Choose Between Conditional and Population-Average Effects

Two Questions, Two Effect Summaries

A regression analysis can answer more than one question about an exposure and an outcome. One question asks how outcomes compare for people with the same measured covariates. Another asks how outcomes would compare, on average, across a specified population. These are different summaries, and neither is automatically the right one in every application.

The distinction matters especially when a model uses a nonlinear link, such as logistic regression. A logistic-regression exposure coefficient describes a conditional odds ratio on the model’s odds scale. It is not, by itself, a risk difference or an odds ratio calculated after averaging predicted risks over a population. The choice of summary should follow the scientific question and the scale that makes its answer useful.

Definition: Let $A$ be a binary exposure, $X$ a set of pre-exposure covariates, and $Y$ an outcome. A conditional effect compares exposure levels at a specified covariate profile $X=x$. A population-average effect averages the exposure-specific outcome contrast over the covariate distribution of a defined target population. For a mean outcome, its standardized contrast is $$ \psi_T = E_T[m(1,X)-m(0,X)], $$ where $m(a,x)$ is the model’s predicted mean outcome at exposure level $a$ and covariate profile $x$, and $E_T$ denotes averaging over the target population’s distribution of $X$.

For a causal interpretation, the exposure-specific predictions must represent the relevant potential-outcome means. This requires assumptions established for observational analyses: consistency (the observed outcome under the exposure actually received equals the corresponding potential outcome), conditional exchangeability (no unmeasured confounding given $X$), and positivity (each exposure level is possible at the covariate profiles in the target population). The outcome model must also estimate the needed conditional means adequately. Without these assumptions, the quantities may still describe adjusted associations, but should not be presented as causal effects.

A conditional effect is tied to a covariate profile; a population-average effect is tied to a target distribution. The latter is not fully specified until the target population is stated. Standardization computes it by predicting outcomes under each exposure level for the target population’s covariate values, then averaging those predictions separately and contrasting the averages.

When the Model Scale Makes the Difference

With an identity-link mean model and a constant exposure coefficient, the conditional and population-average mean differences agree. Suppose $m(a,x)=\beta_0+\beta_Aa+\beta_Xx$. For any fixed $x$, $$ m(1,x)-m(0,x) =(\beta_0+\beta_A+\beta_Xx)-(\beta_0+\beta_Xx) =\beta_A. $$ Averaging this difference over any target distribution leaves $\beta_A$, because it is constant: $$ E_T[m(1,X)-m(0,X)]=E_T[\beta_A]=\beta_A. $$ This equality depends on the model’s additive form and the mean-difference scale; it is not a general property of regression coefficients.

In logistic regression, the model specifies log odds, not risks. A coefficient for a binary exposure represents a conditional log odds ratio when the model includes that exposure as a main effect. Exponentiating it gives the conditional odds ratio, comparing people with the same modeled covariates. To obtain a population-average risk difference, instead average the predicted risks under each exposure level and subtract. Because converting odds to probabilities is nonlinear, averaging and converting do not generally produce the same result as conditioning at a profile.

Worked Example: An Additive Mean Model

Hypothetical example. Suppose a correctly specified regression model predicts a weekly symptom score $Y$, in points, from a binary program $A$ and baseline severity $X$: $m(a,x)=10+2a+0.5x$. Consider a target population with half its members at $X=0$ and half at $X=2$. Assume the causal identification conditions stated above hold for this program and these covariates.

At $X=0$, the predicted means are $m(0,0)=10$ and $m(1,0)=12$, a difference of $12-10=2$ points. At $X=2$, they are $m(0,2)=11$ and $m(1,2)=13$, also a difference of $13-11=2$ points. The conditional exposure coefficient is $\beta_A=2$ points.

The standardized means are $$ \bar m_0 = \frac{10+11}{2}=10.5,\qquad \bar m_1 = \frac{12+13}{2}=12.5. $$ Thus the population-average contrast is $12.5-10.5=2.0$ points. It matches the conditional coefficient because this model gives a constant mean difference at every $X$, not because conditional and population-average effects must always agree.

Logistic Regression: A Conditional Odds Ratio and Standardized Risks

For a binary outcome, write $p(a,x)=P(Y=1\mid A=a,X=x)$. A logistic model represents the log odds as a function of exposure and covariates. If the exposure coefficient is $\beta_A$ and the model has no exposure-by-covariate term, then, for each $x$, the conditional odds ratio comparing $A=1$ with $A=0$ is $\exp(\beta_A)$. This statement concerns odds within covariate profiles. It does not say that the risk difference is constant across profiles, nor that the same odds ratio will result after risks are averaged over a population.

The following example calculates both a conditional odds ratio and several population-average summaries. The target population has two equally common covariate strata. The conditional odds ratio is 2 in each stratum. Assume the fitted conditional probabilities are correct and, for a causal interpretation, that consistency, conditional exchangeability, and positivity hold.

Worked Example: Standardizing a Logistic Model

Hypothetical example. In a target population, half the people are in a lower-baseline-risk stratum and half in a higher-baseline-risk stratum. Under no exposure, their outcome risks are $0.10$ and $0.50$, respectively. Suppose the conditional odds ratio for exposure is 2 in both strata.

For the lower-risk stratum, the unexposed odds are $0.10/(1-0.10)=1/9$. Doubling the odds gives $2/9$, which converts to the exposed risk $ (2/9)/(1+2/9)=2/11\approx0.1818$. For the higher-risk stratum, the unexposed odds are 1; doubling gives odds 2 and exposed risk $2/(1+2)=2/3\approx0.6667$. These conversions verify the stated conditional odds ratio in each stratum.

