Biom. J. 36 (1994) 8, 927 - 934 Akademie Verlag

Symmetric Intervals and Confidence Intervals

CLAUDE BAUDOIN

Unit& de Recherche Cliniques et Epidhiologiques, Villejuif, FRANCE

INSERM U 21 -

JOHN O'QUIGLEY Department of Mathematics, University of California, San Diego, USA

Summary

We show that the symmetric "codidence intervals" of W m m ( 1 9 7 2 , 1976), widely used and referred to in bioequivalence studies, a n not confidence intervals in any acccpted sense. Nevertheless, meaningful symmetric intervals can be constructed in the context of Bayesian or fiducial inference.

Key words : Bayesian interval; Bioequivalence ; Confidence interval ; Fiducial inference; Symmetric interval.

1. Foreword

The statistical problem of showing bioequivalence of drug formulations (equiva- lence of biological or therapeutic effect) has been the subject of much research since 1971, when a committee of the American Statistical Association was formed to study the problem (METZLER 1983). Some of the conclusions of this committee were published by METZLER (1974).

The bioequivalence objective is not to show a difference between formulations, but rather to investigate whether any difference is of practical importance. The classical hypothesis testing framework Seems poorly adapted to this problem since a statistically significant test may reflect a negligible effect whereas a non- significant test cannot be advanced as evidence of an insignificant effect.

The statistical history and the context of the bioequivalence question has been reviewed by METZLER (1983), and the statistical methodology by COM-NOUGUE (1987), STEINIJANS (1990), PIDGEN (1992) and NICOLAS (1993). Initially, alternative statistical approches were based on confidence intervals, either parametric and conventional, (METZLER, 1974; KIRKWOOD, 198 l), parametric and symmetric, (WESTLAKE, 1972, 1976) or non-parametric (STEXWANS, 1983). Later, the idea of a Bayesian approach to the question was introduced, beginning with RODDA

928 C . BAUDOIN. J. O’QUIGLEY: Confidence Intervals

(1980), SELW (1981) and FLUEHLER (1981); one aspect of the Bayesian ap- proach is that it can lead to a generalization of the symmetric confidence interval of Westlake, (MANDALLAZ, 1981). About the same time, a reformulation of the problem in classical terms was proposed, the customary null and alternative hypotheses being interchanged (ANDERSON, 1983; ROCKE, 1984). This leads to some conceptual clarification at the cost of some additional technical complexity; maximization of p-values over the null space (no longer a simple point hypoth- esis) and the use of non-central distributions (ANDERSON, 1983). Special cases have been given particular attention; 2 x 2 contingency tables by D U ” ~ (1977) and ordered categorical data by MEHTA (1984).

The purpose of this paper is to examine the statistical foundation of the symmetric confidence interval of WESTLAKE (1972, 1976, 1979) the most referred to method in the bioequivalence literature. Confidence interval methods are largely recommended by the clinical guidelines published by drug regulatory agencies (WHO, 1986, or APV Guideline, 1987, for example).

2. Background

The conventional method is to calculate a lOO(1 -a)% confidence interval for the point estimate (generally the mean difference) and accept bioequivalence if this confidence interval is completely contained within some defined range of tolerance. The choice of the upper and lower limits of the tolerance region are the concern of the practitioner. (WESTLAKE, 1972, 1976, 1977, 1979, 1981) argues forcefully for the use of confidence intervals, symmetric about some value typically different from the point estimate; this value would be zero when dealing with differences, and one when dealing with ratios.

This symmetrical confidence interval has received considerable attention: its use was exemplified in pharmacological journals by HECK (1979), tables were established to facilitate its determination by SPIUET (1978), its power was com- pared with other kinds of confidence interval by Roc= (1984) and DEHEUVELS (1984), and it set up the results of many studies of bioequivalence by HECK (1979) and ALKALAY (1980) for example. Much controversy (KIRKWOOD, 1981; SHIRLEY, 1976 and MANTEL, 1977) has surrounded the approach, mainly concerning the desirability of symmetrics intervals, the conventional method giving .the shortest confidence interval.

To our knowledge at least, the question as to whether or not a symmetric interval is a confidence interval at all has not been raised.

Biom. J. 36 (1994) 8 929

3. Confidence Intervals

3.1 Symmetric intervals

Let 6 represent a population parameter of interest defined on the interval (aL, 6”). In bioequivalence studies 6 would be the true difference between two treatment preparation means. Denote the point estimate by a one-dimensional sufficient statistic for 6. The cumulative distribution of afor given 6 is written as F ( @ ) .

