UNIVERSITY OF 
 
 ILLINOIS LIBRARY 
 
 AT URBANA-CHAMPAIGN 
 
 BOOKSTACKS 
 
Digitized by the Internet Archive 
 
 in 2012 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/bimultivariatere138joha 
 

 Faculty Working Papers 
 
 
 BIMULTIVARIATE REDUNDANCY MAXIMIZATION 
 
 Johny K. Johansson and R. Narayan 
 
 0138 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana-Champaign 
 
r 
 
FACULTY WORKING PAPERS 
 College of Commerce and Business Administration 
 University of Illinois at Urbana-Champaign 
 December 19, 1973 
 
 BIMULTIVARIATE REDUNDANCY MAXIMIZATION 
 Johny K. Johansson and R. Narayan 
 
 #138 
 
. 
 
BIMULTIVARIATE REDUNDANCY MAXIMIZATION 1 
 
 by 
 
 Johny K. Johansson and R. Narayan 
 University of Illinois 
 
 Introduction 
 
 The relationship between two sets of variables is often analyzed 
 with the help of canonical correlation techniques. The interpreta- 
 tion problems are often severe, however, as soon as more than one pair 
 of variates are significant at the pre-selected level. Stewart and 
 Love (1968) have suggested a method, called "redundancy analysis", to 
 deal with these problems. Miller and Farr (1971) pointed out that the 
 redundancy measure would remain invariant with respect to any ortho- 
 gonal rotation of the complete set of canonical variates, and that, 
 consequently, canonical correlation was only one special case of a 
 general redundancy analysis. 
 
 It can be argued that since the redundancy measure provides a 
 straightforward interpretation of the degree to which two sets of 
 variables covary, one focus of bimultivariate analysis ought to be 
 the maximization of the redundancy contribution from as small a set 
 of variates as possible. As a start in that direction, this paper 
 presents an approach to the maximization of the partial redundancy 
 attributable to the first pair of variates. 
 
 'Several persons contributed to this paper. Jagdish Sheth encouraged us 
 to focus upon the problem and gave valuable feedback throughout; Charles 
 Lewis gave exceedingly helpful assistance on the theory part; and Joseph 
 Kolman and Maurice Tatsuoka contributed many valuable ideas to the testing 
 of the optimizing approach. Funds were made available by the University 
 of Illinois Computer Services Office and the Bureau of Economic and 
 Business Research. The authors want to thank the people involved but 
 also absolve them of the responsibility for any remaining errors. 
 
The Theory 
 
 Miller and Farr (1971) show that the redundancy attributable to the 
 first linear combination of the Y variables Gi i? equal to 
 
 2 
 
 RED, = (L 6 /tr(R YY ))*(5 r GF.) 
 
 where L« stands for the sum o1 of :he Y variables upon 
 
 6j > tr (Ryy) stands for the trace of the correlation matrix of the Y's 
 
 (equal to the number of criterion variables), and the F. , i=l,..., I 
 
 2 
 
 stand for the successive orthogonal factors of the X variables. 
 
 Because these orthogonal factors together span the space of the X's 
 completely, we have 
 
 I 2 * 2 
 
 so that the redundancy becomes a product of the loadings and the squared 
 multiple correlation: 
 
 RED Gi . (L Gi /tr(R YY ))*R 2 SiX 
 
 The canonical correlation technique maximizes the latter component of 
 the product, whereas a principal component analysis of the Y's would 
 maximize the loading part. In general, then, neither of these two 
 approaches would maximize the redundancy measure. 
 
 2 In what follows, we will treat the Y's as "dependent" and the X's as 
 "independent" -- an inverse relationship is dealt with similarly but 
 yields no new insights so is ignored here. Also, in what follows, the 
 redundancy measure will always refer to the first linear combination 
 of Y's unless otherwise stated. Finally, both the Y's and the X's 
 are assumed standardized. 
 
