CHAPTER V THE MULTIPLE CORRELATIONS BETWEEN FRESHMAN MARKS AND THE OTHER DATA COLLECTED The multiple coefficients of correlation computed. Although a brief statement as to what multiple correlations were found was made in Chapter III it seems best to repeat it here in somewhat more complete form. Seventeen subjects from among the 49 carried by ten or more freshmen each were selected for this procedure. All but one or two of these were the subjects carried by the largest numbers of freshmen, these one or two being added because of some especial interest in them. In three of the subjects, chemistry, French, and Latin, two sets of multiple correlations were computed, one for all, or almost all, freshmen carrying the subject and another only for those who had also carried certain high-school subjects. In Spanish, three sets of coefficients were found, two special groupings being made according to the highschool subjects carried. Thus, including the general freshman average, 23 sets of multiple coefficients were computed. The procedure in computing the multiple coefficients was first to select all of the simple coefficients of correlation which seemed worth using, the number so selected varying from two to six, and then to combine these to secure the multiple ones. The two always used were the general high-school average and the point score. In addition to these the high-school average in the group of subjects and the mark in the one or more single subjects most similar to the college freshman subject were also included, except in two or three cases in which no highschool subjects that could be said to be similar had been carried by enough pupils to be worth including. The one marked example of this was physical education, almost none of the freshmen who carried this. subject having marks recorded for any similar work in high school. In a few cases age was used. In a number of instances the computations were begun with more criteria than were carried through to the finish, since, as the work progressed it could be seen that some of those used made no contributions. In each case the work with those included was carried to the point that no further increase as great as .01 was obtained by computing multiple coefficients of higher orders. In the majority of cases the number of criteria required to accomplish this end was three, in three cases four were required and in none more; in one case the highest zero order coefficient could not be increased and in the remaining ones two were all that was necessary. In this connection the reader should again be reminded that many of the criteria used, such as the general high-school average, the high-school science average, the high-school mathematics average, and the high-school foreign language average, were themselves combinations of marks in several different subjects and therefore the simple correlations with them were in a sense multiple, although not computed by multiple methods. Because of this fact the increases above the simple coefficienets were not nearly as great as if no such averages had been used. The multiple coefficients of correlation obtained in this study. The highest multiple coefficients obtained in this study along with related data are presented in Table IV. The first set of four columns therein is for the highest simple coefficient of correlation obtained for each of the subjects mentioned, the second group of four for the highest multiple coefficient and the last group of three for the increase of the multiple over the simple coefficient. Within each of the two groups of four the first column, headed "r" and "R," contains the actual coefficients of correlation, the second, headed "k," the corresponding coefficients of alienation, the third, headed "P.E." the corresponding probable errors of estimate and the fourth the one or more criteria1 used in the correlations. The last three columns contain, in order, the increases in the highest multiple over the highest simple coefficients of correlation and the accompanying decreases in the coefficients of alienation and the probable errors of estimate. For example, taking the first line of the table, the highest simple coefficient of correlation of freshman algebra mark with any single criterion was .52, the corresponding coefficient of alienation was .85, the probable error of estimate 6.4 and the criterion, high-school mathematics average. The highest multiple correlation obtained for algebra was .53, with a coefficient of alienation of .85 and a probable error of estimate of 6.4. It was based on two criteria, high-school mathematics average and high-school general average. The increase in the coefficient of correlation was .01, whereas there was no change in the coefficient of alienation or the probable error of estimate. 'It will be noted that in a few cases in the table the abbreviations for two of the criteria are connected by the word "or." This means that in such cases the correlation based on the two was the same or so nearly the same that it makes no appreciable difference which one is used. For example, in the case of geometry, the simple correlations with high-school geometry mark and high-school mathematics mark differed by only .0004, so that it makes no material difference which one is used. 'It should be remembered that an increase in the coefficient of correlation and decreases in the coefficient of alienation and the probable error of estimate indicate closer relationship or greater accuracy of prediction. TABLE IV. HIGHEST SIMPLE AND MULTIPLE COEFFICIENTS OF CORRELATION BETWEEN FRESHMAN MARKS AND VARIOUS CRITERIA USED IN PREDICTING THEM College Highest simple r2 k P.E.est Criterion3 R k Algebra. .52 .85 6.4 Math... .53 Aver., P. S. .01 .55 .84 3.8 Aver.. .58 .81 3.7 Aver., P. S.. .03 .01 .03 Average. The abbreviations in these columns are of the different high-school subjects and groups of subjects, except "P.S." which is used for point score. 7Spanish I includes all who had carried high-school Latin, Spanish II all who had carried high-school Spanish, and Spanish III all who had carried highschool French. The increases of multiple over simple indices of relationship. A comparison of the multiple with the simple coefficients shows that on the whole there was little increase in the latter. In one case out of the 23 the highest simple coefficient could not be raised by including other criteria. The median increase produced was only .03 and the greatest .07. with corresponding decreases in the coefficient of alienation ranging from zero to .05, over three-fourths of them being .02 or smaller. As regards the probable error of estimate over half of the cases were decreased by .1 and in only one case was the difference more than 2. In other words, the increased reliability of prediction obtained in this study by using the best multiple correlations and regressions is so small that it is very doubtful if it can be said to be worth the additional labor and expense required. The criteria of highest predictive value. In addition to the fact just mentioned probably the one of chief interest in the table is the question of what criteria are most valuable as bases for predicting freshman marks. It was shown in the preceding chapter that in many cases. the high-school average yielded the highest correlation with the freshman mark, whereas in many others some similar subject or group of subjects did so. Of the 253 criteria employed in the simple correlations given in this table, the high-school average appears in 13 cases, the average mark in a similar group of high-school subjects in six, that in a similar high-school subject in five, and the point score only a single time, for physical education. The criteria used in obtaining the multiple coefficients run very similarly except that the point score appears a much larger number of times. In approximately three-fourths of the cases the high-school average is one of the criteria and the same is true of the point score. Single subject marks occur somewhat less frequently and those in groups of subjects in less than half of the cases. It appears that although the simple correlations between the point score and freshman marks are in general decidedly lower than those of the latter with the high-school average and with marks in various subjects and groups of subjects, yet the point score makes a contribution in prediction somewhat distinct from that made by the other criteria mentioned. Summary of this chapter. Multiple coefficients of correlation were computed for about one-third of the freshman subjects, in a few cases more than one set being computed for each subject. These coefficients run from .20 up to .63, most of them being between .40 and 60, al This number includes the two criteria giving practically equal coefficients in geometry and rhetoric. |