Miscellanea 503

Biometrika (1980), 67, 2, pp. 5034 Printed in Great Britain

Efron's conjecture on vulnerability to bias in a method for balancing sequential
trials

BY J. MICHAEL STEELE
Department of Statistics, Stanford University

SUMMARY

Efron (1971) proposed a method for sequential assignment to.treatments or control which is in many ways superior to traditional procedures. To analyse the method's susceptibility to accidental bias a criterion concerning the maximum eigeinvalue of a fundamental covariance matrix was introduced. On the basis of numerical evidence, Efron conjectured an explicit formula for this eigenvalue. This note gives a proof of that conjecture.

Some key words: Balanced experiment; Biased coin design; Covariance matrix; Maximum eigenvalue; Sequential trial.

Suppose that subjects are to be assigned sequentially to either treatment or control. If at the time of arrival of a new subject there have been D more subjects assigned to treatment than control, then Efron (1971) suggests the following:

if D > 0, assign to treatment with probability q and to control with probability p, where
p + q = 1, p > 1;
if D = 0, assign to treatment with probability 1 and to control with probability J;
if D < 0, assign to treatment with probability p and to control with probability q.

This biasedcoin design has several benefits over some traditional procedures such as Student's sandwich plan, and has attracted considerable practical and theoretical attention (Matts & McHugh, 1978; Pocock, 1979; Pocock & Simon,,1975; Wei, 1977, 1978).
Now suppose that N subjects have been assigned to treatment and let Tk be + 1 or  1 accordingly as the kth subject is assigned to treatment or control. The vector P = (T11 ... 1 T.) has mean E(T) = 0, and its covariance matrix will be denoted by Q.
Efron argued persuasively that the vulnerability of a balancing design to an accidental bias is sensibly measured by the maximum eigenvalue of the covariance matrix , and he studied this by considering the maximum eigenvalue AN of the asymptotic covaria9;~,ce of e
h th vector (Th+l, ..., Th+N) as h >. oo. As N >. oo, these AN increase to a finite limit A, and on the basis of considerable numerical evidence, Efron conjectured that A = 1 + (p  q)s.
To prove this consider the asymptotic covariances and the associated spectral density:

Pk=liME(Thph+k), f(oj)= M.Pkeiwk.
h+Go kco

Efron observed that A = maxf(w); this maximum can now be calculated using a general lemma (Katznelson, 1968, p. 22).

LEMMA. Suppose that an even sequence {a.} of positive real numbers tend to zero and
satisAY an+i  2an + an, > 0 for all n > 0, then the series

00
g(x) 1 an eino
n_00 represents a nonnegative function.

504 Miscellanea

Setting g(x) = f(,r) f(x) one expands g in a Fourier series with coefficients {a,,}. Efron's
Theorem 4 shows f(ir) = 1  (p  q)2, so that by the lemma it remains only to cheek the
positivity and convexity of the {a.}' *
Setting r = plq, Efron showed that

Po=l, p2pl=i(r1)2/{r(r+1)2},
and that for k > 1, pk+l  pt is positive and decreasing. This implies that a.+,  2a. + a,., >, 0
for n > 1. To cheek the remaining case n = 0 one computes

%  2a, + a. J(r  1)%r(r + 1)2}  1(r  1)2/{r(r + 1)} + (r  1)21(r + 1)2

(r 1)21{r(r + 1)2},>
,0.

The lemma then shows g(x) 0 and the conjecture is proved.

REFERENCES '
Eirnox, B. (1971). Forcing a sequential experiment to be balanced. Biometrika 58, 40317.
KAUNELSON, Y. (1968). An Introduction to Harmonic Analysis. New York: Wiley.
MATTs, J. P. & McHuGn, B. B. (1978). Analysis of accrual randomized clinical trials with balanced groups in strata. J. Chron. Die. 31o 72540.
Pocoax, S. J. (1979). Allocation of patients to treatment in clinical trials. Biometrics 85, 18397.
PocooK, S. J. & Simox, B. (1975). Sequential treatment assignments with balancing for prognostic factors in controlled clinical trials. Biomstrics 31, 10315.
Wzi, L. J. (1977). Class of designs for sequential clinical trials. J. Am. Statist. Awoo. 72, 3826.
Wicl, L. J. (1978). Adaptive biased coin design for sequential experiments. Ann. Statist. 6, 92100.

[Received October 1979. Revised December 1979]