To derive an expression of the redundancy measure — our objec- 
 tive function -- in the original variables Y and X we proceed as 
 
 follows. Let W r denote the m by 1 vector. of variable weights for 
 
 1 
 the first linear combination G-,. Then we. have 
 
 Y W^ = B, , 
 
 with the dimension of the Y matrix equal to n by m, n denoting the 
 number of observations, m the number of Y's. Then for the loadings 
 we have 
 
 R YG 1 : R YY W G ] 
 
 R again denoting the correlation matrix. Since we want the squared 
 loadings we need 
 
 *il } \ = w g{ r yy r yy w g 1 - 
 
 the T superscript indicating transpose. We also note for future use 
 that 
 
 tr(Ryy) - m . 
 
 As for the squared multiple correlation, we have first 
 
 XB * G 1 , 
 
 as the predicted value of G , with B denoting the I by 1 vector, of '.parar 
 meter weights. Using a least squares fit, we compute B as 
 
 B = R XX R XG 1 
 
4 
 
 To get a measure of the squared simple correlation between the 
 
 «. 
 actual and the predicted G^s -- which is the squared multiple correla- 
 tion we are looking for — we compute first 
 
 V G 1 G ] = B R XX B 
 
 K BR x Gl • 
 where V stands for the variance. Then we get 
 
 VG 1 G 1 
 (R" 1 R W ) T R IL 
 
 - -xx xy g/ xy g 1 
 " w gJ r yy w Gi 
 
 B "g^xy r xx r xy w g 1 
 
 W G YY G 
 1 1 
 
 for the correlation between the predicted and actual G's. The complete 
 objective function cag then be written 
 
 RED G ■ ( W 4 (R V V )2W S1 >< W gJ R XY R XX R XY % > 
 
 "s,"rY \ ' - 
 
 which is to be maximized under a normalizing constraint such as W r W r = 1 , 
 
 G l G l • 
 
 or W<J R YY W Gi * 1 . 
 
The Algorithm 
 
 Since the objective function (1) consists of the product of two 
 
 quadratic functions, for which a gradient procedure might easily stop 
 
 at a local maximum, the algorithm empoyed was a direct search routine 
 
 i 
 (the Hooke-Jeeves algorithm described in detail by Himmelblau, 1972, 
 
 p. 142). 
 
 The basic approach to the maximization routine utilized the fact 
 that the objective function can be written 
 
 RED = F(a)* H(a,b) , 
 with a,b, denoting the weights of the Y-compound and X-compound, respec- 
 tively. This is a direct generalization of the function as stated in (1). 
 Then the dynamic programming "knapsack" approach gives a solution as 
 
 max{RED} = max{F(a) * max{H(a,b)}}. 
 a,b a b 
 
 That is, for a given vector a, find the vector b that maximizes H(a,b); 
 
 then, search over feasible vectors a, maximizing H every time, to find 
 
 the one that maximizes RED. Since for a given a, and thus a given Y- 
 
 compound, the maximum G is obtained by a multiple regression of the given 
 
 Y-combination upon all the X variables, the first maximum can be located. 
 
 Then, considering the constraints, the search can be made over a relatively 
 
 small number of a-values, namely those that lie within the limits -1.0 
 
 to +1.0 for all elements in a. 
 
 Thus, the algorithm iterated a search by first picking the trial 
 
 a's, then getting the loadings of the original Y variables upon the 
 
 generated linear compound, and finally computing the regression of the 
 
 compound upon the X variables. The derived R , multiplied by the 
 
 average squared loadings constituted the trial value of the objective 
 
function. A search then generated new a-values, and another iteration 
 took place. The routine would stop iterating when either one of four 
 preset test values was superceded. 
 
 The strength of the search routine was abetted by the fact that a 
 generally good starting point could be generated (the canonical correla- 
 tion weights) and by the fact that the total redundancy between the two 
 
 2 
 sets was given by the average R between each of the Y's and the X's. 
 
 3 
 Thus, the maximum obtainable solution could be checked against it. 
 
 The constraint used in the runs was W r W r =1. Initially, each 
 
 G ] G 1 
 
 set of trial a's within the [-1, +1] hypercube were scaled so as to ful- 
 fill the constraint, before the value of the objective function was com- 
 puted. This approach impaired the efficiency of the algorithm considerably, 
 however, making it necessary to adopt another approach. The constraint 
 was now tested for, and the a-values scaled, only after the optimal solu- 
 tion had been located. The approximation resulting from this approach 
 
 4 
 was very close to the earlier solution for the problems tested. 
 
