Confounding by Indication in Treatment Comparisons

Why Treatment Comparisons Can Mislead

In clinical care, treatment is rarely assigned at random. A clinician may prescribe a stronger treatment, or begin treatment sooner, because a patient has more severe symptoms or a higher risk of a poor outcome. Those same clinical features may also predict the outcome. As a result, a comparison of treated and untreated patients can make a helpful treatment look ineffective—or even harmful—because the groups differed before treatment began.

This pattern is called confounding by indication. The “indication” is the clinical reason for considering or giving a treatment, such as symptom severity, disease stage, or an elevated risk score. The central question is not simply whether treatment and outcome are associated. It is whether the observed association mixes a treatment difference with pre-treatment differences that helped determine who received treatment.

Definition: Confounding by indication occurs when a pre-treatment clinical indication influences treatment and also predicts the outcome. Because treatment groups then differ in that indication, their unadjusted outcome contrast can combine the treatment-outcome association with differences in clinical need or prognosis.

The Common-Cause Structure

Let \(T\) denote treatment, \(Y\) the outcome, and \(I\) a measured clinical indication recorded before treatment. The basic structure is \(I \rightarrow T\) and \(I \rightarrow Y\): the indication is a common cause of treatment and outcome. For example, greater symptom severity may make treatment more likely and may also predict a longer recovery time. An unadjusted comparison of recovery time by treatment status then compares groups with different severity distributions.

The timing and role of a variable matter. A candidate indication should be measured before treatment if it is to represent a reason for treatment rather than a consequence of treatment. It must also be relevant to the comparison: an indication that predicts treatment but has no relationship to outcome, conditional on other relevant variables, does not create this common-cause path by itself. Conversely, a clinical variable that predicts outcome but does not differ by treatment may be prognostic without confounding the treatment comparison.

When indication is measured, one approach is to compare treated and untreated patients with the same indication values, or to include the indication in a regression model. This changes the question from a marginal comparison between all treated and all untreated patients to a conditional comparison at a fixed indication value. The two associations need not be equal.

Worked Example: Drawing the Indication Structure

Suppose clinicians are more likely to prescribe an intensive therapy to patients with severe symptoms. Severe symptoms also predict a higher follow-up symptom score, where higher scores are worse. Identify the relevant variables and explain why the raw treatment comparison may be misleading.

Let \(T=1\) indicate intensive therapy, \(T=0\) no intensive therapy, \(I\) indicate severe symptoms before treatment, and \(Y\) be the follow-up symptom score. The clinical structure is:

I \longrightarrow T \qquad\text{and}\qquad I \longrightarrow Y

Because severe symptoms increase both the chance of intensive therapy and the expected follow-up score, treated patients may have worse outcomes on average even if treatment improves outcomes for patients with the same initial severity. The raw difference in mean follow-up scores does not separate these two features of the data.

The relevant adjustment variable is the pre-treatment symptom severity. Adjusting for a score measured after treatment would not represent the original indication: it could already have been changed by treatment.

Marginal and Indication-Adjusted Associations

A marginal association compares outcomes across treatment groups without holding indication fixed. For a quantitative outcome, a common marginal summary is the difference in group means, \(\bar{Y}_{T=1}-\bar{Y}_{T=0}\). An indication-adjusted association compares outcomes between treatment groups at the same indication value, or summarizes such comparisons using a model. The treatment coefficient in a suitable regression model is one way to express a conditional association.

For example, with a binary treatment and one binary indication, an additive linear model is:

Y_i=\beta_0+\beta_1T_i+\beta_2I_i+\varepsilon_i

Here, \(Y_i\) is the outcome for patient \(i\), \(T_i\) is treatment status, and \(I_i\) is indication status. Under this model, \(\beta_1\) is the difference in mean outcome associated with treatment among patients with the same indication value. The model assumes that the conditional mean is additive and linear as written: the treatment contrast is the same at each indication value. If treatment differences vary by indication, this model’s single treatment coefficient does not describe both contrasts.

To interpret a fitted treatment coefficient as a conditional association, the model must be appropriate for the outcome and predictors, and there must be treated and untreated patients at the indication values being compared. The coefficient is not automatically a causal effect. A causal interpretation additionally requires that the measured adjustment set adequately controls common causes, that treatment and outcome are defined consistently, and that comparable patients could have received either treatment. Adjustment cannot establish these conditions by itself.

Worked Example: A Crude Difference and a Stratified Difference

Consider a hypothetical comparison of a treatment among 200 patients. The pre-treatment indication is either mild or severe symptoms. The table reports the number of patients and their mean follow-up score; higher scores are worse.

Indication	Treatment	Number of patients	Mean follow-up score
Mild	Untreated	80	4
Mild	Treated	20	5
Severe	Untreated	20	10
Severe	Treated	80	11

First calculate the marginal mean for each treatment group. There are 100 treated patients, so their mean is:

\bar{Y}_{T=1} = \frac{20(5)+80(11)}{100} = \frac{100+880}{100} = 9.8

There are also 100 untreated patients, with mean:

\bar{Y}_{T=0} = \frac{80(4)+20(10)}{100} = \frac{320+200}{100} = 5.2

The unadjusted difference, treated minus untreated, is \(9.8-5.2=4.6\) score points. On its own, this comparison makes treatment appear associated with a substantially worse outcome.

