1. Do variables in moderation have to be normally distributed before centering?
No, normality is not a requirement. However, you could apply a transformation if you felt it was justified, such as when it better reflects the meaning of the variable.
See the following for a discussion of issues in transformations:
Assumptions in moderation are the same as in normal multiple regression. They mainly concern the pattern of the residuals. For a discussion see: http://www.statisticshell.com/multireg.pdf
2. Do I use the R squared or adjusted R squared when reporting a multiple regression?
You can use either or both. The larger your sample, the smaller the difference is between adjusted R square and R-squared. R-squared is a biased estimate.
3. p=0.52, is that still significant?
Jacob Cohen once said something along the lines that surely God must love .06 almost as much as .04.
You can write p = .05
Whether this is statistically significant depends on your alpha. If your alpha is .05, then your obtained p value is not less than .05, and thus, it is not statistically significant. Reflecting the seemingly arbitrary cut-off consequences off null hypothesis significance testing, researchers often write that there was a trend toward significance when p <.10 and >=.05.
4. Would a significance level of 0.06 be slightly significant or not at all?
Trend toward significance is a common label, see above.
5. Do you have any examples of how to write up a moderator regression?
Frazier, P., Tix, A., & Barron, K. (2004). Testing moderator and mediator effects in counseling psychology research. Journal of Counseling Psychology, 51, 115-134.
McAlister, A. R., Pachana, N., & Jackson, C. J. (2005). Predictors of young dating adults' inclination to engage in extradyadic sexual activities: A multi-perspective study. BRITISH JOURNAL OF PSYCHOLOGY, 96(3), 331.
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