 Initially, the algorithm was set up for raw data only, but as can 
 be seen from equality (1), the only d, ta input needed would be the correla- 
 tion rnrtrix of the Y's and the X's. When raw data are input, this correla- 
 tion ~.~tr::: !: c^p'jted at the initial iteration, and the program can 
 u»w.. k y n =*ss this computation in later iterations. 
 
 3 
 
 In addition, an alternative search routine, the Nelder-Mead technique 
 
 of searching successive simplexes (Himmelblau, p. 148), was used for 
 some runs. The optimal solutions located by the two algorithms were 
 the same throughout. 
 
 4 
 This closeness can be attributed to the fact that the contours of the 
 
 objective function in all the cases examined formed a ridge in a radial 
 
 direction from the origin (see Figure 1). 
 
The Results 
 
 Initial runs were made on the TALENT data provided by Cooley and 
 Lohnes (1971, Appendix B). The use of published and thus easily acces- 
 
 
 
 sible data facilitates cross-checks and further analysis. The analysis 
 carried out and their results follow. 
 
 The criterion set of variables chosen consisted of "Physical Science 
 Interest", "Office Work Interest", and "Plans to attend College". The 
 predictor set of variables consisted of test scores on "Information test 
 II", "Mechanical Reasoning", and "Reading Ability", plus the student's 
 "Socioeconomic Status". (For further information on the data and these 
 variables, the reader is referred to the Cooley and Lohnes book). The 
 algorithm was run from two starting points, one provided by the criterion 
 weights of a canonical correlation analysis, the other by a principal 
 component analysis of the Y variables. In all cases the search routine 
 
 Isolated the same maximum of the objective function. The runs were 
 
 c 
 i?,ade separately for males and females. 
 
 The results are presented in Table 1. Overall, they are somewhat 
 surprising in that the redundancy maximization routine only does mar- 
 ginally better than the canonical correlation solution. For the males 
 data the reason is clear: there is wery little additional redundancy 
 to account for once tne first canonical solution has been taken out. 
 In the female data the reason is less clear — but one explanation for 
 the almost zero improvement of the redundancy maximization would be 
 that the data are truly explained by two, rather than one, pair of 
 
 5 
 
 Additional runs were made for males and females combined, as well as 
 for other sets of variables. Since the results were similar to the 
 runs reported here, these other runs are not included. 
 
8 
 
 variates. With these marginal improvements, no great changes are to 
 be expected in the weights — as can be seen, only minor fluctuations 
 away from the canonical solution occur. The principal components solu- 
 tion, on the other hand, is not as close to the optimal as is the canonical 
 solution. The principal components weights accordingly also show a wider 
 divergence from the redundancy solution. 
 
 Since these results were largely reproduced in other runs, it was 
 decided that the objective function be plotted and its behavior more 
 closely examined. As the plotting required one dimension for the func- 
 tion value, plus one additional dimension for each criterion variable, 
 it was decided to plot a case where only two criterion variables were used. 
 Accordingly, the "Office Work Interest" variable was dropped, and the 
 objective function as a function of the ensuing two-element vector a was 
 plotted (the predictor variables remained the same). The values of the 
 resulting objective function for the male data are depicted in Figure .1. 
 As could have been inferred from the earlier runs, the function has a 
 flat ridge around the optimum, making for quite a large near-optimal 
 region. Plots of other runs tended to follow the same pattern. There 
 seems, then, to be a general indication that the canonical solution will 
 quite often be very close to optimizing the redundancy contribution from 
 the first pair of variates. 
 
 The symmetry of the objective function follows from the fact that a 
 change in sign will not affect the optimal property of the weights. 
 For completeness, the redundancy analysis results for this case 
 with two criterion variables are included in Table 4. 
 
