Estimating the Common Odds Ratio

Introduction

In our previous lesson, we explored the Cochran-Mantel-Haenszel (CMH) test, a method to determine if an association exists between an exposure and an outcome after controlling for a confounding variable. The test's null hypothesis was that the association, measured by the odds ratio, is null (i.e., OR=1) in every stratum.

But what if the test tells us an association does exist? The logical next step is to ask, "How strong is it?" This lesson focuses on quantifying that association. We will explore the concept of a common odds ratio, which serves as a single, adjusted summary of the effect's magnitude across all strata. You will learn what this concept means and how it is estimated using the widely-accepted Mantel-Haenszel method.

1. The Concept of a Common Odds Ratio

When we stratify our data, we calculate a separate odds ratio for each stratum (e.g., for each clinic, age group, or country). If the effect of the exposure is consistent across these different strata, the odds ratios will be similar to each other. This situation is called homogeneous association.

STAT 226 Lecture 8 - Section 2.7 Association In Three-Way Tables

This brief excerpt from a lecture slide deck from the University of Chicago defines homogeneous association and explains its meaning.

Please read the sections titled 'Homogeneous Association' and 'Understanding Homogeneous Association' on pages 10-12. Focus on the core idea: homogeneous association means the odds ratios are identical at each level of the stratifying variable Z.

When we can reasonably assume the association is homogeneous, it is very useful to compute a single summary measure that represents this consistent effect. This summary measure is the common odds ratio. It represents the average association between the exposure and outcome after adjusting for the confounding effects of the stratifying variable.

Think of it this way: if a new teaching method increases test scores by about 10% in every school it's tried in, reporting an "average 10% improvement" is a very useful summary. But if it boosts scores by 50% in one school and lowers them by 30% in another, an "average" effect is misleading. The common odds ratio is like that useful average, but it's only meaningful when the effect is consistent.

2. Estimating the Common Odds Ratio: The Mantel-Haenszel Method

The most common way to estimate the common odds ratio is the Mantel-Haenszel (MH) method. The core idea behind this method is to calculate a weighted average of the stratum-specific odds ratios.

Why a weighted average? Because some strata provide more reliable information than others. A stratum based on 1,000 subjects is more informative and precise than a stratum with only 50 subjects. The MH method gives more weight to the larger, more precise strata, so they contribute more to the final estimate.

The following video provides a clear, conceptual explanation of how this weighting works. While it first discusses relative risk, the principle is the same for the odds ratio, which is covered in the second half.

The Mantel-Haenszel method - clearly explained | deal with confounding

The YouTube channel TileStats provides a clear, step-by-step explanation of the Mantel-Haenszel method, focusing on the intuition behind calculating a pooled estimate.

Please watch the segment from 06:59 to 09:00. This part specifically covers the Mantel-Haenszel method for a pooled odds ratio. Pay close attention to how it's framed as a weighted average of the odds ratios from each stratum.

The Mantel-Haenszel Formula

As the video explained, the MH estimate is a weighted average. The actual calculation formula looks a bit different from a simple weighted mean but achieves the same goal. For each stratum $k$ , we have a 2x2 table:

	Outcome +	Outcome -
Exposed	$a_{k}$	$b_{k}$
Unexposed	$c_{k}$	$d_{k}$
Total

The Mantel-Haenszel estimate of the common odds ratio ( $θ^_{M H}$ ) is calculated as:

θ^_{M H} = \frac{\sum _{k} \frac{a _{k} d _{k}}{n _{k}}}{\sum _{k} \frac{b _{k} c _{k}}{n _{k}}}

Let's break this down:

The numerator of an odds ratio is $a \times d$ . The formula sums these products across all strata, but first weights each one by $1/ n_{k}$ . This gives more influence to the $a_{k} d_{k}$ terms from larger strata.
The denominator of an odds ratio is $b \times c$ . The formula does the same for these products.
The final result is a pooled estimate of the odds ratio that is adjusted for the stratifying variable.

3. A Worked Example: Passive Smoking and Lung Cancer

Let's apply this to a real example. The following reading continues with the passive smoking data we saw in the previous lesson, showing how to calculate and interpret the Mantel-Haenszel estimate.

STAT 226 Lecture 8 - Section 2.7 Association In Three-Way Tables

These lecture slides, which we used before, now show the practical application of the Mantel-Haenszel formula.

Please read pages 27-30, titled 'Estimate of the Common Odds Ratio', 'Example: Lung Cancer and Passive Smoking', and 'Confidence Interval for the Common Odds Ratio'. \nFocus on: How the formula is applied to the data from Japan, the UK, and the US. The final calculated value of the common odds ratio (\approx 1.4). The interpretation of this value and its 95% confidence interval.

From the example, the Mantel-Haenszel estimate was $θ^_{M H} \approx 1.4$ . The interpretation is crucial:

After controlling for country, the odds of developing lung cancer for nonsmoking wives with smoking husbands are estimated to be 1.4 times the odds for nonsmoking wives with nonsmoking husbands.

The phrase "after controlling for country" is the key part that reflects the stratified nature of the analysis. The 95% confidence interval of (1.05 to 1.82) tells us that we are 95% confident this true common odds ratio is between 1.05 and 1.82. Since the interval does not include 1.0, the association is statistically significant, which is consistent with the CMH test result from our last lesson.

Conclusion

In this lesson, we moved from testing for an association to estimating its strength. We established that when an association is consistent across different strata, we can summarize it with a single, powerful number.

Key Takeaways:

A common odds ratio is a single summary measure of association for stratified data, representing the average effect after controlling for a confounder.
This concept is most meaningful under the assumption of homogeneous association, where the odds ratio is similar in every stratum.
The Mantel-Haenszel (MH) estimator is the standard method for calculating the common odds ratio.
The MH estimator works by taking a weighted average of the stratum-specific effects, giving more influence to strata that contain more information (i.e., larger sample sizes).

Preview of the Next Lesson:

We've discussed how the common odds ratio is only truly appropriate if the assumption of homogeneity holds. But how do we formally check this assumption? In our next lesson, we will focus on the test for homogeneity of odds ratios (such as the Breslow-Day test). This will complete our toolkit for stratified analysis, allowing us to decide whether to report a single common odds ratio or to describe the effect separately for each stratum.

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