*Learning*

Although the field of learning was my first interest in psychology, I have not been productive in that field. It has always seemed to me that we have missed something essential in learning which is not represented by the ordinary studies of rote learning. My doctor's dissertation on the learning curve equation**[21]**was a very simple study, and a paper on variability in learning**[22]**related to a current controversy at that time. Something more elaborate was developed in a paper on the learning function,**[23]**in which it seemed that the learning curve for rote learning should be S-shaped. Some of these ideas were incorporated in a study of the relation between learning time and length of task,**[24]**in which I was pleased to find that nine experimental studies

(313) in the literature fitted the theoretical expectations according to which the learning time varies as the 3/2 power of the number of rote items in the list. Another study of the error function in maze learning**[25]**was an elaboration of the same theme in another setting.

*Multiple Factor Analysis*

The work on multiple factor analysis was started in 1929, but it did not get under way seriously for another year until completion of other commitments. The original observation equation for multiple factor analysis was written in Pittsburgh before 1922, but it was ten years before I started serious work on the problem. Much has been written on multiple factor analysis, so that this discussion will be limited to some of the incidents and accidents concerned with the development of the main ideas. Our early work was supported by annual grants from the Social Science Research Committee at the University of Chicago. We had a number of research grants from the Carnegie Corporation for research assistants and for the purchase of calculating machines. One of the Carnegie grants was specifically for the development of a matrix multiplying machine. We investigated some of the new calculating equipment that was being designed at Cambridge, but we finally decided to use a modified form of IBM scoring machine which could be adapted for matrix multiplication. The machine was built by the IBM Company in Endicott, and it has been in daily use in our laboratory for many years. As far as I know, that is the only matrix multiplier of this type that has been built. The machine was designed largely by Dr. Ledyard Tucker, and we had the interest and assistance of Professor Eckert of our Physics Department at that time.

When it became evident that the development of multiple factor analysis would require special research grants, I decided to consult Dr. Keppel on one of my trips to New York. I explained to Dr. Keppel that I needed some research funds to develop what I called multiple factor analysis and that it was a big gamble. I told him that I could give him no assurance that this gamble would be successful but that I expected to give my major time to this problem, perhaps for several years. He gave me an initial grant of $5,000, which was a great help. Subsequently we had several grants from the Carnegie Corporation for this work. I did not realize at that time that I would be giving major effort to this problem and its application in identifying primary mental abilities during the next twenty years.

Many of the turning points in the solution of the multiple-factor problem depended on minor incidents. On one occasion, when I was having lunch with Professor Bliss, chairman of the Mathematics Department, and with

(314) the astronomer Bartky, I asked them about some arithmetical operations that I was doing on rectangular tables of numbers. I asked them if there was any kind of mathematics that could be useful in operations of that kind. They both laughed and told me that I was extracting the root of a matrix. When I asked what was meant by a matrix, they suggested that I talk with Professor Barnard, who was teaching three courses in this subject. Professor Barnard took a friendly interest in the problem and helped us a great deal. I appointed Patrick Youtz as a research assistant, and he tutored me in the elements of matrix algebra. Youtz was then a graduate student of mathematics, and he is now on the staff at M.I.T. At a later time Bartky gave valuable assistance when I was working on the principal axis solution. I was trying to solve a problem in least squares with a conditional equation, although I had not at that time put the matter in so simple and direct a manner. He told me that this was an old problem in celestial mechanics and he gave me the solution. Then I discovered that I had myself studied that solution in theoretical mechanics, but I did not think of the solution in connection with my own problem. I described the principal axes solution at an A.A.A.S. meeting in Syracuse in 1932. These incidents illustrate the erratic way in which research can be done, in spite of the limitations of the investigator.

Beginning with Spearman's famous paper in 1904, there was a quarter of a century of debate about Spearman's single factor method and his postulated general intellective factor g. Throughout that debate over several decades, the orientation was to Spearman's general factor, and secondary attention was given to the group factors and specific factors which were frankly called "the disturbers of g." Even now much British writing and some American writing on factor analysis are oriented toward the general factor and the group factors which constantly disturb the general factor g. The development of multiple-factor analysis consisted essentially in asking the fundamental question in a different way. Starting with an experimentally given table of correlation coefficients for a set of variables, we did not ask whether it supported any one general factor. We asked instead how many factors must be postulated in order to account for the observed correlations. At the very start of an analysis we faced very frankly the question as to how many factors must be postulated, and it should then be left as a question of fact in each inquiry whether one of these factors should be regarded as general.

In 1931 and 1932 some of the present writers on multiple-factor analysis were still concerned with the problem of this general factor and with such related problems as the standard error of the tetrad difference. At one time I decided to relate the work on multiple-factor analysis to the earlier work of Spearman, and for this purpose I wrote the tetrad difference equation on a piece of paper. I expected to spend a good deal of time on this problem. As I looked at the tetrad difference equation, it dawned on me

(315) that it was nothing but the expansion of a second-order minor. If all of the second-order minors vanish, the rank is, of course, unity, and immediately. one can then ask the corresponding question about the vanishing of third-order minors, fourth-order minors, and so on. If the question had been asked in that manner, multiple-factor analysis would probably have developed many years earlier. The work in multiple-factor analysis introduced several ideas which extend the earlier work of Spearman. These ideas include the interpretation of Spearman's single factor theorems as a special case of unit rank, the matrix formulation of the factor problem, the communalities, the simple structure concept, the rotation of the reference frame for scientific interpretation, the desirability of interpreting primary factors as meaningful parameters, the use of oblique reference axes, and the principles of configurational invariance. Later work introduced the second-order factors and studies in the effects of selection on the factorial structure. All of these ideas are concerned with methodology.

Throughout this work, the emphasis has been on factor analysis as a scientific method distinguished from problems of statistical condensation of data, which we have considered to be of secondary importance for most scientific work. There are, of course, entirely legitimate problems in which statistical condensation is the essential purpose of a factorial analysis. But this is not the type of problem to which we have given principal attention in the Psychometric Laboratory at Chicago. My first paper on multiple-factor analysis was published in 1931**[26]**and a multiple-factor analysis of vocational interests was published in the same year.**[27]**The principal publications in this field from our laboratory have been *The Theory of Multiple Factors *(1933), *The Vectors o f Mind *(1935), and my APA presidential address of the same title.**[28]**The first volume was rewritten in more extended form with the title *Multiple Factor Analysis *which was published in 1947. In the last fifteen years multiple-factor analysis has attracted the attention of many competent students, so that there are now available a number of texts on this subject. For some reason that I have never been able to understand, the principal concepts of multiple-factors analysis have met severe criticism. Among these the greatest surprise was the criticism and ridicule of the introduction of communalities, the simple structure concept, and the oblique reference frame. These concepts and methods were introduced to resolve troublesome problems of factorial indeterminacy. The striking results that have been obtained in a large number of scientific studies with these methods have reduced the severity of criticism, but these concepts are by no means generally accepted. A curious type of criticism has been made of my attempt to give meaningful

(316) interpretation to the factorial parameters. I can hardly imagine a more absurd type of criticism and yet it is very commonly made.www.psychspace.com心理学空间网