Can I infer the correlation/relationship from Welch's t-test?

I have a binary variable (if athlete is the winner or not) and a non-binary quantitative one (participants per country).

Both variables are non-normal and they are heteroscedastic. I read that the test I can use to compare variables like these is Welch's t test. I did it and the p-value was big so I did not rejected the null hypothesis that the means are the same.

The thing is, I would like to get some conclusion out of this: are the variables actually correlated/related? does this prove it? Because by just calling the result "the means are statistically the same", I can't really see how that explains that there is a relationship between the variables (to the case, i don't see how that explains that the number of participants per country and if the athlete is the winner are related).

Can anyone suggest another method to do this if this isn't the correct one? I looked for one but the Welch test was the closest thing I could find given the characteristics of my data.

• "correlated/related" doesn't mean the same thing as "means are statistically [different]". You might need to clarify (to yourself in any case) what it is you are actually trying to determine. – Sal Mangiafico Jan 5 at 21:28

To say that the mean of a variable in one group differs from the mean of the variable in another group is to say that what group you're in has a relationship with the variable. For example, if men and women differed in average income, you would say there is a relationship between income and gender. So a test that compares the mean of the two groups is a valid test to assess whether your variables are related to each other.

Welch's test is a good test for this, but there are others. The Mann-Whitney U test doesn't require assumptions about normality or homoscedasticity, so it's another good one to use. If you get a large p-value, then you don't have enough evidence to claim there is a relationship between the two variables.

Edit: It seems like you want to see if the number of participants per country predicts whether the country wins or not. In that case, you need to model this relationship. A straightforward way to do this is with logistic regression. You should read up on logistic regression, though, because it's not a simple technique, and if you were just learning about t-tests, it might take some time before logistic regression makes sense (it's usually taught in the second year of graduate programs).

In R, you would run summary(glm(athlete_winner_or_not ~ participants_per_country, family = binomial())). The coefficient for participants_per_country represents the increase in the log odds of whether an athlete won for each additional participant on the team. This value is not easily interpretable, but there are ways to make it more interpretable. The p-value on the coefficient is the test of whether the relationship exists between the two variables. It will likely be similar to the p-value you get from a t-test. You should definitely look into logistic regression tutorials before you attempt this on your own.

The t-test code you wrote will not do what you want. I'm surprised it even runs without an error.

• Okay I understand what you're saying, but I don't think I was analyzing it like that before, maybe you can explain a bit more how can I do that? I was just computing in R t.test(rank(athlete_winner_or_not)~rank(participants_per_country), var.equal=FALSE). I tried it using the rank of each and without it and the result was analogous. Do I need to do something different to find out if the mean of each group is the same or different from the mean of paticipants per country? Also, I would think that participants per country does not have "groups" – confus Jan 6 at 18:42
• I edited my answer. – Noah Jan 6 at 20:48