Note that the U-test (Mann-Withney) is noly testing mean differences when the shaped of distributions are equal AND symmetric in the groups. Under this assumption, the t-test also works wery good (the sample means approximate the normal-distribution very well already for tiny sample sizes when the distrivution of the data is symmetric).

When the distributions of the data in the groups have the same shapes but are skewed, then the U-test is a test on median differences.

When the shapes are different and skewed, the U-test is a test on stochastic differences.

But to your problem:

I would do the following: Model your response depending on the variables you have (group membership, moderation variables, covariables) as a standard linear model.* Then check the distribution of the residuals. If the distribution of these residuals is approximately normal, then everything is fine. If not, try to log-transform the response. You are then assuming that the effects are proportional rather than additive ("multiplicative model"). This is often a reasonable assumption and it may solve your "problems". If you further have covariables (moderators or whatever) on a continuous scale (like time in minutes, weight in kilograms, current in amperes,...) then plot the response versus these variables and try to estimate the kind of relationship (linear, exponential, logarithmic, quadratic, ...). Use a transformation of the variable to get a (more or less) linear relationship between the variable and the response.

*if you have counts or proportions, better use a generalized linear model of the Poisson or binomial family, respectively.

I suggest for you that, as another non-parametric method, Regression Tree Method with CHAID algorithm can be used if continous dependent variable as well as both contionus and discrete independent variables are available. 

Best Regards

Note that the U-test (Mann-Withney) is noly testing mean differences when the shaped of distributions are equal AND symmetric in the groups. Under this assumption, the t-test also works wery good (the sample means approximate the normal-distribution very well already for tiny sample sizes when the distrivution of the data is symmetric).

When the distributions of the data in the groups have the same shapes but are skewed, then the U-test is a test on median differences.

When the shapes are different and skewed, the U-test is a test on stochastic differences.

But to your problem:

I would do the following: Model your response depending on the variables you have (group membership, moderation variables, covariables) as a standard linear model.* Then check the distribution of the residuals. If the distribution of these residuals is approximately normal, then everything is fine. If not, try to log-transform the response. You are then assuming that the effects are proportional rather than additive ("multiplicative model"). This is often a reasonable assumption and it may solve your "problems". If you further have covariables (moderators or whatever) on a continuous scale (like time in minutes, weight in kilograms, current in amperes,...) then plot the response versus these variables and try to estimate the kind of relationship (linear, exponential, logarithmic, quadratic, ...). Use a transformation of the variable to get a (more or less) linear relationship between the variable and the response.

*if you have counts or proportions, better use a generalized linear model of the Poisson or binomial family, respectively.

maybe try ordinal regression?

Please find my papers on using regression tree method in my researchgate page. The method is more flexible!

I agree - both classification and regression trees are great methods.  Just need to be careful not to over-fit the model to the data.

Sarah I agreer with Jochen Wilhelm. Try some normalizing transformations. Try a t-test, this kind of test results are easier to interpret and easier to understand.

Thanks for all your answers! In my case the sample size it's 197 children (boys and girls). First of all, I executed normal distribution analysis (Kolmogorov-Smirnov) for all the variables:

- Facial emotion recognition (FEEL)

- Vocabulary subtest  WISC-IV

- Total aggression (RPQ- Reactive and Proactive Questionarie)

   - Reactive aggression (RPQ)

   - Proactive aggression (RPQ)

- Theory of Mind (Happé)

Only the facial emotion recognition and vocabulary variables had normal distributions FEEL, D(197) = 1.96, p = .21 // WISC, D(197) = 1.15, p = .15.

As Robert Borotkanics said, I think that are outliers influencing the central tendency (extreme values in some test). I will check the distributions of the residuals as Jochen Wilhelm said. 

But there is my problem... I wish to run statistics not only for analyze the mean differences in gender in a single variable. I want analyze the moderating effect of ToM and emotion recognition in the link between aggression and peer victimization. And this implies that I need to take both type of variables (normally and not-normally distributed).

(I never heard about the regression tree method, I'll take a look).

I think sample size plays an important role here. Because of the central limit theorem you can assume normality of the sampling distribution of tzhe differences with a subsample size of at least ni = 30. In this case you wouldn't have to test for normality to ensure proper conduct of the t-test for independent samples because the t-test will react "robust" even when the assumption of normality is violated for one of the (or both ) subpopulations.

