Suppose I have the following 2 way table and I perform a (Pearson) chi-squared test on it. With the null hypothesis: children's goals are independent of gender.

goal      boy girl
  grades  117 130
  popular  50  91
  sports   60  30

I get X-squared = 21.4553, df = 2, p-value = 2.193e-05, so I accept the alternative hypothesis that children's goals is dependent on gender.

For further analysis, how would I find what this dependence might be? Could I look at the column percentages, as in the following table?:

goal         boy   girl
  grades    51.5   51.8
  popular   22.0   36.3
  sports    26.4   12.0
  Total     99.9  100.1
  Count    227.0  251.0

I could say that 51.5 and 51.8 are fairly close and 117 and 130 provide relativity good support to say that. Boys and girls both closely equally represented when having grades as their goal. And similarly I could say that more girls have goals to be popular than boys but more boys have goals to be more sporty than girls.

Would I still be able to make the same statement about boys and girls goals for grades if say there was only 13 boys and 17 girls that have their goal as grades out of 1000 boys and girls total? I would then have a very low support, what conclusion can I then make about that? More importantly how big of a support would I need to be able to make such a conclusion about their goals?

Are there also other things I can do after I the chi-squared test? (For example, if this was all the data I had, and no other variables to work with. I have got the individual responses as well in a data frame.)

  • $\begingroup$ Have you only got this data (the table as you show it), or have you got the detail of responses per participant, i.e. for each response a gender and a goal (or many goals if participants were allowed to select multiple goals)? $\endgroup$ – Arthur Spoon May 22 '18 at 11:48
  • $\begingroup$ @ArthurSpoon Yes I have the responses in a data frame. There are only 3 goals and each boy or girl can only choose 1 goal that they think is most important to them. $\endgroup$ – stats May 22 '18 at 13:13
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    $\begingroup$ Possible duplicate of Understanding $\chi^{2}$ and Cramér's $V$ results $\endgroup$ – Arthur Spoon May 22 '18 at 15:28

There is a really good answer to your question right here, but just to expand a little bit on this and to make it specific to your problem:

In your case, you are dealing with gender which can be considered a grouping variable, when goal is much more a dependent variable as it is a response from your participant, so I would say that in your case, as in the answer I link to, Cramér's $V$ isn't what you want, so you will prefer going for one of the other measures listed in that answer.

What you were intuitively doing by comparing the column percentages was a risk difference.

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  • $\begingroup$ That seems to work using risk difference or odds ratio. From another post I was recommended to try a post hoc test with Bonferroni Adjustment. Would you think that would be more suited as it uses statistical significance? $\endgroup$ – stats May 23 '18 at 1:30
  • $\begingroup$ You could use Bonferroni correction as well, in which case what you want to do is run a $\chi^{2}$ on subsets of your data, and you need to decide in advance how many subtests you are running. In your case, I think you want one test for each row, so three tests in total, and your corrected $\alpha$ would be 0.017. However there are many reasons why you shouldn't be super keen on obtaining $p$-values and significant results (see e.g. this). $\endgroup$ – Arthur Spoon May 23 '18 at 10:02

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