THE USE OF MULTIVARIATE ANALYSIS IN DEVELOPMENT THEORY: A CRITIQUE OF THE APPROACH ADOPTED BY ADELMAN AND MORRIS A. C. RAYNER Introduction, 639. Factor analysis, 639. Discriminant analysis, 644. INTRODUCTION The use of multivariate techniques in development theory has been pioneered in this Journal by Irma Adelman and Cynthia Taft Morris.' They have utilized both factor analysis (FA) and discriminant analysis (DA) to show the interrelationship between economic success and noneconomic variables. While the articles in question were very important in introducing these new tools of analysis, it is appropriate at this juncture to examine their correct usage and their relationship to each other and to multiple regression. The process of discussing these techniques inevitably develops into a critique of the Adelman-Morris articles. This is not the whole purpose of this paper, which aims also to show how and when FA and DA should be used. The paper is in two sections, the first concentrating on factor analysis and the second on discriminant analysis. In each case the procedure adopted is to commence by outlining the general methodology behind the technique, so as to give a background to the following discussion. The discussion first considers the applications of the technique in development theory and then its use in the Adelman- Morris articles. FACTOR ANALYSIS The first and most important comment to make about FA is that it treats all variables equally; the analysis does not attach 1. Irma Adelman and Cynthia Taft Morris, "Factor Analysis of the Interrelationship between Social and Political Variables and Per Capita Gross National Product," this Journal, LXXIX (Nov. 1965). This article is extended in Society, Politics, and Economic Development (Baltimore: The Johns Hopkins Press, 1967), and "Performance Criteria for Evaluating Economic Development Potential: An Operational Approach," this Journal, LXXXII (May 1968).
640 QUARTERLY JOURNAL OF ECONOMICS greater weight to one than to the others? Thus, in the 1968 article, GNP per capita is neither more, nor less, important than, say, the degree of social tension. From the set of observations on the equally important variables, FA proceeds by forming a new artificial variable, which is simply a linear combination of all the true variables. In particular it is that artificial variable, created in this way, having the greatest possible variance. It is termed the first factor and can be shown either as a series of the values it takes for each observation, or, more usually, as the weights attached to the real variables. (In the latter case the value for any observation can be calculated from those weights, using the values of the true variables.) The first factor, therefore, simply summarizes, as far as is possible in one variable, the variance in the whole set of true variables. At this stage it is no more than a statistical device having the property described above, but with no intuitive meaning. The analysis may proceed by calculating another artificial variable, the second factor, in a similar way. Again it has greatest variance, but with the added restriction that it is independent of the first factor. In other words, this means that it has greatest variance after allowing for the variance explained by the first factor. Further factors, independent of all the previous ones, can be estimated in like fashion. The procedure is stopped when a further factor only has less than a predetermined proportion of the total variance of all the true variables. The set of factors are then all independent of each other and summarize the greatest possible amount of the total variance for that given number of variables. It should be remembered that they are still no more than statistical summaries of the true variables and have no direct meaning in themselves. A possible final stage of the analysis, factor rotation, is used as an aid to meaningful interpretation. The linear combinations that formed the original factors are transformed in such a way that the rotated factors remain independent, continue to explain the same total amount of variance, but become more closely allied to some subset of the true variables. Each factor is altered so that the weights attached to some of the variables increase, while those of the 2. It is in fact possible to vary the weighting by permitting the variables to have differing variances. This has a complex effect on the analysis, so that it is not generally desirable to use variances intentionally as weighting devices. Because of the influence of the variance on FA, it is normal practice to standardize the variables in such a way as to give them all constant variance. When this is done, as was the case in this article, each variable does indeed have equal weight in the analysis.
