may be that the assumption of independence may not be valid. In reality, the samples are rarely independent, they are rather dependent, at best equicorrelated. This paper investigates how such intraclass correlation among the training samples affects the misclassification probabilities of the Bayes' procedure.

In this paper we prove that certain estimators and tests of significance used in regression analysis when observations are independent are equally valid in the presence of intra-class correlation. An application of this result is presented for the situation in which several replications of the correlated set of observations are available. As a special case of this application, it is shown that the usual test of ``column effects'' in the analysis of variance for a two-way classification remains valid when rows are independent and columns are uniformly correlated. This latter fact is also pointed out in [3].

This paper is the report in which the author tried to show his interest in the problem of the influence of heredity and environment upon personality, and in the common factors found in the correlation between the personality traits and some other psyco-physical characteristics in twins.The subjects are 50 pairs of twins (including 25 pairs of identical twins, 18 pairs of fraternal twins, 7 pairs of unlikesex twins) and 14 pairs of the control group of unrelated individuals. The latter are made up of pairs of children of likesex and of the same birth dates selected from the same classes in which the twins were enrolled. (Age : from 6 to 12)The experiments were conducted during the period from May, 1952 to Octover, 1952.The personality traits were appraised by using Uchida's Kraepelin Numeral Addition Test, Nagashima's Adjustment Inventory and the author's Moral Judgement Test. In the first place the author examined the problem of the influence of heredity and environment by the method of Verschuer's mean percentile deviation, then calculating the ratio of the hereditary and environmental influences by F. Lenz's formula, (u2/u1)2-1=1. In the second place the author adoped the method of analysis of variance by which one may obtain the intraclass correlation coefficient (r′).The outline of the result is as follows :(1) In the ratio of the heredity and environment obtained by Lenz's formula, the results of Kraepelin Numeral Addition Test, the Adjustment Inventory the Moral Judgent Test are 1.38:1, 1.34:1, 1.45:1 respectively. The intraclass correlation coefficient obtained through the analysis of variance are as follow : The r's based on the result of Kraepelin Test are .68** in ident. twins, .43 in frat. twins, the r's based on the Adjustment Inventory being .48* in ident. twins, .23 in frat. twins, and the r's of the Moral Judgement Test are .63** in ident. twins, .36 in frat. twins, (** 1 percent level * 5 percent level), but these r's are relatively low as compared with the r's of other mental and physical characteristics obtained in twins.(2) The intraclass correlation coefficients which are culculated from concurrence personality traits and intelligence or physical maturity are as follows :The r′ of C. Q. between personality traits and intelligence are .56*-.82** in ident. twins, .20-.62** in frat. twins. The r′ of C. Q. between personality traits and physical maturity are .86**-.88** in ident. twins, .61**-.70* in frat. twins. These results show that the r's of the C. Q. are higher than the r′ of personality traits, and that the above-mentioned C. Q. of the individuals are influenced by heredity, and more remarkably influenced especially when C. Q. is between the personality traits and physical maturity.(3) From the above-mentioned study, it is evidently concluded that the correlations between the personality traits and the intelligence or physical maturity degree are influenced by the hereditary factors which are common in mental and physical development, to which the author refers as“general maturity degree”.

