Returning to restate our objectives, we find that what we are looking for is an approximation to the accuracy, or inaccuracy, due to statistical considerations, of the coefficient of mother-son resemblance. We are keeping two places in the estimate. It would not seem to be stretching the canons of probability too violently to state this estimate as .06. If these methods be allowed as feasible and approximate (though admittedly lacking in theoretical rigor), we may continue by adopting the geometric mean as perhaps the best of the approximations considered. The formula for the actual determinations, given the refractory N, is that appropriate to the correction for the attenuation used, involving the corrected and uncorrected r's and the respective reliability coefficients,10 by the application of which we may arrive at the following values of the probable error for the parental coefficients : Mother-Son Mother-Daughter Father-Son Father-Daughter 1 2 3 4 5 6 7 8 9 10 11 .07 .07 .07 .06 .06 .07 .08 .06 .06 .05 .07 .07 .06 .07 .07 .06 .08 .06 .07 .06 .06 .06 .07 .07 .09 .07 .07 .08 .07 .07 .06 .06 .08 .07 .07 .07 .07 .08 .07 .08 .07 .07 .07 .08 The fraternals offer more difficulties, but may be attacked, perhaps, on the same principles. We have the following brother-brother and sister-sister populations: با The effective N, after the analogy of the parental case, obviously lies between the number of families as a lower limit and the number of pairs as an upper. We may experiment on Test 4 again: the number of families (brotherbrother) is 44; the number of pairs, 66. The arithmetic mean is 55, the geometric mean, 53.9; one-third of the way from 44 to 66 is 51.3; two-thirds, 58.7. The corresponding probable There could hardly be any objection to the value .09 from a practical standpoint, it would seem; we shall accordingly use the geometric mean. This yields the following values for the probable errors of the fraternals, and completes the empirical solution of the problem of their estimation: Brother-Brother Brother-Sister .07 1 2 3 4 5 6 7 8 9 10 11 .07 .08 .09 .09 .07 .09 .09 .09 .07 .07 .09 .10 .07 .09 .09 .09 .09 .11 .07 .08 .08 .07 .07 .06 .07 .06 .07 .08 .07 .06 .07 .07 .07 The coefficients are repeated here for convenience in comparing them with their probable errors. Their means have already been given, and pages 261 to 264 have been devoted to a discussion of the values. IV MEANING OF THE RESULTS Difficulty of Isolating Extra-Chance Factor. Comparison by rigorous statistical methods between the tests and between the different kinds of correlations is virtually impossible, since the populations involved are overlapping but not identical; the making of reasonable approximations leads to the conclusion that any of the differences found may easily have arisen by chance. For example: Suppose standard errors of .11 and a correlation between errors11 of .30; the standard error of the difference evaluates to .13. Drop the r to .15, keeping the stand d r.r 12 13 error at .11; σ becomes a little over .14. Drop the standard error to .10 and the r r..r. 12 13 to .25; the value sought becomes almost exactly .14. Let the standard error drop to .09 and the r rise to .40; the criterion value falls to .10. Suppose a virtually impossible case let the standard errors be .14 and the correlation between errors vanish entirely; the standard error of the difference is still slightly under .20. And so on. Attacking the problem thus empirically, few of the criterion values d/od (d=difference between correlation coefficients to be compared) reach 2.0, and none go over 2.5; such are certainly not conclusively extra-chance. The Fisher Coefficient of Environment. These facts, together with the small numbers, 12 the similarity (as between tests) of the mean and the standard deviation curves and the high (4 to 9) intercorrelations, make it desirable for pur "Were the "1" populations completely independent, this figure would be 0; were they completely identical, it would be given by Kelley's formula 129; since they are neither (by reason of the indefinite family groups) it is slightly less than that given by the formula; but just how much less is (theoretically) uncertain. Number of mothers involved, about 100; fathers, about 90; all children, about 400. This does not mean, of course, an average of four children per family-that figure is near two. poses of summarization, to consider all the tests as comprising slightly different manifestations of the same ability, and to use the averages of the familial coefficients of the same kind on the eleven tests as fairly reliable measures of the relationships in that ability. This yields a marital coefficient of .44, a fraternal coefficient of .42 and a parental coefficient of .35; the corresponding probable errors have not been computed, but are probably not far from .03. The technique on which a brief notation was promised above is the Fisher "coefficient of environment" (C1).13 This is defined (op. cit., p. 420) as "a constant equal to the ratio of the variance (2) with environment absolutely uniform to that when difference of environment also makes its contribution." Its derivation depends upon certain reasonable hypotheses and much intricate logic and cannot be reproduced here. Its value (which depends, apart from the derivation, only on the obtained correlations) is, with the figures given above, .54. That is, granted the soundness of Dr. Fisher's argument, in the case of mental-test ability environment operates not quite to double the variability as measured by the square of the standard deviation. Whether this is equivalent to saying that inheritance and environment are approximately equal determinants of mental-test ability is a question we only suggest here. Summary of Findings. The investigation has resulted in (a) two positive contributions, viz., that familial coefficients of resemblance in mental-test abilities lie at about .4, and that the curves of performance for these abilities show a sharp rise ("growth curve") and a gradual decline; and (b) a tentative suggestion that the effect of environment in determining these abilities (measured, however, in a manner not easily interpretable) is equal to or possibly a little less than that of inheritance. (b) is so tentative as not to make necessary any further re 13Cf. R. A. Fisher, "The Correlation between Relatives on the Sup position of Mendelian Inheritance," Tr. Roy. Soc. Edinburgh, Vol. 52, pp. 399-430. |