I am conducting a study of 126 organizations. Each organization has one or more registered domain names. Each domain name needs to conform with specific cybersecurity requirements. Some organizations outsource their email to external providers, which they determine on a per-domain basis. I want to determine:
- Whether outsourcing email correlates with better or worse conformance to the cybersecurity requirements. In other words, the null hypothesis is that there is no correlation between outsourcing email and conforming to the cybersecurity requirements.
- Whether this correlation is significant at the 5% threshold, i.e., $p<0.05$.
- If it is significant, which values of which variables are significant.
Here is the data crosstab in which the values represent the number of organizations whose domains (have outsourced email, don't, or a mix in which some domains are and some aren't) * (conform with the cybersecurity requirements, don't, or have a mix of some that conform and some that don't):
| Mixed | Not Outsourced | Outsourced | Sums | |
|---|---|---|---|---|
| Conforming | 13 | 9 | 59 | 81 |
| Mixed | 11 | 0 | 3 | 14 |
| Nonconforming | 4 | 0 | 27 | 31 |
| Sums | 28 | 9 | 89 | 126 |
Attempts so far:
- My go-to for this type of analysis is a $\chi^2$ test; however, with the "expected count in at least 80% of the cells being 5 or greater" threshold not being met, I believe I cannot use this test.
- There is a Freeman-Halton extension to the Fisher exact test, but this has a limitation of $N\leq90$ and my $N=126$, which also rules this test out.
- I found a process by Arthur Ghent that finds the crosstab significant such that $p=9.926\times10^{-15}$, but there is no information in this same article about post-hoc analysis to know which values of which variables are significant and which are not.
Regarding related questions here on StackExchange:
- Appropriate stat test for multiple 3x3 tables seems to start asking this question, but then branches and remains incomplete and thus, is unable to be answered.
- 3x3 contingency table: what to use instead of chi-squared test? looks promising because of its title, but it does not relate to significance.
- Can we compare the difference of two specific proportions in a 3x3 contingency table? would work if I could use a $\chi^2$ test.
- How do I perform a follow up on a chi-squared table 3x3 or larger? asks about the post-hoc analysis that I will need to figure out how to do, but it is unanswered and a comment that provides an "answer" links to a server that is no longer online.
How can I proceed? Is there a better approach?
Please note that I do not have any statistics software but have been using a spreadsheet. Thank you.
