# Interpreting the effect of treatment: the difference between conditional and marginal models

Adjusting for covariates in a multivariate model is a common practice in both randomized (to increase the accuracy of estimates) and observational studies, in order to take into account a skewed distribution of covariates and confounders.
However, the results of this correction must be correctly interpreted.

There is one thing to take into account.
You can, from your analysis, get either “conditional” treatment effects (i.e. the average effect on the individual patient) or “marginal” effects (where the effect you estimate is the effect on the population).

Let’s take the case of a randomized clinical trial

In the case of a randomized trial, randomization should ensure equal distribution of covariates and confounding factors in randomized groups, whereby using a multivariate model might be useless. However, it is known that when adjusted, the estimated coefficients of the treatment effect for covariates and confounders would be absolutely identical but would have greater precision (more narrow confidence intervals).

Warning: this is true when the effect you are going to measure is a continuous variable and you use a conventional regression model. The same happens when you want to evaluate a relationship between risks (relative risk). These measures have the property of “collapsibility“, that is, their value would not change if you used a simple regression or introduced a covariate into the model.

Let us take into account that we are talking about a randomized trial, where the confounders and covariates are equally distributed in groups.

Other types of association measures, such as the Odds Ratio and the Hazard Ratio, have different characteristics. Their value, even in a randomized study, is sensitive to any adjustment by covariates in the model. The value of the Odds Ratio would change if you used a simple logistic regression compared to the use of a multiple logistic regression.

The two calculated values are interpreted differently. The correct Odds Ratio for covariates and confounders is the “conditional” Odds Ratio that you would observe on the individual subject. The incorrect Odds Ratio is that which refers to the entire population if it were treated.

For example, if you get the adjusted Odds Ratio of a treatment equal to 0.7, it means that the risk of an event will be lower by 30% for an individual (such as the patient who does not know if to take the drug or not).

If the value of Odds Ratio is 0.7 but obtained without adjusting for potential covariates and confounders, it is interpreted as the observed effect on the whole population.

Let us take a case of an observational study.

It is almost a must introducing covariates and confounders when analyzing an observational study, given the predictable imbalance between the comparative groups.
Therefore the observed Odds Ratio or Hazard Ratio can be interpreted as conditional and referable to the individual subject. If, on the other hand, we wanted to extend the estimated effect of a treatment on the entire reference population, then other techniques are to be preferred in order to obtain the “marginal” estimates of the effect. For example, the Propensity Score or the Targeted Maximum Likelihood Estimation, or simply use non-collapsible estimators such as Risk Ratio in log-binomial models.

Further considerations.

The divergence between the conditional Odds Ratio and the marginal Odds Ratio depends mainly on two factors:

first, on the association between the covariates or the confounders and the outcome. The bigger is the association, the bigger is the divergence between conditional and marginal estimates. This can also be understood intuitively: if you interpret the effect in the light of the subject’s characteristics, the bigger is the impact of the characteristics on the effect, the bigger is the bias introduced with respect to the population estimates (all subjects).

Second, the marginal estimate is always closer to the null effect than the conditional one. So, the conditional effect overestimates the true effect of the treatment and the adjustment itself in a multivariate model, meaning that the overestimation is also more precise (narrower confidence intervals).

# What to do in practice?

Define the objective of the analysis: we are interested in estimates on the individual subject (for example in a clinical context) or on the population (for example in a context of health policies).

Choose the appropriate analysis. As anticipated above, it is possible to use techniques in which the estimate is non-collapsible to avoid bias due to the discrepancy between marginal and conditional estimates (for example, using a log-binomial regression instead of a logistic regression, in order to obtain the Relative Risk instead of the Odds Ratio).

Commenting the results in the light of the above: the effect of the treatment obtained could be “overestimated” by the multivariate adjustment compared to the true one.

THAT’S IT!