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I have a dataset composed by 5000 observations. Each observation contains the income per year of a person (from 50 to 50.000.000) and the fact of having a car (yes/no).

I would like to check if a correlation exists between these two features. Which test I should run?

thanks in advance

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As always, it depends on the data generating mechanism you have in mind. I should note, however, that it would be customary to use a logit model. In this, we assume that the odds of having a car is linear in the explanatory variables, i.e.

$\log(\frac{\Pr(Car)}{1-\Pr(Car)})=\beta_0+\beta_1income$

One can then test whether or not $\beta_1=0$, which is a test of the hypothesis that income influences car ownership.

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You can use several non-parametric tools to investigate the correlation:

First you can use the Pearson correlation coefficient and do a permutation test, i.e. you permute one of your vector and calculate the Pearson coefficient for each permutation, it gives you the distribution under the null hypothesis. Then you compute the Pearson coefficient for your actual data and see if it is in the extreme 5% (if yes then the correlation is significant). I'm just quickly explaining in case you don't know permutation tests, but you should seek further information (for example starting here http://en.wikipedia.org/wiki/Resampling_%28statistics%29#Permutation_tests)

You can also use the Kendall Tau : $$\tau = \frac{c - d}{\binom{n}{2}}$$ where $c = $ number of concordant pairs and $d = $ number of discordant pairs. Under the null hypothesis (of independence), $\tau$ is approximately normally distributed. Further information here: http://en.wikipedia.org/wiki/Kendall_tau_rank_correlation_coefficient#Significance_tests

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