The effect of nonzero second-order interaction on combined estimators of the odds ratio

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1 Biometrika (1978), 65, 1, pp Printed in Great Britain The effect of nonzero second-order interaction on combined estimators of the odds ratio BY SONJA M. MCKINLAY Department of Mathematics, Boston University, Massachusetts STTMMABY In combining odds ratios across strata, the assumption of zero second-order interaction is usually made in order to simplify estimation. The implications for the use of the general logistic model, when this assumption is not met, are considered. Using Monte Carlo techniques, the effect of nonzero second-order interaction on five available combined estimators is considered and an attempt made to provide results applicable in a variety of research situations. Some key words: Contingency table; Logistic model; Monte Carlo method; Odds ratio; Relative risk; Stratified sample. 1. intbodttcmon The odds ratio is one of the earliest measures of association in x contingency tables (Yule, 191; Goodman & Kruskal, 1959). However, it was not until the 1950's that techniques for combining estimates from several x tables were proposed. One of the earliest combined measures (Woolf, 1955) consisted of a sum of the logarithms of the odds ratios, weighted by the inverse of their variances. But, as Woolf points out, this combined measure is only valid in the absence of second-order interaction, equivalent to absence of 'heterogeneity' in his paper, and a suitable test for this heterogeneity is provided. The combined measure, proposed by Biroh (1964), modified by Goodman (1969), and the conditional maximum likelihood estimator itself (Gart, 1971) have all assumed a constant odds ratio. Possibly the only exception has been the measure given by Mantel & Haenszel (1959). These authors state (p. 735) that "... the assumption of a constant relative risk can be discarded as usually untenable'. Indeed, researchers frequently want a summary measure of association from several fourfold tables even though there is heterogeneity among the individual odds ratios. Certainly, in order to combine tables, the second-order interaction should never be so marked that there are important reversals in direction, some tables having odds ratios less than and others greater than unity. This was noted by Birch (1964, p. 30). Heterogeneity between tables generally results from the differential effect of one or more factors or covariates, extraneous to the investigation, and it is often hoped that by combining estimates, the effect of these covariates is reduced. Indeed, stratification is sometimes employed with the specific purpose of reducing covariate effects, in order to estimate the constant component of an association. Two reasons for combining estimates have been mentioned here: (a) to provide a summary statistic, and (b) to remove the effects of specific covariates from the comparison. In fact, it is difficult to differentiate between (a) and (b) except, perhaps, by sampling design. In general, when samples sizes are fixed within predetermined strata, summary estimates are required. In contrast, when the strata are formed during the analysis, from fixed total samples, elimination of covariate effects is usually the motivation.

2 19 SONJA M. MCKINIAY In this paper, the use of a general logistic model is considered, with reference to the combination of odds ratios from K strata. The estimators to be used in the Monte Carlo study are then described and the related simulation methods are presented. The final section discusses and interprets findings of the Monte Carlo investigation for application in research.. THE GENERAL MODEL Consider a binary response variable Y if with parameter p if, for the tth individual in the jth population. Assume, for simplicity that the covariate X is univariate, and assume further that (^) -«,+/»,*«,. (-1) Now, for the jth population containing N t individuals, we have a vector of independent binary response variables {Y tj } for i = 1 N }, with corresponding vectors of probabilities {Pa) an< i covariables {X.,}. Assume that the p if are independently distributed as in (-1) and that the X ii are independently and identically distributed within the jth population, so that one can write (X. y ) = E(X j ) for all i. Then, taking the expectation of the logits with respect to the X,, we have say, and the odds ratio between the two populations is defined as xft = e Qi ~ Qi, or alternatively log^ = K-oJ + ft^xj-ft^x,). (-) Now define K subpopulations or strata on the distributions of X l and X. In general, the»th individual from population 1, and the hth individual from population will be included in the fcth stratum if, for an ordered sequence of constants {a k } (k = 1,...,K), Ofc-i< «. x k% < fc- Then we have, for the jfcth stratum, where u) k = {x ij : a k _ 1 <x ii <a k } (t = 1,...,N t ; j = 1,). (-3) If the constant term say, then it is clear that in general log 0 4= log 0 A =t= log 0'. Indeed, both log</«and log^r fc consist of the sum of the common element log^r' and a variable or bias component dependent on the distribution of the covariate. In order either to provide a summary statistic or to reduce the effect of the covariate, we must reduce the size of this variable component in relation to logi/r'. For stratification to be effective in reducing the bias, the following relationship should hold for almost all k: Define L* = G^log^,...log^) so that L* represents an average of the log^r fc for a suitably chosen function (?