A theorem is presented relating the squared multiple correlation of each measure in a battery with the other measures to the unique generalized inverse of the correlation matrix. This theorem is independent of the rank of the correlation matrix and may be utilized for singular correlation matrices. A coefficient is presented which indicates whether the squared multiple correlation is unity or not. Note that not all measures necessarily have unit squared multiple correlations with the other measures when the correlation matrix is singular. Some suggestions for computations are given for simultaneous determination of squared multiple correlations for all measures. © 1972 Psychometric Society

Cooper, Lee G

Meredith, William

Tucker, Ledyard R

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UCLAUCLA Previously Published WorksTitleObtaining squared multiple correlations from a correlation matrix which may be singularPermalinkhttps://escholarship.org/uc/item/1g6159d1JournalPsychometrika, 37(2)ISSN0033-3123AuthorsTucker, Ledyard RCooper, Lee GMeredith, WilliamPublication Date1972-06-01DOI10.1007/bf02306772Copyright InformationThis work is made available under the terms of a Creative Commons Attribution-ShareAlike License, availalbe at https://creativecommons.org/licenses/by-sa/4.0/ Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaPS’~CHOM]~FRIKA--VOL. ~7, lqO. ~¯ u~., 1972OBTAINING SQUARED MULTIPLE CORRELATIONS FROM ACORRELATION MATRIX WHICH MAY BE SINGULAR*LEDYARD R TUCKERUNIVERSITY OF ILLINOISLEE G. COOPERUNIVERSITY OF CALIFORNIA, LOS ANGELESWILLIAM MEREDITHUNIVERSITY OF CALIFORNIA, BERKELEYA theorem is presented relating the squared multiple correlation ofeach measure in a battery with the other measures to the unique generalizedinverse of the correlation matrix. This theorem is independent of the rank ofthe correlation matrix and may be utilized for singular correlation matrices.A coefficient is presented which indicates whether the squared multiple corre-lation is unity or not. Note that not all measures necessarily have unit squaredmultiple correlations with the other measures when the correlation matrix issingular. Some suggestions for computations are given for simultaneous deter-ruination of squared multiple correlations for all measures.For over thirty years the squared multiple correlation of a measure withthe other measures in a battery has been known to be the lower bound of thecommunality of that measure [Roff, 1936; Guttman, 1940]. For a numberof years the squared multiple correlation of each measure with the othermeasures has been used as an estimate of the measure’s communality. Whenth~ matrix of intercorrelations, R, among the measures is non-singular,these squared multiple correlations are easy to compute from the inverseof the correlation matrix. Let c~ be the squared multiple correlation of thek’th measure with the other measures in the battery and r ~ be the k’thdiagonal entry in 2-~, then(1)c~1This procedure is not possible, of course, when R is singular. An alternativeprocedure, but little used due to amount of time required for the solutions,* The research reported in this paper was supported by the Personnel and TrainingBranch of the Office of Naval Research under Contract Number 00014-67-A-0305-0003with the University of Illinois.143144 PSYCHOMETRIKAhas been to make a separate solution of the regression system for each measurein turn as predicted from the remaining measures. In each such solution astep-wise procedure might be used which discarded measures from the pre-dictor set which were dependent on measures already employed. This reportconcerns a theorem which provides a basis for simultaneous computationof the squared multiple correlations for all measures in a battery even whenthe correlation matrix is singular.Let R be the n X n correlation matrix for n measures and be of rank rgreater than zero and equal to or less than n. Let k be any one of the measuresand j represent all other measures. Also, let R;; be the matrix of intercorrela-tions of the measures j, R~ be a column vector of correlations of measures jwith measure k, R,; be the transpose of R;, . When k is the n’th measurematrix R may be represented as:LR~In general, R. may be obtained by deleting the k’th row and column of R;R~, may be obtained from the k’th column of R by deleting the k’th element.NormM equations for prediction of the k’th measure from the i measuresmay be written as(2) R~.2~ = R~kwhere ~ is a column vector containing the regression weights. When R~ issingular, the normal equations are consistent and solutions for (2) exist;however, ~ is not uniquely determined. The squared multiple correlation,c~, is given by(3) Rk,~ = c~and is unique even when R~ is singular. Every solution for ~ yields thesame value of c~ . Consider ]~ to be the n’th measure. A permissible con-struction is given in (4).(4) ;; = c~ = - c’ "The construction of (4) to any measure k by letting B be an n X matrix with ~.ero diagonal entries,(5) b~ = 0,and with the/~ in the remaining cells of the k’th column. Let C2 be a diagonalmatrix containing the squared multiple correlations c~ . Equation (4) generalized to any measure k by (6).(6) RB~ = R~--I~+~LEDYARD R TUCKER, LEE O. COOPER AND WILLIAM MEREDITH 145where B~ , R~ , I~ and C~ are the k’th columns of the indicated matrices.Before stating the basic theorem of this report two constructions must bedefined. Let the matrix R (with unities in the diagonal cells) be factored an n × r matrix F of full column rank so that:(7)and(s)R = FF’IF’F] ~ O.Let matrices P and Q be defined as in (9) and (10).