A Numerical Example of Confounding by a Common Cause

Why the Marginal and Adjusted Associations Can Differ

Suppose a hypothetical evaluation records whether students attended a study workshop, their score on a later assessment, and their level of preparation before the workshop was offered. Students with stronger prior preparation may be more likely to attend and may also score higher. In that situation, the overall difference in scores between attendees and nonattendees can reflect both the workshop–score association within preparation levels and the different mix of preparation levels in the two attendance groups.

This tutorial calculates that difference in a small dataset. The purpose is to make the distinction between a marginal association and an adjusted association concrete. Neither estimate, by itself, proves that workshop attendance causes a score change.

A Hypothetical Dataset

Let \(X=1\) mean that a student attended the workshop and \(X=0\) mean that the student did not. Let \(Z=1\) indicate high prior preparation and \(Z=0\) indicate low prior preparation. The outcome \(Y\) is the assessment score, measured in points. The following eight observations are hypothetical.

Preparation is associated with both variables in this dataset: three of the four highly prepared students attended, compared with one of the four students with low preparation; and the scores are higher in the high-preparation group. These patterns make preparation a plausible common cause in the example. The calculations that follow describe associations in these eight observations; the table alone cannot establish the causal relationships behind them.

Calculating the Adjusted Association

For this numerical example, use the additive linear regression model

Here, \(\beta_1\) is the model’s adjusted association between workshop attendance and score, in score points, for students with the same preparation value. The additive model assumes this conditional difference is the same at both preparation levels; it contains no \(X Z\) interaction. The coefficient \(\beta_2\) describes the model’s score difference by preparation level at a fixed workshop status.

The model is used as a description of conditional mean differences. For standard regression interpretation, observations are treated as independent, the conditional mean is represented by the stated additive form, and the errors have conditional mean zero given \(X\) and \(Z\). For inference using the usual small-sample \(t\) procedures, additional distributional and variance assumptions would be needed; no such inference is required for these calculations. To interpret \(\beta_1\) causally would require further design and causal assumptions that these data do not establish.

The cell means show the within-preparation comparisons directly. Among students with low preparation, the mean score is 10 points for nonattendees and 12 points for attendees. Among students with high preparation, it is 20 points for nonattendees and 22 points for attendees. Both within-level differences are 2 points. Because they agree, the additive model can represent all four cell means exactly.

Why the Estimates Differ: A Numerical Decomposition

The difference can be understood by separating the within-preparation association from the unequal composition of the attendance groups. For a binary exposure and binary covariate, write \(\mu_{xz}\) for the mean outcome among observations with \(X=x\) and \(Z=z\). The marginal mean for each exposure group averages its covariate-specific means using that group’s own covariate distribution. Adding and subtracting the same term gives

The first sum uses the within-preparation mean differences, weighted by the preparation distribution among attendees. The second sum reflects the fact that the preparation distributions differ between attendees and nonattendees. This identity is an arithmetic decomposition of the marginal difference; it does not by itself establish a causal interpretation.

A Common Preparation Distribution Gives Another View

One way to display a covariate-adjusted comparison is to calculate predicted means for both exposure levels using the same covariate distribution. Here, give each preparation level weight \(1/2\), the proportion with each level in the full dataset. This creates a standardized comparison in which the two exposure groups are compared using an identical preparation mix.

Interpretation and a Common Pitfall

The numerical example makes the role of the common cause visible. High preparation is more common among workshop attendees, and high preparation is also associated with higher scores. Consequently, the marginal comparison of 7.0 points combines a 2-point within-preparation difference with a 5-point composition component. Adjusting for preparation changes the comparison to 2 points in the additive model.

A common mistake is to say, “The marginal difference is 7 points, so attending the workshop raises scores by 7 points,” or to assume that the adjusted estimate must be causal simply because a covariate was included. The first statement mistakes an unadjusted association for a causal effect. The second overlooks whether the measured covariate is an appropriate common cause, whether important causes remain unmeasured, and whether the model is suitable. Here, “adjusted” has a precise statistical meaning: the model compares attendance groups at the same value of the measured preparation variable. Causal conclusions require additional evidence and assumptions.