Average the risks separately over the two equally common strata: $$ \bar p_0 = \frac{0.10+0.50}{2}=0.30, \qquad \bar p_1 = \frac{2/11+2/3}{2} =\frac{14}{33}\approx0.4242. $$ The standardized risk difference is $$ \bar p_1-\bar p_0 =\frac{14}{33}-\frac{3}{10} =\frac{41}{330}\approx0.1242. $$ The estimated population-average risk is therefore about 12.42 percentage points higher under exposure than under no exposure in this target population.

The marginal odds ratio, calculated from the standardized risks, is $$ \frac{\bar p_1/(1-\bar p_1)}{\bar p_0/(1-\bar p_0)} =\frac{(14/33)/(19/33)}{0.30/0.70} =\frac{14/19}{3/7} =\frac{98}{57}\approx1.719. $$ It differs from the conditional odds ratio of 2. The standardized risk difference, the marginal odds ratio, and the conditional odds ratio are three distinct summaries, each with its own scale and interpretation.

This difference can occur even when the conditional odds ratio is the same in every stratum and the causal identification assumptions hold. It is not necessarily evidence of confounding or a calculation error. The odds ratio is non-collapsible: in general, an odds ratio formed from risks averaged across covariate strata need not equal the common conditional odds ratio. In this example the risks are averaged first; their odds are then calculated. That calculation is not the same as averaging the stratum-specific odds ratios.

The Target Population Is Part of the Answer

Conditional model predictions can be used to standardize to different target populations. If those populations have different covariate distributions, their standardized risks and risk differences can differ, even when the conditional model is held fixed. This is not a flaw in standardization: it reflects that the population-average question has changed.

Worked Example: Changing the Target Population

Hypothetical example. Keep the two strata and conditional probabilities from the logistic example, but now define a target population with 80% of people in the lower-risk stratum and 20% in the higher-risk stratum. The conditional odds ratio remains 2 in each stratum. Assume the model and causal identification conditions apply to this target population.

The standardized risk without exposure is $$ \bar p_0 = 0.80(0.10)+0.20(0.50)=0.08+0.10=0.18. $$ Under exposure, the stratum-specific risks are $2/11$ and $2/3$, so $$ \bar p_1 = 0.80\left(\frac{2}{11}\right) +0.20\left(\frac{2}{3}\right) =\frac{8}{55}+\frac{2}{15} =\frac{46}{165}\approx0.2788. $$ The standardized risk difference is $$ \bar p_1-\bar p_0 =\frac{46}{165}-\frac{9}{50} =\frac{163}{1650}\approx0.09879. $$ For this target population, the risk is about 9.879 percentage points higher under exposure. This differs from the 12.42-percentage-point contrast for the equally divided target population because the populations assign different weights to the two strata.

The corresponding marginal odds ratio is $$ \frac{(46/165)/(119/165)}{0.18/(1-0.18)} =\frac{46/119}{9/41} =\frac{1886}{1071}\approx1.7619. $$ It is still distinct from the conditional odds ratio of 2. The calculation also shows why a population-average result should name its target population: changing the covariate distribution can change the answer without changing the conditional model.

Choosing and Reporting the Summary

The best summary depends on the question, not on which number appears most directly in a regression table. A conditional coefficient can be useful when the question concerns comparisons among people with the same covariates, or when the model’s conditional parameter is itself of scientific interest. A standardized contrast is useful when the question concerns the average outcome difference or risk in a specified population.

For a binary outcome, state the measure explicitly. A risk difference is in percentage points; a risk ratio compares risks; an odds ratio compares odds. Do not describe one as another. In particular, a logistic-regression coefficient is on the log-odds scale, and exponentiating it gives a conditional odds ratio under the specified model. It is not a risk difference. To report a population-average risk difference, calculate and contrast the standardized risks.

For either summary, report the exposure contrast, covariates used for adjustment, target population, outcome scale, and assumptions supporting the interpretation. If the estimate is causal, explain that its causal interpretation relies on the identification assumptions; adjustment by regression alone does not establish them.

Common mistake: “The logistic-regression coefficient gives the population-average change in risk.” This is incorrect: exponentiating the coefficient gives a conditional odds ratio, not a population-average risk contrast. For a population-average risk difference, predict risk under each exposure level for the target population’s covariate profiles, average each set of predictions, and subtract the averages.

Why the Distinction Matters

A conditional coefficient and a standardized contrast may both be correct while answering different questions. Treating them as interchangeable can lead to a misleading interpretation, an inappropriate comparison across studies, or a report that omits the population to which the estimate applies. This is especially important when a nonlinear model is used, because the coefficient’s scale and the averaging operation affect the result.

Before communicating a regression estimate, ask: Is it conditional on covariates or averaged over a target population? What is its scale? Which population distribution was used? Answering these questions makes the reported effect interpretable and helps readers distinguish model parameters from population-level contrasts.

Check Your Understanding

Use the distinctions and calculations in this tutorial to answer the following questions.

In an identity-link mean model with a constant exposure coefficient and no exposure-by-covariate term, why does averaging the conditional mean difference leave the coefficient unchanged?
In the logistic example, what does the conditional odds ratio of 2 compare, and what two risks are used to calculate the standardized risk difference?
Why is the marginal odds ratio of approximately 1.719 not a contradiction of the conditional odds ratio of 2?
In the changed target population example, what covariate distribution was used, and what was the standardized risk difference in percentage points?
What additional assumptions are needed to interpret an adjusted regression contrast causally in an observational study?