THE USE OF MULTIVARIATE ANALYSIS 641 others decrease. The rotated factors have the same properties as the original ones, but, because of the association with a particular set of true variables, may have some interpretation in behavioral terms. It should be noted that, although factor rotation preserves the total variance explained by the factors, it does not preserve the properties whereby the first factor has the greatest possible variance, or successive factors each have smaller variance than the preceding ones. The rotated factors are simply a summary of the true variables. What, then, is their use in development theory? There are three possibilities: (1) as a ranking device, (2) as a descriptive device, and (3) as a tool for further analysis. Suppose that an economist is faced with the situation where he has a large number of socioeconomic variables, which he considers to be related to the "success" of an underdeveloped country, but where he is not prepared to consider any one to be a more important indication of this success than any other. Suppose, further, that he wants to rank the countries from the most successful downward. Since he is not prepared to choose among the variables, he cannot use any one by itself to rank them, but must instead take some weighted index of them all. The weights he chooses should be such as to form an index that distinguishes among the countries to the greatest extent possible and thus has maximum variance. The index having that property is precisely the first factor, before rotation. The first factor can therefore be used as an index of success. Notice that it only gives an ordering. The economist must decide which end of the scale measures success and which, failure; he cannot escape making some value judgments A possible limitation to the acceptability of this use of FA stems from the fact that the variables all enter the analysis with equal importance. Further, although the weight attached to each true variable in the first factor is not constant, this only results froin the similarity in the variation of that variable when compared with the heavily weighted variables in the factor.4 The actual weights need not bear any resemblance to the subjective weights of the economist. Provided GNP per capita had not been included in the list of 3. In Table I of the "Factor Analysis" article, p. 562,. GNP per capita appears with a positive coefficient in the first factor. A high positive score therefore presumably implies success rather than failure. 4. In the first factor of Table I, we find that GNP per capita does not have greatest weight amongst the variables. It is possible that such may have been the case before rotation.
642 QUARTERLY JOURNAL OF ECONOMICS variables, there might have been agreement on treating the remainder in a completely equal way. However, it is very probable that most economists would want to give a greater weight to the level of GNP than to the other variables, when they are used as measures of success. Some might even go a step further and say that it, or the change of GNP, is the only real measure of "success." If the first of those positions is held, some variables can be given greater weight by increasing their variances. This is both an arbitrary operation and one whose effects is complex. If, on the other hand, GNP is considered to be the one measure of success, it clearly cannot be included on equal terms along with the other variables and therefore cannot appropriately be included with them in FA. The second use of FA is purely descriptive. The series for each true variable are easy to interpret individually, but when they all vary together, they may merely give a confusing picture. A large part of the variance may be explained by a small number of factors. If these artificial factors can be linked with some meaningful economic or sociological concepts, then they can be used as a descriptive device in explaining the differences between countries. As stated earlier, factor rotation, by linking each factor with particular true variables, makes that interpretation easier. This use of FA probably comes closest to the approach of the authors. It is always a matter of skill and ingenuity to choose a name for the factor, by examining the variables that have greatest weight. Thus "this factor obviously portrays the social and cultural changes accompanying urbanization and industrialization." 5 FA, when used as a descriptive device in that way, is of value. It may well be interesting and useful to know that a certain, large percentage of the variance of countries' individual characteristics can be explained by a composite variable having some particular interpretation. At this stage it is necessary to make some critical comments about the authors' aproach. If their intention was simply to use the factors as a descriptive device, then they should have included the variable, GNP per capita, in the first factor. Since all variables are assumed equal, the analysis clearly shows it to be in this factor. If it is included, then a reinterpretation of the factor is required. If the variable was excluded from the factor on the grounds that it was of a different type from the others, then it should not have been included along with them in the FA. The analysis could have pro- 5. "Factor Analysis," p. 563.
