Corrections to Correlation Coefficients on Account of Homogeneity in One Variable

Abstract

Very often it is necessary to use a sample that includes several grades or several age levels in order to find the correlation between two other traits, say X and Y. This may happen because the sample in one grade or age class is too small to determine the cor relation with the desired degree of accuracy. If the correlation in the large range is known, an estimate of the correlation in the small range may be had through the use of well known formulas, provided the standard devi ation of one of the variables in the small range is available. The partial correlation technique is not applied here either because the number of classes in the third variable is usually small or because the regressions are not linear. Sometimes the situation is reversed. The correlation of X with Y within a grade or a single age class is known and it is desired to estimate the correlation in a range of several grades or age classes. The usual procedure is to make the necessary assumptions of linear ity and homoscedasticity and apply one of the formulas available for the purpose, if the standard deviation of one of the variables in the large range is known. The writer has pointed out some of the difficulties encountered in applying the formulas for estimating the correlation in one range from that obtained in a different range when rectilinearity and homoscedasticity are not present in the data.1 In some cases the absence of these properties from the correla tion chart may be due to the fact that the distribution of one variable in the large range is homogeneous while that of the other vari able is heterogeneous. As used here, the term implies that the means and Standard deviations are equal in all the grades or ages. This quality was observed while finding the correlation between an arithmetic test and the school marks in the subject. When the marks A, B, C, D and F are turned into numerical equivalents, the means of the several grades are approximately equal to each other, and the standard deviations are almost exactly equal, while the test distribution is heterogeneous throughout the grades. The same situation prevails whenever the same rating scale is used separately to classify the children in each grade. Thus, homogeneity is usually observed when each of several classes is rated on the basis of a second trait through an identical rating scale, and then the rec ords are mingled into a single total group. However, this property may also result from direct measurement, as may be observed in a chart showing the correlation between two traits within a given age range, one of which traits continues to grow throughout the en tire age range, while growth in the other trait has ceased before reaching the lower limit of the age range, development in this trait re maining practically stagnant throughout the In this case the means of the age sub groups will increase with the age in the grow ing trait, while the means and standard devia tions of the stagnant trait will be approx imately equal in all the subgroups. As far as the writer knows, none of the existing formulas is suitable for transmuting correlations from one range to another in the presence of the above described conditions. Let RX7 be the correlation of X with F ob tained by mingling the records of n subgroups into a total single record. Then, we have,2 1 Casanova, T. A test of the assumptions of linearity and homoscedasticity made in estimating the correlation in one range from that obtained in a different range. J. Exp. Educ, VII, 245-249, 1939. 2 For this formula in more explicit form, that is, without the summational notation, see Dunlap, J. Combinative prop erties of correlation coefficients. J. Exp. Ed., 1937, V, 286 288.