Use of regression tree methods (CHAID or exhaustive CHAID algorithm) against outlier is more flexible. For parent node : child node, you must take 30:15. In SPSS program, Click Analyze > Classify > Tree....

you must recognize that your data have normal distribution or not.

we can not say this data have moderator normal. we say data are normal or not normal you can used from kolomogrov-sirnov test to know this.if p-value less than 0.05  then data are not normal. also you  can transform your data to have normal behavior . at end if your data or error of your model not have be normal distribution. you must used nonparametric methods to do your test

if p value of Kolmogorov Smirnov for  the data is less than 0.05, the data are non-normal, and you can use regression tree method a non-parametric method for the data contamined with outlier. you have two opinions:

1-SPSS program allows you to specify the data with a dependent (continous) variable and independent variables (continous and discrete). Classify > Tree> CHAID.... 

1-SPSS program allows you to specify the data with a dependent (categorical=discrete) variable and independent variables (continous and discrete). Classify > Tree> CART....

I hope that this explanation will be more useful for you.

Sarah, I am not sure I completely understand the question. If you are comparing means of random variables, and the data distributions are not highly skewed, t-test should work for moderate t large sample sizes; else you can use non-parametric methods. If you are fitting models, then the residuals from the model need to be normally distributed for inference on the parameters of the model - normality of variables is not required. 

Several myths are being repeated here. 

The first is that there is some value in formal statistical tests for normality. As AK Singh points out, it is the errors that should be normally distributed, not the variable. Furthermore, tests like the KS are so sensitive to departures from normality that a significant result means nothing. You are better off examining normal quantile plots.

The other myth is that the t-test requires large samples. It does not. And, indeed, works well under conditions where the Wilcoxon Mann-Whitney simply cannot be statistically significant because the sample size is so small. 

See these useful papers

Fagerland, M. W. (2012). t-tests, non-parametric tests, and large studies--a paradox of statistical practice? BMC Medical Research Methodology, 12, 78. doi:10.1186/1471-2288-12-78

Fagerland, M. W., Sandvik, L., & Mowinckel, P. (2011). Parametric methods outperformed non-parametric methods in comparisons of discrete numerical variables. BMC Medical Research Methodology, 11(1), 44. doi:10.1186/1471-2288-11-44

Rasch, D., & Teuscher, F. (2007). How robust are tests for two independent samples? Journal of Statistical Planning and Inference, 137(8), 2706–2720.

It depends on the end goal.  If you are trying to find the best-fitted model, I'd follow Jochen's suggestion, with the twist of applying the Fisher's Z transform to use t test on sqrt of the R-Square and this way finding the model that fits best.  You can also try the Box-Cox transformation.

 If you know the shape of the linear (in parameters) regression, but are not sure if the data are normal, use quantile regression for enough percentiles to reasonably cover the range of the data.  

If you are analyzing an experiment, you'd be best off normalizing the data (again, Box-Cox?) and then running the results through the DOE software.  Alternatively, you can use nonparametric equivalent of the F- and t- tests on unnormalized data to achieve the same outcome.

Cheers!

Just a quick comment on t-test and sample sizes.  You can relatively easily extract the minimum sample size from the T formula, which makes it very attractive. All you need is the standard deviation, confidence level, and variation of the values that you are trying to detect.

Alex, I am not sure which data should be normalized - response, predictors, or both? I have seen researchers normalizing predictors and that confuses me - there is no such requirement for fitting a regression model. 

I would normalize both: least-squares regression will want to draw the line so as to leave normal residuals, and if your response variable is not normal, you may end up with incorrect predictions...

The assumption of normality is not on the response variable (see Ronán' comment above) - in regression, one does not have iid observations, just independent observations - inference is based on the MODEL being fitted. 

I agree with Dr. Singh and Dr. Conroy. Additionally, if normality assumption is violated, one has to fix it, otherwise OLS will be biased. Very often, I have encountered both normality and heteroskedasticity are violated.

I also want to emphasize that normality assumption is applied on the model residuals or error terms. There is no need for data to be normally distributed. Even residuals can be non-normal, that is why we have those other distributions such as t, GED, skewed t, and skewed GED. For more details, please refer to my working paper:

https://www.researchgate.net/publication/269701303_Comprehensive_Time-Series_Regression_Models_Using_GretlU.S._GDP_and_Government_Consumption_Expenditures__Gross_Investment_from_1980_to_2013

or http://ssrn.com/abstract=2540535

Article Comprehensive Time-Series Regression Models Using Gretl—U.S....