Yes you can. First of all, only the residuals (errors) are assumed to be normally distributed in regression analysis. Second, you can also use robust estimation techniques (e.g., robust maximum likelihood and/or bootstrapping) when using path analysis to estimate the regression coefficients and robust standard errors for your mediation model.

Expanding on the response from Professor Christian Geiser , estimates of the standard errors in your analyses *may* be biased when your data are non-normal.

For example, say you have a regression coefficient of 2.00 with a standard error of 1.00. That means the 95% confidence interval is taken to be in the +range 2.00-1.96*SE to 2.0+1.96*SE, or +0.04 to +3.96. So, we would decide that the coefficient is "statistically significant" (just). Now, if the data are skewed then there is a chance that the error terms around the coefficients also will be skewed. So, the "correct" 95% confidence interval might be something like in the range of -0.01 to +3.90, and you would have to declare that the coefficient is "not statistically significant at the 95% confidence level" (just).

This is why it is helpful to calculate robust standard errors with bootstrapping, or a similar repeated-sampling approach, as professor Geiser suggests.

Note also that:

(1) non-normal data does not necessarily mean that standard errors will also be non-normal, and

(2) bias in the standard errors is only a problem at the boundary between "significant" and "not significant". If an estimate is definitely significant or definitely not significant then robust estimation will not change this.

Hume F. Winzar additionally, we already expect that the distribution of the indirect effect (the product term) is not normally distributed and hence MacKinnon, Lockwood & Williams (2004) proposed to use bootstrappig for inference, which resulted in more accurate CIs in their simulations.

MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate behavioral research, 39(1), 99-128.

Thank you all

Much appreciated