So I'm a bit of a novice at stats. I'm trying to determine statistically whether there is an underrepresentation of ethnic minorities in referrals to a particular type of service.
I wanted to compare the % of ethnic minorities in a locality against the number of referrals to a service in that locality that are ethnic minorities.
eg. let's say one locality, Manchester, has a 25% population of ethnic minorities. The service in Manchester has received 1200 referrals, of which 50 were ethnic minorities (4.16%). *
Then let's say we have Rochdale, and it has 30% population of ethnic minorities. The service in Rochdale has received 500 referrals, of which 70 were ethnic minorities (14%). *
So it's almost like each locality is a participant, with a "score" of overall ethnic minorities and service referral based ethnic minorities. I have this data on 10-15 localities. I was thinking of converting the referral data for each locality into it's percentage (like the 4.16% and the 14% above) and then seeing if there's a significant difference in means of overall population % vs overall service % through a t-test. But then I'm reading that we shouldn't do t-tests with percentages.
I was also hoping to look at whether there are any inter group differences - eg. certain localities come under certain Trusts so did certain Trusts have significantly better representation or not.
Other papers that have done work around ethnic underrepresentation have used chi squared test but I don't think my data would fit into that model because it isn't really categorical.
*made up numbers
Your data *are* categorical. You observe the ethnicity, which is categorical, and you observe the existence of referrals, which is also categorical.
Categorical data is analyzed statistically by the frequencies whith which the different categories were observed. This is done be Chi²-tests, or with Poisson, quasi-Poisson or negative-binomial models, when covariables (like SES, education, gender distribution etc.) need to be considered.
For the simple case, take Manchester, you want to compare the counts of (50, 1150) to the proportions (0.25, 0.75). This can be done by chi-square goodness-of-fit test.
The code here is in R, which can be run at https://rdrr.io/snippets/ without installing software. But these tests are common in software packages. In general, just be careful about whether the software expects, e.g. counts of 50 and 1150, or 50 out of 1200.
Observed = c(50, 1150)
Theoretical = c(0.25, 0.75)
chisq.test(Observed, p=Theoretical)
A similar approach for a binomial observation is a binomial test:
Minority = 50
Total = 1200
Population = 0.25
binom.test(Minority, Total, Population)
The output for this test gives a confidence interval for the observed proportion. This is useful to report. The 95% confidence interval for the observed proportion in this case is from about 0.03 to 0.05, obviously not including the proportion in the population of 0.25.
If you conduct such tests on multiple localities, understand the principle of type-I error inflation with multiple hypothesis tests.
However, if you want to consider localities together, or incorporate Trusts, you would need a more sophisticated approach.
I think t tests are for continuous, comparing mean of two samples. You have a percentage. I think the answers above are more appropriate. Good luck.
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