Conclusions and Extensions 
 
 Although these initial data runs pointed in the other direction, it 
 is clearly too early to dismiss the possibility that significant changes 
 in the weights — and hence of the interpretations — of the original 
 variables can occur when the redundancy attributable to the first pair 
 of variates is maximized rather than its canonical correlation. The 
 theory is unequivocal: the canonical solution will in general not be 
 optimal. In what type of particular data structures it will be approxi- 
 mately optimal remains to be investigated further. One thing seems 
 already quite clear: If only one canonical root is significant at the 
 pre-selected level, chances are that a redundancy maximization will 
 make very little difference. 
 
 Although in this paper redundancy was maximized with reference 
 only to the first pair of variates, a straightforward generalization 
 to further variates is easily made. For the optimal linear Y-compound, 
 the loadings of the separate Y-variables are first computed. Using the 
 fundamental factor theorem the amount of variation in the original Y- 
 variables explained by the optimal compound is then derived. The unex- 
 plained variation in the Y-variables is what then remains to be explained 
 by a second Y-compoun<J. Similarly, the residual variation in the X- 
 variables after the first X-cornpound is extracted can be derived. The . 
 second redundancy maximization can then take place using the residual 
 variations in the Y and X variables. 
 
10 
 
 List of variables : for TALENT DATA (Males and Females) 
 
 Y] Plan College full time 
 
 1 . Definitely will go 
 
 2. Almost sure to go 
 
 3. Likely to go 
 
 4. Not likely to go 
 
 5. Definitely will not go\ 
 
 Y2 Physical Science Interest Inventory 
 
 Y3 Office Work Interest Inventory 
 
 X-j Information Test Part II 
 
 X2 Reading Comprehension Test 
 
 X3 Mechanical Reasoning Test 
 
 X4 Socioeconomic Status Index 
 
11 
 
 TABLE 1 
 
 MALES : Total Redundancy = .180 
 
 h 
 
 CC 
 -.187 
 
 PC 
 
 .085 
 
 RED 
 - -.161 
 
 b 2 
 
 -.173 
 
 .182 
 
 -.184 
 
 b 3 
 
 -.112 
 
 .137 
 
 -.124 
 
 b 4 
 
 -.303 
 
 .250 
 
 -.299 
 
 a l 
 
 .715 
 
 -.599 
 
 .688 
 
 a 2 
 
 -.636 
 
 .673 
 
 -.636 
 
 a 3 
 
 .292 
 
 .435 
 
 .312 
 
 Redundancy 
 
 .171 
 
 .136 
 
 .179 
 
 FEMALES : Total Redundancy = .145 
 
 bl 
 
 cc 
 
 .204 
 
 PC 
 .211 
 
 RED 
 .207 
 
 b 2 
 
 .131 
 
 .141 
 
 .136 
 
 b 3 
 
 .144 
 
 .101 
 
 .129 
 
 b 4 
 
 .188 
 
 .197 
 
 .193 
 
 *1 
 
 -.690 
 
 -.651 
 
 -.690 
 
 a 2 
 
 .651 
 
 .515 
 
 .609 
 
 a 3 
 
 -.316 
 
 -.558 
 
 -.404 
 
 Redundancy 
 
 .121 
 
 .118 
 
 .120 
 
12 
 
 TABLE 2 
 
 CANONICAL 
 
 CORRELATIONS 
 
 MALE DATA 
 
 
 
 
 Function 
 
 Eigenvalue 
 
 Correlation 
 
 Wilfcs Lambda 
 
 Chi -Square 
 
 DF 
 
 1 
 
 0.3727 
 
 0.6105 
 
 0.6017 
 
 116.8313 
 
 12 
 
 2 
 
 0.0339 
 
 0.1842 
 
 0.9592 
 
 9.5756 
 
 6 
 
 3 
 
 0.0071 
 
 0.0842 
 
 0.9929 
 
 1.6373 
 
 2 
 
 
 
 FEMALE 
 
 DATA 
 
 
 
 
 Function 
 
 Eigenvalue 
 
 Correlate 
 
 )n 
 
 Wilks Lambda 
 
 Chi -Square 
 
 DF 
 
 1 
 
 0.2460 
 
 0.4960 
 
 
 0.6851 
 
 100.9900 
 
 12 
 
 2 
 
 0.0876 
 
 0.2960 
 
 
 0.9086 
 
 25.6011 
 
 6 
 
 3 
 
 0.0042 
 
 0.0645 
 
 
 0.9958 
 
 1.1130 
 
 2 
 
13 
 
 TABLE 3 
 
 Sample Size = 234 
 
 
 
