What “Holding Other Predictors Fixed” Means

Why “Holding Fixed” Needs a Precise Meaning

A multiple regression includes more than one predictor, so a comparison involving one predictor can be hard to interpret if other predictors also change. Suppose a model relates a course outcome to study time and a prior score. Comparing two students who studied for different amounts of time is not a clean illustration of the study-time comparison if their prior scores differ too. The model’s fitted values reflect both differences.

The phrase “holding other predictors fixed” describes a particular way to set up a comparison: choose two predictor profiles that have different values for the predictor of interest and identical values for every other predictor included in the model. The paired profiles make clear which model input has changed and which inputs have not.

“Other predictors” means the other predictors actually included in the model. It does not mean every characteristic that could possibly be relevant. Nor does it imply that the two people, places, or other units represented by the profiles are identical in every way. The comparison is defined by the listed model inputs.

A Simple Model and a Matched Pair

Consider an additive linear regression with outcome \(Y\) and two predictors, \(X_1\) and \(X_2\). The model for the conditional mean is

Here \(x_1\) and \(x_2\) are specified predictor values, \(\beta_0\) is the intercept, and \(\beta_1\) and \(\beta_2\) are the model coefficients. This is an additive model: it has no interaction term, so the coefficient for \(X_1\) does not vary with the value of \(X_2\). The fitted version uses estimated coefficients, often written \(b_0,b_1,b_2\), and gives a fitted response \(\hat{Y}\) for each profile.

For a comparison focused on \(X_1\), a matched pair of profiles might be \((x_1,x_2)=(3,70)\) and \((5,70)\). The \(X_1\) values differ, while both profiles have \(X_2=70\). By contrast, \((3,70)\) and \((5,76)\) do not hold \(X_2\) fixed: both predictors change. The second pair does not isolate a comparison in which only \(X_1\) changes.

Why Matching the Other Values Matters

The reason for matching the other predictor values can be seen directly from the fitted model. For profiles A and B, suppose the fitted model is \(\hat{Y}=b_0+b_1X_1+b_2X_2\), and both profiles have \(X_2=68\). Their fitted values are

Subtracting one fitted value from the other cancels the shared intercept and the shared \(X_2\) contribution:

This calculation shows what the matching accomplishes in this additive model: the \(X_2\) term contributes equally to both fitted values, so it cannot account for their difference. If the \(X_2\) values were unequal, their contributions would not cancel. Matching is therefore not just a verbal qualification; it is a way to ensure that the comparison has the structure described by “changing \(X_1\) while keeping \(X_2\) fixed.”

The same logic applies when a model includes several other predictors. For a comparison focused on one predictor, check each remaining predictor in turn. Each must have the same value in both profiles. For categorical predictors, matching means using the same category; for numerical predictors, it means using the same recorded value at the precision relevant to the comparison.

More Than Two Predictors

With several predictors, it can help to treat a profile as a checklist rather than relying on an impression that two cases are “similar.” Suppose a model includes exercise time, age, and smoking status, and the comparison of interest is exercise time. A valid matched pair must have:

• different exercise-time values;
• the same age value; and
• the same smoking-status category.

The pair need not represent two actual people in the observed data. A profile can be a set of predictor values selected for a model-based comparison. Whether the model can reasonably be used at those values is a separate question; matching alone does not establish that the profiles are well supported by the observed data.

What “Fixed” Does—and Does Not—Claim

In this setting, “fixed” describes the comparison: the other listed predictor values are set to be the same across the profiles. It does not mean those predictors are constant in the dataset, that they have no relationship with the outcome, or that they have been physically held constant in an experiment.

A matched profile comparison is also not automatically a causal comparison. In observational data, two people with the specified profiles may differ in unmeasured ways. A regression model adjusts for the predictors it includes, subject to how well the model represents the relationship, but matching on included predictors does not remove every possible source of difference. Therefore, describe the result as a model-based adjusted comparison unless the study design and assumptions justify a causal interpretation.

The additive model above also matters. If a regression includes an interaction between \(X_1\) and \(X_2\), the relationship associated with changing \(X_1\) can depend on the value of \(X_2\). The profiles still need to match on other predictors for a comparison that changes only \(X_1\), but the interpretation may depend on the particular matched value. Always check what terms the stated model contains.

Common Mistake

• Wrong reasoning: “These two profiles are close in age, so age is effectively held fixed,” or “They match on most of the other predictors, so the comparison is adjusted.”
• Correction: For a matched comparison, compare the specified predictor values directly. Every other included predictor must have the same value or category in both profiles. If even one differs, the pair changes more than the predictor of interest.

A second mistake is to interpret “holding fixed” as “controlling for everything.” The correction is to name the scope precisely: the comparison holds the other included model predictors fixed. It says nothing by itself about variables not in the model.