A new model for the length of stay of hospital patients

Abstract

Hospital Length of Stay (LoS) is a valid proxy to estimate the consumption of hospital resources. Average LoS, however, albeit easy to quantify and calculate, can be misleading if the underlying distribution is not symmetric. Therefore the average does not reflect the nature of such underlying distribution and may mask different effects. This paper uses routinely collected data of an Italian hospital patients from different departments over a period of 5 years. This will be the basis for a running example illustrating the alternative models of patients length of stay. The models includes a new density model called Hypergamma. The paper concludes by summarizing these various modelling techniques and highlighting the use of a risk measure in bed planning.

Appendix section

A function $g:\mathbb {R}\rightarrow \mathbb {R}$ is a probability density function of a continuous random variable if

• g is a non-negative Lebesgue-integrable function;

• ${\int }_{-\infty }^{+\infty }\, g(x)\, dx=1$.

Then we have the following result

Proposition 1

Let g be a probability density function. For every α ₁ , α ₂ , …, α _n ≥0 with α ₁ +α ₂ +⋯+α _n =1, and ł ₁ , ł ₂ , …, ł _n > 0, the function

$$h(\tau)=\sum\limits_{i=1}^{n}\, \alpha_{i} g(-\l_{i}\tau)\l_{i}$$

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is a probability density function.

Proof

It suffices to prove that ${\int }_{\mathbb {R}}\, h(\tau )\, d\tau =1$. In fact, we have

$$\begin{array}{@{}rcl@{}} {\int}_{\mathbb{R}} h(\tau) d\tau = \sum\limits_{i=1}^{n} \alpha_{i}{\int}_{\mathbb{R}}g(-\tau \lambda_{i}) \lambda_{i} d\tau. \end{array}$$

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For any integral, in the sum, we apply the change of variable τ = −s/λ _i . Thus

$$\sum\limits_{i=1}^{n} \alpha_{i}{\int}_{\mathbb{R}}g(-\tau \lambda_{i}) \lambda_{i} d\tau = \sum\limits_{i=1}^{n} \alpha_{i}{\int}_{\mathbb{R}}g(s) ds = \sum\limits_{i=1}^{n} \alpha_{i}=1.\\$$

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In the same way it is possible to evaluate the mean and the variance of a random variable distributed according to a Hypergamma density function:

$$\begin{array}{@{}rcl@{}} &&\mathbb{E}[X] = \frac{k}{\theta}\, \sum\limits_{i=1}^{n} \frac{\alpha_{i}}{\lambda_{i}}, \end{array}$$

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$$\begin{array}{@{}rcl@{}} &&\mathbb{E}[(X-\mathbb{E}[X])^{2}] = \frac{(k^{2}+k)}{\theta^{2}}\, \sum\limits_{i=1}^{n} \frac{\alpha_{i}}{{\lambda_{i}^{2}}}-\biggl(\frac{k}{\theta}\, \sum\limits_{i=1}^{n} \frac{\alpha_{i}}{\lambda_{i}}\biggr)^{2}, \end{array}$$

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$$\begin{array}{@{}rcl@{}} &&\mathbb{E}[e^{Xt}] = \sum\limits_{i=1}^{n} \alpha_{i} \biggl(1- \frac{1}{\theta} \frac{t}{\lambda_{i}}\biggr)^{-k}. \end{array}$$

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Here the moment generating function is well defined on the interval t < 𝜃min(λ ₁, λ ₂, . . . , λ _n ).