Must every main effect be significant before an interaction is included?

No. Hierarchical inclusion and individual coefficient significance are different issues.

Does centering remove an interaction?

No. Correct recoding changes lower-order coefficients but preserves the represented interaction and predictions.

Are there exceptions to marginality?

Yes. Scientifically justified constraints and particular model structures can motivate exceptions, but they should be explained rather than chosen solely from isolated p-values.

AI Glossary / Statistics and probability / Advanced

Principle of marginality

Definition

The principle of marginality is a model-specification guideline that generally retains lower-order terms when a model includes higher-order terms built from them. For example, a regression containing an x-by-z interaction usually also contains x and z, and a quadratic polynomial usually retains its linear term. This hierarchy supports coherent interpretation and protects the model from arbitrary changes caused by recoding or shifting predictors. It is a modeling principle rather than an exceptionless rule that overrides a justified scientific constraint.

What it means

An interaction says that the relationship involving one variable depends on another. Removing a main-effect term merely because its coefficient is not statistically significant can impose a strong constraint at a particular reference value. That constraint may change if the variable is centered differently. Similarly, a quadratic curve without a linear term fixes its symmetry around the chosen zero point. Marginality encourages analysts to consider the whole function and the meaning of coding, rather than treating every coefficient as an unrelated feature that can be independently kept or deleted.

How it works

Identify which model terms are formed from others and specify the intended hierarchy. When an interaction is included, usually retain the relevant constituent terms; for a higher-order interaction, consider the associated lower-order interactions as well. For a polynomial, retain the lower powers under the usual hierarchical formulation. Interpret effects through predicted values, contrasts or conditional slopes at meaningful predictor values. Use model comparisons that respect the chosen hierarchy. If a substantive theory calls for a reduced or constrained form, explain the constraint and check that its interpretation is preserved under the intended variable definitions.

Three practical examples

Product experiments: retain the constituent factor terms when modeling an interaction between interface design and device type.
Process analysis: keep the linear term when fitting an unconstrained quadratic response over temperature.
AI feature modeling: apply a documented hierarchy when selecting interaction features, distinguishing a predictive regularization choice from a scientific claim that a lower-order effect is exactly zero.

Think of it this way

Think of describing a joint mechanism together with its supporting parts. A combined term can change meaning when the reference point for one part changes. Retaining the relevant terms lets the model represent the same overall relationship after a sensible change of coordinates, rather than accidentally changing the scientific claim with the coding.

Worked example

Consider an original artificial regression surface y_hat = 2 + 3x + 4z + 5xz. The conditional slope in x is 3 + 5z: it is 3 at z = 0 and 8 at z = 1. Now center x by defining u = x - 10, so x = u + 10. Substitution gives y_hat = 32 + 3u + 54z + 5uz. The predictions are unchanged, even though the intercept and z coefficient differ. If a model contained only an intercept and the product xz, shifting x would generate both uz and a z term because xz = uz + 10z. Omitting the lower-order z term after recoding would therefore change the set of functions the model can represent. This calculation explains the hierarchy's invariance motivation. It does not say that every scientifically justified model must include every algebraically conceivable term.

Illustrative case study

In a fictional case, Vale Operations finds an interaction between workload and staffing in a regression. An automated selection routine removes staffing's main effect because its p-value at the current workload reference is large. Analysts review the specification and recognize that the coefficient describes staffing at workload zero, which is outside the relevant operating range. They retain the hierarchical model and report predicted differences at meaningful workload levels. The explanation focuses on the conditional relationship rather than declaring staffing irrelevant from one coefficient test.

Common misconceptions and mistakes

Marginality does not mean that lower-order coefficients must each be statistically significant. A main-effect coefficient in a model with interaction is usually conditional on reference values, not a universal average effect. Changing centering can alter that coefficient and its test without altering the fitted response surface. Hierarchy also does not prove causality or guarantee better out-of-sample prediction. Some constraints follow from physical laws, nested designs or specific parameterizations and can justify departures. The important issue is whether the resulting model and hypothesis match the substantive question, not mechanical obedience to a term-selection rule.

Business use

Businesses should document the full prediction function and explain effects at realistic settings. This reduces confusion when analysts use different reference categories or units and appear to report different main effects. When automated selection is used, make the hierarchy rules explicit and evaluate stability. For predictive systems, compare the chosen specification through valid evaluation; for explanatory models, ensure the tested contrasts answer the intended questions. If a lower-order term is deliberately omitted, state the restriction it imposes rather than presenting the omission as a harmless simplification.

Check your understanding

In y_hat = 2 + 3x + 4z + 5xz, is the effect of increasing x by one always 3?

Answer: No. Holding z fixed, the predicted change is 3 + 5z, so the relationship depends on z.

Related terms

Polynomial regression
Centering
Conditional effect
Contrast
Model selection
Interaction
hierarchical model specification

Learn next

Learn how recoding transforms an interaction model, then practice interpreting predicted contrasts and comparing hierarchical models using questions defined in substantive rather than coefficient-only terms.

Sources

The marginality principle revisited: Should higher-order terms always be accompanied by lower-order terms in regression analyses?: https://pmc.ncbi.nlm.nih.gov/articles/PMC7615369/ R Core, add1 and drop1, model terms and hierarchy: https://stat.ethz.ch/R-manual/R-devel/library/stats/html/add1.html

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