The present paper is a report of a study of heredity and the influence of the environment on the degree of physical maturation, intelligence proficiency in the language in twins.The subjects studied are 50 pairs of twins, -25 pairs of identical twins, 18 pairs of fraternal twins, 7 pairs of unlike-sex twins, and 11 pairs of the control group of unrelated individuals. The last group is made up of 11 pairs of children, aged from 6 to 12, of the like-sex and of the same date of birth, and were picked up from the classes in which the selected twins were enrolled in school. The intelligence was measured by the non-language group intelligence test. The proficiency of the Japanese language was examined by the present author's test for each grade. The rough scores of the tests were converted to standard scores. The degree of bodily maturation was determined by the degree of ossification, which is evaluated by the measurement of the X-ray pictures of carpal bones, and is represented by Growth Quotient, G. O. = (the total of the areas of the subject's wrist bones) ÷ (the standard area obtained by the author's formula)×100.In the first place the author calculated the correlation between each of the I. Q′. s or proficiency in the language (Z-Score) and the bodily maturation degree (G. Q.). In the second place the author tried to determine how much influence heredity and environment have on each of the G. Q.′s and the I. Q.′s, and Z-score and calculated the ratio of heredity and environmental influence on the Concurrence Quotients (C. Q.) which are the index standing for the proportion of concurrence of I. Q. and G. Q. (I. Q./G. Q.×100), of Z-score and G. Q. (Z-score/G. Q.×100), or of Z-score and I. Q. (Z-score/I. Q.×100).As a statistical procedure the writer adopted, first, the method of calculating the degree of resemblance of twins by the percentile deviation (D = A - B/A + B×100) of measured values or index, and the ratio of heredity and influence of environment on twins based on F. Lenz's formula and others. Then the author resorted to the method of analysis of variance by which one may obtain the intraclass correlation (r′).The correlation coefficients and the correlation ratios thus obtained are as follows (Table 1) :Table 1** 1 percent level.Table 2The resemblance of twins calculated from the mean percentile deviation (e= ΣD/N) and the ratio of heredity and influence of enviroment obtained by the Lenz's formula follow (Table 2) : Since the homogeneity of variances as regards the trains under consideration is the fundamental assumption underlying the analysis of varance by which one may obtain the intraclass correlations, L-tests are run for the comparsions of properties for the twins and the control group. All of the L-values obtained indicate the homogeneity of variance at 1 percent level. The author then proceeded with the test of hypothesis that the trails of individuals are independent of the particular twin pair to which the individual belongs, by calculating the F-ratio. Then, when the result thus obtained is significant at a specified level of significance, the author calculated the intraclass correlation. The intraclass correlation coefficients (r′) thus calculated on G. Q., I. Q, , Z-Socre, and each C. Q. are as follows (table 3) :From the author's observation based on his above-mentioned study, it is evident the heredity exerts on influence not only on each realm of intelligence, proficiency in the language and the physical maturation degree but also on the degree of concurrence of each pair of the two properties among them. Then it is also observed that the correlation between intelligence or proficiency in language and the bodily maturation degree is based upon the hereditary factor which is someTable 3** 1 percent level * 5 percent levelcommon trait in mental and physical development.

THE concept of heritability is well established among geneticists and practical breeders, but the information on the heritability of production traits is still quite fragmentary, especially in turkeys. This paper reports heritability estimates of 25-week body weights in Broad-breasted Bronze and White Holland turkeys. A survey of the literature reveals only a few estimates of the heritability of body weight in turkeys. An estimate of about 0.23 was calculated for 26-week old male turkeys by Kentucky researchers (1950). This estimate was obtained by the intrasire regression method after correction was made for age (date of hatch?). Abplanalp and Kosin (1952), utilizing data from Broad-breasted Bronze and Beltsville Small Whites, calculated estimates by intraclass correlation and offspringdam regression methods at 4, 8, 14 and 26 weeks of age. Their estimates vary considerably because of age, sex and environmental (climatic) factors, but their most reliable restimates were in the medium to high …

This paper, the purpose of which is mainly expository, gives an elementary presentation of the effects of correlation of observations on the standard test for linearity of regression amongst a series of measurements on the same subjects, each observed at a number of different #x2018;control#x2019; levels. Subsequent to a review of this problem in the light of the classical linear transformations, an appropriate modification of the usual test is presented.

N SURVEYS involving natural clusters, I such as families, school classes, sororities, villages, health districts, and the like, the problem of intraclass correlation is frequently ignored, with the likely consequence of overestimating the precision and statistical significance of results obtained. The purpose of this paper is to review briefly some circumstances in which correlation within clusters has been encountered and the various statistical methods that have been used to deal with the problem. Special reference will be made to a study of the housing environment and family life,*1-4 a longitudinal assessment of the effects of housing on morbidity and mental health. In this study, with which the authors are associated, the problem of intraclass correlation arose in the statistical analysis of morbidity data and made a substantial

This paper is a critical survey of methods for analyzing the similarity between profiles. Among the measures of similarity examined are intraclass correlation, Cattell's rp, Cronbach and Gleser's distance measure, analysis of variance, Hotelling's T2, discriminant function, and Mahalanobis' D. An evaluation is made of each, some illustrative examples are given, and the relationships among several of them are presented.