Adjusted Contrasts Among Three Rehabilitation Programs

Why Compare Adjusted Program Outcomes?

Suppose patients enter one of three rehabilitation programs with different baseline severity. Comparing the programs' raw average outcomes may mix two things: differences in program assignment and differences in who began with more severe symptoms. Regression can describe each program's fitted outcome at the same baseline value, making the comparison explicit. The resulting comparison is an adjusted contrast: a difference between fitted means under a stated model, not automatically proof that one program caused better outcomes.

The key is to compare like with like. We will model a quantitative recovery score using a baseline severity measure and an indicator for program membership. We will use a common baseline slope, so the adjusted difference between programs does not depend on the baseline value at which the fitted means are reported. That assumption is useful for a focused comparison, but it should be considered rather than taken for granted.

Model and Interpretation

Let \(Y_i\) be patient \(i\)'s recovery score at follow-up, with higher values indicating better recovery. Let \(x_i\) be that patient's baseline severity score, and let \(z_i=x_i-50\), so that \(z_i=0\) corresponds to a baseline score of 50. Let programs A, B, and C be the three possible program assignments. Taking A as the reference program, define \(I_{Bi}=1\) for a patient in B and 0 otherwise, and define \(I_{Ci}\) similarly for C.

For a patient with baseline value \(x^*\), the fitted means are obtained by inserting \(z^*=x^*-50\) and the relevant program indicators. In particular, the fitted means for A, B, and C are \(\beta_0+\beta_1z^*\), \(\beta_0+\beta_1z^*+\beta_B\), and \(\beta_0+\beta_1z^*+\beta_C\). Subtracting these expressions gives the pairwise contrasts:

These differences are constant across baseline values because the model has a common slope and no program-by-baseline interaction. That is a model property, not a general fact about rehabilitation outcomes.

Assumptions for the Comparisons

The regression describes the conditional mean of the outcome as a straight-line function of baseline severity within each program, with the same slope in each program. The errors have mean zero given the predictors, are independent across patients, and have constant variance. For the usual exact small-sample \(t\)-based confidence intervals and tests, the errors are also assumed normally distributed. The data must contain patients from each program across a useful shared range of baseline severity; otherwise, fitted means at a common value may rely on extrapolation.

The common-slope form is especially important here. If program differences change with baseline severity, a single adjusted contrast cannot summarize the difference at every baseline value. Also, adjustment alone does not establish causation. A causal interpretation requires a suitable study design and additional conditions; in an observational comparison, unmeasured differences between patients may remain.

Uncertainty for a Pairwise Contrast

A contrast estimate has sampling uncertainty because the regression coefficients are estimated from data. Its standard error depends on the variances of the coefficients involved and, when a contrast combines coefficients, their covariance. For a linear combination \(a\hat{\beta}_B+b\hat{\beta}_C\), its estimated variance is

For a 95% confidence interval using the usual regression \(t\) procedure, the estimate is paired with its standard error and the appropriate \(t\) critical value with residual degrees of freedom \(df\). The interval describes uncertainty in the population contrast under the model and its assumptions. It is not a statement that there is a 95% probability that a fixed parameter lies in this particular calculated interval.

Changing the Reference Program

A reference category is a coding choice. It determines which program's fitted mean is represented by the intercept and which differences appear directly as coefficients. It does not determine which pairwise comparisons are scientifically meaningful. To see this, take C rather than A as the reference. Write the model as

Now \(\alpha_0\) is the fitted mean for C at baseline 50; \(\alpha_A\) is A minus C, and \(\alpha_B\) is B minus C. From the first coding, the coefficient values transform as follows:

The intercept changes from 68 to 70, and the displayed program coefficients change from 5 and 2 to -2 and 3. But the comparisons are recovered directly: B minus C is \(\hat{\alpha}_B=3\), C minus A is \(-\hat{\alpha}_A=2\), and B minus A is \(\hat{\alpha}_B-\hat{\alpha}_A=3-(-2)=5\) points.

Common Pitfalls and Why the Details Matter

• Reading each coefficient as a comparison with every other program. With A as the reference, \(\beta_B\) is B minus A, not B minus C. Calculate B minus C as \(\beta_B-\beta_C\).
• Ignoring covariance when subtracting coefficients. The variance of a difference includes \(-2\widehat{\operatorname{Cov}}(\hat{\beta}_B,\hat{\beta}_C)\). Adding the two variances alone gives the wrong standard error unless the covariance is zero.
• Calling adjusted means raw averages. They are model-based fitted means at a specified baseline value, not necessarily the observed average score in each program.
• Assuming adjustment proves a treatment effect. Regression adjustment describes a conditional association. Causal conclusions require suitable design and assumptions beyond the fitted equation.
• Changing the reference and thinking the result changed. Individual coefficient values depend on coding, but correctly formed comparisons between the same two programs remain the same.

Adjusted contrasts are useful because they express program differences in outcome units, at a shared baseline value, with uncertainty attached. A clear report states which subtraction is being made, the covariate value used for fitted means, the model assumptions relevant to that comparison, and the interval or test result. That makes the direction, size, and precision of each comparison interpretable without confusing coding choices with substantive conclusions.