Is it possible to get high AUC while the correlation between predictor and response is very low, around 0.01?

Question

My data has three continuous predictors and one binary response. I built a logistic regression model but AUC is only 0.52..it's almost like the model did nothing.. Then I calculated the correlation between the response and predictor. The correlation are all very low, around 0.01, all less than 0.05. That is to say, the predictors and response are actually independent. My question is:

1.is it possible to get high AUC while the predictors and response are almost independent

2.Since my response is binary data 0/1, is the correlation a good measure on independence? If not, what is a better way to test independence?

Thank you everyone!

Answer 1

Regarding your first question: 1) Low correlation does NOT necessarily mean the variables are independent, it just means they aren't related linearly. 2) I sure hope that independence between the DV and the IVs implies a low AUC! However, you have 3 IVs, so the strength of your model is not entirely given by correlations. The regression analysis is controlling for other variables. Correlations do not do this. If the IVs are related to each other, this could make a big difference.

Regarding your second question: Instead of correlations, I would look at parallel box plots of the 3 IVs against the DV. If there is enough data, I would look at trellis plots.

Answer 2

1) Suppose a variable $v$ take values from 0-100 and completely determines the class in the following way:

• $00 \le v < 25$ ==> class A (0)
• $25 \le v < 75$ ==> class B (1)
• $75 \le v < 100$ ==> class A (0)

The correlation between $v$ and the class label will still be zero, even though $v$ perfectly predicts the labels! This could even happen in a "smooth" way (e.g., if y ~ $x^2$).

2) Correlation is useful if you expect a linear relationship between two variables. For more complicated scenarios, I'd follow Peter Flom's suggestion and start by plotting your data; I doubt there's a reliable test that could rule out all possible relationships between your predictors and DVs.