From a Coefficient to a Fitted Difference
A multiple-regression coefficient is often reported as a change in the fitted outcome for a one-unit change in one predictor, with the other predictors held fixed. But a question may ask about a change of several units, a change in the opposite direction, or a two-category predictor. In each case, the same idea applies: scale the coefficient by the size of the specified change.
This tutorial focuses on a contrast: the difference between two fitted outcomes when one predictor changes and all the others stay the same. It is a difference in the model’s fitted values, not automatically a difference caused by changing someone’s predictor value. The distinction matters when interpreting regression results in context.
The Additive Multiple-Regression Model
For an outcome \(y\) and \(p\) predictors, consider the fitted additive linear equation
Here, \(\hat{y}\) is the fitted outcome, \(b_0\) is the fitted intercept, \(b_j\) is the fitted coefficient for predictor \(x_j\), and \(x_j\) is the value of that predictor. Each \(b_j\) is expressed in outcome units per unit of its predictor. For example, if the outcome is measured in points and \(x_j\) in hours, \(b_j\) has units of points per hour.
The calculation uses the fitted coefficient from the stated model. To interpret it as a fixed-coefficient contrast, the model must represent the predictor’s contribution linearly and additively over the values being compared. If the model includes an interaction involving \(x_j\), the contrast can depend on the values of other predictors; the simple calculation \(b_j\Delta x_j\) alone may then be incomplete. The contrast is also limited to values for which using the fitted model is reasonable.
No assumption about the distribution of the regression errors is needed just to subtract two fitted values. Such assumptions matter for other tasks, such as assessing uncertainty in an estimated coefficient. Here, the result is a calculation from the specified fitted equation, not a claim that the coefficient is exact in a population.
Why Multiplication Gives the Contrast
Let two predictor profiles have the same values for every predictor except \(x_j\). Call the original value \(x_j\), and let the changed value be \(x_j+\Delta x_j\). The fitted equation at the original profile contains the term \(b_jx_j\); at the changed profile, it contains \(b_j(x_j+\Delta x_j)\). Subtracting the first fitted value from the second cancels the intercept and every unchanged predictor term.
This subtraction explains two useful features. First, the original value of \(x_j\) does not appear in the final expression; only the size and direction of the change matter. Second, all unchanged predictor terms cancel. That cancellation is the algebra behind “holding the other predictors fixed.”
Worked Example: A Three-Hour Increase in Study Time
Suppose a hypothetical fitted model predicts a quiz score in points from weekly study time in hours and a prior assessment score in points:
where \(x_1\) is weekly study time and \(x_2\) is the prior assessment score. What fitted score difference corresponds to increasing study time by 3 hours while keeping prior assessment score fixed?
The study-time coefficient is \(b_1=2.4\) points per hour. The specified change is \(\Delta x_1=3\) hours, so
Thus, the fitted quiz score is 7.2 points higher for the profile with 3 additional weekly study hours, at the same prior assessment score. The units work out because hours cancel, leaving points. This is a model-based adjusted contrast; it does not by itself show that increasing study time causes a score increase.
The calculation uses the model’s additive linear form: the study-time coefficient does not change with prior assessment score. Both profiles must also be within a range where applying this fitted equation is reasonable. The calculation itself does not establish that those conditions hold in real data; they are part of how the model is specified and used.
Direction, Units, and Predictor Coding
A contrast is directional: it is defined as the fitted outcome after the change minus the fitted outcome before it. A positive result means the fitted outcome is higher after the specified change; a negative result means it is lower. If the predictor change is negative, the sign of \(\Delta x_j\) must be negative too. Keeping the direction explicit prevents a common sign error.
Units provide a useful check. Multiplying a coefficient in outcome units per predictor unit by a predictor change in predictor units should leave outcome units. For a change from one category to another in a binary predictor coded 0 and 1, the change is \(1-0=1\). The coefficient therefore gives the fitted difference for category 1 relative to category 0, with the other predictors held fixed. Reversing the comparison changes the sign.
