An Equivalence Test between Mantel-Haenszel Rate Ratio and Standardized Mortality Ratio

In the analysis of stratified person-time follow-up data, the multiplicative incidence rate model requires the rate ratio homogeneity across strata. This homogeneity assumption is statistically convenient but biologically stringent. When we do not have any strong biological models on incidence rate ratio homogeneity, we cannot justify using the estimators for the common (homogeneous) incidence rate ratio without checking the assumption. Given that the homogeneity test has a low power, we at least compare two summary estimators : one based on the homogeneity assumption, and the other not. This paper proposes an equivalence test between the Mantel-Haenszel rate ratio, which is an estimator for the common incidence rate ratio, and the standardized mortality ratio (SMR), which does not require the homogeneity assumption. The proposed test gives a practical choice between these two summary measures. J Epidemiol, 1994; 4 : 133-136.

In the analysis of stratified person-time follow-up data, one often employs the Mantel-Haenszel rate ratio as a summary effect estimate for a dichotomous exposure1,2).Because of its simple form and high efficiency, it has been extended to a multi-level exposure variable3.4).Like any multiplicative models, the Mantel-Haenszel rate ratio requires the assumption of homogeneity of stratum-specific incidence rate ratios.Usually we do not have any biological reasons for believing such a strong assumption, and we need to examine the incidence rate ratio homogeneity based on data).For evaluation of this assumption, several tests of homogeneity have been proposed1,2).Unfortunately, the homogeneity test usually has low power (i.e., the test tends to give a non-significant result, even when incidence rate ratio heterogeneity, effect modification, exists) in typical epidemiological situations6).
When the homogeneity assumption is violated, the alternative summary rate ratio would be given through standardization1,7).When we choose the exposed population as the standard, for example, the standardized mortality ratio (SMR) is interpreted as the proportionate change in the average incidence rate of the exposed produced by exposure 8).This interpretation of SMR does not require the incidence rate ratio homogeneity.Without strong prior information on biological reasons that sustain the homogeneity assumption, one may consider that some effect modification is present.In that case, the Mantel-Haenszel rate ratio (or other estimates for the common rate ratio) should be used only when effect modification lies within a specified level ; and otherwise, SMR may be used for presentation.In such a case, the logical choice is an equivalence test rather than a significance (homogeneity) test.This discussion is parallel to testing confounding, where, in some extent, confounding is almost always present in nonrandomized studies9).
In this paper, I propose an equivalence test between the Mantel-Haenszel rate ratio and SMR.The proposed test has a closed-form and is based on the confidence interval approach.

PROPOSED TEST
We use the following notation in a stratum from a follow-up study of dynamic populations : For simplicity, no stratum index is used.The Mantel-Haenszel estimator of the common incidence rate ratio is T .Sate defined as MH=MH1/MH2, where MH1=YMX/T, MH2=*NY/T, and the summations are across all strata.When we choose the person-time distribution of the exposed population as a standard, the standardized mortality ratio is defined as SMR=O/E, where O=EX, the observed cases in the exposed, and E=*NY/M, the expected cases in the exposed if the exposed had not been exposed.Associated confidence interval methods have been proposed for the common rate ratio and for the true standardized mortality ratio 1,2,10).
The proposed test is based on the ratio of MH to SMR, RRR=MH/SMR.
The null hypothesis for the significance test is that the large sample expectation of MH is equal to that of SMR, in other words, the true RRR= 1. Instead of testing this hypothesis, we will test the null hypothesis that the true RRR$1 against the alternative hypothesis that the true RRR=1.
According to Hauck and Anderson"), an a-level equivalence test is obtained by the following steps : 1. Specify the equivalence interval for the true RRR, i.e., 0.9-1.1.The lower and upper ends should be close enough to one, so that one may consider the two summary rate ratios are equivalent.2. Calculate the 100-2a percent confidence interval for the true RRR. 3. If the 100-2a percent confidence interval in the step 2 falls inside the interval specified in the step 1, conclude that the Mantel-Haenszel rate ratio and SMR are equivalent.
If the test result is significant, that the confidence interval falls entirely within the prespecified equivalence interval, one will report the Mantel-Haenszel rate ratio and the associated confidence interval.If not, one will report SMR with the associated confidence interval.
The 100-2a percent confidence interval for the true RRR is obtained by and z is the 100-a percentage point of the standard normal distribution.Derivation of the above interval is given in the Appendix 1.

