significant differences between groups in gender

Question

I have a cell array containing the gender e.g.

GENDER = {'Male';'Female';'Male';'Male';'Male';'Female';'Female';'Female';'Male';'Male'};

and another array containing the respective groups e.g.

GROUPS = [1; 2; 1; 1; 3; 3; 1; 2; 2; 3];

I would like to find whether there are significant differences between the 3 groups in the number of male and number of female.

Or maybe, even better, the differences between the ration male/female in the 4 groups.

I think that the GENDER variable could be assumed Bernoulli-distributed so using the ANOVA should be statistically incorrect.

If instead of the GENDER I've to analyse the difference in eye colour what is the best approach?

EYE_COLOR = {'Blue';'Blue';'Brown';'Black';'Black';'Blue';'Brown';'Black';'Black';'Black'};

Can I assume the EYE_COLOUR a categorical distribution?

Thanks a lot

Answer 1

Both variables are categorical without natural ordering. A $\chi^2$-test of independence (and its versions) is usually the test of choice in such situation. It tests the null hypothesis that the true proportions of females (and males) are equal across eye color versus the alternative that there are some differences.

(Of course, a non-significant result does not mean that the proportions are equal.)

If you compare the p value of the $\chi^2$-test with the F-test of the one-way ANOVA with 0-1 coded gender as response, you will get about the same. So your intuition was quite right!

Edit: The $\chi^2$-test is symmetric in its two variables. So you can as well check the null hypothesis that the color distribution of males equals the color distribution of females.