45. Testing for independence. Let X = (Xa;). i = 1•. .. , P. a = 1,. . . , N. be a sample from a p-variate normal distribution; let q < p , max(q, P - q) N; and consider the hypothesis H that (Xli' . . .• Xl q ) is independent of (XI q+ I • ... ,Xlp)' that is. that the covariances ajj = E(Xaj - U(Xaj - j) are zero for all i s q, j> q. The problem of testing H remains invariant under the transformations X~ = Xaj +

b, and X· = XC, where C is any nonsingu-

lar p x p matrix of the structure 497 C= (Cll 0) o C22 with Cll and Cn being q X q and (p - q) X (p - q) respectively. (i) A set of maximal invariants under the induced transformations in the space of the sufficientstatistics X' i and the matrix S, partitioned as ( Sll S = S21 SI2) , S22 are the q roots of the equation IS12Sn1S21 - ASlll = o. (ii) In the case q = 1, a maximal invariant is the statistic R2 = SI2S221S21/Sll ' which is the square of the multiple correlation coefficient between Xll and (X12 , • • • , X1p ) ' The distribution of R2 depends only on the square p2 of the population multiple correlation coefficient, which is obtained from R2 by replacing the elements of S with their expected values 0 i j ' (iii) Using the fact that the distribution of R2 has the density [see for example Anderson (1984») (1 _ R2)~( -P-2)(R2) ~(P -l)-I(1 _ p2) ~(N l) r[t(N - 1)]r[t(N - p)] X f (p2)h(R2)h r 2[HN- 1) + h] h-O hlr[Hp -l)+h] and that the hypothesis H for q = 1 is equivalent to p = 0, show that the UMP invariant test rejects this hypothesis when R2 > Co . (iv) When p = 0, the statistic R2 N- P 1 - R2 • P - 1 has the F-distribution witn p - 1 and N - P degrees of freedom. [(i): The transformations X* = XC with C22 = I induce on S the transformations (Sll' SI2 ' Sn) -. (Sll ' CllSI2' CllSnCJ1) with the maximal invariants (Sll' SI2S221S21)' Application to these invariants of the transformations x* = XC with Cll = I completes the proof.)