(.). Then we expect that one of two conditions is satisfied as K is increased, where L It is also clear from the presentation of the model that log^ = log^' only when either Pi = 0i = 0 or & = /3 a #= 0 and the central moments of X x and X t are identical. The first situation may be approximated if the correlation between X, and Y f is small and a summary (-4)

3 Combined estimators of the odds ratio 193 statistic is desired. In this case log^r t =^=log0' also. If, however, the aim is to reduce the effect of Xj, then the correlation, and hence the values of y8 x and /3, will clearly be nonzero and relatively large numerically. In the second case, identical covariate means can generally only be assumed in an experimental situation when two treatments are randomly assigned to subjects within one homogeneous population, and not in observational studies involving two distinct populations. Moreover, a model in which /Jj = j9 and E(X 1 ) = E(X i ) implies a zero second-order interaction, which may pertain in an experimental situation, but seldom, if ever, for observational data. The discussion thus far has focused on the log of the odds ratio rather than the odds ratio itself. The reasons for this are two-fold. First, log ip is a linear function of X and, therefore, direct estimates of this value from unstratified samples will be unbiased. If we assume, for simplicity, that j3 x = /? = /? in (-3) and define T = 6, where 6 = log^, then where E(b 0 ) = (a^-aj), E^) = fi and e r ~#(O,CTf,)- Clearly E(T) = 6. However, it can be easily demonstrated that e T is a biased estimator of ip. Given that we are interested in estimating the constant component, equivalent to ip', it seems preferable to consider estimators in a scale which produces at least one unbiased estimator, T. Secondly, the main part of this investigation involves the comparison of Monte Carlo values. Provided the number of samples generated is sufficiently large, these values should be good approximations to the corresponding expected values of the estimators. However, the assumption is implicitly made that the distribution of an estimator is approximately normal so that the Monte Carlo mean is an unbiased estimate of the expected value. Now, the distributions of the estimators of the odds ratio tend to be noticeably positively skew unless the samples are very large, while the distributions on the log scale are approximately normal, even for moderate samples. To use estimators of the odds ratio itself in a Monte Carlo study would, therefore, distort the estimates further by introducing an unknown distortion due to the skewness of the distribution. To study the relationship between logi/r', log tp and L*, values of log^t were first computed using (-3) with /? x = /5 = /?, and X j ~N(jiprf). To provide a wide range of situations, two values were used for each of log tjj', /} and (ji v of; /j. t, of), yielding eight different versions of the model; see Table 1 for these values. Within each of these, four values of K were considered (1,,5,10). Of course, for K = 1, log^r fc = log0, which is the parameter for unstratified samples. To demonstrate the positive bias of the odds ratio mentioned above, values of ip k were also calculated, using the exponent of (-3) for the ith individual in population 1 and the /tth in population and integrating over the range of values for each stratum. The strata were denned with the Monte Carlo investigation in mind. For K = 10, a k~ a t-i = 3*0 in all strata with the exception that a 1 a 0 = a 10 a i = co. The central value, a s, was set at 4, given the values of ^ and /i chosen for the investigation. To form the strata for K = and 5, adjacent strata were then combined. An unweighted arithmetic mean of the \ogifi k was used as a value for L*. The use of equal stratum weights has been investigated by Cox (1957) and found to be nearly as efficient as optimal weighting for a normally distributed variate, so that this simple function would appear to be a reasonable choice. However, the fact that equal weight is given to the extreme strata, with infinite boundaries, reduces the effectiveness of this function, so that the L* as defined in this paper cannot be considered optimal. Similarly, ip* was defined as the unweighted geometric mean of the ip k. The values of log ip', log tp and i* as well &sip',<p and <p* are given in Table 1 for each of the eight situations defined above. For equal covariate means, (i x = fi % = 4, and given that