THE USE OF MULTIVARIATE ANALYSIS 643 ceeded on the basis of the other variables alone and the resultant factors used in the descriptive way adopted by the authors. There is further evidence that the authors do indeed think that GNP per capita is a variable of a distinctive type. One of the side products of FA is to calculate the multiple correlation coefficient between all the factors and each of the true variables. Thus, as the authors correctly point out, the value of the square of this coefficient for GNP per capita is 0.661. 66.1 per cent of the variance in GNP per capita is therefore explained by the four factors they calculated. (That is quite a small proportion, when compared with the explanation of the other true variables.) Why do the authors emphasize this variable? If it is because they are particularly interested in explaining the variance of this one variable, then they must surely think of it as different from the others. If it is considered as all important, if the main aim is to explain its variance rather than that of any other variable, then FA should not have been the statistical tool used. The explanation of the variance of this variable, given by the four factors, is only incidental to the factor analysis, which aims, rather, to describe as much as possible of the variance of all the variables. In order to maximize the explanation of the variance of the one variable, GNP per capita, it must be treated as an endogenous variable in multiple regression analysis. The third possible use of FA in development economics is concerned with just this situation. In order to perform further statistical analysis, such as multiple regression, it is often convenient, or necessary, to reduce the original number of exogenous variables to a smaller number, which are preferably independent, and which together explain a large proportion of the original variance of all the variables. FA does precisely this. If the authors did want to explain the variable GNP per capita, itself accepted as being the one measure of success, in terms of other social and political variables, they could have regressed it directly on all the other variables. The results would have been difficult to interpret and probably largely insignificant. The alternative approach would have been to perform FA on all the variables other than GNP per capita, identify the resultant factors in terms of some meaningful economic and sociological concepts, and then regress GNP per capita on the four factors. There is no reason why the estimated parameters of this regression, or the multiple correlation coefficient, should have been the same as those given in Table I of the article, although one would expect some similarity.
644 QUARTERLY JOURNAL OF ECONOMICS The three uses of FA in development economics are all distinct. The authors appear to have combined the second two. The significant question is whether we know what is "success." If it is known, as for instance when measured by the level of GNP per capita, then the first use of FA is not needed. The second is still valid as a descriptive device, provided the success variable is not included. The third use is then an extension of the second as a preliminary to multiple regression analysis. This, in contrast to FA, does distinguish between the one "success" variable and the others. DISCRIMINANT ANALYSIS While DA is similar to FA in some ways, there is one extremely important difference. That difference is that the analysis starts with observations that have already been grouped according to some criterion. In the context of this discussion, "success" is already determined before the analysis is undertaken. It is interesting to note that the authors have largely used growth in GNP per capita as their measure of success.6 This adds a further complication to their ambivalent treatment of the level of GNP per capita in the earlier paper. DA, then, commences with observations that have been grouped. The analyst has at his disposal series of true variables, as in the case of FA, showing the various characteristics of each observation. The analysis proceeds by first choosing one of the true variables that has the following property. Of all such variables it is the one that has greatest variance between its mean value for each group, relative to its variance within the groups. It is thus the variable "closest" to the one on which the original grouping was based. The next stage searches for a second variable that, when combined with the first in a linear combination, forms an artificial variable explaining, as before, as much as possible of the variance between group means. The procedure stops when the addition of a further variable adds less than a certain proportion to the explanation of that variance. The proxy variable, formed in this way, discriminates as far as possible between the groups. The discriminant function, which expresses the new artificial variable as a linear combination of some of the true variables, can be used to obtain the numerical value of the discriminant for each observation. If the discriminant is good, the value of this variable will be similar for each observation in 6. "Performance Criteria," p. 261.
THE USE OF MULTIVARIATE ANALYSIS 645 any particular group, but will differ for observations in different groups. The perfect discriminant is, of course, the original basis of the grouping, the original measure of "success." The estimated discriminant function is based on different variables and attempts to get as close as possible to the original measure of success. Thus there are two methods of grouping, with the discriminant function, of necessity, being the worse. Is there any value in having a second way of discriminating between groups? The answer is largely dependent on whether the original measure of success is difficult to obtain. If we have a sample whose success grouping is known (perhaps by academic or medical examination in other uses of DA), and if we then perform DA on their other characteristics, the discriminant can be used as a grouping device for individuals who were not in the sample and whose success is unknown.' It is difficult to see the value of this procedure in development economics. If the original measure of success is always at least as easy to measure as the discriminant, which is based on certain true variables, and such was certainly the case in the authors' paper, then the easiest and best way to group is by the true measure of success. The discriminant function is sometimes described as being useful as a predictive device. It is important to be careful about the meaning of the word "predictive" in this case. The discriminant is predictive only in the sense that, from the value of the discriminant function, it is possible to predict the grouping according to the original criterion. But this is only saying that they are alternative methods of grouping and, as stated above, this form of prediction by the discriminant is of no interest if the original method of grouping is itself as easy to use. The discriminant does not usually predict into the future. Thus, in the authors' article, a country with a high value for its discriminant function, based on certain variables, is only likely to be successful contemporaneously with the observations on those variables. There is no evidence whatsoever that it will continue to be successful in the future. To be able to state this, it would be necessary to show that present growth continues into the future. The point is in contrast to the position taken by the authors. "This method 7. Thus age and sex may be combined to form a good discriminant for the incidence of cervical cancer in a sample of the whole population. As a result, instead of examining everybody for the disease, the population can be put into risk and nonrisk groups by the discriminant alone.
