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1 Some Applications of Exponential Ordered Scores Author(s): D. R. Cox Source: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 26, No. 1 (1964), pp Published by: Wiley for the Royal Statistical Society Stable URL: Accessed: :49 UTC JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at Royal Statistical Society, Wiley are collaborating with JSTOR to digitize, preserve and extend access to Journal of the Royal Statistical Society. Series B (Methodological)
2 1964] 103 Some Applications of Exponential Ordered Scores By D. R. Cox Birkbeck College, University of London [Received August Revised September 1963] SUMMARY There are simple standard tests for the comparison of samples assumed to arise from exponential distributions, but the properties of these tests are known to depend severely on the assumption of exponential form. However, the observations can sometimes usefully be ranked, and then replaced by the corresponding expected values of order statistics in sampling the unit exponential distribution, before calculating the appropriate test statistic. Some special tests of this type are examined. The procedure is analogous to the use of Fisher and Yates's scores in normal theory. Savage (1956) gave the test of this type for comparing two samples, in the course of a general study of rank tests for the two-sample problem. 1. INTRODUCTION IN 1938, in the first edition of their Statistical Tables, Fisher and Yates gave the expected values of the order statistics in sampling a standardized normal distribution, the so-called normal scores (Fisher and Yates, 1957, Table 20). They suggested that these scores should be used for significance tests involving comparisons with ranked data, assumed to be derived from an underlying normal distribution. For example, to compare the locations of two samples, the ranks are replaced by the corresponding normal scores and, as an approximation, the standard t-test is then applied. Similar methods apply for testing null hypotheses about problems of regression, serial correlation, etc. involving ranks. The intuitive basis of these procedures is, of course, that the normal scores provide a good reconstruction of the underlying variate values on a standardized scale, the null hypothesis being assumed true. Much later, mdre formal justifications of the use of normal scores were given. The point of view taken in this work is that, although the variate values may be available, it is required to test a non-parametric null hypothesis with a rank-invariant test. In particular, Terry (1952) showed that among all rank-invariant tests for the two-sample location problem the one based on normal scores is locally most powerful for detecting a difference in the means of two normal populations. Chernoff and Savage (1958) showed that for detecting shifts in location of an arbitrary distribution, the test based on normal scores has asymptotically a power greater than or equal to that of the usual large-sample normal test using variate values. In particular, if the populations are normal there is asymptotically no loss of power from using normal scores rather than variate values. Lehmann (1959, pp ) has discussed this work. The use of normal scores is only a special case of a general treatment. The above application of expected order statistics is distinct from that to the analysis of censored data; see, especially, Plackett (1958, 1959) for such use both for normal and logistic distributions.
3 104 Cox - Some Applications of Exponential Ordered Scores [No. 1, Now there has been much recent interest in statistical methods associat the exponential distribution, for instance in connection with application testing; see, for example, Maguire et al. (1952), Epstein and Sobel (1953) and Birn (1954). The simplicity of these methods, and the ease with which they are exten censored data, make them attractive, but unfortunately the methods are s to the assumption of exponential form (Zelen and Dannemiller, 1961). Th taken with the properties sketched above of procedures based on normal suggests the following procedure for significance tests involving the comp exponential distributions: rank the observations, replace the ranks by the c sponding expected order statistics in sampling the unit exponential distribu then calculate the usual exponential theory test statistic. In this way we ca to lose very little when, in fact, the underlying distribution is close to the exp form, while retaining control over the test if there is an appreciable departure the exponential distribution. The special case of this procedure for the comparison of two samples was given by Savage (1956) in the course of a general study of rank tests for the two-sample problem. The formal justification for the procedure, contained in the general work referred to above, is strong. When the true distributions are exponential, the test is a locally most powerful rank test with no asymptotic loss of power. If the true distributions are not exponential, the exact exponential theory test will, except in very special cases, be invalid, whereas the test based on exponential scores is distribution-free. It seems very likely that the same justification applies not only to the two-sample problem, but also to the other applications of this paper. Unfortunately, the loss of information in