Appendix A The Case of Sir Cyril Burt Verily, it is easier for a camel to pass through the eye of a needle than for a scientific man to pass through a door A. S. Eddington In 1972 Professor Arthur Jensen began to reanalyse data relevant to the con­ troversy concerning the inheritance of intelligence, and among others reanalyse data published by Sir Cyril Burt (.:.1.966). He discovered (and later published) twenty cases where Sir Cyril had reanalysed twin data and other data several times, adding new cases each time; thus the number of cases included in the analyses differed. However, some of the results (e. g. the correlations between twins) were identical from analysis to analysis, even to the third decimal. This is so unlikely as to be practically impossible. As Jensen says: "Any particular instance of an invariant r despite a changed N can be rationalised as being not too improbable. But 20 such instances unduly strain the laws of chance and can only mean error, at least in some cases. But error there surely must be." (Jensen, 1974.) Jensen concluded that for further analysis and theory-testing, Burt's data could no longer be relied upon. They had to be rejected as useless, a conclusion which it is diffult to fault. Jensen did not suggest that Burt's data were in any sense faked or fraudulently obtained; he simply suggested error or possible carelessness. What were the reasons for these errors? Jensen writes as follows: "The reporting of kinship correlations at times with and at times without noting the sample size, the rather inconsistent reporting of sample sizes, the higher than ordinary rate of misprints in Burt's published tables . and the quite casual description of the tests and the exact procedures and methods of data analysis all stand in quite strange and marked contrast to the theoretical aspects of Burt's writings in this field, which were elegantly and meticulously composed, with profound erudition and impressive technical sophistication. It is almost as if Burt regarded the actual data as merely and incidental backdrop for the illustra­ tion of the theoretical issues in quantitative genetics, which, to him, seemed always to hold the centre of the stage." This is well said, and probably suggests the right explanation. One must bear in mind that at the time of data collection and calculation, standards of evidence were less strict than today. Furthermore, Sir Cyril did not regard his data so much as proving a case which most psychologists at the time would have consi­ dered as proven already, but rather as being used to illustrate new methods of analysis which he was putting forward, and which marked a great improvement 229 on methods previously used. He was thus more concerned with the didactic elements in his papers rather than with the substantive ones. This may in part explain, although it does not excuse, his apparent carelessness in the treatment of data. The question of whether, in addition to treating his data with almost criminal carelessness, Burt actually faked at least some of these data is still unresolved. Some eminent experts, like Jensen, have concluded that the evidence is in­ sufficient; others, equally eminent (e. g. A. D. B. Clarke, Anne Clarke, and J. Tizard) believe that the case is proven. Professor L. Hearnshaw is preparing a biography of Burt, and is in possession of all the written and verbal evidence; until the appearance of his book it is probably best to avoid further speculation on this point. Perhaps it is best to remember the first principle of English justice, namely that the accused should be presumed innocent until proven guilty. Such proof is obviously difficult, particularly when the accused is no longer with us; nevertheless the possibility cannot be ruled out that circumstantial evidence may in due course be sufficiently strong to require us to return a different verdict. Investigations currently going on suggest that Burt was certainly guilty of same depree of deception; whether this amounts to actual "faking" of data is another question (Dorfman, 1978). What is important to consider is the degree to which the exclusion of Burt's data makes any difference to the conclusions which we may draw from the remaining evidence on the genetic contribution of phenotypic 10 differences. In this book we have looked at the evidence that remains, and find that the conclu­ sions to be drawn are not materially affected by this exclusion of Burt's data. The same conclusion was drawn by Rimland and Munsinger (1977), when they plotted the position of Burt's averaged results in a diagram made up from the results of almost 100 studies of twins, family relations, and adoption results. It is clear that Burt's results are very similar to those reported by numerous other workers, and that while in their time they were of considerable importance, at the moment they are not needed to buttress the case for heredity as a major determinant of 10 differences. 230 Appendix B The Mathematical Basis of Factor Analysis Et harum scientarum porta et davis est Mathematica. (Mathematics is the door and the key to the sciences.) Roger Bacon The basic equation of factor analysis may be stated as follows: (1) where X is an m x n data (score) matrix having m variables and n subjects, A is the m x k factor pattern matrix having k common factors, and F denotes the k x n common factor score matrix. Au is the m x m unique factor matrix, F u the m x n unique factor score matrix. The factor analytic model states that a given data matrix X can be analyzed according to equation 1, with k < m, the rows of the supermatrix (2) being linearly independent (i. e. F being of rank k + m), and Au being a diagonal matrix. This being the case, the Gramian matrix of the row vectors of X is solely a function of the k common factors. If we now normalize the row vectors of Equation 2, assigning to them length n, then we obtain the generalized Garnett equation for the case of orthogonal unique factors: 1.- XX' = ACtA' + A2 (3) n u where Ct == 1.- FF'. The way data are scaled does not affect the application of n this model to data matrices, and if the scores are in standard form, the left-hand side of Equation 3 becomes the matrix of test intercorrelations, Cf, the matrix of common factor intercorrelations, and I - A~ = IF denotes the diagonal matrix of test communalities. In determining the number of factors to be extracted the Guttman criterion is widely used. According to this, the number s of eigenvalues of the correlation matrix (with unities in the principal diagonal) equal to or greater than one is the lower bound to the number k of common factors s~k (4) 231 if s is the number of eigenvalues 0 of R fulfilling (5) Other criteria are available, and the Guttman criterion is not universally used by factor analysts nowadays. The model assumes that a set of m variables can be decomposed into k common factors and m unique factors, all k + m factors being linearly inde­ pendent vectors, and the k common factors being in addition orthogonal to the m unique factors. (In Thurstone's model we actually need, in addition to the k common factors, m error factors and m specific factors, making a total of k + 2m linearly independent factors.) These restrictions make the common factors uncontaminated by errors of measurement, but unfortunately this property of the common factors is lost when the restrictions are violated as they may be (and often are), e. g. when the row vectors of X cease to be linearly independent, or when unique variance components are correlated between variables. 232 Appendix C Algorithm for Speed - Persistence - Error Theory of Intelligence (SPET) After White (1973b) I often say that when you can measure what you are speak­ ing about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind William Thomson (Lord Kelvin) A set of n problems has been administered to each of N subjects, problems being indexed by the subscript j and subjects by the subscript i. Having been presented with problem j, subject i responds after some time 1ji = tp, either by putting forward an attempt at solution, which may be either correct or incorrect, or by abandoning it. We define the observed data in terms of the following relationships; these define the data to which a model requires to be fitted. { 1, correct response (1) J0i = xji = 0, otherwise { 1, abandoned (2) Jji = YjI = 0, otherwise 1ji = tji = response time (3) We next list the unobserved quantities of the model, and state some constraints imposed on them. For each subject, we assume three unobservable random variables: Si (speed), Pi (persistence), and ai (accuracy.) For each problem, we assume two unknown parameters: dj (difficulty level) and Dj (discriminating power.) It is assumed that speed, accuracy, persistence, and discriminating power are all positive quantities, and that speed has an upper limit of unity. These constraints are stated fonnally in equations (4) to (7) (4) i = 1, 2, ... , N (5) D>O } (6) j = 1, 2, ... , n . ~x<d<+xJ (7) The aim is to develop a mathematical function which will express the probability of observed J0i' Jji combinations (given the observed response time) as a func- 233 tion of the speed, persistence and accuracy of the subject, and of the difficulty level and discriminating power of the problems.