646 QUARTERLY JOURNAL OF ECONOMICS of analysis clearly has a strong potential as an operational guide for forecasting development prospects of underdeveloped countries." 8 Prediction for the future, using DA, would be possible provided the original success grouping was based on performance at a later date than that of the variables used to form the discriminant function. The situation would then be a perfect example of a case where the true measure of success is more difficult to obtain than the variables in the discriminant. The problem then becomes very similar to that of prediction in multiple regression: that is, given a sample of observations on both the endogenous variable and the exogenous variables, to predict a value for the endogenous variable from a further set of observations on the exogenous variables. The exogenous variables entering the predicting equation may be limited to those that are significant, in the same way that there is some limit on the variables entering the discriminant. There is little purpose in predicting in either case, other than as a test of the estimate, if the actual value of the endogenous variable, which is being predicted, is known or can easily be discovered. The main difference in the two methods of analysis is that DA predicts the group that an observation belongs to, while multiple regression gives a numerical value to the observation. As an alternative to being used as a grouping device, DA can be used for descriptive purposes, showing the characteristics that most distinguish the class of successful countries from that of the unsuccessful ones. In this case one can consider the original method of grouping to be an endogenous variable, which is described by a linear combination of the exogenous variables. If description is the purpose for which DA is being used, then it again seems reasonable to take the logical next step and use multiple regression directly on the exogenous variables or their factors. It is difficult to see what is gained by starting with a measure of success, blunting this by grouping, and then using discriminant analysis, rather than multiple regression, to explain the grouping in terms of exogenous variables. It is true that the original measure of success may be considered vague, but this is equivalent to saying that there is error in the endogenous variable in multiple regression analysis, which is not a serious problem. Either approach results in the measure of success being expressed as a linear combination of the exogenous variables. Multiple regression, however, has optimal properties that are not shared by DA. There would be strong justification for using DA as a descriptive device when success or failure naturally fall into groups. 8. "Performance Criteria," p. 280.
THE USE OF MULTIVARIATE ANALYSIS 647 However, there is no such justification in this case, since the grouping is derived from a continuous variable. In addition to the preceding general comments on the value of DA in development economics, there are two specific complaints about the use to which the authors put it in their article. First, the whole purpose of the analysis is to produce a discriminant that is an alternative to the original method of grouping. Clearly nothing is gained by using the same variable to group originally and then allowing it to appear in the discriminant function. Apart from the question of the rationality of the procedure, there is a great likelihood that the function will give considerable weight to the variables that were used in the grouping. If exactly the same single variable was used to group and was then allowed in the discriminant, the results would be ludicrous. Although the authors did not go this far, they did allow variables that entered the analysis to have some influence in determining membership of the first group. 9 It is therefore not surprising that some of them then appeared in the discriminant functions. Second, because the original grouping is supposed to be given when using DA, it is not correct to regroup on the basis of the discriminant. The authors did this, in a small way, as a preliminary to estimating a new discriminant.' The discriminant is only an approximation to the original grouping. There is little purpose in regrouping until the stage is reached when one gets perfect discrimination by the discriminant, unless the original grouping is not considered to be correct. If such was the case, so that the variables in the discriminant were used as an alternative means of grouping, rather than simply as a substitute, then the situation was one in which the measure of success was not certain. If this was so, then the arguments of the first part of this paper would suggest using factor analysis to rank, or group, the countries, rather than using discriminant analysis in this mixed way. Finally, a two-fold conclusion may be drawn from the foregoing discussion. First, there are definite rules to be observed in the correct application of the tools of multivariate analysis. Second, while factor analysis and discriminant analysis may be of some value in development economics, their novelty should not hide the fact that multiple regression will often be a more appropriate technique. UNIVERSITY OF CANTERBURY 9. Ibid., p. 261. 1. Ibid., p. 277.