I wonder if someone can help me. I'm running a multiple mediation (parallel model) using the preacher and hayes process script. I am no expert but it seems to be that the $a$ paths (IV to mediators) are essentially correlations, whilst the $b$ paths (mediators to DV controlling for the IV) are essentially a linear regression. I have checked this using correlations/linear regressions and the coefficients obtained are identical.
The snag in my data is that the dependent variable has a number of outliers and is not normally distributed. When running the simple regression it is only when my data is transformed and the outliers are removed that I meet the assumption of normality (of my residuals). What I am wondering, therefore, is whether I should be transforming this data (just the dependent variable, as I know there is no assumption of normality across $a*b$) before I run my multiple mediation? I believe that because it is based on bootstrapping these assumptions are not required, but the results obtained with the raw data (full mediation) are very different to the transformed/outliers removed data (akin to 'partial' mediation).
Question: Given this I am wondering what is most appropriate to use, the raw data or the transformed/cleaned data?
Background: The outliers are falling about 3-4 standard deviations above the mean and the overall data pattern is that the majority of people are good on the task (hence the negatively skewed distribution). And @gammer you are absolutely right the model we are using is $X \to M \to Y$ with 8 mediators (so 8 M's). We initially ran this as a multiple regression (hence why I had transformed my data to meet the assumptions of normality of residuals/removal of outliers) but I am now unsure whether to use normalised/raw data for my multiple mediation. All advice much appreciated.