4 194 SONJA M. MCKINLAY Pi = Pi> no second-order interaction is present, so that a priori we expect log^ = log^'. Moreover, because the strata are symmetrical about the covariate means, and because L* is a simple average, L* log^r' also. For unequal covariate means, there is some second-order interaction, especially for the higher correlation, /} = 0-10, because the value of differs considerably between strata. Table 1. Expected values of the odds ratio and the log odds ratio for unstratified and stratified samples, compared toith the constant component, if/' or log^r', for different sets of covariate distributions and values of /? Number of strata Log odds ratio logf log ^4 L* L* L* Odds ratio \lf V* Covariate distribution, (/x lt o\; fh> l) (4, 64; 4, 36) (8, 64; 0, 36) P = A The relationship (-4) holds consistently for all the situations considered in Table 1. Moreover, for unequal covariate means and across all situations, for the odds ratios themselves, the combined value approaches the constant term, L*-*logifi', or \fi*-*-i]t', as K is increased. In other words, increasing the number of strata decreases the bias. The Monte Carlo study described in the following sections compares the approximate expected values of the logarithms of five well-known combined estimators of the odds ratio with the values of L* and log ifi' given in Table 1 in order to assess their usefulness as summary measures. 3. THE ESTIMATORS Several approximate statistics have been derived for combining the within-stratum estimates, assuming the odds ratio is constant across strata, ifj k = ip', for all k. Four of these are relatively well known (Woolf, 1955; Mantel & Haenszel, 1959; Birch, 1964; Goodman, 1969) and have been the subject of recent investigations by Gart (196, 1970, 1971). Fleiss (1973, Chapter 10) has provided a good general description. Woolf's estimator, with modifications by Gart & Zweifel (1967), and those of Mantel & Haenszel, Birch and Goodman are given below, with the simple estimator for unstratified samples, in the notation used by Birch and others. Let n iik represent the ith response (t = 1,) in the jth population {j = 1,) and the &th stratum (k = 1 K). Summation is represented by a period in the usual way. The unstratified estimator of log <ji, from two independent samples n #1, and n A is simply

5 Combined estimators of the odds ratio 195 A constant of $, added to each of the four frequencies, has been suggested by Haldane (1966) to reduce the bias in estimating log^r and remove the possibility of an infinite value. This correction was not used here because the overall frequencies were never zero, and, given the generally large samples, the addition of this constant would have had a negligible effect on the estimate or its variance. However, for smaller samples the correction is certainly advised for both the log odds ratio and its variance. Woolf (1955) combined the logarithms of the observed within-stratum odds ratios, using as weights the inverses of the asymptotic variances. Because the within-stratum variances are a simple function of cell frequencies, this estimator is equivalent to a weighted arithmetic mean of the within-stratum odds ratios. The inclusion of the constant,, in both L k and w k provides the least biased estimates of both the difference in logits and its variance as shown by Gart & Zeifel (1967): L v = p k L k where Mantel & Haenszel (1959) used an indirectly weighted sum of the odds ratios themselves, combining the numerators and denominators separately. As noted in their paper, these weights are of the same order as the appropriate weights for a difference constant on the logit scale (Cochran, 1954; Radhakrishna, 1965). In order to provide a comparable estimator, the logarithm of this statistic was used in this investigation: 4»A = log {S^u TWnjtVS^nmnWn.*)}. More recently Birch (1964) proposed an approximation to the maximum likelihood estimator, valid if ip = ifi'^ 1, using the first two terms in the expansion o E(n llk \ip'), a method proposed earlier by Cox (1958), considered valid under the hypothesis of independence: Goodman (1969) subsequently modified L b for use when ifi = ifi' 4= 1, through the inclusion of further terms in Cox's expansion and obtained L a = {(G + 3M) L b - HOL\ + H}/(0 + BM- H*), where The conditional maximum likelihood estimator, suggested by Gart (1971), requires an iterative solution, and a computer routine has been developed (Thomas, 1975) which calculates both exact and asymptotic values. The conditional estimation of the constant difference uses a considerably simplified form of the likelihood, assuming that the second-order interaction is zero, ip k = ip'. The standard error of the estimate of the odds ratio should be smaller than its unconditional counterpart. As a further point of interest in this investigation, the first approximation used in the iteration is ^;ma. The logarithm of the asymptotic version, tyani in Gart's notation, will be used in this paper because it is equivalent in most instances to the exact estimator to at least the second decimal place. The notation L M will be used in this paper for the logarithm of the asymptotic maximum likelihood estimator.

