# #StackBounty: #mixed-model #biostatistics #asymptotics #auc #asymptotic-covariance Mixed Model in a repeated measurement design and AUC

### Bounty: 50

I have data on healthy patients and patients with cancer. My goal is to predict the cancer risk for each patient based on certain biological markers. Since I have repeated measurements, I was told to use a mixed model strategy, i.e. I assume that $$mathbb P(text{patient i has cancer}mid x_i) = frac{1}{1+ exp(-x_icdotbeta – mu_i)},$$
where $$x_i$$ is the vector of biological markers of patient $$i$$, $$mu_i$$ is the random effect to account for repeated measurements, and $$beta$$ is the (unknown) coefficient vector. Here $$cdot$$ denotes the usual Euclidean inner product.

The quality of my model is assessed by the $$AUC$$. I know how to compute the $$AUC$$ and in a previous question here on CV, it was clarified why I can expect the $$AUC$$ to be asymptotically normal. However, all proofs for asymptotic normality of the $$AUC$$ assume independent observations, which is not the case here due to repeated measurements.

I suppose that the normality argument still holds as there are CLTs for dependent data. However, I could not find any proofs for asymptotic normality of an $$AUC$$ in such a setting. This made me think about whether I would even have to worry as I account for repeated measurements in my model, which is used to obtain the $$AUC$$. I am very confused (mainly because I spent thinking about this issue for the last couple of days). So my key questions are:

1. Can I expect the $$AUC$$ to be asymptotically normal in the given setting, and if so why.
2. How would I account for the repeated measurements in the variance of the $$AUC$$? Do I even have to bother given the fact that I account for repeated measurements in the model.

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