Same Cohort, Two Odds Ratios: Non-Collapsibility Without Confounding

อ่านฉบับภาษาไทย (Thai version)

Abstract

Two analyses of the same cohort, a logistic regression adjusted for covariates and a correctly implemented weighting analysis such as inverse probability weighting (IPW), can give different treatment odds ratios with no coding error in either. The usual reason is not confounding but the question each method answers. Adjusted regression usually targets a conditional odds ratio: the odds ratio among patients with the same covariates, such as age, sex and severity. Weighting usually targets a marginal odds ratio: the average comparison over the whole population. For the odds ratio these two need not be equal even when confounding is absent, a property called non-collapsibility. In a randomised hand example, the odds ratio is 2.0 at low severity (risk 10% versus 18.2%) and at high severity (50% versus 66.7%), yet the whole-trial odds ratio is 1.72. This article works through that example and ends with one rule: same cohort, two correct methods, different odds ratios, so first ask whether they target different estimands.

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One cohort, two analyses, two odds ratios

A hospital team wants to know whether a treatment changes the odds of an outcome in a cohort of inpatients. Treated and untreated patients differ in age, sex and disease severity. So the team analyses the same patients in two ways.

One analyst fits a logistic regression of the outcome on treatment, adjusted for age, sex and severity. Another uses inverse probability weighting (IPW). IPW weights each patient by the inverse of the probability of the treatment they actually received, so that treatment no longer depends on the measured covariates. The guide to IPTW builds these weights step by step.

Both analysts checked their code, and neither found an error. Yet the two treatment odds ratios differ. Before deciding who is wrong, it is worth asking whether the two analyses answer the same question at all.

The trap: no confounding, so the odds ratios should match?

The words that mislead here are "confounding is absent". They tempt the thought: if nothing is confounded, the two odds ratios should be equal. For the odds ratio, that is not true.

The odds ratio has a property called non-collapsibility. The odds ratio for a whole population can differ from the odds ratio inside groups of similar patients, even when nothing is confounded [1, 2]. The guide to collapsibility and the change-in-estimate approach places this property among other effect measures.

Two methods, two estimands

The two methods may be answering different estimands. An estimand is the exact quantity an analysis aims to estimate: its population, its comparison and its summary measure.

• Adjusted logistic outcome regression usually gives a conditional odds ratio. It asks: "Among people with the same covariates, what is the odds ratio for treatment?" For example, treatment versus control with age, sex and severity held fixed.
• Weighting such as IPW usually gives a marginal odds ratio. It asks: "If the whole population is compared, treatment versus control on average, what is the odds ratio?" [3]

Why "must coincide" is wrong

A common statement reads: "Conditional and marginal odds ratios must coincide whenever confounding is absent." The words must coincide are the error.

Suppose treatment is perfectly randomised, so there is no confounding. But one variable, severity, is strongly related to the outcome. Patients who share one level of severity form a stratum (plural strata), and the table below gives the risk of the outcome in each.

The hand example: the same odds ratio in each stratum

Step by step: 2.0 in each stratum, 1.72 overall

Hand example. Odds are risk divided by one minus risk. Low and high severity each make up half of both treatment groups, as randomisation gives on average.

1. In each severity stratum

   \[ \mathrm{OR}_{\text{conditional}} = 2.0 \]

   At low severity the odds are 0.10/(1 - 0.10) = 0.111 for control and twice that, 0.222, for treatment, a risk of 18.2%. At high severity the odds are 1.0 and 2.0, a treatment risk of 66.7%.

2. Control overall risk

   \[ \frac{0.10 + 0.50}{2} = 0.30 \]

   With equal halves, the overall risk of the control group is the plain average of its two stratum risks.

3. Treatment overall risk

   \[ \frac{0.182 + 0.667}{2} \approx 0.424 \]

   The same average for the treatment group.

4. Marginal OR

   \[ \mathrm{OR}_{\text{marginal}} = \frac{0.424/(1 - 0.424)}{0.30/(1 - 0.30)} \approx 1.72 \]

   The treatment odds are about 0.737 and the control odds about 0.429. Their ratio is the odds ratio for the whole trial.

Result: $\mathrm{OR}_{\text{conditional}} = 2.00$, but $\mathrm{OR}_{\text{marginal}} \approx 1.72$, although there is no confounding at all. This is non-collapsibility of the odds ratio.

Pooling the strata averages the risks, not the odds. The odds of the averaged risks do not give back the common odds ratio, so 1.72 is not an average of the two stratum values.