Worked Example: A Reduction in Medication Dose
Consider a hypothetical additive model for a measured outcome in millimetres of mercury (mmHg), with dose measured in milligrams and age included as another predictor. Suppose the fitted dose coefficient is \(-0.85\) mmHg per milligram. What is the fitted outcome contrast for reducing dose by 4 mg while keeping age fixed?
The change is \(\Delta x=-4\) mg, not \(+4\) mg. Therefore,
The fitted outcome is 3.4 mmHg higher for the profile with a dose 4 mg lower, at the same age. The positive contrast follows from multiplying two negative quantities. Equivalently, increasing dose by 4 mg would have a fitted contrast of \((-0.85)(4)=-3.4\) mmHg.
This calculation assumes that the specified fitted model uses a constant linear dose coefficient over the dose change and that age is unchanged. It describes the model’s fitted contrast, not a guaranteed outcome change for an individual and not evidence on its own of a causal dose effect.
Worked Example: Comparing Two Categories
Suppose a hypothetical additive model predicts a test score in points using a binary indicator \(x\), coded 0 for one program category and 1 for another, along with a baseline score. The fitted coefficient for \(x\) is \(5.6\) points. Find the fitted score contrast for category 1 compared with category 0 at the same baseline score.
For this comparison, \(\Delta x=1-0=1\). The contrast is
The model’s fitted score is 5.6 points higher for category 1 than for category 0, holding baseline score fixed. If the comparison were stated in reverse—category 0 compared with category 1—the change would be \(0-1=-1\), giving
The additive model assigns the same fitted category contrast at each baseline score. The conclusion is about an adjusted fitted difference between categories; it is not, without further design and assumptions, a causal effect of program membership.
When This Calculation Is Useful
A coefficient is reported per one unit of its predictor, but the scientifically meaningful comparison is often larger, smaller, or reversed. Converting the coefficient to a specified contrast makes it easier to communicate a result in the units and scale relevant to a question. It also allows direct comparison of changes such as “three additional hours” or “four fewer milligrams” without confusing the coefficient’s per-unit scale with the contrast being asked about.
The calculation is intentionally narrower than calculating complete fitted outcomes for two profiles. It needs only the relevant coefficient and the predictor change, because the other predictor terms cancel. If a question asks for the fitted value for each profile, those values require substituting the full predictor values into the model; that is a different calculation.
A fixed-other-predictors contrast is also not the same as an unadjusted comparison between two groups. It compares the model’s fitted outcomes at the same values of the other included predictors. This description is conditional on the model and its adjustment set; it does not claim that the two groups are identical in every respect or that adjustment removes all possible differences.
Common Mistake
- Wrong reasoning: “The coefficient is 2.4, so any change in study time corresponds to a 2.4-point fitted difference.” This treats a per-hour coefficient as the contrast for every possible change.
- Correction: Multiply the coefficient by the signed change in hours. A 3-hour increase gives \(2.4\times3=7.2\) points; a 3-hour decrease gives \(2.4\times(-3)=-7.2\) points, with other predictors fixed.
Another frequent error is to describe an adjusted fitted contrast as a causal effect. The arithmetic establishes what the fitted equation says for the specified comparison. A causal interpretation requires additional justification and is not supplied by multiplication alone.
Check Your Understanding
For each question, state the contrast’s direction and include the outcome units.
- A fitted additive model has a coefficient of 1.8 points per hour for practice time. What is the fitted contrast for a 2.5-hour increase, holding the other predictors fixed?
- A coefficient is \(-0.4\) kg per unit of a predictor. What is the fitted outcome contrast for decreasing that predictor by 3 units?
- A binary predictor is coded 0 and 1 and has a fitted coefficient of 7 points. What does that coefficient say about the fitted contrast for category 1 versus category 0, at fixed values of the other predictors?
- Why does subtracting the fitted equations eliminate the terms for predictors that do not change?
- Does calculating an adjusted fitted contrast by itself establish that changing the predictor causes the outcome to change? Explain briefly.