Now compare patients within each indication group. Among those with mild symptoms, the difference is \(5-4=1\) point. Among those with severe symptoms, it is \(11-10=1\) point. In this hypothetical example, the treatment-outcome difference is consistently 1 point higher for treated patients within each indication category. The crude difference is much larger because 80% of treated patients had severe symptoms, compared with only 20% of untreated patients.

The arithmetic makes the source of the discrepancy clear: treatment and indication are associated, and indication predicts the outcome. The data show different marginal and conditional associations; they do not, without further assumptions, establish that treatment caused a 1-point increase.

Why Adjustment Changes the Comparison

The marginal mean in each treatment group is a weighted average of the indication-specific means. The weights are the proportions of that group in each indication category. If those proportions differ by treatment, the marginal contrast combines within-indication differences with differences in the mix of indications.

For two indication groups, write \(p_t\) for the proportion with severe symptoms among patients with treatment status \(t\). Let \(\mu_{t,S}\) and \(\mu_{t,M}\) be the mean outcomes for treated or untreated patients with severe or mild symptoms. Then:

\bar{Y}_{T=t} = p_t\mu_{t,S}+(1-p_t)\mu_{t,M}

If \(p_1\) and \(p_0\) differ, the treated and untreated marginal means use different weights. This is why a marginal difference can differ from the comparisons made within indication groups. Adjustment aims to compare treatment groups at the same indication values, rather than allowing their different clinical mixes to determine the comparison.

Key takeaway: When indication predicts both treatment and outcome, the unadjusted treatment-outcome association may reflect clinical differences that existed before treatment. Comparing or adjusting for indication changes the comparison, but does not by itself prove a causal treatment effect.

Worked Example: Interpreting an Adjusted Regression Coefficient

Suppose a hypothetical regression analysis uses the model \(Y_i=\beta_0+\beta_1T_i+\beta_2I_i+\varepsilon_i\), where \(Y_i\) is a follow-up pain score in points, \(T_i=1\) for a new treatment, and \(I_i=1\) for high pre-treatment pain. A fitted model reports \(\hat{\beta}_1=1.4\) and \(\hat{\beta}_2=3.0\). Assume for this interpretation that the additive model is suitable and that both treatment groups include patients with low and high pre-treatment pain.

For a patient with low pre-treatment pain, the model’s fitted mean is \(\hat{\beta}_0\) without treatment and \(\hat{\beta}_0+1.4\) with treatment. Their fitted difference is:

(\hat{\beta}_0+1.4)-\hat{\beta}_0=1.4 \text{ pain-score points}

For a patient with high pre-treatment pain, the fitted means are \(\hat{\beta}_0+3.0\) without treatment and \(\hat{\beta}_0+1.4+3.0\) with treatment. The fitted difference is again:

(\hat{\beta}_0+1.4+3.0)-(\hat{\beta}_0+3.0)=1.4 \text{ pain-score points}

Thus, \(\hat{\beta}_1=1.4\) is an adjusted conditional association: within the model, treated patients have a predicted follow-up pain score 1.4 points higher than untreated patients with the same pre-treatment pain category. Since higher scores represent more pain, this is an association with a worse score—not evidence on its own that the treatment caused harm. The coefficient is conditional on the measured indication and the model assumptions.

When Adjustment Is Not Enough

Clinical records may not capture all the information that guided treatment. A clinician might consider a subtle examination finding, prior treatment response, or a patient’s overall frailty even when those features are absent from the analysis. If such factors also predict the outcome, adjusting for a recorded indication may leave differences between treatment groups. The adjusted coefficient then remains an association that may reflect more than treatment.

Even a well-measured indication can be modeled poorly. If the treatment association differs between mild and severe patients, a model that assumes one common treatment coefficient may conceal that variation. In that case, the conditional comparisons need to be described by indication level rather than summarized as though a single coefficient applied to everyone. Likewise, a comparison is not supported at a severity level where one treatment group is absent.

These cautions do not make adjusted comparisons useless. They clarify what the calculation says: it compares outcomes at the same measured indication values under a specified model. Clinical knowledge is needed to judge whether those indications are sufficient and whether the model represents the comparison of interest.

Common Mistake

Wrong reasoning: “The treated group had worse outcomes, so the treatment must be harmful.” This treats a marginal association as a treatment effect, ignoring that patients may have received treatment precisely because they were sicker.
Correction: Check whether pre-treatment indications predict both treatment and outcome. Compare patients with similar measured indications, and describe the result as adjusted or conditional. Do not claim causation unless the assumptions needed for a causal interpretation are also justified.
Another mistake: Adjust for every available clinical variable automatically. Instead, use the timing and causal role of each variable to decide whether it is an appropriate adjustment variable. A post-treatment measurement is not the pre-treatment reason for treatment.

Check Your Understanding

Answer each question using the direction and units specified.

A treatment is more common among patients with advanced disease, and advanced disease predicts a higher follow-up score. What common-cause structure could explain a difference between the marginal and adjusted treatment associations?
In the worked table, calculate the treated-minus-untreated mean difference among patients with mild symptoms and among those with severe symptoms. How do these compare with the marginal difference?
In a model with treatment coded 1 for treated and 0 for untreated, what does a treatment coefficient of \(-2\) mean for a quantitative outcome, conditional on the included indications?
Why must an indication used for adjustment be measured before treatment if it is meant to represent a reason for treatment?
Does including a measured clinical indication in a regression model prove that the treatment coefficient is causal? State one additional concern that may remain.