COMPUTERS

AND

improved

BIOMEDICAL

25, 75-84 (1992)

RESEARCH

and Extended Exact and Asymptotic the Combination of 2 x 2 Tables DONALD

Biostatistics

Branch.

G.THOMAS National

Cancer

for

J. GART

ANDJOHN Institute,

Methods

Bethesda,

Maryland

20892

Received December 3, 1990

Exact and approximate methods for analyzing the common odds ratio in the combination of 2 x 2 tables are programmed. The approximate methods are improved by incorporating bias and skewness corrections in testing and estimation. Exact methods are done more efficiently by employing network theory. This makes more feasible the exact analyses of sparse data, large numbers of 2 x 2 tables each based on small numbers. Additional tests of interaction, particularly for sparse data, are added. Another feature is the point and interval estimation of the attributable risk. The various aspects of the program are illustrated in three numerical examples. An executable version of this program, for IBM compatible PCs, requires about 3 14K for up to 500 tables. It is available from the authors upon submission of a PC diskette formatted with MS-DOS. o 1992 Academic PKSF. IX.

1. INTRODUCTION This program extends and improves on the analyses of the combination of 2 x 2 tables given in Thomas (I). The approximate (or asymptotically justified) methods are improved by incorporating the corrections for bias and skewness given by Gart (2). The exact analyses are performed more efficiently by using modified network theory of Mehta, Patel, and Gray (3), particularly as it applies to sparse data, that is, large numbers of 2 x 2 tables each based on very small numbers. Vollset, Hirji, and Elashoff (4) have recently shown how this graph theoretic analysis may be more simply formulated in algebraic terms. Numerous improvements which save computer time and facilitate analysis of larger datasets on a microcomputer are employed in the exact analysis. Some of these are: 1. automatically optimizing the order in which the 2 x into the network analysis, 2. eliminating redundant vectors, making more efficient 3. exploiting the cyclical nature of the computations so portion of memory is required for a given problem, and 4. computing an exact conditional test of the hypothesis 75

2 tables are entered use of memory, that a much smaller of equal odds ratios OOlO-4809/92 $3.00

Copyright 0 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.

76

THOMAS

AND

GAKl

over strata (no interaction) using efficient direct enumeration methods. This is computationally a more difficult problem than testing for no main effect. Seth of tables with equal marginal totals (when they exist) are treated as one table. This is more efficient than previously published methods see Mehta. Patel. and Wei (51, also Thomas (I). New approximate methodology for additional aspects of analyses iz also included. Tests of the validity of summing over tables, or pooling the data. due to Armitage (6), are added, see also Gart (7). Two additional tests of homogeneity of odds ratios, or interaction, are given. One groups the 2 x 3 tables by ;I covariable such as age, and tests for variation in the odds ratio over this variable. The second, which applies only to matched stud& with tables of a single case or test subject versus a constant number of controls. is due to Ejigou and McHugh (8). see also Gart (9). Finally when the 2 ) 2 tables arise from stratifying population based case-control studies. the program give4 two methods of point and interval estimation of the attributable risk. see Gart and Thomas (10). This program may be used to compare simple or stratified proportions of an)’ kind arising from independent binomial distributions. The corrected maximum likelihood estimators of the odds ratios and their associated confidence intervals are more efficient than those based on the Mantel-Haenszel estimators. In the context of epidemiologic studies, the corresponding point and interval estimators of the attributable risk are also to be preferred to those based on the Mantel-Haenszel estimator as well as those based on making no assumptions of a common odds ratio. An executable version of this program, for IBM compatible PCs. 13 available from the authors upon submission of a PC diskette formatted with MS-DOS. 7. NOTATION

