In a factorial experiment the definition of the main effect of a factor A with two levels, 0 and 1, is the difference in outcome means between the two level averaging over all other factors, $\bar{y}_{A=0} - \bar{y}_{A=1}$.

What is the extension of this definition to factors with more than two levels?

Some possible ideas are

  • The difference between a factor level and the grand mean giving us one main effect for each factor level.
  • The difference in mean outcome between each pair of factor levels.
  • The anova sum of squares associated with the factor giving us one main effect per factor.

Is there a formal definition of 'main effect' when factors have more than two levels?

Reference: https://methodology.psu.edu/media/techreports/12-120.pdf

  • $\begingroup$ The Wikipedia page on main effects is surprisingly sparse. https://en.wikipedia.org/wiki/Main_effect According to Wikipedia the term "main effect" refers to the overall factor and not specific factor levels, i.e. one main effect per factor. $\endgroup$
    – Adam Black
    Nov 12, 2017 at 19:26

1 Answer 1


This response is not a formal definition of a multilevel main effect but it was too long for a comment. Regardless, it is included here as a hopefully useful, intuitive, non-formal discussion of the issues. With a little luck, the ruling CV participants will be flexible enough to permit such a discussion without demanding rigid fealty to the OPs precise question. If not, I have no problem deleting this response.

Main effects are most typically defined as the constant effect of one variable across all values of another variable (e.g., Aiken and West, Multiple Regression, p. 38). In classic OLS ANOVA evaluation of these effects, particularly categorical factors, rests on decisions about the presence (absence) of an intercept.

For models with an intercept -- the most common parametrization -- continuous main effects are evaluated at the conditional mean of the independent variables while dummy (0,1) variables represent distance or deviations from the conditional grand mean (the intercept), as defined by the model. For multilevel factors, the presence of an intercept necessitates introduction of a zero (0) or base level against which the other levels are evaluated in the cross-products matrix -- failure to include a zero or base level will exhaust the degrees of freedom for that factor. For models with an intercept, the factor coefficients represent distances or deviations from that pre-specified base level. The choice of which base level to use is an analyst decision, e.g., the last alphanumeric value in the factor based on EBCDIC-type rankings is one common approach.

For models without an intercept the interpretation changes as continuous main effects are now evaluated conditionally at zero wrt the other variables in the model. Dummy variables still represent the distance or deviation of that variable from the 'grand mean' conditional zero as defined by the model. For multilevel factors, the absence of an intercept eliminates the need for a zero or base level. The factor coefficients now capture distance or deviations for each level from the 'grand,' conditional zero defined by the model. 'Evaluation at zero' can lead to interpretive problems wrt nonnegative variables. Mean centering of independent variables in the case of models without an intercept is one recommended solution to this problem.

  • $\begingroup$ Thanks. It sounds like the term "main effect" for a multilevel factor refers to the effect of a single factor level on the outcome. This is the difference between whatever the intercept is and mean response at the factor level of interest. Just for clarification: "conditional grand mean" refers to the mean outcome when all variables are zero and "grand mean conditional zero" simply refers to an intercept that has been fixed to zero (i.e. no intercept). Also when interactions are present dummy coefficients won't represent factor-level deviations from the intercept (main effects). $\endgroup$
    – Adam Black
    Nov 12, 2017 at 19:17
  • $\begingroup$ My wording is regrettably awkward but I was trying to distinguish between a nonzero, conditional grand mean without interactions versus a conditional grand mean of zero with interactions. That said, centering of main effects is a surprisingly contentious issue in statistical modeling with, basically, polar opposite positions for and against it. This recent post on AndrewGelman.com discusses this in the context of collinearity and hierarchical modeling ... andrewgelman.com/2017/11/12/… >> $\endgroup$ Nov 13, 2017 at 12:53
  • $\begingroup$ -ctd. Singer and Willett's book, Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence (chap. 5.4), has a careful discussion of the impact of centering on the interpretation of the intercept, a key focus of their approach. At the end of the day, there are multiple issues to consider with the choice left to the analyst. The fact is, given the nearly total lack of consensus in the literature, virtually any decision can be motivated and supported with peer-reviewed references. $\endgroup$ Nov 13, 2017 at 12:56

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