 
 
 
 Correlation Matrix 
 
 
 • 
 
 MALE DATA 
 
 
 
 XI 
 
 X2 
 
 X3 
 
 X4 
 
 Yl 
 
 Y2 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 1 1.00000 
 
 
 
 
 
 
 2 0.79089 1, 
 
 .00000 
 
 
 
 
 
 3 0.50389 0, 
 
 ,44324 
 
 1.00000 
 
 
 
 
 4 0.48847 0. 
 
 ,36824 
 
 0.27985 
 
 1.00000 
 
 
 
 5 * -0.43111 -0. 
 
 ,43956 
 
 -0.28882 
 
 -0.44597 
 
 1.00000 
 
 
 6 0.43540 0. 
 
 ,36442 
 
 0.34489 
 
 0.36142 
 
 -0.47276 
 
 1.00000 
 
 7 -0.04110 0.00452 
 
 0.03433 
 
 -0.01529 
 
 -0.11191 
 
 0.29653 
 
 MEANS 
 
 STANDARD DEVIATION 
 
 
 
 "" »n 09 
 
 0.17882D 02 
 
 
 
 
 2 0.33585D 02 
 
 0.95132D 01 
 
 
 
 
 3 0.13568D 02 
 
 0.35819D 01 
 
 
 
 
 4 0.98543D 02 
 
 0.94442D 01 
 
 
 
 
 5 0.29017D 01 
 
 0.15596D 01 
 
 
 
 
 6 0.21321D 02 
 
 0.92181D 01 
 
 
 
 
 7 0.12291D 02 
 
 0.790T*3D 01 
 
 
 
 
 Y3 
 
 7 
 
 1 .00000 
 
TABLE 3 (con't) 
 Sample Size =271 
 
 
 
 Correlation Matrix FEMALE DATA 
 
 
 
 XI X2 X3 
 
 X4 Yl 
 
 Y2 
 
 Y3 
 
 1 2 3 
 
 4 5 
 
 6 
 
 7 
 
 1 1.00000 
 
 
 
 
 2 0.66609 1.00000 
 
 
 
 
 3 0.50726 0.57799 1.00000 
 
 
 
 
 4 • 0.30383 0.23813 0.16453 
 
 1.00000 
 
 
 
 5. -0.32493 -0.33711 -0.19538 
 
 -0.32406 1.00000 
 
 
 
 6 0.31734 0.27348 0.39062 
 
 0.13056 -0.28241 
 
 1.00000 
 
 
 7 -0.24229 -0.20473 -0.11889 
 
 -0.18551 0.32471 
 
 -0.13402 
 
 1.00000 
 
 MEANS STANDARD DEVIATION 
 
 
 
 1 0.73786D 02 0.17970D 02 
 
 
 
 • 
 
 2 0.33860D 02 0.89382D 01 
 
 
 
 
 3 0.90627D 01 0. 34981 D 01 
 
 
 
 
 4 0.98531 D 02 0.10733D 02 
 
 
 
 
 5 0.32362D 01 0.16963D 01 
 
 
 
 
 6 0.12066D 02 0.769J8D 01 
 
 
 
 
 7 0.25317D 02 0.97818D 01 
 
 
 
 
15 
 
 TABLE 4 
 Males (Two Criterion Variables): Total Redundancy = .2581 
 
 b l 
 
 CC 
 .133 
 
 PC 
 -.152 
 
 RED 
 .140 
 
 b 2 
 
 .203 
 
 -.183 
 
 .196 
 
 b 3 
 
 .121 
 
 -.130 
 
 .124 
 
 b 4 
 
 .302 
 
 -.293 
 
 .299 
 
 a l 
 
 -.808 
 
 .707 
 
 -.774 
 
 a 2 
 
 .590 
 
 -.707 
 
 .634 
 
 Redundancy 
 
 .2568 
 
 .2562 
 
 .2574 
 
 CANONICAL CORRELATIONS : 
 Function Eigenvalue 
 
 1 .3507 
 
 2 .0289 
 
 Correlation 
 .5922 
 .1700 
 
 Wilks Lambda Chi -Square DF 
 .6305 106.32 8 
 
 .9711 6.76 3 
 
• o 
 
 O 
 
 • •— 
 
 O 
 
 
 *3 
 
 o 
 
 " 
 
