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I am going to compare means of two groups (Male, Female) from my complex stratified sample.

I have scores of 3 subjects of more then 3000 respondent for each subject. I checked for normality and get the following picture: enter image description here
i.e data for groups isn't normal in all three cases. Therefore I can not use the tests of hypothesis of homogeneity which assume normality, also randomness of sample. Nevertheless I want to compare who is better boys or girls? How can I do this correctly and without bias?

Edit: By the way there are no intersections in the three set of respondents. I visualize the three because there is the one problem for all three subjects.

May be I ask the silly question but I guess when normality and randomness conditions are violated, it is incorrect to use such methods. Am I not right?

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  • $\begingroup$ can you provide some kind of figure? $\endgroup$ – D_Williams Mar 24 '17 at 15:52
  • $\begingroup$ Sorry, what do you mean? $\endgroup$ – Evgeny Kuznetsov Mar 24 '17 at 15:54
  • $\begingroup$ What are these data? What are your variables? What do you mean by "complex stratified sample"? We are going to need more information to help you. $\endgroup$ – gung - Reinstate Monica♦ Mar 24 '17 at 15:56
  • $\begingroup$ A visualization of the data. Also, is it correct that the sample size is 3? $\endgroup$ – D_Williams Mar 24 '17 at 15:58
  • $\begingroup$ It is education research, the variable I analyze is score and gender. The respondent was sampled by strata then school ten class, the corresponding weights are provided. I am just asking for method I should use to compare groups. $\endgroup$ – Evgeny Kuznetsov Mar 24 '17 at 16:01
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Visualization would help, often the eye can see and understand more through graphics than the numbers. I can picture what your outcomes look like, but understanding what the score comes from would help.

The min/max and mean match appropriately. The small adjustment made to the median let us know the sample is skewed, but since the adjustments are small, the skew isn't too influential (though it has shown to be significant by some rule of thumb ideas). The Kurtosis is the biggest problem it seems like. I picture a bell that has a peeked top possibly with a divit in the middle, the drops rapidly and goes into two low population tails, with one of them skewed. This suggests that an overwhelming majority of people test within a small range, and the other groups differ significantly. It may help to identify what differentiates the small groups the are significantly above and below the mean. It may make sense to take them out of the sample, with the explanation that you want to assess differences in individuals, and to avoid the undue influence of X and Y you have constrained your sample to not include them. I'm guessing you would then have a sample with a smaller range, but a more normal distribution. Without more information on the measures, it is difficult to say more than that. Look up tests that are robust to violations of the assumption of a normal distribution, such as generalized estimating equations.

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  • $\begingroup$ You offer to cut off outliers. I'll try. In general data is the sort which must be normal. I'll try to cut off some respondents. $\endgroup$ – Evgeny Kuznetsov Mar 24 '17 at 18:16
  • $\begingroup$ Randomly cutting off extreme scores always rubs me the wrong way, but identifying the extreme scores, and providing a good rationale for cutting them out seems like an important part of analyses. I would recommend only cut the extreme scores that you can justify with a reason, don't just cut them out because they don't fit the mold. Another option is to show the analysis before and after cutting the outliers. Or else google analytic models that are robust to violations of (fill in the blank). Good Luck. $\endgroup$ – Mprante Mar 24 '17 at 23:07
  • $\begingroup$ I tried to cut outliers as follows: Compute z-scores and cut out of interval (-2.68, 2.68) But I still nave no normality. $\endgroup$ – Evgeny Kuznetsov Mar 25 '17 at 11:52
  • $\begingroup$ I would not cut any scores. if your data is clearly not normal, chose a different response distribution. Use a multilevel model with a Student-t response or a skew normal response. These have to be implemented in Stan, but not to difficult. You can also look into frequentist robust methods. If you must alter the data, see the wr2 package, although I prefer approaches that model the observed data. $\endgroup$ – D_Williams Mar 25 '17 at 14:56

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