6 196 SONJA M. MCKIPTLAY 4. MONTE CABLO COMPABISON The model defined in (-1) and (-3), with parameter values given in Table 1, was used for a Monte Carlo study to compare the effectiveness of the five estimators outlined in the previous section, relative both to each other and to the corresponding expected values summarized in Table 1. Parameter values, stratum sizes and numbers of strata remained as for Table 1. The values were so chosen that, besides generating situations similar to those encountered in practice, the samples almost never involved zero marginal totals in the stratum tables. This ensured comparable sampling situations for all the estimators. Sample sizes were determined from the results of a preliminary investigation using the odds ratios, in which it was found that the distributions of the estimators were affected by both relative and absolute sample sizes (McKinlay, 1975). The following four pairs (n L > n.t) f r total sample sizes were chosen; (n^,n_ a ) = (1,1); (1,5); (5,1); (5,5). The second pair (1,5) represents the most inefficient combination, with the smaller sample selected from the population with the larger covariate variance. The sampling schemes were chosen to approximate most closely the practical situations as outlined in the introduction, (a) the combining of estimates from strata of size fixed during the design and (b) the combining of estimates from strata, determined at the analysis, in order to reduce the effect of covariables. For scheme (a), the stratum samples (w^-.n^) are considered fixed for all k and contribute nothing to the variability of the estimator. The second scheme allows for variable stratum samples depending on the distribution of the covariate in the original sample. This introduces a variation component which may be considerable for those estimators which use stratum samples primarily as weights. Cochran (1954) and Radhakrishna (1965), for example, have noted that an assumption of constancy of the difference between proportions on the logistic scale implies the use of weights dependent on stratum sample sizes only. The second sampling design was most easily adapted to Monte Carlo methods, and it was found that the general distribution of units across strata remained comparable among samples generated, although actual numbers within strata varied. The first scheme required prior specification of the n jk. A simple, but somewhat extreme design was chosen, with n ik = 0-ln f, for all k (K = 10) and samples generated separately within strata. This meant a slight change in stratum boundaries to increase the probability of obtaining values in the extreme strata, a modification which had a negligible effect on the expected values L* while reducing computing time to practicable limits. The number of samples generated for each scheme was, fixed stratum size, and 10, variable stratum size. These numbers were determined by both the computing time required and the reproducibility of estimates. Preliminary simulations using these numbers produced estimates which did not differ from each other by more than 0-04, for n a = 1, and 0-01, for n L = 5, with comparable stability for the root mean squared error, indicating reasonable stability of the Monte Carlo results. 5. RESULTS Before considering results for the two sampling schemes, some general remarks are in order concerning the Monte Carlo values obtained. A comparison of the combined estimators yields the following relationships, consistent through all the results. First, the logarithm of the maximum likelihood estimator, L^, is in almost all instances equivalent to, or more biased and less precise than, the logarithm of Mantel & Haenszel's statistic, used as its initial approximation; the same relationship holds

