Analyze by Assignment Rather Than Treatment Received

Why Analyze Participants by Assignment?

Randomization creates comparable groups by assigning participants, not by ensuring that every participant follows the assigned plan. Some participants may not start the intervention, may stop it, or may receive it despite assignment to the control group. The analysis should preserve the comparison that randomization created: compare participants according to their assigned groups.

This is called an intention-to-treat analysis. Its central estimate is the effect of assignment to the intervention rather than the effect of actually receiving it. That distinction matters: assignment can affect outcomes through treatment uptake and through other parts of the intervention policy, while treatment received is a post-randomization behavior that may be related to participants’ characteristics and outcomes.

For a causal interpretation, the assignment groups must be defined clearly, and the randomization must be carried out as intended. It is also necessary that the recorded outcome correspond to the assigned condition and follow-up being studied. In the usual potential-outcomes description, each participant has a potential outcome under each assignment; randomization makes assignment independent of those potential outcomes, subject to the trial’s design. The assignment effect is therefore a causal effect of the randomized assignment policy in the trial population, not automatically the effect of taking the treatment.

Regression With Randomized Assignment

For a quantitative outcome \(Y_i\), a simple regression uses the assignment indicator as its predictor:

Here \(\beta_0\) is the fitted mean outcome for participants assigned to control, and \(\beta_1\) is the difference between the fitted mean for participants assigned to intervention and the fitted mean for participants assigned to control. With only this indicator in the model, the ordinary least squares estimate of \(\beta_1\) is exactly the difference between the two sample means. The regression form is useful because it expresses the randomized comparison as a coefficient and can be extended to include suitable baseline covariates.

The point-estimate identity does not require normally distributed errors. For conventional model-based inference, additional conditions may be used; the choice of standard error and inference method should also respect the trial design. The regression coefficient alone does not resolve missing outcomes, departures from the randomization procedure, or a poorly defined intervention.

Why the Regression Coefficient Is the Difference in Means

The least-squares fit chooses both coefficients jointly to minimize the sum of squared residuals. When \(Z_i=0\), the fitted value is \(\beta_0\); when \(Z_i=1\), it is \(\beta_0+\beta_1\). Write these two fitted group means as \(m_0=\beta_0\) and \(m_1=\beta_0+\beta_1\). This is a one-to-one re-expression of the two regression coefficients: any pair \((m_0,m_1)\) corresponds to \(\beta_0=m_0\) and \(\beta_1=m_1-m_0\).

The residual sum of squares can therefore be minimized over \(m_0\) and \(m_1\) independently, one for each assigned group. In a group with observations \(Y_i\) and sample mean \(\bar{Y}\), the sum of squared deviations around any proposed fitted value \(m\) satisfies the identity below. Since the first term does not depend on \(m\), the sum is smallest at \(m=\bar{Y}\).

Thus, minimizing jointly sets the control fitted value to the control sample mean and the intervention fitted value to the intervention sample mean. Subtracting them gives the assignment coefficient:

The joint minimization matters. If one arbitrarily fixed \(\beta_1\) first, the two fitted group values would be constrained to differ by that fixed amount and would not generally equal both group means. The difference-in-means result follows because the two coefficients are estimated together.

What Changes When Participants Do Not Follow Assignment?

An intention-to-treat analysis retains each participant in the group to which randomization assigned them. Someone assigned to intervention remains in that group even if they never receive it. Someone assigned to control remains in control even if they access the intervention elsewhere. The ITT estimate consequently reflects the effect of the assignment strategy as implemented, including the uptake and crossover that occurred.

Replacing \(Z_i\) with an indicator for treatment actually received changes the comparison. Receipt happens after randomization and may depend on motivation, health, access, prognosis, or early response. These factors can also be related to the outcome. Randomization guarantees comparability by assigned group, not comparability by the groups participants later form through treatment uptake. As a result, the received-treatment coefficient is generally an association, not a randomized estimate of the treatment’s causal effect.

Assignment Effects Are Not the Same as Effects of Receipt

The ITT effect is often directly relevant to a practical question: what happens, on average, when eligible participants are assigned or offered an intervention under trial-like conditions? When adherence is incomplete, its estimate can be smaller in magnitude than an effect among participants who actually receive treatment, but this is not guaranteed. Variation in uptake, crossover, outcome effects, and chance can all affect the observed estimate. One should not divide the ITT estimate by an adherence percentage and call the result a treatment effect without additional assumptions and an appropriate method.

If the scientific question concerns receipt rather than assignment, the answer cannot generally be obtained just by substituting a receipt indicator into the randomized regression. That change discards the protection provided by randomization. More specialized causal methods require additional assumptions and are outside this assignment-based comparison.

Baseline adjustment can be used to improve precision when chosen and specified appropriately, but the randomized assignment indicator remains the treatment predictor for an ITT analysis. If a model includes assignment-by-baseline interactions, the assignment coefficient alone may describe a conditional contrast rather than the overall assignment effect; the relevant model-based contrasts must be interpreted accordingly. A post-randomization variable such as treatment receipt is not a baseline covariate.

Common Mistakes and a Practical Check

• Reclassifying participants by what they received. This changes the randomized groups and can introduce differences related to adherence or access. Keep the original assignment indicator for the ITT comparison.
• Calling the assignment estimate the effect of taking treatment. The estimate concerns assignment to the intervention strategy. Uptake and crossover are part of how that strategy operated.
• Assuming nonadherence invalidates the ITT comparison. Nonadherence may affect the size and practical meaning of the assignment effect, but it does not justify regrouping participants by receipt.
• Treating an assignment coefficient as meaningful without checking the model. With only an assignment indicator, it is the difference in group means. In models with interactions or other terms, interpret the coefficient according to the model rather than assuming it is always the overall effect.

Before reporting an assignment-based regression, check that \(Z_i\) records the randomized group, that the outcome and follow-up are defined consistently, and that participants are still grouped according to assignment. Then state the estimated contrast in the outcome’s units and describe it as an effect of assignment. These steps keep the statistical calculation connected to the comparison the trial was designed to make.