Defining the Association Before Choosing Adjustments

Start With the Association You Want to Describe

A regression model can include many predictors, but the meaning of its exposure coefficient depends on the question being asked and on what else is in the model. Before choosing adjustments, specify what outcome is being described, which exposure is being compared, whose outcomes are of interest, and which exposure comparison matters. Otherwise, a coefficient can be calculated correctly yet answer a different question from the one intended.

In this tutorial, exposure means the variable whose association with the outcome is of interest. It need not be a treatment, and calling it an exposure does not imply that it causes the outcome. A regression association describes how outcomes vary with exposure in the data; by itself, it does not establish a causal effect.

Definition: To define an association question, specify (1) the outcome \(Y\), including its units and measurement time; (2) the exposure \(X\), including the values or contrast to compare; (3) the target population, the population whose association is of interest; and (4) whether the comparison is marginal or conditional on selected covariates.

For example, “Is exercise associated with blood pressure?” leaves important details unresolved. One could specify systolic blood pressure in millimetres of mercury measured at a particular visit, weekly exercise hours, a population of adults attending a set of clinics, and a comparison of people reporting 1 versus 3 hours per week. One must still decide whether to compare overall mean blood pressures at those exposure levels or compare them among people with the same age and smoking status.

Marginal and Conditional Associations

A marginal association compares outcomes across exposure values without conditioning on selected covariates. For a quantitative outcome and two exposure values \(x_1\) and \(x_0\), one such comparison is the difference in population mean outcomes:

\Delta_{\text{marginal}} = E[Y\mid X=x_1]-E[Y\mid X=x_0].

This describes the difference in mean outcomes between members of the target population who have the two exposure values. It does not, on its own, compare people who are alike in other characteristics. For a continuous exposure, one may instead describe a change in mean outcome per unit of exposure, if a linear summary is suitable.

A conditional association describes the exposure-outcome comparison at specified values of one or more covariates, denoted collectively by \(Z\). For example, a conditional mean difference at covariate value \(z\) is:

\Delta_{\text{conditional}}(z) = E[Y\mid X=x_1,Z=z]-E[Y\mid X=x_0,Z=z].

The covariates in \(Z\) must be named: “adjusted association” is incomplete unless it is clear what the model conditions on. A conditional comparison may vary across \(z\). If the exposure association differs by age, for instance, a single conditional difference without an age value may not fully describe it.

In a linear regression model with a common exposure slope,

E[Y\mid X,Z] = \beta_0+\beta_1 X+\beta_2 Z,

\(\beta_1\) is the expected difference in \(Y\) for a one-unit difference in \(X\), comparing observations with the same \(Z\). This interpretation uses the model’s linearity and common-slope form. In a model with an \(X\)-by-\(Z\) interaction, the conditional exposure association can depend on \(Z\), so there is not one common slope for all covariate values.

Why the Two Comparisons Can Differ

A marginal mean averages over the covariate values found among people at each exposure level. A conditional mean comparison instead fixes the covariate value before comparing exposure levels. The averaging can be expressed using the law of total expectation. For a categorical covariate \(Z\) with possible values \(z\):

E[Y\mid X=x] = \sum_z E[Y\mid X=x,Z=z]P(Z=z\mid X=x).

The weights \(P(Z=z\mid X=x)\) describe the covariate distribution at exposure level \(x\). They can differ across exposure levels. Therefore, a marginal difference is not generally the same quantity as a conditional difference at a particular \(z\), or as a weighted average of conditional differences using a separately chosen common distribution. This is a mathematical distinction between summaries; it does not, by itself, explain why the covariate distributions differ or establish a causal interpretation.

Worked Example: Defining a Marginal Comparison

Suppose a hypothetical survey describes adults in a specified region. The outcome is mean systolic blood pressure at the survey visit, measured in mmHg. The exposure is reported weekly exercise time. Consider adults reporting 1 hour per week and adults reporting 3 hours per week. The survey summaries give mean blood pressures of 142 mmHg and 130 mmHg, respectively.

The marginal mean difference, defined as the mean for 3 hours minus the mean for 1 hour, is:

\Delta_{\text{marginal}} = 130-142=-12\text{ mmHg}.

In this hypothetical survey, adults reporting 3 hours of weekly exercise had a mean systolic blood pressure 12 mmHg lower than adults reporting 1 hour. This is a descriptive marginal association for the specified population and comparison. It is not a claim that increasing an individual’s exercise would lower that individual’s blood pressure by 12 mmHg.

The outcome, exposure, target population, and contrast are now explicit. No covariates were used to define this marginal comparison. Adding age or another covariate would change the question to a conditional or otherwise adjusted comparison, and that change should be intentional.

Regression Coefficients Need a Comparison in Context

Before interpreting a regression coefficient, state the model and the conditions under which its coefficient has the intended meaning. In the following linear model, suppose \(Y\) is a continuous outcome, \(X\) is the exposure, and \(Z\) is one selected covariate:

Y_i = \beta_0+\beta_1X_i+\beta_2Z_i+\varepsilon_i, \qquad E[\varepsilon_i\mid X_i,Z_i]=0.

For \(\beta_1\) to describe one common conditional linear association, the conditional mean must be adequately represented by this additive form over the exposure and covariate values being considered. In particular, the form assumes a linear mean relationship in \(X\) and no \(X\)-by-\(Z\) interaction. The observations must be measured in a way that supports the comparison, and the data must include relevant exposure variation at the covariate values where the model is interpreted. Independence and the sampling design also matter for uncertainty calculations and for generalizing beyond the observed data.