7 Combined estimators of the odds ratio 197 for the odds ratios themselves. This means that, for the models considered here, even in the absence of second-order interaction, fi nil is never preferable as an estimator of tft', the simply computed ^r mh being always at least as good, in terms of both bias and precision. Secondly, the two approximations L b and L g exhibit remarkably similar trends, with few exceptions, even for log \\>' + 0. In those few instances where these are the least biased estimators, therefore, the extra computation involved in Goodman's modification is seldom justifiable. Thirdly, Woolf 's estimator, while usually one of the most precise, shows the greatest sensitivity to sample size, and marked increases in bias for larger K. Increasing the number of strata also produces some consistent trends among the estimators. As expected from prior work (Cox, 1957; Billewicz, 1965; Cochran, 1968; McKinlay, 1975), K = 5 results in near minimal residual bias, the gains in using more strata being negligible or even negative. In other words, in many situations, increasing the number of strata merely reduces the variation, while leaving the bias unchanged or even increasing it. The first sampling scheme, fixed within-stratum samples, produced generally precise estimates with negligible bias for equal covariate means, representing no second-order interaction. This was anticipated given the expected values in Table 1 and the use of fixed, equal, stratum weights in the form of the constant within-stratum samples. For unequal covariate means, the biases of the various estimators followed consistently the same pattern as for the second sampling scheme, to be discussed below. The only difference was that in all cases the biases were smaller, relative to log^r'. None of the results for this sampling scheme will, therefore, be considered in detail here. The second sampling scheme, fixed total samples only, is, perhaps, of more interest and wider applicability with considerable variation in within-stratum sample sizes, both between and within the samples generated. Table presents a summary of results for equal covariate means, equivalent to no secondorder interaction, and for j3 = 0-10 only. The results for = 0-05 were equivalent, with smaller biases and somewhat greater precision, and are therefore not presented in detail. For log ip' = 0, any bias, both in L o and in the combined estimators was generally negligible, with the outstanding exception of L w which showed large biases for unequal sample sizes. The dependence of this estimator solely on n afc and n ^ as stratum weights, combined with the use of the constant,, dampened the positive bias of the uncorrected odds ratio too severely in the inefficient case, samples of (1,5), and not enough in the converse situation, sample of (5,1). For a moderately large odds ratio, log^' = or rp' = 4-05, L w again exhibited marked instability, with a large negative bias, increasing with K for n^ = 1. The negative bias remained but was much less marked for n L = 5. Even though the odds ratio was not near unity, L b and L g had identical means and standard errors for equal samples, n A n A. For samples (1,5), both estimators had negative biases, particularly L b, while for samples (5,1), L b showed an equally large positive bias, and the negative bias of L g increased. In other words, both of these estimators exhibited considerable instability and biases at least as marked as those of L w in some situations. The only estimators which appeared to be almost unbiased and stable for different samples as well as increasing strata, were L mh and L^. Moreover, as noted above, because the difference between them appeared to be negligible, both in bias and in standard error, L ma would appear to be the preferred estimator in most situations in which negligible second-order interaction can be assumed. In Table 3 corresponding results are summarized for different covariate means, which implies that second-order interaction is present. This was most clearly detected for j3 = 0-10,