(9) P = F(F’F)-~F’;(10) Q = RP = F(F’F)-’F’.P is the unique generalized inverse of R as defined by Penrose [1955]. (Fordiscussion of generalized inverses see Searle, [1966, pages 144-145], andGraybill, [1969, chapter 6].) The basic theorem may be stated in terms ofthe diagonal entries p~ and q~ of the matrices P and Q.Theorem:(11) Ifq~ < 1, c~ = 1;(12) if q~ 1, c~1P~A first step in the proof of the theorem is a demonstration that the theoremcovers all possible values of q~. The two parts of the theorem will be provenby development of possible column vectors B~ which satisfy (6) and establish-ment of the values of c~ determined by these column vectors B~.Two properties of the matrix Q are important in the proof of the theorem.From (7) and (10), Q is symmetric, idempotent.(13) Q = Q’,(14) QQ’ = QQ = Q,(15) RQ = QR = R.Proof that the theorem covers all possible values of q~ involves (14)from which(16)or(17)iq. (1 - .146 PSYCHOMETRIKANote that the term on the left of (17) will be negative for values of q~ lessthan zero or greater than unity and will be non-negative between thesevalues of q~ . However, the term on the right of (17) necessarily is non-negative, being the sum of squared real numbers. Therefore, the permissiblerange of values for q~ is:(18) o =< q~ _-< 1.Consequently, the theorem covers all possible values of q~.Consider the first case when q~ < 1. Define a diagonal matrix D~containing the diagonal entries of Q. Then, the matrix (Q - D~) has zerodiagonal entries. A column vector of regression weights, B~ , may be de-veloped from the following series of operations.(19) R(Q - D~) = RQ - RDo = R -- RD~ = R(I -- in which the transition between the second term and the third term is by(15). Let (Q - D~)~ be a column vector composed of the k’th column (Q - D~). Then by (19)(20) R(Q - D~)~ -= R(I - Do)~ R~(1 - q~where (I -- Do)~ is the k’th column of (I - Do). Since (I - Do) is a diagonalmatrix, (I -- D~)~ contains all zero entries except for the k’th entry which (1 -- q~,), this observation resulting in the transition from the second third term in (20). Then(21)Comparison f (21) with (6) leads to the possibility of defining1B, = (Q- Do), (1 - q**)(22)(23)which yields(24)--I~+~ = 0This completes the proof of the first part of the theorem.Consider the second case when q~ = 1. A first point is that p,~ mustbe greater than zero. Let F, be a row vector composed of the k’th row of F.Note that F, can not be a null vector since its length is unity, the diagonalentry in R. Note also that (F’F)-2 is positive, definite since (F~F) is non-singular. The product F, (F’F)-2F~, is a quadratic form which must be positivefor a non-null F, and positive, definite (F’F)~. This product, however, byLEDYARD R TUCKER~ LEE G. COOPER AND WILLIAI~ I~EREDITH 147(9), is p~ . Consider the column vector (I~ = P~(1/p~)) where P, is thek’th column of P. Note that the k’th entry is zero. Note that(25) R(I,- p, 1) = R,- Q,1. p~Equation (10) was used in establishing this equality. A further point derived from (17). When q~ -- 1, the left term of (17) equals zero; sequently, the sum of squares on the right equals zero and all q~ forequal zero. Then, Q~ is the k’th column of an identity matrix, I~. Equation(25) becomes(26)Comparison of (26) with (6) leads to the possibility of definingB~ --(27)with(28)This yields(29)--h + C~ = --I~---1.P~2 1cA--This completes the proof of the second part of the theorem.Some interesting results when the factoring of R is by a principal axesmethod. Let V, be an n X r section of an orthonormal matrix containing thefirst r characteristic roots of R, and let A, be a diagonal matrix containingthe first r characteristic roots in decreasing order. Since R is of rank r, allroots noQincluded in A, are zero and(30)Then(31) Fand(32) F’FThis factoring results in(33) P(34) O = V,Y~’.148 PSYCHOMETRIKA "Then(35)(36) ~2q~ = V~v .Thus the test coefficient, q~ , is a sum of squared coefficients in the char-acteristic vectors. The coefficient p~ needs be calculated only when q~ isless than unity.Implementation of calculations for the squared multiple correlationsusing principal axes of R may involve use of two very small values, e, and ~,,set according to computational inaccuracies of the computer in use. Thefirst problem involves the number of principal axes factors to be utilized.Let the roots be in decreasing algebraic order. A suggestion is to establish rsuch that(37)A suggestion for the test coefficient is to compare it with 1 -- ~ . Thus,(38) when q~r< 1-~,, c~ = 1;~ 1(39) when q~ => 1 -The computational scheme based on the theorem is applicable whetheror not the correlation matrix is non-singular. It would not be as rapid asthe procedure involving the inverse of the correlation matrix for non-singularmatrices. However, it has an advantage over the separate solutions for thevarious measures when the correlation matrix is singular. The theorem doesyield a way to compute all of the squared multiple correlations in a singlesolution and should involve fairly rapid calculations with modern computers.REFERENCESGraybill, Franklin A. Introduction to matrices with applications in statistics. Belmont,California: Wadsworth Publishing Company, 1969.Guttman, Louis. Multiple rectilinear prediction and the resolution into components.Psychometrika, 1940, $, 75-99.Penrose, R. A. A generalized inverse for matrices. Proceedings of the Cambridge PhilosophicSociety~ 1955, $I~ 406-413.Roff, Merrill. Some properties of the communality in multiple factor theory. Psychometrika,1936, 1, No. 2, 1-6.Searle, S. R. Matrix algebra for the biological sciences. New York, Wiley, 1966.Manuscript received 8/~/70Revised manuscript received 5/21]71

Obtaining squared multiple correlations from a correlation matrix which may be singular

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