# #StackBounty: #r #probability #maximum-likelihood #roc How do I evaluate the likelihood of the binormal smoothed ROC curve?

### Bounty: 50

As I understand the binormal model for ROC curves assumes that the decision variable can be monotonically transformed so that both the case and control values are normally distributed. Under this assumption there is a simple relationship getting the sensitivity from the specificity:

$$phi^{-1}(SE) = a+bphi^{-1}(SP)$$

Where $$phi$$ is the normal CDF, and SE/SP is sensitivity and specificity. The R package pROC (Robin et al 2011) fits a linear model to the observed SE/SP values to get $$a$$ and $$b$$ and then calculates the smoothed curve from that.

My question is, how do you evaluate the likelihood of this ROC curve on some holdout points (test set)? As an example of this, suppose we fit Kerned Density Estimates to the case ($$bar{D}$$) and control ($$D$$) points, call the resulting pdfs $$k_{bar{D}}$$ and $$k_{D}$$ with hyperparameters $$Theta$$. We could then evaluate (I think) the likelihood of the overall smoothing on holdout sets $$X$$ and $$bar{X}$$ as:

$$mathcal{L}(X,bar{X}, Theta) = prod_X k_{D}(x_i)prod_bar{X} k_{bar{D}}(bar{x}_i)$$
and compare different options for $$k$$.
I’m not sure how to do the same thing for the binormal model.

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