8 198 SONJA M. McKlNLAY Table. Means and standard errors from 10 samples for the simple log odds ratio and five combined estimators, for equal covariate means and fixed total samples, sample sizes (n^n % ), K strata, [(^.of; ^o») «(4,64; 4,36),j3 = 0-01] Expected value Samples Estimator 1, 1 L o L w L b Zi g 1,6 5,1 5,5 Expected 1,1 1,5 5, 1 6,5 A«J A> L w Jjw, & L o L w L g L mjt A* A) L m L b h t Am» Art value A) L m A. A»J Ai L m L b L mh A.i Ai A. A A.» A A A. A«* = 1 (0-60) (0-38) (0-43) (0-3) 1-4 (0-44) 1-6 (0-38) 1- (0-3) 1-3 (0-4) K = = log^r Oil -Oil -Oil -Oil (0-17) (0-48) (0-48) (0-51) (0-51) (0-39) (0-39) (0-38) (0-39) (0-39) (0-36) (0-38) (0-40) (0-44) (0-44) (0-3) (0-3) (0-3) (0-3) (0-4) log./.' = 119 (0-44) 118 (0-40) 118 (0-40) 1-18 (0-43) 1-30 (0-43) 118 (0-39) 0-99 (0-45) 113 (0-36) 1-31 (0-37) 1-30(0-37) [-9 (0-9) 1-59 (0-44) 1-0 (0-40) 7 (0-31) 8 (0-31) 1-6 (0-) 1-0 (0-5) 1-0 (0-5) 1-9 (0-1) ] 8 (0-1) = K == 6-1 (0-38) - (0-48) -0-0 (0-48) - (0-51) - (0-51) -0-4 (0-39) (0-38) -8 (0-37) -0-0 (0-39) -0-0 (0-39) -0-1 (0-38) (0-39) -3 (0-40) (0-4) (0-4) -0-0 (0-0) (0-) -3 (0-) -3 (0-) (0-) 1-17 (0-4) 1-3 (0-36) 1-5 (0-36) 1-39 (0-44) 1 (0-45) (0-4) (0-40) (0-3) (0-38) (0-37) 1-38 (0-7) 1-68 (0-48) 107 (0-35) 1-36 (0-9) 1-37 (0-9) 1-3 (0-19) 1-5 (0-1) 1-6 (0-0) 1-37 (0-19) 1-38 (0-19) K = (0-46) (0-38) (0-38) 0- (0-39) 0 (0-40) (0-3) (0-49) (0-49) (0-6) (0-54) (0-48) (0-39) (0-40) (0-4) (0-43) (0-19) (0-) (0-) (0-) (0-) 1-05 (0-47) 1-5 (0-37) 1-6 (0-37) (0-47) 8 (0-49) 0-9 (0-55) 107 (0-39) 1- (0-31) (0-39) 3 (0-39) 1 (0-5) 1-68 (0-49) 1-08 (0-34) 1-38 (0-30) 0 (0-30) 1-9 (0-0) 1-6 (0-0) 1-8 (0-19) 1-39 (0-19) 0 (0-19)

9 Combined estimators of the odds ratio 199 Table 3. Means and standard errors from 10 samples for the simple log odds ratio and five combined estimators, for unequal covariate means and fixed total samples, sample sizes (n±,n±), K strata, [(^.of; ^,a\) = (8,64; 0,30),/3 = 0-01] Expected value Samples Estimator 1, 1 L o 1,5 5, 1 5,5 Art Art A, A, Art Art A, Expeoted value 1, 1 A, A> 1, 5 5, 1 5,6 L b A, Ai* A. A, Art K = (0-89) 0-7 (0-85) 0-64 (0-7) 0-68 (0-71) (0-79) -01 (0-75) 1-99 (0-64) - (0-63) K = (0-55) 0-3 (0-57) 0-4 (0-59) 0-6 (0-63) 0-6 (0-6) 0-14 (0-41) 0- (0-44) 0- (0-48) 0-8 (0-54) 0-8 (0-54) 0-4 (0-4) 0-17 (0-41) 0-17 (0-39) 0-15 (0-40) 0-16 (0-40) 0-1 (0-31) 0-0 (0-31) 0-0 (0-31) 0-1 (0-31) 0-1 (0-3) logf (0-43) 3 (0-35) 4 (0-35) 1-6 (0-65) 1-64 (0-55) 7 (0-39) 1- (0-8) 1-35 (0-6) 63 (0-50) 64 (0-50) 57 (0-3) 76 (0-49) 36 (0-19) 54 (0-3) 1-56 (0-3) 1-58 (0-7) 6 (016) 6 (0-16) 1-59 (0-8) 1-60 (0-9) = (0-43) 0-07 (0-54) 0-07 (0-65) 0-10 (0-58) 0-09 (0-58) -0-14(0-38) 0-03 (0-43) 0-03 (0-44) 0-08 (0-48) 0-08 (0-48) 0-4 (0-39) 0-0 (0-36) 0-0 (0-36) 0- (0-37) 0- (0-37) 0-06 (0-3) 0-04 (0-4) 0-04 (0-4) 0-04 (0-4) 0-04 (0-5) (0-40) 1-9 (0-37) 1-31 (0-37) 8 (0-6) 1-51 (0-5) 1-16(0-41) 1-1 (0-37) 1-5 (0-33) 7 (0-46) 8 (0-46) 6 (0-7) 1-68 (0-38) 1-8 (0-4) (0-9) 4 (0-9) 1 (0-0) 1-30(0-18) 1-33 (018) 5 (0-) 6 (0-) K = (0-38) 0-04 (0-55) 0-04 (0-55) 0-07 (0-59) 0-07 (0-60) -0-6(0-40) 0- (0-44) 0- (0-44) 0-05 (0-48) 0-06 (0-48) 0-40 (0-48) (0-36) (0-36) -0-03(0-37) -0-03(0-38) 0-04 (0-) 1 (0-4) 0-01 (0-4) 0-0 (0-4) 0-0 (0-5) (0-41) 1-7 (0-38) 1-9 (0-38) 6 (0-53) 1-5 (0-65) 0-97 (0-51) 1-10 (0-38) 1-3 (0-34) 4 (0-47) 6 (0-47) 6 (0-5) 1-65 (0-36) 1-7 (0-5) 1 (0-9) 4 (0-30) 1-34 (0-0) 1-8 (0-0) 1-31 (0-19) (0-) 4 (0-)