Example
For illustration, we use the data from the Montana smelter workers study on arsenic exposure and respiratory cancer death (2, p. 111, Table 3.14).The data are stratified into 13 strata defined by age-class and calendar-period.The Breslow-Day test of incidence rate ratio homogeneity2) gave X2=12.9 with 12 degrees of freedom and p=0.378, non-significant.The Mantel-Haenszel rate ratio and the 95 percent confidence interval1O> are MH=3.14 and 2.01-4.89.However, even when excluding two extreme values (zero and indeterminate), stratum-specific incidence rate ratios ranged from 0.868 to 14.0.Thus, a non-significant test result alone is inadequate to report the Mantel-Haenszel rate ratio.We set 5 percent as the a-level and 0.9-1.1 as the equivalence interval.Since SMR=3.18 (the 95 percent confidence interval1), 2.05-4.95),we obtained RRR=0.986 and the 90 percent confidence interval 0.965-1.01.The confidence interval fell well inside 0.9-1.1, and we conclude to report the Mantel-Haenszel rate ratio.

DISCUSSION
The assumption of homogeneity of an effect parameter is widely accepted (e.g., the Mantel-Haenszel odds ratio or the logistic regression in the analysis of case-control data), because it is statistically convenient.However, this assumption is questionable in a biological sense when we do not have a strong biological model on the exposurecovariate-disease relationship.In such a situation, the standardized rate ratios are the alternative (or even the first) choice over the common rate ratio estimator.Given that the homogeneity test has usually low power, the equivalence test proposed in this paper seems natural for us to choose SMR or MH.
Although the proposed test compares between the Mantel-Haenszel rate ratio and SMR, the exposed population as the standard (using SMR) is not always justifiable.The choice of the standard should depend on the population about which inferences will be made.When the unexposed is the target population in the previous example, the standardized rate ratio in the unexposed is 2.93, and RRR and the 90 percent confidence interval are 1 .07and 0.978-1.18.Unlike the previous result based on SMR, the test is not significant and we will report the standardized rate ratio in the unexposed with the associated 95 percent confidence interval (e.g., 1.88-4 .58).The formula of the proposed test is given in the Appendix 1 for any valid standard distributions.
It should be noted that the proposed test is not strictly testing the homogeneity assumption or , equivalently, effect modification.Even when severe effect modification exists, RRR could become one.To illustrate this point, we use the next data set from the British doctors study on smoking and coronary death (1, p. 184, Example 12-1 ; 2, p. 112, Table 3.15).Table 1 shows that stratum-specific rate ratios decrease with age, and the Breslow-Day homogeneity Table 1.Data from the British male doctor's study on smoking and coronary death 1.2).* 95% confidence interval * ` 90% confidence interval test gave a small, significant p-value (p = 0.025, the last line in Table 1).However, the Mantel-Haenszel rate ratio and SMR in Table 1 give virtually the same value, and RRR= 1.01 (the 90 percent confidence interval, 0.983-1.03).
According to the equivalence test result, we would choose to report the Mantel-Haenszel rate ratio although the homogeneity assumption is violated.The last column in Table 1 shows stratum-specific fractions of smokers' person-years.Except for the youngest age class 35-44, the age-specific person-year smoking prevalences are similar, and it makes the Mantel-Haenszel rate ratio and the standardized mortality ratio equivalent regardless of severe effect modification.In this constant person-time exposure prevalence case, as shown in the Appendix 2, the variance estimate of the Mantel-Haenszel rate ratio becomes identical to that of the standardized mortality ratio.Thus, the equivalence test result is still valid in this example.In this paper, I have proposed the equivalence test between the standardized mortality ratio and the Mantel-Haenszel rate ratio.Two cautions will be made.First, the proposed test is for the large sample one, and may not be used for small samples.Second, since we consider using the standardization methods, we should carefully examine all the cautions for standardization") : the standard should be chosen to represent the proper target population : the stratum-specific incidence rate ratios should be examined.