AND METHODS

For the most part we shall consider J independent binomial variateb. .\!,. based on sample sizes, n, ;, with parameters, pi,, for i = 0. 1 and,j = I. 2. J. Usually a common odds ratio is assumed. $, = (p,Ic~,~,i)i(poiq,,) -$. for allj. where y,i = I - plj for all i and j. We find point and interval estimates of I,!Jas well as a test of no main &tct, that is, $ = I. Tests of the model, $, = $, that is. $-homogeneity or 170 infev~~ction in the sense of Bartlett, are also considered. A test of whether the tables may be legitimately pooled, called p-homogeneity by Gart (II). is a test of the null hypothesis: p,, = p, and p,), = I-I,,. for all j. When considering the estimator of the attributable risk, N. the FZ~,‘s are assumed to be multinomial variates based on a fixed sample size II, . the total number of cases in a population-based case-control study. In these analyses. $-homogeneity may or may not be assumed to be valid. Exact analyses are based on permutations with fixed marginals. Exact intervals for $J and exact tests of JJ = 1 are based on permutations of generalized multivariate hypergeometric distributions based on conditioning on x0, + x,, = s , , for all j. Interaction or $-homogeneity tests are additionally conditioned on

COMBINATION

77

OF 2 x 2 TABLES

Cj~,j = x, . Because of the discrete character of the problem the results are exact in the sense that the exact confidence coefficient is always greater than or equal to 1 - E. the nominal level. 3. CORRECTED

TESTS AND ESTIMATION

OF THE COMMON

ODDS RATIO

This program reproduces the asymptotic or approximate analyses given in Thomas (1) derived from Gart (12). It goes on to give corrected analyses which adjust for bias of the point estimators to the order ~~7’ as well as correcting the main effect test and confidence intervals for skewness to order ti(;“?. As has been widely noted, see, e.g., Breslow (13). the unconditional maximum likelihood estimator of I,!J, when I/J # 1, is typically biased away from unity. McCullagh (14, and subsequently Levin (15), arguing from the conditional distribution, find a bias correction for $. Using Bartlett’s general theory for the unconditional theory, Gart (2) finds a similar bias correction which was subsequently found to be virtually identical with Levin’s correction, see Gart (16). Levin and Kong (I 7) show that this bias correction is a saddlepoint correction to the conditional distribution. Although this correction is to the order n(~ ’ , Breslow and Cologne (18) show that it also corrects for bias for sparse data locally, that is, for $ not far from unity. This bias correction is employed in point and interval estimation as well as tests of interaction. As the bias correction vanishes for $ = 1, it does not affect the main effect test. A skewness correction, based on Bartlett’s general theory, is derived in Gart (2). This is employed in conjunction with the Cornish-Fisher expansion to find corrected confidence intervals and tests for r/~. Unlike the bias correction, the skewness correction does not necessarily vanish for $J = 1 unless n,j = n,j for allj. The skewness corrected methods, particularly for small sample sizes and high confidence coefficients such as 99%, may not always converge. In such instances the exact analyses are preferred. A second test of interaction is added for the case where the tables may be grouped by a covariable, see Breslow and Day (19, p. 173). If the covariable, say age, has K levels then the resulting approximate x2 which employs the bias correction has K - 1 degrees of freedom. 4. TESTS FOR POOLING

Other added features of the present program are tests of p-homogeneity, i.e., = 17 2, . . . 3 J. This is tantamount to assuming that stratification is not needed, and the data may be legitimately summed into a single table. Two tests, both due to Armitage (6), are given. One is based on Pearson’s xz, and the second, xt, is a quadratic form. Gart (7) shows the two statistics are identical when sample sizes are balanced, i.e., noj = ~n,~, for c a constant for all j. It should be noted that even when p-homogeneity holds, the stratified or multiple table analyses are not necessarily less efficient than the pooled analyses, except for sparse data. However, incorrectly pooling data from balanced

Plj = Pi, POj = PO, forj

78

THOMAS

AND TABLE

GAR’I I

ANALYSES OF A BIOASSAY IN MICE OF AVADEX Strain-sex

Controls

A-M A-F B-M B-F

Stratified

211338

analyses

Approximate

Common odds ratio. 4 Test of main effect J, = 1 (one tailed) Tests of interaction Overall: x’ (3 d.f.1 Covariable A vs B x’ (I d.f.) (cont. corrected) Confidence limits: + 95% 99%

x; = 7.169. xi = 7.171.

Analyses

4.933 3.000 1.286 I.881 -.