10 SONJA M. MCKINLAY but, as for ^ /xj = 4, in Table, the results for js = 5 were essentially equivalent and are omitted here. The first point to note is that the bias of the unstratified estimator was of the same order as its standard error, at least 80% of the standard error in all cases. Moreover, even the most inefficient estimators removed at least 80% of this large initial bias in absolute terms for K = 5, with the exception of L w for samples (1,5). For \ogifi' = 0, the results were equivalent to those already discussed for Table. Only L w retained a large bias which increased with K. Of the remaining four estimators, L b and L g performed marginally better than L,^ and L^, although the differences in both bias and precision are so small that these four could be considered equivalent. For iogip' =, the estimators again showed trends similar to those evident in Table. The only notable change was in Goodman's estimator, L g. This statistic appeared to be least biased and clearly the most precise for all four sample pairs, provided K =. As in Table, L b and L g were equivalent for n x ==» a. Unfortunately, for K>, the bias in both L g and L b increased rapidly in absolute value, generally in a negative direction. In contrast, L rroi and L^ steadily removed bias with increasing K, and with reasonable precision. For log^' = and unequal covariate means, the differences between these two estimators were largest, with L^ always being less biased than, and of equivalent precision to, Lnj. Table 4 summarizes the Monte Carlo findings by giving the optimal statistics, in terms of both bias and precision. Where two or more estimators were equivalent or nearly so, that one involving the least computation or greatest precision is given first, with the best alternative in parentheses. Samples 1, 1 1,5 6, 1 6,5 Table 4. Optimal estimators for different sample sizes strata, K, and parameter values K > > > > (A) Equal covariate means (Hi = /A,, = 4-0) log (^') = log(f)= w ()* MTT W(B) B () (W) w () w () w () MTT (n^.n^), (B) Unequal covariate means 80, /X, = 0-0) log (if,') = 0-0 log (</.') = 1- B B W(B) B MTT B () B () B () w () MTT o () MTT o () * The estimator in parentheses, while not the least biased and most precise, provides a closely competing alternative. Estimators: B, Birch; G, Goodman;, Mantel-Haenszel; w, Woolf. MTT Some of the Monte Carlo findings were unexpected. The phenomenon of a maximum likelihood estimator being no less biased nor more precise than its initial approximation, even when the assumption of no second-order interaction was met, is difficult to explain satisfactorily, although the closeness of these two estimators has been observed by Gart in the analysis of real data sets (Gart, 1970, 1971). Another unexpected finding is the variable performance of Goodman's modification of the null hypothesis estimator, first proposed by Birch. As expected, for log >(i' = 0, both statistics have almost identical means and variances. For log tfi' 4=0, however, Birch's statistic does surprisingly well, provided the samples are approximately equal and second-order inter-

