1. ## (non-)parametric test dilemma

I want to compare means of 6 variables and to do correlation analysis. I did Kolmogorov-Smirnov test and it showed that 3 of them are normally distributed and the other 3 are not. It's familiar to me that Pearson's correlation and ANOVA are used when data has normal distribution and that in the opposite case Spearman's correlation and Kruskal-Wallis test (or Friedman's ANOVA) are used, but what would be the most appropriate statistical procedure in this situation, when some variables are normally distributed and some aren't?

2. I think it depends to some degree on the degree of departure from normality (check the skewness and kurtosis values). Very minor departures won't affect Pearson's & ANOVA too badly. More serious ones will.

If the departures from normality in your 3 non-normal variables are quite serious, I'd suggest using non-parametric analyses for all of them - based on the argument that the non-parametric analyses are still valid for the normally distributed variables (albeit with some data reduction involved), while the parametric analyes would likely NOT be valid for the non-normally distributed variables. It'd certainly be less convoluted than doing parametric tests for the relationships between some pairs of variables and non-parametric tests for other pairwise comparisons.

Just a sidenote: on the off chance your 6 variables represent measurements of some kind of the same subjects, you would need to use repeated measures analyses rather than Kruskal-Wallis etc.

Hope that helps!

3. Thank you for answering :-)

How can I check the skewness and kurtosis values?

SPSS doesn't give that in the output. It gives only significance level.

Just a sidenote: on the off chance your 6 variables represent measurements of some kind of the same subjects, you would need to use repeated measures analyses rather than Kruskal-Wallis etc.
Yes, all those 6 variables are measurments of the same subjects. So, you suggest that ANOVA would be better even though some variables aren't normally distributed?

4. Originally Posted by rmil

How can I check the skewness and kurtosis values?

SPSS doesn't give that in the output. It gives only significance level.
Go Analyze > Descriptive and then select the kurtosis and skewness options

Originally Posted by rmil
Yes, all those 6 variables are measurments of the same subjects. So, you suggest that ANOVA would be better even though some variables aren't normally distributed?
Not quite I'm saying that you probably need to use a repeated measures/dependent samples test (but not necessarily repeated measures ANOVA!) Some options here. For pairwise group mean comparisons you might want to run these tests could help.

5. I have checked skewness and kurtosis values.

Skewness values (for every single variable) are out of the range.

But, there's only one kurtosis value that's out of the range. It seems quite contradictory. How' s that possible?

P.S.
I've calculated ranges using instructions from this web site --> http://www.une.edu.au/WebStat/unit_m...skew_kurt.html

6. ## To RMIL

A better way to check whether skewness and kurtosis indicate significantly skewed distributions is to take the skewness and kurtosis values, and divide them by their respective standard error. This creates a z-value which must be compared to a cut-offs of -1.96 to 1.96 (based on normal distribution for p <. 05). If the z-values are within the range -1.96 to 1.96, then the variable is normal, if outside this range it is skewed. If you want to test for severe skewness, then use the cut-offs between -2.5 to 2.5. The only problem with this method that very large sample sizes have smaller standard errors, making it more likely to have significant z-values indicating skewness. A more useful guideline is to use a range for skewness and kurtosis of -1 to +1, within this range is normal. See this online book from Google:

7. Originally Posted by rmil
I have checked skewness and kurtosis values.

Skewness values (for every single variable) are out of the range.

But, there's only one kurtosis value that's out of the range. It seems quite contradictory. How' s that possible?

P.S.
I've calculated ranges using instructions from this web site --> http://www.une.edu.au/WebStat/unit_m...skew_kurt.html
Skewness and kurtosis are quite different things - it is possible and very plausible that your variables have skewed distributions that are only slightly kurtotic. The good news is that skewness tends to be a somewhat less serious assumption violation. You might like to try some simple transformations (e.g. square root) to see if you can reduce the skewness and kurtosis values. I agree with Dr D though that a simple rule of thumb such as -1 to +1 being acceptable (-2 to +2 is often used too) is better than methods based on the standard error that test for "statistically significant" levels of skewness or kurtosis. It's the extent of skewness and kurtosis you're interested in, not whether you can reject the null hypothesis of zero population skewness and kurtosis values (which will depend heavily on sample size).

8. An interesting thing has occured. I did another Kolmogorov-Smirnov test, but this time I used explore command (Analyze --> Descriptive Statistics --> Explore). The results indicate that all distributions are not normal. Q-Q plots also show certain deviations in every distribution.

Z-scores for skewness are: 6.28; 3.32; 3.61; -2.53; -2.09; 3.93

Z-scores for kurtosis are: 4.85; -0.43; 0.51; 1.45; 0.57; 1.51

I guess that this means that distributions are definitely non-normal.

By the way, sample size is 123.

Thank you, guys, for helping me!

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