The subtlety of the confidence interval concept derives from the logical transition of a probabilistic statement concerning a fixed population to one concernjng an infinite family of populations. Specifically the expression

pr(6 + I , 5 if< 6 + I , ) = F(6 + I , 16) - F(6 + I, 16) 2 1 -a

p r @ - I, 5 6 52- I , ) > 1 -a

(1)

(I*)

is straighforward and unambiguous, as is

which is just an algebraic rearrangement of equation 1. The interpretation of (l*), from which the notion of a confidence interval arises, is of course very different from that of (1). The frequentist interpretation of a confidence interval is possible because (1) and (1*) are algebraically the same.

What happens when we follow Westlake’s procedure to try and symmetrize (1*) about some value 8 (in general 8 = 0). We observe and on the basis of this and some additional calculations, fairly simple ones in the case of a normality assumption, we can replace I, and I, by It and 1% such that (1*) becomes symmetric about 8. The calculations are not guaranteed to have a solution but will do so in many cases.

The point to note though is that, unlike I, and I,, 1; and 1% are more or less complicated, functions of & (WESTLAKE, 1976). For clarity we ought now write (I*) as

p r ( i f - I ~ ( d ) 5 6 < 6 - 1 ~ ( ( d ) ) > 1-a (2 *)

This expression may not trouble us unduly until we write down the “mirror image” of 2*

This equation makes no sense as becomes apparant when we replace 1, and 1, in the right hand side of the equation by Ii(n) and e(4. It would seem to be impossible to regroup the random elements of the equation together between the inequality signs without making a trivially useless statement. Furthermore, if F ( - ) gives the cumulative distribution function for & then some function of 2 is, in general, going to have a distribution other than F(- ) .

Another approach to confidence interval construction consists in, after having observed performing a test via a uniformly most powerful test procedure, at

930 C. BAUDOIN, J. O'QUIG'LEY: Confidence Intervals

level a. Different null values of b are selected and all those such that the null hypothesis is not rejected constitute a 100(1 -a)% confidence interval for b. Any attempt to amend the procedure, in order to obtain symmetric intervals, encoun- ters the same problem suffered by the more familiar approach outlined above. In conclusion then, confidence intervals cannot be symmetrized and remain confi- dence intervals.

3.2 Operating characteristics

In order to study the properties of the symmetric interval, a Monte Carlo study was carried out (table 1 and figure 1). To compute each symmetric interval, loo00 values of zand sa were randomly drawn from a normal distribution (PRESS et al., 1989), where 6 = 5 and was incremented from 0,l to 10. The 95% symmetric confidencd interval was calculated using the two equations (SPRIET, 1978):

z I , + I , = 2- sa

12 . 11

J F(u)~u- F ( ~ ) d ~ = 0 . 9 5 -OD - W

(3)

(4)

The values I, and I, were determined iteratively as follows: Initially a value of pr(I,) was arbitrarily fixed. The corresponding value of I, was determined by the computation of the percentage points of the normal distribution (BEASLEY, 1977), the value I, by equation (3) and the value of pr(12) from the normal integral (HILL, 1973). The iterations were stopped when pr(l,) f pr(l , ) = 0.95 f O.oooO1, otherwise another value of pr(I,) we fixed by dichotomic searches. The confi- dence coefficient was calculated as the proportion of intervals which included 6.

Table 1 shows that when the two sided confidence interval for 6 includes the value zero, the confidence coefficient of the symmetric intervals is always equal to 100%. Figure 1 shows that confidence coefficient increases to 100% very rapidly. Consequently, the construction of the symmetric interval is only close to the nominal significance levels when the mean is significantly different from zero. In this case, Table 1 shows that when the confidence coefficient of the symmetric interval is close to the nominal value of 95%, the symmetric interval is equivalent to a one-sided confidence interval. This is because one of the two terms in the equation (4) is zero when the mean is significantly different from zero. These simulations quantify and illustrate the two criticisms which have been described theoretically about the symmetric interval : by WESTLAKE (1976) who said "the confidence coefficient is 100 percent for b = 0, decreasing monotonically to 95 percent as 161 tends to infinity" and by KIRKWOOD (1981) who underlined that the symmetric interval converges towards a unilateral confidence interval when a,- is decreased or 161 increased.