These are modeling conditions, not guarantees supplied by the word “adjusted.” If the relationship is curved or the exposure slope differs with \(Z\), the stated coefficient may be an inadequate summary. Even a well-fitting model describes a conditional association; the model form alone does not turn that association into a causal effect.

Worked Example: Interpreting a Conditional Slope

Consider a hypothetical analysis of systolic blood pressure \(Y\), in mmHg, against weekly exercise \(X\), in hours, and age \(Z\), in years. Suppose the fitted model is:

\widehat{Y}=150-0.50X+0.30Z.

The target population is adults represented by the hypothetical data, and the specified comparison is the conditional association between exercise and mean blood pressure at the same age. The estimated exposure coefficient is \(-0.50\) mmHg per additional weekly exercise hour, conditional on age.

For a comparison of 3 versus 1 exercise hour per week at the same age, the fitted difference is:

\widehat{Y}(3,z)-\widehat{Y}(1,z) = [150-0.50(3)+0.30z]-[150-0.50(1)+0.30z] = -1.00\text{ mmHg}.

The intercept and age term cancel because the comparison holds age fixed. Under the additive linear model, the fitted mean blood pressure is 1.00 mmHg lower for a two-hour-per-week difference in reported exercise, at the same age. For this interpretation, assume the conditional mean is approximately linear in exercise and age over the relevant range, that a common exercise slope is reasonable, that relevant exercise values occur at the ages being compared, and that observations and measurements are appropriate for the intended population. These assumptions would need support from the design, subject knowledge, and model assessment; the coefficient alone cannot verify them.

Working Through a Difference in Covariate Mix

A comparison of marginal means can differ from a conditional comparison simply because the marginal means average over the covariate distributions present at each exposure level. The arithmetic below makes that distinction visible without assigning a causal explanation to it.

Worked Example: Averaging Across Two Covariate Groups

In a hypothetical data set, \(Z\) indicates a lower or higher age group. The table gives the number of observations and mean outcome in each exposure-by-age group. The outcome is a score in points, and \(X\) has values 0 and 1.

Exposure \(X\)	Age group \(Z\)	Number of observations	Mean score (points)
0	Lower	90	10
0	Higher	10	20
1	Lower	10	12
1	Higher	90	22

Within the lower age group, the conditional mean difference for \(X=1\) versus \(X=0\) is \(12-10=2\) points. Within the higher age group, it is \(22-20=2\) points. The marginal mean for \(X=0\) weights the two age groups according to their counts:

\bar{Y}_{X=0} = \frac{90(10)+10(20)}{100} = 11\text{ points}.

For \(X=1\), the corresponding marginal mean is:

\bar{Y}_{X=1} = \frac{10(12)+90(22)}{100} = 21\text{ points}.

Thus the marginal difference is \(21-11=10\) points, whereas each age-specific conditional difference is 2 points. The marginal comparison uses different age-group weights at the two exposure levels. That is why these summaries differ; this calculation alone does not say what produced the different group composition.

A model that includes age would summarize an association conditional on age, given its model form. It would not report the same quantity as the raw marginal difference. If the goal were a population summary that averages conditional predictions over a chosen common age distribution, that distribution would need to be stated too. “Adjusted” does not uniquely name a single comparison.

Specify the Question Before Selecting Covariates

The order of reasoning matters. First state the association of interest. Then decide whether the target comparison is marginal or conditional, and identify exactly which covariates define the conditional comparison. Only then choose a model capable of representing that quantity. Covariates should not be added merely because they are available, nor omitted merely because they do not change a preferred coefficient.

For a marginal description, the question concerns the overall exposure-specific outcome means in the target population. For a conditional description, the question concerns exposure comparisons at stated covariate values or within stated covariate groups. A regression model can also support a summary formed by averaging predictions over a declared distribution, but that is a distinct target from a raw marginal comparison and should be described explicitly.

This distinction matters for communication as well as analysis. A result should name the outcome and units, the exposure contrast and units, the population, and the conditioning set or averaging distribution. A coefficient without these details can invite readers to interpret a conditional association as an overall comparison, or an observational association as a causal effect.

Common Mistakes

Calling every exposure coefficient an overall association. In a model with covariates, the coefficient usually describes a comparison conditional on those covariates, subject to the model form.
Using “adjusted” without naming what was adjusted for. State the covariates and whether the reported comparison is conditional at particular values or averaged over a specified distribution.
Assuming marginal and conditional comparisons must agree. They can differ because they average across covariates differently, as the numerical example shows.
Interpreting a descriptive association as an intervention effect. A regression coefficient alone does not establish that changing the exposure would change the outcome.
Leaving the comparison vague. Specify meaningful exposure values or a unit change, the outcome scale and units, and the population to which the description refers.

Key takeaway: Define the outcome, exposure, target population, and exposure comparison before choosing covariates. A marginal association and an association conditional on selected covariates are different summaries; the model and the wording of the result must make clear which one is being described.

Check Your Understanding

Try these questions using the definitions and distinctions in this tutorial.

For a study of weekly exercise and systolic blood pressure, name a complete outcome, exposure, target population, and exposure contrast.
In a regression of outcome \(Y\) on exposure \(X\) and age \(Z\), what does the exposure coefficient describe under an additive linear conditional-mean model?
Why can a marginal mean difference differ from a conditional mean difference even when both are calculated correctly?
What additional detail is needed to make the phrase “adjusted association” precise?
Does a conditional regression coefficient by itself establish that changing the exposure would cause a change in the outcome? Explain briefly.