Why the Terms Belong Together
An interaction allows the relationship between a quantitative predictor and an outcome to differ by group. But an interaction term does not, by itself, give each group a freely varying fitted line. The lower-order terms determine the lines' intercepts and baseline slope; dropping one of them imposes a particular restriction on the fitted structure.
Consider two groups, coded \(G_i=0\) for the reference group and \(G_i=1\) for the other group, and a quantitative predictor \(x_i\). A model with a group-by-predictor interaction and both corresponding main effects is
Here \(Y_i\) is the outcome for observation \(i\), and \(\varepsilon_i\) is its error. For the usual linear-model interpretation, assume the errors have mean zero conditional on the predictors, are independent, and have a common variance \(\sigma^2\). These assumptions support standard regression inference; the structural comparisons below follow directly from the equations.
Substituting each group code into the model gives the fitted lines:
Thus \(\beta_0\) and \(\beta_1\) are the reference group's intercept and slope. The other group's intercept differs by \(\beta_2\), and its slope differs by \(\beta_3\). This group-specific-line representation is the key to seeing what an omitted term does.
What Omitting a Main Effect Constrains
If the group term \(G_i\) is omitted but the interaction is retained, the model becomes \(Y_i=\beta_0+\beta_1x_i+\beta_3x_iG_i+\varepsilon_i\). For \(G_i=0\), the fitted line is \(\beta_0+\beta_1x_i\); for \(G_i=1\), it is \(\beta_0+(\beta_1+\beta_3)x_i\). The groups are forced to have the same fitted outcome at \(x_i=0\), because both lines have intercept \(\beta_0\). Their slopes may still differ.
If instead \(x_i\) is omitted, the model is \(Y_i=\beta_0+\beta_2G_i+\beta_3x_iG_i+\varepsilon_i\). The reference group's fitted line is constant at \(\beta_0\): its slope is forced to zero. The other group's line can have slope \(\beta_3\). Omitting both main effects imposes both restrictions: the reference group is flat, and the two groups have the same fitted outcome at \(x_i=0\).
These are restrictions on the model's possible fitted values, not merely changes to the wording of a coefficient. Unless a restriction is appropriate, it can prevent the fitted lines from representing the relationship in the data. The interaction term alone does not compensate for a missing main effect.
Worked Example: Reading the Restrictions From the Equations
Suppose \(x\) is a dose measured in milligrams, and group A is coded 0 while group B is coded 1. A fitted model with both main effects is
For group A, substitute \(G=0\): \(\widehat{Y}_A=30+2x\). For group B, substitute \(G=1\): \(\widehat{Y}_B=35+3.5x\). At zero milligrams, the fitted outcomes are 30 and 35 units, respectively; the slopes are 2 and 3.5 outcome units per milligram.
Now omit \(G\) but keep \(xG\). The model is \(\widehat{Y}=a+bx+cxG\). Its group A line is \(a+bx\), and its group B line is \(a+(b+c)x\). Both lines predict \(a\) at zero milligrams. This model cannot reproduce the full model's distinct predictions of 30 and 35 at zero dose.
Alternatively, omit \(x\) but keep \(G\) and \(xG\). The group A line is \(\widehat{Y}=a\), with zero change per milligram. Group B's line is \(a+d+cx\), which may slope upward or downward. This model excludes a nonzero dose slope for group A, even if the observed data show one.
A Numerical Illustration of the Fitted Change
The restrictions affect estimates and predictions, not only the set of coefficients in a table. The following hypothetical data have two groups observed at the same three predictor values.
| Group | \(x\) | Observed \(Y\) |
|---|---|---|
| A | 0 | 10 |
| A | 1 | 12 |
| A | 2 | 14 |
| B | 0 | 14 |
| B | 1 | 17 |
| B | 2 | 20 |
Worked Example: Omitting the Group Main Effect
Code group A as \(G=0\) and group B as \(G=1\). The full model fits group A's line \(10+2x\) and group B's line \(14+3x\) exactly. In coefficient form, \(\beta_0=10\), \(\beta_1=2\), \(\beta_2=4\), and \(\beta_3=1\). The two groups differ by 4 outcome units at \(x=0\), and their slopes differ by 1 outcome unit per unit of \(x\).