small samples when the true distribution is exponential, while very probably small, is not known. The present paper investigates some tests based on exponential scores. Of course, in practice, formal tests of significance are likely to be only part of the whole analysis. An important thing, for instance, is likely to be graphical analysis of the survivor curves of the separate sets of data. 2. THE EXPONENTIAL SCORES In this Section we examine some properties of the finite population consisting of the expected order statistics for a random sample of size n from the distribution e-v. Denote by trn the expected value of the rth observation in increasing order of magnitude. It is well known that so that in particular trn = /n+ + 1/(n-r+ 1) (r =1,...,n), (1) tin = 1/n, t2n = 1/n + 1/(n- 1), (2) tnn = 1/n +11(n -1)+%+ 1. For n not too large, these are easily The moments of the finite populatio sums n Sin = trinn r=1
4 1964] Cox - Some Applications of Exponential Ordered Scores 105 Now the triangular scheme (2) for sample size n is derived from that for sample size n -1 by adding the first column of 1/n's. There follow easily recurrence relations such as the following: 51n = 51,n-1 + 1, S2n = 52,n-1 + (2/n) s1,n-1 + 1/n, S3n = S3,n-1 + (3/n) S2,n-1 + (3/n2) s1,n-1 + I /ni J These can be solved subject to the initial condition sil = 1. In particular, n Sin = n., S2n = 2n - (I /r) = 2n -tnn, r=1 After some calculation, we have, for Fisher's K parameters for the finite population, Kln= 1, K27, = nn-tn = 1 logn+y-1 0(lo( Kln 2n -n-i n n where y = is Euler's constant. The leading term in the asymptotic expansion for each cumulant is the corresponding cumulant of the unit exponential distribution. Table 1 gives some values of K2n and of the approximation 1- (log n + y - 1)/n. TABLE 1 Some constants connected with the exponential scores n K2n K2 n ln ln (approx.) (approx.) P f P f P P132 1P f863 0f f897 0f901 1P082 1P P067 1P f930 1P057 1f COMPARISON OF Two SAMPLES Suppose that we have two independent samples of sizes ml, m2 with sample means il, Y2. To examine the null hypothesis that the population means are equal, assumin the distributions to be exponential, we test j13/2 in the F-distribution with (2ml, 2m2) degrees of freedom. Now follows the procedure sketched in Section 1. Let the whole set of n = ml+ m2 values be ranked and replaced by the scores {trn} o We assume for simplicity that there are no ties. Let 4l., T2 be the sample mean scores. A first approximation is then to test [1/12, again as F with (2ml, 2m2) degrees of freedom. The "exact" test procedure, however, is to regard the set {trn} of scores as a finite population. Under the null hypothesis, the scores arising in, say, the first sample are a random sample of ml drawn without replacement from the finite population. Since ml 1 + m2 t2 = Sin and is fixed, a critical region based, say, on large i1/i2 is equivalent to one based on large ii., or, of course, on ul = ml f1/s1n. Hence we can regard the test as based on one sample mean.
5 106 Cox - Some Applications of Exponential Ordered Scores [No. 1, From the theory of sampling a finite population (Kendall and Stuart, 1958, pp. 301, 302), we have, from (3) and (4), that under the null hypothesis E(ul) = ml/n, var (ul) = (ml M2) (n - t..)/{n3(n - 1)}. (5 Now the simplest test based on ul would be to apply the F test to i1/t2, just as if we were working with the original observations. This is equivalent to treating ul having the beta distribution xmi-l(1 - x)m2-'/b(ml, i2). (6) For this the first two moments corresponding to (5) ar ml/(mi+m2) and mlm2/{(ml?m?)2 A natural modification to improve the agreement T-/T2 as having an F-distribution with (21ml,21m2 Corresponding to (7), we have a mean and variance ml/(ml + M2) and ml m2/{(ml + m2)2 (Ml + 1m and these agree with (5) if I is chosen as a function 1~n-2n+t ~~ y-r2+logn (log n In =n( _n - tnn) + n +O( kln). (8) The last two columns of Table 1 give some values for the factor In approximation 1 + (y -2 + log n)/n. We could test the above procedure by finding higher moments of the te Instead a simple example has been worked out in detail. Example. Suppose that we have for comparison two samples of five o each. The first step is to rank the whole set of ten observations and t ranks by the scores (2), namely 0.10, 021, 034, 0-48, 065, 0O85, 1.10, 1P43, 1P93, We now calculate the sample mean scores tl, i2 and base our test on -l/i2 lently, on i1. We consider four methods of computing significance limits. Method 1. There is an "exact" permutation test, since under the null the first sample is equally likely to be any of the 252 distinct sets of 5 num can be formed from the finite population of 10. The resulting distribution been enumerated, with a grouping interval of 0-1, and interpolated va 25, 10, 5, 2i upper per cent. points are given in Table 2. The dist symmetrical with mean 5, so that the lower tail need not be examined. Method 2. We can use a crude approximation treating the exponen as variate values. Then F-= T1/T2 = T1/(2 - i1) is tested in the variancebution with (10, 10) degrees of freedom. Resulting values of 5T1 = 1O given in Table 2. Method 3. This is the same as Method 2 except that modified degrees of freedom are used in accordance with (8). Since 11 = 1-173, the new degrees of freedom are (11.7, 11.7). Table 2 gives the revised limits for 5T1.