I Ii65 Bias and/o1 skewness corrected

EXXl

3.093 Z = 2.628 P = .0042Y

3.049 Z =- 2.408 P = .00801

P

,007 IX

x2 = P =

yJ -P =

.X651 .8338

P

.9379

$ =

.‘4Sl

P =

.6205

.864Y .833Y

:.I)48

I.239 5 I/, 5 7. I73 .92X c 4 P 9.085

I.243 5 JJ 5 7.131 ,944 5 t$Jl-5 Y.OSI

Bias and/or skewness corrected

Exact

d.f.)

P = .0667 P = .0666

of pooled

results

Odds ratio, I$ Test of main effect (one tailed) Confidence 9% 99%

416 2116 ‘t/I8 l/IS

1.294 5 $ 5 7.311 1.025 5 1 5s 9.241

1e.st.t for possiblepooling(3

TUMORS

Estimated odds ratios

Treated

5179 3187 I0190 3182

Pooled

FOR PUI.MONARY

limits:

Approximate

3.064 2 = 2.436

3.064

3.075 Z = 2.671 P = .00378

P = .00743

P -=

1.304 5 I/I 5 7.162 I.037 5 l) 5 9.019

1.249 5 s,ll5 7.129 ,936 5 +!I 5 9.081

I.250 I- rl, 1 7.102 ,957 5. rir c 8.988

.0065Y

C

designs leads to pooled estimators unity, see Gart (II). 5. EXAMPLE

of the odds ratio which are biased toward

I: MODERATE

SAMPLE

SIZES

Consider the data in Table 1, see (I), on pulmonary tumors in four sex-strain combinations of mice wherein the test animals were fed a fungicide. The sample sizes are moderate, varying from 15 to 90. The approximate, corrected. and

COMBINATION

OF 2 x 2 TABLES

79

exact analyses are given in the three columns of Table 1. It is seen that the bias-corrected estimator of $ is virtually identical to the conditional maximum likelihood estimator to four significant figures. The skewness-corrected test for main effect has a P-value quite close to the exact result. The 95% corrected confidence limits are virtually identical to the exact limits. The tests of p-homogeneity are marginally significant, but the analyses for the stratified data differ little from those of the pooled data. Clearly the corrected analyses are adequate for this example without the need to employ exact methods.

6. SPARSE DATAORMATCHED

STUDIES

When the data consist of a large number of tables, each based on very small numbers, the usual approximate analyses are often unreliable. Typically the point and interval estimators are biased away from unity and the x2 statistics for the tests of interaction are consequently unreliable. Only the test of no main effect is usually reasonably accurate. The corrected analyses are usually a considerable improvement, but not necessarily adequate unless I,!Jis not too far from unity, 6 < JI < 5, see (18). For matched studies with a single case or treated subject versus a fixed number of controls per stratum, say R, the program also can compute the test for interaction of Ejigou and McHugh (8); see also (9). This is based on the bias corrected estimator and is an approximate x2 variate with R - 1 degrees of freedom. Both this test and the one based on a covariable are more reliable for sparse data than Bartlett’s general interaction test.

7. EXAMPLE 2: SPARSE DATA, 1 TO R MATCHING Table 2 gives data of a matched study of endometrial cancer and estrogen use among women over 55 years of age, see Breslow and Day (19, p. 175). Each of the 63 cases has four matched controls and a covariable of age divides the data into three groups: 55-64, 65-74, and 75 or more. It is seen that the approximate point and interval estimates of the odds ratio greatly exceeds the exact conditional estimates. This in turn results in a highly significant approximate test of interaction (P - .OI), while the exact test yields P = .7195. Only the approximate test of $ = 1 is reasonabIy accurate. The corrected analyses are a great improvement on the approximate analyses. All these tests of interaction are not significant and the point and interval estimates are reasonably close to the exact results. Although the exact method is preferred here, where $ is far from unity, when I,!J is closer to unity the corrected analyses may be adequate even for sparse data such as these. As the design is balanced the tests for pooling are identical, but not significant. Once again it is seen that poohng has little effect on the width of the confidence intervals.

