QUANTILES OF THE DISTRIBUTION OF THE SQUARE OF THE SAMPLE MULTIPLE-CORRELATION COEFFICIENT

Algorithm AS 261: Quantiles of The Distribution of The Square of The Sample Multiple-Correlation Coefficient

Author(s): Cherng G. Ding and Rolf E. Bargmann

Source: Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 40, No. 1 (1991), pp. 199-202

Published by: Wiley for the Royal Statistical Society

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199

Algorithm AS 261

Quantiles of the Distribution of the Square of the Sample Multiple-correlation Coefficient

By Cherng G. Dingi

National Chiao Tung University, Taipei, Republic of China

and Rolf E. Bargmann

University of Georgia, Athens, USA [Received April 1989]

Keywords: Illinois method; Multiple-correlation coefficient; Secant method Language

Fortran 77

Description and Purpose

Let X1,..., Xp be distributed as Np(it, E) and R be the sample multiple-correlation coefficient between Xi and the other p - 1 random variables based on a sample of size N. The cumulative distribution function (CDF) of R2 is

R 2

M(R2; p, N, p2)= g(t) dt, 0

R2 1

(1)

0

where p denotes the population multiple-correlation coefficient and g(R2) is the density of R2 (see Anderson (1984) for its expression). For given values of m (0 < m < l),p (> 1), N(>p) andp2

(

Numerical Method

LetX=R2 andf(x) = M(x;p, N, p2) - m. The equationf(x) = 0 is to be solved.f

is a strictly increasing function and the solution is unique. A modification of the secant method, called the Illinois method (see Dowell and Jarratt (1971) and Kennedy and Gentle (1980)), is used to find the root.

Given two values xi and xi- 1, the next approximation xi,1I to the root is determined by

Xi+ lI x XI - f(x fx i)( I(x - _{fxi -} - i- ')} ₎ (2)

The end points unity and zero serve as two starting values xo and xl. Iterations are performed according to the following rules.

tAddressfor correspondence: Institute of Management Science, National Chiao Tung University, 4F, 114 Chung- Hsiao W. Road, Section 1, Taipei, Taiwan, Republic of China.

200 DING AND BARGMANN

(a) Iff(xi+ 1) f(xi) < _{0, then (xi- 1, f(xi- 1)) is replaced by (xi, f(xi)).}

(b) Iff(xi+ 1) f(xi) > 0, then (xi- 1, f(xi -1)) is replaced by (xi- 1, f(xi- 1)/2). After these two rules have been applied, (xi+1, f(xi+1)) replaces (xi, f(xi)). The convergence criterion is based on I -xi+ -xi and If(xi+ 1) (relatively).

Structure

REAL FUNCTION SQMCQ(CDF, IP, N, RHO2, IFA UL T) Formal parameters

CDF Real input: the cumulative probability m (at which

the quantile is desired)

IP Integer input: the number of random variables p

N Integer input: the sample size N

RHO2 Real input: the square of the population multiple-

correlation coefficient p2

IFA UL T Integer output: a fault indicator:

= 1 if there is no convergence after n iterations in SQMCOR;

=2 if p< 1, p>N, p2<o or p2> 1;

= 3 if m<O or m> 1; = 0 otherwise

Auxiliary Algorithms

The auxiliary routine SQMCOR (Ding and Bargmann, 1991) is invoked by

function SQMCQ to evaluate the CDF of R2. SQMCOR calls algorithm CACM 291

(Pike and Hill, 1966) and algorithm AS 63 (Majumder and Bhattacharjee, 1973).

Constants

The variable EPS in the DATA statement represents a small real number to indicate the convergence criterion. The value given here is 1.0 x 10-6.

Precision

Double-precision operation may be performed by changing REAL to DOUBLE

PRECISION in the REAL statement. The real constants in the DATA statements also need to be in double precision, and the value of EPS may be changed from 1.0 x 10-6

to 1.0 x 10- 12 to increase the accuracy. The auxiliary routines must also be converted to double precision.

Time

No absolute timings are given here since the execution time depends entirely on the values of the input parameters.

Acknowledgements

The authors thank the referee and the Algorithms Editor for their valuable comments that led to substantial improvements in the manuscript.

STATISTICAL ALGORITHMS 201 References

Anderson, T. W. (1984) An Introduction to Multivariate Statistical Analysis, 2nd edn, p. 145. New York: Wiley.