 • rt • 
 
 rt 
 
 • C- 
 
 • IN 
 IN 
 
 in 
 
 • ft 
 
 IN 
 
 • rt 
 
 IN 
 
 • rt 
 IN 
 
 • rt 
 
 IN 
 
 •0 
 
 •8 
 
 • o 
 
 B 
 
 rt 
 
 *© 
 
 o 
 
 O 
 
 rt 
 • o 
 
 rt 
 
 •'■ 
 
 00 
 
 • 0* 
 
 00 
 
 • M 
 
 IN 
 
 10 
 
 IN 
 
 •A 
 
 • rt 
 
 IN 
 
 • rt 
 
 IN 
 
 10 
 
 • rt 
 
 IN 
 
 rt 
 
 . rt 
 IN 
 
 
 «o 
 
 ? 8 
 
 eg 
 
 •s 
 
 •8 
 
 0» 
 • rt 
 
 o 
 
 • en 
 
 O 
 
 0- 
 
 O 
 
 • o 
 
 IN 
 
 V 
 
 • rt 
 
 IN 
 
 . ft 
 IN 
 
 • rt 
 
 IN 
 
 « 
 
 • IA 
 IN 
 
 IN 
 
 • rt 
 IN 
 
 ■a 
 • ■» 
 
 IN 
 
 'IN 
 
 M 
 
 rt 
 
 • o 
 
 O 
 
 ■ 
 •8 
 
 © 
 
 O 
 
 .3 
 
 e 
 
 © 
 
 ^*> 
 
 as 
 
 • o 
 
 IN 
 
 4 
 
 IN 
 
 • rt 
 
 IN 
 
 • ft 
 
 IN 
 
 o 
 
 • rt 
 IN 
 
 rt 
 • .» 
 
 IN 
 
 >0 
 • rt 
 
 IN 
 
 o 
 
 • rt 
 
 o 
 
 rt 
 • IN 
 
 e 
 
 " O 
 
 ■ 
 
 •8 
 
 n 
 • n 
 
 
 
 •« 
 
 lA 
 
 ■ 
 
 rt 
 
 • ft 
 
 IN 
 
 • rt 
 CN 
 
 m 
 
 «N 
 
 • rt 
 M 
 
 CO 
 
 • CN 
 
 IN 
 
 e 
 
 • IN 
 IN 
 
 f 
 
 e 
 
 .s 
 
 e 
 
 •© 
 
 • rt 
 
 e 
 
 .8 
 
 e 
 
 • to 
 
 o 
 
 n 
 
 rt 
 
 in 
 
 • IA 
 IN 
 
 M 
 
 rt 
 • IN 
 
 M 
 
 rt 
 IN 
 
 rt 
 
 • e 
 
 M 
 
 10 
 M 
 
 rt 
 
 • IN 
 
 rt 
 
 M 
 • in 
 
 pd 
 
 
 m 
 • © 
 
 e 
 
 o 
 • <o 
 
 o 
 
 • o 
 
 o 
 
 F-l 
 
 • <A 
 
 M 
 
 rt 
 IN 
 
 •W 
 
 • ^ 
 
 *•* 
 
 IN 
 
 • rN 
 
 
 .3 
 
 ' rt 
 
 •-• 
 
 ^ 
 
 ' th 
 
 r. 
 