X0

AND GAKT

THOMAS

TABLE ANALYSES

OF A MATCHED

CASE-CONTROL STUDY OF ENDKOMETRIAL ESTROGEN USE

Covariable -age

Control case 0 I Case 0 1 Case 0 I

50-64 65-74

75 + Pooled

Controls:

Matched

1171252:

Cases.

analyses

Common odds ratio, Test of main effect I,!I = I (one tailed) Tests of interaction: Overall x2 (57 d.f. I Ejigou-McHugh

4 z p

Confidence

13.677 5.438 .27

= =

y’ = 60.17 P .ihl6 k2 = 3.92

-

Analyses

5 $4 3.809 5 rl, 5

Limits:

4 I 2 0 2 0 I

Exact

Py

7195 -

.26Y8 .7906

.6735

39.927 52.923

3.709 3.030

c IfI c- 22.7x0 5 J 5 32.795

3.431 2.772

‘; rl, c 21.546 5 II! 5 3n.i.s.5

(62 d.f.) P = .I461

results

Common odds ratio. Test of main effect 4) = 1 (one tailed) Confidence 95%’ 99%

:

x: =

p = 4.981

of pooled

: (1 2 I IO 0 i

$

95% 99%

Tesrs fix pooling ~1 = xi = 73.739.

2 0 4 I 10 0 2

AND

>: lo-‘

x1 = x3.97 P = .0116 -

x2 (2 d.f. I limits:

I I 7 I IO ’ ;

Bias and/or skewness corrected

P Covariate

0 0 (1 0 I 0 1

CANCER

.S6/63

Approximate

x2 (3 d.f.)

2

Bias and/or skewness corrected

Approximate 7.874

Q!I

Exact

7.830

z = 5.387 P = .?6 x

lo-’

z = 5.524 P= .I7

7.830 x

lo-

P -=

.47

y

IO x

3.386

5 I.II 5 21.162 5 J, 5 29.606

$I 3.295 2.609

8. ANALYSES

5 $I 5 19.705 5 II, 5 25.060

3.368 2.709

i 11,i 21.583 5 I// 5 31.035

OF THE ATTRIBUT-ABLE

2.737

RISK

When the cases are from a population-based study then the n,,i, stratified by a covariable such as age and/or sex, are considered to be multinomial variates. The nOj’s are either assumed fixed or multiples of the n,,i’s. The attributable risk in a given stratum, assuming pli > poj, is

COMBINATION

TABLE ANALYSESOFACASE-CONTROL

81

OF 2 x 2 TABLES 3

STUDYOF

BLADDERCANCERANDSMOKING

Age

Controls

Cases

Odds ratios

4

50-54 55-59 60-64 65-69 70-74 15-79

22126 35139 38/41 42157 51179 32152

24125 35137 31136 46153 60173 39153

4.364 2.000 ,489 2.347 2.534 1.741

.7400 .4730 - .0709 .4981 .4976 .3132

Pooled

2201294

2351277

Stratified analyses

Tests for possible

pooling

Exact

1.976 Z = 2.991 P = .00139 x* = 4.811

1.963 Z = 2.995 P = .00137 x2 = 4.811

1.963 P = .00131 -

P

P =

P =

=

.4394

1.254 5 J, 5 3.119 1.097 5 + 5 3.566 (5

-

Bias and/or skewness corrected

Approximate

Common odds ratio, $ Tests of main effect II, = 1 (one tailed) Test of interaction Overall x2 (5 d.f.) Confidence Limits: J, 95% 99%

-

.4394

1.253 5 $ I 3.109 1.098 5 I) % 3.583

.4840

1.252 5 JI 5 3.110 1.097 5 l/l 5 3.579

d.f.)

x: = 29.791, P = .OOOO x; = 29.793, P = .OOOO Analyses of attributable risk. (Y No assumption of constant odds ratio & = .4071, SE(h) = .0944 Confidence limits

95%

99%

Ordinary Transformed

.2221 % 01 5 .5921 .2074 5 a 5 .5143

.1640 5 (Y5 .6502 .I398 5 (Y5 .6191

Assumption of constant odds ratio, $ = 1.963 iu = .4160, SE(&) = .1034 Confidence limits Ordinary Transformed

95%

99%

32129 5 a 5 .6190 .1948 5 a 5 .5969

.1491 5 (Y 5 .6828 .1196 5 cy5 .6443

82

THOMAS

AND

GAR-I

18.11

a, = 1 - 4,,/%l,. If a common

odds

ratio is assumed, this is equivalent a .I = p,, ($

The overall attributable

1,i$.

