Linear regression does not require predictors or dependent variables to be normally distributed. (You can see this from the fact that regression allows you to use, e.g., binary/dichotomous dummy variables as predictors, which could never be normally distributed.) The normality assumption is for the errors (sometimes referred to as "residuals"). You can examine this assumption by looking at the distribution of the error scores in your sample using a histogram or normal probability plot (https://en.wikipedia.org/wiki/Normal_probability_plot#:~:text=In%20a%20normal%20probability%20plot,line%20suggest%20departures%20from%20normality.). Most software programs for multiple regression such as SPSS or JASP will allow to to easily generate a variety of such error/residual plots.

Christian Geiser thank you very much for this. I did examine the variable using a histogram and q-q plot. From the q-q plot, I can see that some points are aligned with the diagonal line, whereas others do deviate quite significantly, possibly reinforcing the positive skew.

With this in mind, is this likely to be problematic for a moderating variable? I'm thinking about how this may influence the results in trying to see if the relationship between X and Y changes at different levels of the moderator if most of the value for the moderator variable are low? Apologies if this is a silly question or I am missing the point completely!

What you should primarily check is normality of the errors (residuals) of the regression. The moderator variable itself does not have to be normally distributed for valid OLS regression results.

Christian Geiser Right! When looking at the ZPRED ZRESID scatterplot for the regression, I can see that this looks normal. I did wonder whether it would be helpful to instead make the moderating variable dichotomous to help with this issue. However, it sounds as though it is acceptable to run the analysis as long as it is heavily caveated re: power.

Thank you for your help!

Dichotomizing the moderator would likely cause an even greater loss of information and/or power. I would keep the variable as is.

An alternative, if you know what the residual distribution should be, is to run a piecewise structural equation model (PSEM), using generalized linear models (GLMs) as paths. PSEMs are very flexible with which kinds of regression paths you can fit, and because GLMs allow you to choose the residual family, it is not a requirement to have Gaussian residuals. The downside to this would be that you need to manually bootstrap the standard errors (particularly with the indirect effects), but this can be achieved using the boot package with any model estimate you choose.

Christian Geiser You wrote "Sure, an extremely skewed variable may limit your power/ability to show an effect (difference/moderation)."

I do not think this is correct. Generally skew and (especially leptokurtosis) in predictors increase statistical power to detect moderator effects. This is because extreme observations have high leverage and carry more information about the regression slope. This can overcome the downsides of having a lot of error (large residuals) when looking at polynomial effects or interactions with continuous predictors because the product term for the interaction is extreme when both x1 and x2 are extreme and this is very rare with normally distributed predictors. (Plus including the interaction should reduce the error term if there is a true interaction to be detected). So non-normal predictors quite often increase power relative to a normal distribution. Power is increased when x1 and x2 are leptokurtotic relative to the normal distribution (i.e., have heavier tails).

See:

McClelland, G. H., & Judd, C. M. (1993). Statistical difficulties of detecting interactions and moderator effects. Psychological Bulletin, 114(2), 376–390. https://doi.org/10.1037/0033-2909.114.2.376

Thom Baguley : Interesting! I did not know that. Thank you for pointing this out and for providing the reference. I have edited/corrected my previous post accordingly.

You can use a non-normally distributed variable as a continuous moderator in moderation analysis. The assumptions for moderation analysis (especially with tools like PROCESS) are less stringent about the distribution of the moderator. However, since your moderator is positively skewed, you may want to consider transforming the variable (e.g., using a log transformation) to improve its distribution, which can help reduce any potential impact of the skewness on the analysis.