The situation in plain words

Go back to the opening situation and say it plainly. The researchers took one set of patients and analysed it in two ways:

1. a regression that adjusts for the covariates;
2. weighting, such as IPW.

They got a different treatment odds ratio from each. Neither analysis has a coding error, so blaming the code is ruled out.

The real question is therefore not which odds ratio is right. It is: why do two analyses that are both done correctly give odds ratios that are not equal?

What each method is asking

In the simplest terms, the two methods may be answering different statistical questions.

Regression asks, roughly: "Comparing people with the same age, severity and so on, how large is the effect of treatment?" That is a conditional effect.

Weighting asks, roughly: "Looking at the effect of treatment across the whole population, on average, what is it?" That is a marginal effect.

So even with the same patients, and with both analyses correct, conditional OR ≠ marginal OR can happen.

The same idea without statistics

Picture two rehabilitation programmes, A and B, after hip surgery, and two questions about the same patients.

First question: "Among patients who started with the same level of mobility, how much better does programme A do than programme B?"

Second question: "Averaged over all the patients, how much better does programme A do than programme B?"

The patients are the same, but the questions are not. So the numbers that answer them need not be the same either.

Common misreadings and their fixes

• "Without confounding, the conditional and marginal odds ratios must coincide."

  "Must coincide" means "must be equal", and for the odds ratio that is wrong. In the randomised hand example nothing is confounded, yet the conditional odds ratio is 2.00 and the marginal odds ratio is 1.72.

  Fix: Expect the two to differ whenever a covariate is strongly related to the outcome, because the odds ratio is non-collapsible, even with no confounding.

• "The two odds ratios differ, so one analysis has an error or leftover confounding."

  When both analyses are done correctly, the gap can come from the estimands alone.

  Fix: Before hunting for a coding error, check whether one analysis estimates a conditional odds ratio and the other a marginal one.

Glossary

non-collapsibility (ความไม่ยุบรวมได้)

The property of the odds ratio whereby the odds ratio for a whole population can differ from a common odds ratio within groups of similar patients, without any confounding.

conditional odds ratio (odds ratio แบบมีเงื่อนไข)

The odds ratio among patients with the same covariates, for example the same age, sex and severity; adjusted logistic regression usually gives it.

marginal odds ratio (odds ratio ระดับประชากร)

The odds ratio comparing the whole population treated versus untreated, on average; weighting such as IPW usually gives it.

estimand (ปริมาณเป้าหมายของการประมาณ)

The exact quantity an analysis aims to estimate: its population, its comparison and its summary measure.

inverse probability weighting (IPW) (การถ่วงน้ำหนักด้วยส่วนกลับของความน่าจะเป็นในการได้รับการรักษา)

Weighting each patient by the inverse of the probability of the treatment they received, so that treatment no longer depends on the measured covariates.

stratum

A group of patients who share one value of a variable, such as one level of severity; the plural is strata.

References

1. Greenland S, Pearl J, Robins JM. Confounding and collapsibility in causal inference. Stat Sci. 1999;14(1):29-46. doi:10.1214/ss/1009211805 https://doi.org/10.1214/ss/1009211805
2. Whitcomb BW, Naimi AI. Defining, quantifying, and interpreting 'noncollapsibility' in epidemiologic studies of measures of 'effect'. Am J Epidemiol. 2021;190(5):697-700. doi:10.1093/aje/kwaa267 https://doi.org/10.1093/aje/kwaa267
3. Austin PC, Stuart EA. Moving towards best practice when using inverse probability of treatment weighting (IPTW) using the propensity score to estimate causal treatment effects in observational studies. Stat Med. 2015;34(28):3661-3679. doi:10.1002/sim.6607 https://doi.org/10.1002/sim.6607

Key takeaways

• Two correct analyses of one cohort, an adjusted logistic regression and a weighting analysis such as IPW, can give different treatment odds ratios.
• Adjusted regression usually targets a conditional odds ratio, for patients with the same covariates; weighting usually targets a marginal odds ratio, for the whole population on average.
• No confounding ≠ conditional OR = marginal OR, because the odds ratio is non-collapsible.
• In a randomised hand example, an odds ratio of 2.0 in each of two equal severity strata gives a whole-trial odds ratio of about 1.72.
• Same cohort, two correct methods, different odds ratios: first ask whether they estimate different estimands, not which method is better.

Related in the wiki: [[repeated-measures-modeling-guide]] [[collapsibility-effect-change-guide]]