Inter-DRG resource dynamics in a prospective payment system: a stochastic kernel approach

Abstract

This paper empirically investigates the resource distribution dynamics across Diagnosis Related Groups (DRGs) of elective surgery patients, in a continuing Prospective Payment System (PPS). Existing econometric literature has mainly focussed on the impact of PPS on average Length of Stay (LOS) concluding that the average LOS has declined post PPS. There is little literature on the distribution of this decline across DRGs, in a PPS. The present paper helps fill this gap. It models the evolution over time of the empirical distribution of LOS across DRGs. The empirical distributions are estimated using a non parametric “stochastic kernel approach” based on Markov Chain theory. The results for inlier episodes suggest that resource redistribution will increase capacity and expected number of admissions for DRGs having increasing waiting times. In addition, adjustments in relative cost weights are perceived as price signals by hospitals leading to a change in their casemix. The results for high outlier patients reveal that improved quality of care is one of the factors causing reduction in high outlier episodes.

Appendix

1.1 A.1 Estimating stochastic kernels

For estimation purposes, stochastic kernel can be written as [2]:

$$\label{eq7} \phi_{t+n}(S) = \int_{0}^{\infty} f_{n}(S|S')\phi_t(S)dS $$

(7)

where S is the share in period t + n and S′ is the share in period t. f _n (S|S′) is the conditional density which describes the probability that a DRG moves to a specific state of share, given the share in period t. Thus a stochastic kernel can be expressed as a conditional density and its estimator can be derived from the estimation of conditional density. A nonparametric estimator for the conditional density as proposed by Rosenblatt [29] is given by:

$$\label{eq8} \hat{f}_n(S|S') = \frac{\hat{g}_n(S',S)}{\hat{h}_n(S')} $$

(8)

where the estimator for the joint density $\hat{g}_n(S',S)$ is given by:

$$\label{eq9} \hat{g}_n(S',S) = \frac{1}{Jab}\sum\limits_{j=1}^{J} K\left(\frac{||S'-S'_j||}{a}\right)\left(\frac{||S-S_j||}{b}\right) $$

(9)

and the estimator for the marginal density $\hat{h}_n(S')$ is given by:

$$\label{eq10} \hat{h}_n(S') = \frac{1}{Ja}\sum\limits_{j=1}^{J} K\left(\frac{||S'-S'_j||}{a}\right) $$

(10)

where a and b are bandwidth parameters controlling the smoothness of fit, K is the chosen kernel function and ||S′ − S′ _j || and ||S − S _j || are the Euclidian metrics. Substituting Eqs. 9 and 10 in Eq. 8 the conditional density estimator can be rewritten as:

$$ \hat{f}_n(S|S') = \frac{1}{b}\sum\limits_{j=1}^{J} w_i(S') K\left(\frac{||S-S_j||}{b}\right) $$

where

$$ w_i(S') = K \left(\frac{||S'-S'_j||}{a}\right)/ \sum\limits_{j=1}^{J} K \left(\frac{||S'-S'_j||}{a}\right) $$

The above kernel estimator is the Nadaraya-Watson kernel regression estimator. It shows that a conditional density can be obtained by the sum of J kernel functions in S space weighted by the w _i (S′) in S′ space.

1.2 A.2 Kernel choice and bandwidth selection

Throughout this appendix f is kernel estimate, K is kernel and h is bandwidth. Given that we have used general weight function estimate as the kernel estimate for any x, the expected value and variance of such an estimate is given by:

$$ E[\hat{f}(x)] = \int \frac{1}{h} K \left(\frac{x-y}{h}\right) f(y)dy $$

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$$\begin{array}{lll} var[\hat{f}(x)] &=& \displaystyle\frac{1}{n}\int \displaystyle\frac{1}{h^2} K \left(\displaystyle\frac{x-y}{h}\right)^2 f(y)dy \\ &&-\left\{\int \displaystyle\frac{1}{h} K \left(\displaystyle\frac{x-y}{h}\right) f(y)dy\right\}^2 \end{array} $$

