The topic that I have chosen, “A Superior Rotational Method in Factor Analysis or Psychometricians in Government Service,” may puzzle you at first. But there is an explanation. A study in progress at the University of North Carolina requires the interpretation of 16 factors obtained from a centroid analysis of 66 variables. The task that was approached with greatest apprehension was the location of a meaningful reference frame. With so many vectors protruding into a space of such a large number of dimensions, the problem appeared practically insoluble. Hence we sought a rotational method that would obviate the necessity for the numerous graphs and the matrix multiplication associated with the commonly accepted techniques.

Perhaps some of you have had the experience of solving a difficult mathematical problem in a state of sleep and later being able to transcribe the complete solution immediately. A relatively painless approach to mathematics, it is the one which enabled me to pass a course in Mathematical Probability years ago. I scarcely dared presume that I could evoke it in the present emergency. To my astonishment, however, it worked. One day the project staff had had a particularly trying session during which we tried in vain to stretch the available funds to cover the rotational work. Upon falling asleep in my office chair, I dreamed a solution so simple that an ordinary statistical clerk, upon mere inspection of the score matrix, could visualize the most interpretable reference frame in up to an infinite number of dimensions. This was beyond our most extravagant hopes, for it rendered obsolete both the correlation matrix and the centroid solution.

This paper gives maximum-likelihood estimators for certain parameters in a truncated trivariate normal distribution when the values of the other parameters are known. The estimators are functions of a random sample. Approximate variances and covariances of the estimators, when the sample size is large, are also given. The type of truncation considered is merely restriction of the range of one of the variates, whose true mean and variance are assumed to be known. Two cases of such restriction are treated: (a) (δ ≤x < + ∞); (b) (- ∞ <x ≤ δ'), where δ and δ' are arbitrary “cutoff points” which are assumed to be known. A precise statement of the estimation problem is given in Section 1. Section 2 contains preliminary calculations. The estimators appear in Section 3. The asymptotic variances and covariances of the estimators are given in Section 4. The estimators and their asymptotic variances and covariances can be easily specialized to be suitable for the case of a certain truncated bivariate normal distribution (Sections 3 and 4).

The theoretical basis for the Johnson-Neyman Technique is here presented for the first time in an American journal. In addition, a simplified working procedure is outlined, step-by-step, for an actual problem. The determination of significance is arrived at early in the analysis; and where no significant difference is found, the problem is complete at this point. The plotting of the region of significance where a significant difference does exist has also been simplified by using the procedure of rotation and translation of axes.

Scores from three mathematics tests of the College Entrance Examination Board were examined in order to determine the effect on the scores of (1) choice of test, (2) amount of training in mathematics, and (3) recency of training in mathematics. Groups of candidates were paired in a number of comparisons and matched by means of a regression technique which is described. On the average, students of similar ability made comparable scores on the mathematical section of the Scholastic Aptitude Test and on the Comprehensive Mathematics Test. The scores of candidates who took the Intermediate Mathematics Test averaged substantially higher than those of comparable students who took either of the other two tests. A greater amount of mathematical training and more recent training were both found to be positively related to scores on the mathematical section of the Scholastic Aptitude Test and on the Intermediate Mathematics Test, but the effect of recency appeared to be less than one might expect.

For any fixed total time of testing it is possible, through proper item-and-time allotment, to combine tests into a battery so that the multiple correlation with a pre-assigned criterion will be maximized. By holding constant the ratio of the length in number of items to the time length for each test, a set of general equations has been derived which will yield this maximum value of the multiple R and will enable one to determine, in any given case, the optimal fraction of total testing time that should be devoted to each type of test under consideration. The set of general equations is applied to a two-test-battery problem to obtain the optimal length of each type of test for one hour total testing time. If two other tests had been selected for the two-test sample problem, different subdivisions of the total time would generally occur. The manner in which the results would change when using other tests with different initial reliability, validity, and intercorrelation values is briefly presented. Some general implications of this method of battery development are also discussed.

Elsewhere in this issue is an article by Dr. Calvin Taylor entitled “Maximizing Predictive Efficiency for a Fixed Total Testing Time.” The purpose of this note is to point out that the formulation developed by Dr. Taylor is mathematically equivalent to one presented by me in a previous issue of this journal.

We start from Taylor’s equation (36) .

When K tests are given to N individuals, and for each individual there are two criterion measures, then (1) the multiple regression weight to be applied to the standard score for each test to predict the criterion-difference score equals the difference of the weights for predicting each criterion separately; (2) the difference between the predicted scores equals the predicted difference (each test being assigned the appropriate multiple regression weight); (3) the square of the multiple correlation between predicted and actual criterion-difference scores equals the sum of squares of the multiple correlations of the battery with each criterion less the product of these correlations and the correlation between predicted scores all divided by twice the quantity one minus the criterion intercorrelation; and (4) the variance of errors of estimating the criterion-difference score equals the sum of the variances of errors of estimating each criterion score minus twice the criterion intercorrelation, plus twice the correlation between predicted scores multiplied by the product of the square root of one minus the variance of errors of estimating one criterion and the corresponding square root for the second criterion.

It should be possible to make a test a more accurate measuring instrument by discarding some of the items which have been found, by item analysis, to be inefficient. If this is so, fresh and more reliable scores from the already marked scripts would be obtained by neglecting the discarded items, and, in the future, the abbreviated test would be used. The present article offers two solutions, based on slightly different premises, to the problem of which items to discard when the test has been designed to measure a single ability. In a practical example these yield almost the same result, which is also the result obtained by a less rigorous grouping method described, which is by no means laborious to use.

Alternative procedures for calculating discriminant function coefficients have been illustrated in reported research. One method proceeds from data which has been expressed in deviation score units, whereas the other method implies that the data has been expressed in standard units. It is shown algebraically that both procedures yield identical discriminant function coefficients, and therefore, the method involving less mathematical manipulation and computation is preferable.

The P50-discriminant has been reported elsewhere in connection with its use in predicting whether selective service registrants if inducted would become normal operative soldiers or would commit offenses causing their imprisonment. The standard error of the P50-discriminant is a good measure to use in determining how far to the side of this statistic a particular case falls. The standard error formula itself has also been published elsewhere; but its derivation, as the variance error, is given here.

Rapid computational routines are presented for calculating x2 from frequency data in the following cases: (1) test of goodness of fit between an observed and a theoretical distribution; (2) test of independence of distributions displayed in an r ✗ c table; (3) test of independence of distributions displayed in an r ✗ 2 table. A rapid method of computing the contingency coefficient also follows from the procedure used in the second of these cases.