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	\subsection{Variance decomposition}\label{subsec:variance-decomposition}

For each model, we estimate the proportion of variation explained by the regressors, hospital performance, and state performance or coding differences, and the proportion of variation that remains unexplained.

Estimates of the standard deviation of our models' hospital effects \((\sigma_{H})\) are obtained directly from the output of each model.
By construction, the variance of the residual term of a logit model is \(\pi^2/3\).
To estimate the proportion of variation in outcomes explained by the regressors, we use the pseudo-\(R^2\) proposed for binary dependent variable models by McKelvey and Zavoina\footnote{McKelvey and Zavoina, 1975}:

\[R_{\text{MZ}}^{2} = \frac{\frac{1}{N}\sum_{i} (x_{i}\hat{\beta} - x\hat{\beta})^{2}}{\pi^2/3 + \sigma_{H}^{2} + \frac{1}{N}\sum_{i}^{}(x_{i}\hat{\beta} - x\hat{\beta})^2}\]

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