Is 'partial moderation' a 'thing'? I am conducting a moderation analysis between x and y, whilst controlling for the effects of several covariates.
x is a significant predictor of y before the moderator is inserted into the regression model.
When the moderator is included, x is still a significant predictor of y, whilst the moderation effects are also significant.
I know in mediation analysis, that if M explains some (but not all) of the relationship between x and y, that this is referred to as partial mediation.
Is having direct and moderator effects known as partial moderation, or is there an alternative term used for this (I am struggling to find much when searching this term online)?
Many thanks
 A: "Partial moderation" is not a thing because you can either detect a statistically significant moderating effect or you  fail to do so -- no partial moderations.
Now, assuming moderating effect exists, the coefficient and p-value of X could change depending on the level of the moderator -- which depends on how you code / where you center your moderator variable. Please see this thread for more detailed explanations: Why could centering independent variables change the main effects with moderation?
On a related note, since you stated that

I know in mediation analysis, that if M explains some (but not all) of the relationship between x and y, that this is referred to as partial mediation.

I would like to point out that many researchers have been calling for the abandonment of full vs. partial mediation language and instead advocated for the testing of indirect effects. Please refer to Hayes (2017, p.119 - "A Critique of Complete and Partial Mediation") or Preacher & Kelley (2011, p.96 -- "Verbal Descriptors").

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*Hayes, A. F. (2017). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. Guilford publications.


*Preacher, K. J., & Kelley, K. (2011). Effect size measures for mediation models: quantitative strategies for communicating indirect effects. Psychological methods, 16(2), 93.