to. see Miettinen

(20). IX.21

risk is

where rr, is the fraction of cases in stratumj. and Xjnj = 1. The ri may be either estimated from the data or may be assumed to be known from population data. With no assumption of constant I/J. the estimators of 6, follow by letting g,, = (I?,, - s,,)/rz,, in [8. I] for i = 0, 1 and allj. The r,, are either known or estimated by n,jlrz,, yielding the overall estimator & = C(n,j&,i/n, ). The $-constant estimator employs in [8.2] the corrected estimators of 4 and ,Q,, denoted by 4 and J?,,,, respectively. to yield the stratified estimators, iu,. The corresponding overall estimator is Cr = Xnri(Yillrl.) = /j,, (4 - I)/$, where ~7, = (C~z,,~?,~)in, Both these estimators require modification when $, , < &, for some j, or 12,~1. 1. see (10). Whenp,.,n < p,),, * the estimated attributable risk is redefined to be the negative of the preuct~tiue,~a~tiotz, see Miettirzerr (20), that is. u; = -(I

- li,Jli,,).

18.31

Similarly. if 17/< I, l/t$ and au are substituted for I$ and p,,. respectively, in [8.2]. Further, the sign of the resulting estimator is reversed, that is, 6~;< 0 for allj. Analogous changes are made in the definition of c?. so that Cu< 0 With these definitions, - 1 5 6,. Cu,5 + 1, for allj and thus -~ 1 5 &. Er 5 4 I. Clearly if $ = I, Cu= 0, and if 4 > 1, & > 0. The program gives point and interval estimators of Q which are reasonably accurate for moderately large sample sizes. but are not necessarily valid in sparse data, see (10) for simulation results. If $-constancy holds, the results based on Cwmay be considerably more efficient than those based on &. particularly when the pO,‘s vary widely over the strata. Even when the variation in the 4’s is modest, the methods using & are quite robust. Each ordimq confidence interval is found by taking the point estimator plus and minus the appropriate multiple of the standard error, SE(&) or SE(&). Transformed intervals are found by estimating the standard error of Fisher’s z-transform. (1/2)ln{(l + &)l( 1 ~ ;Y)}. by SE(&)41 - &“;‘), with an analogous result for the z-transform of &. Typically. as Gart and Thomas (10) found, transformed limits are narrower than ordinary intervals and usually achieve the nominal confidence coefficient, Whittemore (21) and others have suggested using the logit transformation for LY. Unlike the z-transform, this transformation is undefined for iu or & 5 0. Furthermore, for Cu> 0 or iu > 0, the corresponding logit limits always exclude zero regardless of the significance level of the test of hypothesis: ti = 1. which implies ff = 0. Greenland (22) advocates using the Mantel-Haenszel estimator rather than $Jin [8.2]. Gart and Thomas (10) show that such confidence intervals typically fall short of the nominal confidence coefficient and tend to be wider on the average than those based on the transformed form of &.