 
 
 
 
 1 
 
 '"I 
 1 
 
 •0 
 1 
 
 1 
 
 •j 
 
 1 
 
 -1 
 
 •4 
 
 n 
 
 rt 
 
 .3 
 
 rt 
 
 •dj 
 
 94 
 
 ,m 
 
 .8 
 
 * ** 
 
 TC 
 
 • IA • 
 IN 
 
 rt 
 1 
 
 09 
 • O 
 
 o 
 
 .2 
 
 e 
 
 rt 
 
 e 
 
 m 
 • o 
 
 r* 
 
 ■a 
 
 H 
 
 IN 
 • IN 
 
 vd 
 
 rt 
 
 •IN • 
 
 rt 
 
 rt 
 
 rt 
 • O 
 
 M 
 
 rt 
 M 
 
 IA 
 M 
 
 IN 
 
 10 
 M 
 
 IN 
 
 i 
 
 A. 
 
 e 
 
 as 
 
 *8 
 
 •> 
 " o 
 
 •z 
 
 .5 
 
 e 
 
 NT 
 
 e 
 
 rt 
 
 •8 
 
 O 
 • M 
 
 M 
 
 
 
 • IN 
 
 M 
 
 M 
 
 • NT 
 •N 
 
 ■0 
 
 • in 
 rt 
 
 rt 
 
 • n 
 
 M 
 
 B 
 • <~ • 
 
 IN 
 
 ia 
 
 • IN 
 
 r* 
 
 rt 
 • rt 
 
 e 
 
 m 
 *8 
 
 art 
 
 *s 
 
 in 
 
 • •! 
 
 e 
 
 • rt 
 
 o 
 
 •> 
 •8 
 
 M 
 
 rt 
 
 • •* 
 
 M 
 
 .5! 
 
 d> 
 
 • w-i 
 
 N 
 
 • ia 
 in 
 
 4 
 
 • ■» 
 
 IN 
 
 • o 
 
 IN 
 
 1 
 
 e 
 
 .3 
 
 o 
 
 o 
 
 • IN 
 
 o 
 
 do 
 
 •8 
 
 *> 
 
 • o 
 
 o 
 
 .3 
 
 IA 
 •IN 
 
 e 
 
 IN 
 
 N 
 
 • rt 
 
 M 
 
 ■a 
 
 • rt 
 
 in 
 
 • rt 
 id 
 
 IS 
 
 • ia 
 •s 
 
 % 
 in 
 . rt 
 
 IN 
 
 IN 
 
 • O 
 
 IN 
 
 1 
 
 0* 
 
 • OS 
 
 c 
 
 0- 
 
 • rt 
 
 o 
 
 N> 
 
 • r* 
 
 e 
 
 •9 
 • O 
 
 o 
 
 
 
 '8 
 
 • —* 
 
 e 
 
 rt 
 • m 
 
 in 
 
 • rt 
 M 
 
 i* 
 
 • rt 
 
 IN 
 
 lA 
 
 • rt 
 in 
 
 •a 
 
 • •» 
 
 N 
 
 *p 
 • « 
 
 IN 
 
 • 
 
 — 
 1 
 
 • ft 
 
 rt 
 
 • o 
 
 • ft 
 
 o 
 
 ■» 
 
 • IN 
 
 o 
 
 rt 
 
 ' S 
 
 00 
 
 •8 
 
 '8 
 
 ■0 
 
 • ia 
 
 M 
 
 • rt 
 r* 
 
 <• 
 • rt 
 
 •1 
 
 in 
 
 M 
 
 IN 
 
 • IN 
 
 IN 
 
 IA 
 
 -* 
 1 
 
 m 
 
 «A 
 
 ' 3 
 
 •a 
 
 *3 
 
 N 
 
 • IN 
 
 o 
 
 IN 
 
 •9 
 
 '8 
 
 
References 
 
 Cooley, W. W. and P. R. Lohnes, Multivariate Data Analysis , New York: 
 Wiley, 1971. 
 
 Hiramelblau, D. M., Applied Non-Linear Programming , New York: McGraw- 
 Hill, 1972. 
 
 Miller, J. K. and S. D. Farr, "Bimultivariate Redundancy: A Comprehen- 
 sive Measure of Interbattery Relationship", Multivariate Behavioral 
 Research , July 1971. 
 
 Stewart, D. and W. Love, "A General Canonical Correlation Index", Psycho- 
 logical Bulletin , September, 1968.