(12)

The expected value of $\hat{f}$ is a smoothed version of the true density, obtained by convolving f with the kernel scaled by the band width. Thus the density estimate is of the form: smoothed version of true density + random error. The smoothed density depends on the choice of parameters through kernel and band width choice. Such a choice is critical for the discrepancy of density estimator $\hat{f}$ from the true density f. Most widely used measure of such a discrepancy which reflects the global accuracy of $\hat{f}$ as an estimator of f is the mean integrated square error (MISE) defined as:

$$ MISE(\hat{f}) = E\int\{\hat{f}(x)-f(x)\}^2 dx $$

(13)

which can be rewritten as:

$$ MISE(\hat{f}) = \int\{E\hat{f}(x)-f(x)\}^2 dx + \int var \hat{f}(x)dx $$

the sum of the squared bias (bias _h ) and the variance at x. Assuming kernel K is symmetric function satisfying: $\int K(t) dt =1$, $\int tK(t)dt=0$, and $\int t^2K(t)dt=k_2\neq 0$ and replacing y = x − ht the integrated square bias is given by:

$$ \int bias_h(x)^2 dx \approx \frac{1}{4} h^4 k_2 \int f^{''}(x)^2 dx $$

and the variance is given by:

$$ \int var \hat{f}(x) dx \approx n^{-1} h^{-1} \int K(t)^2 dt $$

Given the bias and variance, approximate value of MISE will be

$$\label{mise} MISE \approx \frac{1}{4} h^4 k_2 \int f^{''}(x)^2 dx + n^{-1} h^{-1} \int K(t)^2 dt $$

(14)

The above equation reveals that there is a trade-off between the bias and variance terms: the bias can be reduced at the expense of increasing the variance, and vice versa, by adjusting the amount of smoothing or the bandwidth. Suppose h is chosen to minimise MISE. Choosing very low h will eliminate the bias but increase the integrated variance. On the other hand choosing high value of h will reduce the random variation (measured by variance) but will increase the bias or the systematic error. Thus choice of bandwidth implies a trade-off between random and systematic error.

Silverman [32] shows that the optimal value of bandwidth h that minimises approximate MISE is given by:

$$\label{hopt} h_{opt} \!=\! k_2^{-2/5}\left\{ \int\!\! K(t)^2 dt\right\}^{1/5}\left\{\int\!\! f^{''}(x)^2 dx \right\}^{-1/5} n^{-1/5} $$

(15)

It should be noted that the optimal value of bandwidth depends on the unknown kernel density estimate. We first discuss the choice of this unknown kernel.

Choice of kernel:

Substituting the value of h _opt (Eq. 15) back into formula for approximate MISE (Eq. 14) shows that if h is chosen optimally, then the approximate value of MISE will be

$$ \frac{5}{4} C(K)\left\{ \int f^{''}(x)^2 dx \right\}^{1/5}n^{-4/5} $$

where the constant C(K) is given by:

$$ k_2^{2/5}\left\{ \int K(t)^2 dt\right\}^{4/5} $$

Above formulae show that minimising MISE means choosing kernel K which minimises C(K) which in turn reduces to minimising $\int K(t) dt$. The kernel which minimises this term is the Epanechnikov kernel.

Choice of Bandwidth

As discussed earlier, Least square cross-validation method, a completely automatic method based on the dataset on hand, is widely used for choosing bandwidth. It is based on minimizing integrated square error which for any estimator $\hat{f}$ can be written as:

$$ \int (\hat{f} - f)^2 = \int\hat{f}^2 -2\int\hat{f}f +\int f^2 $$

Since last term of above equation does not depend on $\hat{f}$, the optimal choice of bandwidth corresponds to the choice which minimises $\int\hat{f}^2 -2\int\hat{f}f$. The basic principle of least square cross-validation method is to construct an estimate of this term from the data themselves and then to minimise this estimate over h to give the choice of bandwidth. Thus the optimal value of bandwidth depends on the data set used and is not chosen subjectively.