Ding, C. G. and Bargmann, R. E. (1991) Algorithm AS 260: Evaluation of the distribution of the square of the sample multiple-correlation coefficient. Appl. Statist., 40, 195-198.

Dowell, M. and Jarratt, P. (1971) A modified regular falsi method for computing the root of an equation. BIT, 11, 168-174.

Kennedy, W. J. and Gentle, J. E. (1980) Statistical Computing, p. 74. New York: Dekker.

Majumder, K. L. and Bhattacharjee, G. P. (1973) Algorithm AS 63: The incomplete beta integral. Appl. Statist., 22, 409-41 1.

Pike, M. C. and Hill, I. D. (1966) Algorithm 291: Logarithm of gamma function. Communs Ass. Comput. Mach., 9, 684.

REAL FUNCTION SQMCQ(CDF, IP, N, RHO2, IFAULT) C

C ALGORITHM AS 261 APPL. STATIST. (1991) VOL. 40, NO. 1 C

C Returns the quantile of the distribution of the square C of the sample multiple correlation coefficient for C given values of CDF, IP, N, and RHO2

C

C A modification of the secant method, called the C Illinois method, is used

C

C The auxiliary algorithm SQMCOR, used to compute the C C.D.F. of the square of the sample multiple correlation C coefficient, is required

C

INTEGER IP, N, IFAULT REAL CDF, RHO2

REAL DIFF, EPS, FO, Fl, F2, XO, Xl, X2, ZERO, ONE, TWO REAL SQMCOR

EXTERNAL SQMCOR

DATA ZERO, ONE, TWO / 0.0, 1.0, 2.0 /

DATA EPS / 1.OE-6 /

C

SQMCQ = CDF IFAULT = 2 C

C Perform domain check C

IF (RHO2 .LT. ZERO .OR. RHO2 .GT. ONE .OR. IP .LT. 2 .OR.

* N .LE. IP) RETURN IFAULT = 3

IF (CDF .LT. ZERO .OR. CDF .GT. ONE) RETURN IFAULT = 0

IF (CDF .EQ. ZERO .OR. CDF EQ. ONE) RETURN C

C Use ONE and ZERO as two starting points for the C Illinois method C XO = ONE FO = ONE - CDF Xl = ZERO Fl = -CDF C

C Continue iterations until convergence is achieved C

10 DIFF = Fl * (Xl - XO) / (Fl - FO)

X2 = Xl - DIFF

F2 = SQMCOR(X2, IP, N, RHO2, IFAULT) - CDF IF (IFAULT NE. 0) RETURN

202 MAKUCH, ESCOBAR AND MERRILL c

C Check for convergence C

IF (ABS(F2) .LE. EPS * CDF) GO TO 20 IF (ABS(DIFF) .LE. EPS * X2) GO TO 20 IF (F2 * Fl .GE. ZERO) THEN

FO FO / TWO ELSE X0 = Xl FO Fl END IF Xl = X2 Fl = F2 GO TO 10 C 20 SQMCQ = X2 RETURN END Algorithm AS 262

A Two-sample Test for Incomplete Multivariate Data

By Robert W. Makucht and Michael Escobar

Yale University, New Haven, USA

and Samuel Merrill Ill

Wilkes College, Wilkes-Barre, USA [Received July 1 987. Final revision November 1 989]

Keywords: Censored data; Gehan test; Incomplete data; Log-rank test; Multivariate test Language

Fortran 77

Description and Purpose

Wei and Lachin (1984) described a family of asymptotically distribution-free tests for equality of two multivariate distributions based on censored data. These tests are natural generalizations of the log-rank test (Mantel, 1966) and the generalized Wilcoxon test of Gehan (1965), both used extensively in the comparison of time-to- failure distributions between two groups. These methods properly take into account the possibly censored nature of events that contain only partial information about the random variables of interest. The generalized tests are obtained on the basis of the

commonly used random censorship model (Kalbfleisch and Prentice, 1980), where

the censoring vectors for each subject are mutually independent and also are independent of the underlying failure time vectors.

Censored multivariate time-to-event data are encountered frequently in clinical

tAddress for correspondence: Department of Epidemiology and Public Health, Yale University School of Medicine, 60 College Street, New Haven, CT 06510, USA.