11 Combined estimators of the odds ratio 01 action is absent or negligible. Only for marked second-order interaction does L g clearly outperform L b for unequal samples and, even then, is optimal only for K =. Although generally one of the most precise, Woolf's estimator exhibited considerable instability in its mean value, with rapidly increasing bias as stratification was increased, a phenomenon probably due, as noted earlier, to the addition of to cell frequencies, and consistent with prior results on both simulated and real data (McKinlay, 1975; Gart, 1970). For this reason, L w should be avoided as an estimator, especially for K>. Mantel & Haenszel's statistic, formulated under the assumption of some second-order interaction being present (Mantel & Haenszel, 1959, p. 735) is consistent in reducing bias, with adequate precision in all situations considered. Certainly, for large samples and for K > this is the estimator of choice, whether or not the odds ratio is constant. Unfortunately, no usable estimate of variance is available for this simply computed statistic. However, from the Monte Carlo results presented here, it seems that, provided at least one of the samples is larger than 1 and log^'4=0, the variance estimator for L w of \[L k w k, from 3 provides an approximate estimate of variance for L ma. Finally, note that the results are presented on the log scale. When Monte Carlo values were computed for the odds ratio estimators themselves, the relationships among the statistics remained, but with increased positive and decreased negative biases, as well as more widely varying precision. This was to be expected from the discussion in and the values of Table 1. Thanks are due to Nathan Mantel, John Gart and editor and referees for helpful comments on earlier drafts, as well as to Donald Thomas for making available his program for the calculation of the conditional maximum likelihood estimator. REFERENCES BILXEWICZ, W. Z. (1965). The efficiency of matched samples: An empirical investigation. Biometrics 1, BIBCH, M. W. (1964). The detection of partial association. I: the x case. J. R. Statist. Soc. B 6, COCHBAN, W. G. (1954). Some methods for strengthening the common ^* test. Biometrics 10, COCHBAN, W. G. (1968). The effectiveness of subclassification in removing bias in observational studies. Biometrics 4, Cox, D. R. (1957). Note on grouping. J. Am. Statist. Assoc. 5, Cox, D. R. (1958). The regression analysis of binary sequences. J. R. Statist. Soc. B 0, FLEISS, J. L. (1973). Statistical Methods for Rates and Proportions. New York: Wiley. GABT, J. J. (196). On the combination of relative risks. Biometrics 18, GABT, J. J. (1970). Point and interval estimation of the common odds ratio in the combination of x tables with fixed marginals. Biometrika 57, GABT, J. J. (1971). The comparison of proportions: A review of significance tests, confidence intervals and adjustments for stratification. Rev. Inst. Int. Statist. 39, GABT, J. J. & ZWEIFEL, R. (1967). On the bias of various estimators of the logit and its variance with application to quantal bioassay. Biometrika 54, GOODMAN, L. A. (1969). On partitioning x 1 and detecting partial association in three-way contingency tables. J. R. Statist. Soc. B 31, GOODMAN, L. A. & KBTJSKAI, W. H. (1959). Measures of association for cross classifications II: further discussion and references. J. Am. Statist. Assoc. 54, HALDANE, J. B. S. (1956). The estimation and significance of the logarithm of a ratio of frequencies. Ann. Hum. Gen. 0, MANTEL, N. & HAENSZEL, W. (1959). Statistical aspects of the analysis of data from retrospective studies of disease. J. Nat. Cancer Inst., MCKINLAY, S. M. (1975). The effect of bias on estimators of relative risk for pair-matched and stratified samples. J. Am. Statist. Assoc. 70, RADHAKBISHNA, S. (1965). Combination of results from several x contingency tables. Biometrics 1,

12 0 SONJA M. MCKINLAY THOMAS, D. G. (1975). Exact and asymptotic methods for the combination of x tables. Computers and Biomed. Res. 8, Woou, B. (1956). On estimating the relation between blood group and disease. Ann. Hum. Gen. 19, YULB, G. U. (191). On methods of measuring association between attributes. J. R. Statist. Soc. 75, [Received June Revised September 1977]