6 1964] Cox - Some Applications of Exponential Ordered Scores 107 Method 4. We can treat 5i1 as normally distributed with the mean and variance of the exact permutation distribution. The method is a very special case of (11) of the next Section, in which v = 5!1 and xl =... = X5 = 1, X6 =... = X10 =O. approximation is to treat 54- as normally distributed with mean 5 and standard deviation 1(10K10/4) = 1P40. TABLE 2 Comparison of significance limits for a two-sample problem Probability "Exact" Crude F Modified F Normal level limit approximation approximation approximation per cent. (Method 1) (Method 2) (Method 3) (Method 4) j The general conclusion from Table 2 is that even with these extremely small sample sizes, both F approximations and the normal approximation are reasonably good and that in the tail there is a worth-while improvement from using the F-distribution with the correcting factor I.. It is not reasonable in this example to expect the approximations to apply much beyond the range of Table 2, since there are only 252 possible distinct samples. However, the extreme tail can easily be found exactly. Thus the two highest values of 5 il are * = 8-24, P = 8&04, each with probability 1/252. Thus the value attained or exceeded with probability 1/126 is (The approximate 1 per cent. point from the crude F approximation is 8-29, and 8 10 for the modified approximation.) TABLE 3 Comparison of significance limits for total offirst sample, for two-sample problem with sample sizes 3, 9 Probability "Exact" Crude F Modified F Normal level limit approximation approximation approximation per cent. (Method 1) (Method 2) (Method 3) (Method 4) Lower * P Upper *88 6*
7 108 Cox - Some Applications of Exponential Ordered Scores [No. 1, A more severe test of the approximations is provided by a problem with very unequal sample sizes. Table 3 summarizes some calculations on the comparison of two samples of sizes 3, 9, for which the permutation distribution has 210 points. On the whole, there is a substantial improvement from using the correcting factor I., especially in the tails. The normal approximation makes no allowance for the skewness of the distribution of the first sample total and is rather poor. It is very reasonable to expect that as sample sizes increase and all the distributions get more nearly normal, the two approximations provided by the F-distribution will improve. The two specimen examples involve very small sample sizes, and the satisfactory approximations obtained in Tables 2 and 3 suggest that the approximations hold for all reasonable practical purposes. 4. REGRESSION ANALYSIS Suppose that we want to test the observations y1,...,y for regression on the fixed quantities xi,..., x. It is worth outlining briefly the procedure to be followed if Yi'...,Yn are assumed exponentially distributed. In the absence of specific reason to the contrary, it is sensible to consider a simple regression model chosen for mathematical convenience. This is that the yj are independently exponentially distribute with means I/(oQ + fixj). The null hypothesis is that / = 0. The likelihood is exp (- oyyj - /xj yj) II (o + fixj), the sufficient statistic is (Zyj, Zxjyj) and the "exact" test is based on the conditi distribution of Ex1y1 given Xy, which is independent of the nuisance parameter oz. Vincent (1961) has discussed this test in the different context of tests on sample estimates of variance. An interesting special case arises when xj =j; if the xj are interpreted as intervals between events in a point process, the null hypothesis is that we have a Poisson process, whereas under the alternative process there is a trend with the serial number of the event. This can be compared with the model and test considered by Cox (1955) of a time-dependent Poisson process in which the probability rate of occurrence at time -r is oze8. The two significant tests of the null hypothesis /3 = 0 are almost identical; the models are, of course, different when go 0, although equivalent in a deterministic approximation. Another special case is the two-sample problem of Section 3 obtained when the xi takes values 1, 0 only. Turning now to the test based on scores, we rank the yj, replace them by the appropriate scores and consider the test statistic v = Xx. t., where the random variable tj is the score attached to the jth observation. Under the null hypothesis, all permutations of the scores {trn} are equally likely. We shall not develop the theory of v beyond the simplest normal approximation, for which we need the mean and variance of v over all permutations of {trn}. Now E(v) is bilinear in {xj} and {trn} and is symmetric in the two sets separately. Hence E(v) = a. sl 2xj =a. nyxj, where an depends only on n. Consideration of the special case xl =... = = 1 shows that a = 1/n and that E(v) = Zxj. (9)
