Interpreting Mantel-Haenszel Adjusted Estimates

From Multiple Strata to a Single Estimate

After stratifying our data by a confounder like age, we are left with several stratum-specific odds ratios (e.g., an OR for the "young" group, an OR for the "middle-aged" group, and an OR for the "old" group). If we've determined that confounding is present (and another phenomenon called effect modification, which we'll discuss soon, is not), our goal is to synthesize these into one overall, adjusted estimate.

You might first think of simply averaging the stratum-specific ORs. However, this approach is too simplistic. Some strata might contain far more people and data than others. A stratum with 1,000 participants provides a more precise, reliable estimate than a stratum with only 50 participants. Therefore, we need a method that gives more weight to the more informative strata. This is precisely what the Mantel-Haenszel method does. It calculates a pooled, weighted average of the stratum-specific estimates.

The following video provides an excellent illustration of this principle. It shows a situation where a crude relative risk suggests a strong association, but after stratifying by age, the association vanishes within each age group. This highlights the need to combine the stratum-specific results into a single, adjusted number that reflects the true, unconfounded relationship.

The Mantel-Haenszel (M-H) Calculation in Practice

The Mantel-Haenszel (M-H) procedure is the standard method for calculating this adjusted summary estimate from stratified data. It is used when we've determined that there is confounding but no significant effect modification.

Let's walk through a complete example. The following resource demonstrates the entire process, from calculating the crude OR to finding the final, M-H adjusted OR. This process operationalizes the flowchart we saw in the last lesson.

Interpreting the M-H Adjusted Estimate

After following the example, we have three key numbers:

• Crude OR = 1.93
• Stratum-specific ORs ≈ 1.5
• M-H Adjusted OR = 1.51

The crucial part is knowing how to interpret and use this adjusted result. The M-H adjusted OR of 1.51 is our best estimate of the association between obesity and CVD, free from the distorting influence of age.

The correct interpretation would be:

"After adjusting for age, the odds of cardiovascular disease were 1.51 times higher among obese individuals compared to non-obese individuals."

Notice the critical phrase: "After adjusting for age..." or "Controlling for age...". When you present an M-H estimate, you must always state which variable(s) you have adjusted for. This tells your audience that you have acknowledged and analytically controlled for a potential confounder. The M-H estimate (1.51) is much closer to the stratum-specific "truth" (~1.5) than the biased crude estimate (1.93). Therefore, the adjusted estimate is the one you report.

The "Information" Behind the Weights

You might be looking at the M-H formula and finding it a bit opaque.

Where $i$ represents each stratum from 1 to $k$, and $n_i$ is the total number of subjects in stratum $i$.

While you don't need to perform these calculations by hand in practice (statistical software does it for you), understanding the logic is key for interpretation. As someone with a background in information systems, you can think of the M-H procedure as an algorithm that weights each stratum based on the amount of "information" it contains. But what constitutes information in this context?

A stratum is more informative, and thus given more weight, if it has:

1. A large sample size (T): More data leads to a more stable estimate.
2. Good balance: A stratum with 500 exposed and 500 unexposed individuals is more informative than one with 999 exposed and 1 unexposed.
3. A sufficient number of outcomes: If the disease is extremely rare, it's hard to get a stable estimate of its risk.

The following video provides a fantastic intuitive explanation of how the M-H weighting scheme captures these three components of information.

The Big Picture: Revisiting Our Framework

Let's place the Mantel-Haenszel procedure back into our overall analytical framework for dealing with third variables.

In the last lesson, we focused on Steps 1, 2, and the decision in Step 3. Today, we've explored what happens when we answer "Yes" to the question "Are they similar?" in Step 3. This leads us directly to Step 4: Calculate an adjusted measure like the Mantel-Haenszel OR. The M-H estimate is the tool we use when the evidence points to confounding. Step 5 then involves comparing the M-H adjusted OR to the crude OR to confirm that confounding was indeed present, as we did in the obesity-CVD example (|1.93 - 1.51| / 1.51 ≈ 28%, which is well over the 10% threshold).

Conclusion

In this lesson, we completed the analytical pathway for addressing confounding. We moved from simply detecting confounding to calculating a single, summary measure that corrects for its biasing effect.

Here are the key takeaways:

• The Mantel-Haenszel (M-H) estimate is a weighted average of stratum-specific measures of association.
• It provides a single adjusted estimate (e.g., adjusted OR or RR) that controls for the effect of a confounding variable.
• The weights used in the M-H calculation are based on the amount of "information" in each stratum, ensuring that more reliable strata contribute more to the final estimate.
• When reporting an M-H estimate, you must state that it is adjusted for the confounding variable. This adjusted estimate is considered a more accurate representation of the true association than the crude estimate.

So far, all our work has been based on the assumption that the stratum-specific measures are similar. But what if they are not? What if the effect of the exposure is genuinely different across the strata of the third variable? This scenario takes us down the "No" path in our flowchart and introduces the concept of effect modification, which will be the focus of our next lesson.