Fit the reduced model that omits \(G\): \(\widehat{Y}=a+bx+cxG\). It allows separate slopes, \(b\) for A and \(b+c\) for B, but requires a shared intercept \(a\). The least-squares slope for each group can be found from its within-group centered values. For group A, the mean \(x\) is 1 and the mean \(Y\) is 12, so
For group B, the mean \(x\) is 1 and the mean \(Y\) is 17. Its slope is
With these slopes, the intercept each group would have on its own is 12 minus \(2(1)=10\) for A, and 17 minus \(3(1)=14\) for B. The reduced model requires one shared intercept. Because both groups have three observations, least squares sets it to their average: \(a=(10+14)/2=12\). Its fitted lines are therefore \(\widehat{Y}_A=12+2x\) and \(\widehat{Y}_B=12+3x\).
The reduced model has shifted group A's fitted line up by 2 and group B's down by 2 relative to their exact full-model lines. Its residuals are \(-2\) for each A observation and \(2\) for each B observation, giving a residual sum of squares \(6(2^2)=24\), rather than 0 for the full model. The slopes remain different, but the reduced model has forced the fitted group difference to zero at \(x=0\).
The Reference Point and Centering
The restriction created by omitting \(G\) is tied to the point where \(x=0\). If \(x\) is recentered, zero refers to a different predictor value, so the same omission imposes equality there instead. Centering can make an intercept or group difference more interpretable, but it does not remove the restriction created by omitting the group term.
Worked Example: Omission After Centering
Suppose \(x\) is study time in hours, and group A and group B have fitted lines
The difference between the lines at \(x=4\) hours is \((26+2(4))-(20+1.5(4))=34-26=8\) outcome units. Now center the predictor at 4 hours by defining \(z=x-4\). Then \(x=z+4\), so the same lines become
If the group main effect is omitted from a model using \(z\), the fitted lines must agree at \(z=0\), which is \(x=4\) hours. But the specified lines differ by 8 units there. Thus centering changes where the equality restriction applies; it does not make that restriction disappear.
A Short Justification and a Practical Rule
The model's fitted structure follows simply by substituting the group codes. With both main effects included, setting \(G=0\) removes the group terms and leaves the reference line. Setting \(G=1\) adds \(\beta_2\) to the intercept and \(\beta_3\) to the slope. If \(\beta_2G\) is removed, both substituted equations have the same intercept. If \(\beta_1x\) is removed, the \(G=0\) equation has no \(x\) term and therefore has slope zero. This establishes exactly which structures each omission rules out.
For that reason, a common model-building principle is to retain lower-order terms whenever their interaction is included. This is sometimes called the hierarchy principle. It is a useful default, not an unbreakable algebraic law: an omission can be defensible when the corresponding restriction is genuinely part of the scientific model or design. For example, if theory requires the two groups to have equal expected outcomes at a meaningful zero point, omitting the group term may encode that claim. The restriction should be stated and justified, rather than treated as harmless simplification.
Include the quantitative predictor, the group term, and their product.
Substitute each group code to identify the fitted intercept and slope for every group.
Check whether it forces equal fitted outcomes at the zero point or a zero slope for the reference group.
If the restriction is not scientifically intended, retain the lower-order term.
For more than two groups, the same idea applies: a group-by-predictor interaction should ordinarily be accompanied by the group terms and the predictor term. The group terms allow group-specific intercept differences, while the predictor term supplies the reference slope under reference coding. Other contrast codings express these same model dimensions with different coefficient meanings; the important question is whether the model permits the intended group-specific lines.
Common Mistake
A frequent error is to argue, “The interaction is in the model, so the groups can differ; the main effect is unnecessary.” This confuses permission to have different slopes with permission to have different intercepts. When the group term is omitted, the lines can have different slopes but must meet at the predictor value coded zero. Likewise, an interaction term does not allow a nonzero reference-group slope if the predictor term is omitted.
Another mistake is to assume a large or statistically significant interaction makes the omitted term irrelevant. The interaction coefficient describes a slope difference within the specified model; it does not undo the constraints imposed by missing terms. Check the group-specific equations and the predictor value at which any equality is being imposed before interpreting the fit.
Check Your Understanding
Answer each question by writing or interpreting the group-specific fitted lines.
- In \(Y=\beta_0+\beta_1x+\beta_2G+\beta_3xG+\varepsilon\), what are the intercept and slope for the group coded \(G=1\)?
- What restriction on the fitted lines is imposed by omitting \(G\) while retaining \(x\) and \(xG\)?
- What restriction on the reference group's fitted line is imposed by omitting \(x\) while retaining \(G\) and \(xG\)?
- If \(x\) is centered at 10 units, at what original predictor value do lines in a model omitting \(G\) have to agree?
- Give one circumstance in which omitting a lower-order term could be appropriate, and state the restriction it imposes.