Biom. J. 36 (1994) 8 931

Table 1 Results of the Monte Carlo study of the confidence coefficients of the symmetric interval, for 6 = 5 and increasing values of UJ. The limits of the symmetric interval, the one sided and two sided confidence intervals were calculated.

limits of limits of limits of two 1 -a symmetric one sided sided CI

Qi 4 I, pr(1,) pr(1,) observed interval C1 minimal maximal ~~

0.1 101.645 -1.645 0.OOO 0.050 0.9507 5.16 5.16 4.80 5.20 1 11.645 -1.645 O.Oo0 0.050 0.9513 6.64 6.65 3.04 6.96 2 6.645 -1.645 O.Oo0 0.050 0.9531 8.29 8.29 1.08 8.92 2.25 6.089 -1.645 O.Oo0 0.050 0.9586 8.70 8.70 0.59 9.41 2.50 5.645 -1.645 O.Oo0 0.050 0.9789 9.11 9.11 0.10 9.90 2.75 5.281 -1.645 O.Oo0 0.050 1.oooO 9.52 9.52 -0.39 10.39 3 4.978 -1.645 O.OO0 0.050 1.oooO 9.93 9.94 -0.88 10.88 4 4.145 -1.645 O.Oo0 0.050 1.oooO 11.58 11.58 -2.84 12.84 5 3.646 -1.646 O.Oo0 0.050 1.oooO 13.23 13.23 -4.80 14.80 6 3.316 -1.649 O.OO0 0.050 1.oooO 14.90 14.87 -6.76 16.76 7 3.083 -1.655 0.001 0.049 1.oooO 16.58 16.52 -8.72 18.72 8 2.912 -1.662 0.002 0.048 1.oooO 18.30 18.16 -10.68 20.68 9 2.783’ -1.672 0.003 0.047 1.oooO 20.04 19.81 -12.64 22.64

10 2.681 -1.681 0.004 0.046 1.oooO 21.81 21.45 -14.60 24.60

Fig. 1. Results of the Monte Carlo study of the confidence coefficients of the symmetric interval for 6 = 5 and increasing values of UJ

932 C. BAUDOIN, J. O'QUIGLEY: Confidence Intervals

4. Alternative Symmetrical Intervals

Denoting by @(a) the prior density of 6, (see FLUEHLER, 1981) for a choice of appropriate priors) then, having observed 8 a lOO(1 -a)% Bayesian interval for 6 is obtained by solving the equation

for the constants I, and I,. For 6 with support on the real line (-a, co) and F(61 u) continuously differentiable, we can always solve the equation

{jm @(u)dF(~Iu) j - l 'y2 @(u)dF(~Iu)= 1-a 8-72

for y to obtain Bayesian intervals, symmetric about the value 8. Following FISHER (1956, p. 70) and assuming that

lim F(ZIS)=O and lim F(ZI6)= 1 d + d L a+au

where for each given 8 F(Zl6) is a decreasing differentiable function of 6, then the fiducial density function for 6 is given by

YA6) = - dF(Z1 6)/d6

and we can define a lOO(1- a)Yo fiducial interval for 6 by solving

11

j YZ(u)du= 1-a 11

As for the Bayesian interval, if (dL, 6,) = (- 00, co) then we can always solve the equation

e + 7 2

e-72

j !Pdu)du=I-a

for y to obtain fiducial intervals symmetric about the value 8. For other values of (aL, 6,) solutions may or may not exist. KENDALL (1979, Ch 21) make the further stipulation for fiducial intervals that YAZ,) = YdZ2). This stipulation appears to be motivated by a desire to obtain unique intervals and does not seem essential to the approach. Clearly, if maintained, it makes the construction of symmetric intervals impossible outside of some very restrictive special cases. As a final point note that if there exist transformations of Zand 6 to 2 and fl where B is a location parameter for 2, then the fiducial argument is equivalent to the Bayesian argument with a uniform prior distribution for /3, (LINDLEY, 1958).

Biom. J. 36 (1994) 8 933

5. Conclusion

Statistical point estimates of population parameters are usually accompanied by an interval of some kind to indicate the amount of imprecision we can associate with such estimates. Different philosophical approaches to statistical inference will generally give rise to different intervals so that Bayesian, fiducial and classical confidence intervals do not in general coincide. In practice confidence intervals enjoy far more use than do their competitors derived from the Bayesian and fiducial standpoints. The reason for this is undoubtedly the firm frequentist framework upon which their interpretation rests. Users need periodic reminding that in choosing to work with confidence intervals rather than Bayesian or fiducial intervals they are choosing to abandon an attractively intuitive approach to inference, with some occasionally paradoxical and arbitrary behavioural features, in favour of one, more objectively based, and for which frequentist statements can be made.

Only Bayesian intervals and fiducial intervals, under a slight relaxation of the definition of a fiducial interval, can be made symmetric. The symmetric “confi- dence interval” of Westlake is not a confidence interval at all in the usual frequentist sense; it is a new quantity which, for the present, does not skem to be based on any known accepted statistical principles.

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Dr. CLAUDE BAUDOW INSERM U21 Unite de Recherches Cliniqucs et Epidhiologiques 16, avenue P. V. Couturier, 94807 Villejuif Cedex France