8 1964] Cox - Some Applications of Exponential Ordered Scores 109 Similarly, we can show from considerations of degree and invariance that var (v) = bn 2(xj-x)2 K2, (10) where bn depends only on n. If we consider the we find on substituting into (10) that bn = 1. Thus the large-sample test is to take (v - Xj)I{(Xj-)2 -K2n}- (11) as normally distributed with mean zero and unit variance. In special cases, as in Section 3, it may be possible to obtain a better approximation fairly easily. 5. DisCUSSION There are a number of further points which will be discussed only very briefly. First, a number of further problems can be tackled by a direct extension of the above methods. One is the testing of the serial correlation coefficient. Secondly, if, say, in the two-sample problem both samples are censored at the same time point, we can adapt the above procedure in a simple way. If there are, say, p uncensored observations, we can attach scores tin,..., t1n to these as before, and the same score tp+l,n to all the censored observations. Now for exponentially distributed observations, the estimate of the population mean for the first population is (total time at risk)/(number of failures). Thus if there are r1 failures and ml-rl censored observations in the first sample, the corresponding statistic based on scores is {rl il + (ml - rl) t +1 n}/r1, (12) where il is the mean score of the failed individuals. The test sta two samples is the ratio of two statistics of the form (12). Approximately, this will have, under the null hypothesis, an F-distribution with (2r1, 2r2) degrees of freedom. Next, it is possible in principle to obtain confidence limits by the procedures of this paper. Thus in the two-sample problem of Section 3, we can test the hypothesis that the second population mean is po times the first population mean, for given po' by dividing the observations in the second sample by po before ranking and applying the test of Section 3. The set of all values po not rejected in such a significance test forms the required confidence interval. The practical procedure is to compute the standard normal deviate corresponding to the level of significance for a few trial values of po, and to interpolate to find the critical values determining the confidence limits. Finally, the procedures given here will have asymptotic optimum properties for the usual exponential theory alternatives, and therefore also for alternatives obtained by monotonic transformation from the exponential theory alternatives. In particular, the test of Section 3 is asymptotically optimum for testing the equality of two Weibull distributions, with distribution functions of the form 1-exp{-(o(x)f8}, when the alternative hypothesis is that the distributions have the same index f but different means.
9 110 Cox - Some Applications of Exponential Ordered Scores [No. 1, ACKNOWLEDGEMENT I am grateful to Miss B. M. Waters of the Scientific Computing Service, who did most of the calculations. REFERENCES BIRNBAUM, A. (1954), "Statistical methods for Poisson processes and exponential populations", J. Amer. statist. Ass., 49, CHERNOFF, H. and SAVAGE, I. R. (1958), "Asymptotic normality and efficiency of certain nonparametric test statistics", Ann. math. Statist., 29, Cox, D. R. (1955), "Some statistical methods connected with series of events", J. R. statist. Soc. B, 17, EPSTEIN, B. and SOBEL, M. (1953), "Life testing", J. Amer. statist. Ass., 48, FISHER, R. A. and YATES, F. (1957), Statistical Tables for Biological, Agricultural and Medical Research, 5th ed. Edinburgh: Oliver and Boyd. KENDALL, M. G. and STUART, A. (1958), Advanced Theory of Statistics, 1. London: Griffin. LEHMANN, E. L. (1959), Testing of Statistical Hypotheses. New York: Wiley. MAGUIRE, B. A., PEARSON, E. S. and WYNN, A. H. A. (1952), "The time intervals between industrial accidents", Biometrika, 39, PLACKETT, R. L. (1958), "Linear estimation from censored data", Ann. math. Statist., 29, (1959), "The analysis of life test data", Technometrics, 1, SAVAGE, I. R. (1956), "Contributions to the theory of rank order statistics: two-sample case, I", Ann. math. Statist., 27, TERRY, M. E. (1952), "Some rank order tests which are most powerful against specific parametric alternatives", Ann. math. Statist., 23, VINCENT, S. E. (1961), "A test of homogeneity for ordered variances", J. R. statist. Soc. B, 23, ZELEN, M. and DANNEMILLER, M. C. (1961), "The robustness of life testing procedures derived from the exponential distribution", Technometrics, 3,
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