COMBINATIONOF

9. EXAMPLE

3: POPULATION-BASED ATTRIBUTABLE

83

x 2TABLES STUDY WITH RISK

ESTIMATION

OF THE

Table 3 gives the data of a large case-control study of smoking and bladder cancer by Cole et al. as given in Whittemore (21). We assume the cases are a representative sample of incidence by age groups; that is, the n,j’s are multinomial variates based on a sample of 11,. = 277, the fixed number of cases. Thus rj’s are estimated by n,jln,. The numbers of controls are approximately matched to the numbers of cases by age group. The usual analyses of I) must be considered to be conditional on the nIj’s and noj’s being fixed. In this situation with relatively large numbers in each group, the approximate analyses of I) are seen to be quite close to the exact analyses. The corrected analyses are virtually identical to the exact analyses. The tests for pooling are highly significant, P < .00005, so the pooled results are not given. Although there is no evidence of interaction, the program gives estimates of the attributable risk both with and without this assumption. As the poj’s vary only between .6 and .9, the method based on & is relatively efficient. Thus results based on & and & are quite similar, but are quite different from those of Whittemore (21). The latter found an overall estimate of (Yto be about. 30 rather than .41. This is a consequence of the failure to redefine Aj, as in [8.3], for the age group 60-64. Here she used & = - .8981 rather than - .0709 as shown in Table 3. REFERENCES 1. THOMAS, D. G. Exact and asymptotic methods for the combination of 2 x 2 tables. Comput. Biomed. Res. 8, 423 (1975). 2. CART. J. J. Analysis of the common odds ratio: Corrections for bias and skewness. Bull. Znf. Statist. Inst. 45, Book 1, 175 (1985). 3. MEHTA, C. R., PATEL, N. R., AND GRAY, R. Computing an exact confidence interval for the common odds ratio in several 2 x 2 contingency tables. J. Am. Statist. Assoc. 80, 969 (1985). 4. VOLLSET, S. E., HIRII, K. F.. AND ELASHOFF, R. M. Fast computation of exact confidence limits for the common odds ratio in a series of 2 x 2 tables. J. Am. Statist. Assoc. 86, 404 (1991). 5. MEHTA, C. R., PATEL, N. R., AND WEI, L. J. Computing exact significance tests with restricted randomization rules. Biometrika 75, 295 (1988). 6. ARMITAGE, P. The chi-square test for heterogeneity after adjustment for stratification. J. R. Statist. Sot. B 28, 150 (1966): Addendum 29, 197 (1967). 7. CART, J. J. The identity of Armitage’s two tests for heterogeneity of proportions for proportional subclass numbers. J. R. Statist. Sot. B 52 (1992). 8. EJIGOU, A., AND MCHUGH, R. Testing the homogeneity of the relative risk under multiple matching. Biometrika 71, 408(1984). 9. CART, J. J. Testing for interaction in multiply-matched case-control studies. Biometrika 72,468 (1985). 10. GART, J. J., AND THOMAS, D. G. Point and interval estimation of the attributable risk in stratified case-control studies. Submitted for publication. Il. GART, J. J. Pooling 2 x 2 tables: Asymptotic moments of estimators. J. R. Statist. Sot. B 52 (1992). 12. GART. J. J. Point and interval estimation of the common odds ratio in the combination of 2 x 2 tables with fixed marginals. Biometrika 57, 471 (1970).

84 1. 14.

IS. Ih. 17. IH.

IV. -70. 21. 22.

THOMAS

AND

GAR-I

N. Odds ratio estimators when the data are sparse. Biornerrika 68, 73 (1981). P. On the elimination of nuisance parameters in the proportional odds model. J. R. Statist. Sot. B 46, 250 (1984). LEVIN. B. Simple improvements on Cornfield’s approximation to the mean of a noncentral hypergeometric random variable. Biornetrikn 71. 630 (1984). GART. J. J. The equivalence of two corrections to the approximate mean of an entry In a contingency table. Bionlrtrikn 74. 661 (1987). LEVIN. B.. AND KONG. F. Bartlett’s bias correction to the profile score function IS a saddlepomt correction. Biomrfriku 77, 219 (1990). BRESLOW. N., AND COLOGNE. J. Methods of estimation in the log odds ratio regression models. Biometric.\ 42, 949 (1986). BRESLOW, N. E., AND DAY. N. E. Statistid Merhdr irr C‘UII(~CI. RL’.~~~~II.~~I~.I’d. I 7’1~ At/tr/),.vri of Casu-c~ntr.o/ Sl&es. IARC Scientific Publications No. 32. Lyon. 1980. MIETTINEN. 0. S. Proportion of disease caused or prevented by a given exposure. rrait ot intervention. Am. J. Epidemiol. 99, 325 (1974). WHITTEMORE, A. S. Estimating attributable risk from case-control studies. AHI. J. Epirlc~mwi, 117, 76 (1983). GREENLAND. S. Variance estimators for attributable fraction estimators consistent in both large strata and sparse data. Stntisr. Med. 6, 701 (19871. BRESLOW.

MCCULLAGH.

Improved and extended exact and asymptotic methods for the combination of 2 x 2 tables.

Exact and approximate methods for analyzing the common odds ratio in the combination of 2 x 2 tables are programmed. The approximate methods are impro...
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