Stochastic Frontier Analysis

Stochastic Frontier Analysis in Hospital and Health System Efficiency

Stochastic frontier analysis (SFA) fits a best-practice production or cost frontier by regression and then asks how much of each hospital's distance from that frontier reflects inefficiency rather than chance. In health economics SFA is the main parametric alternative to data envelopment analysis, used to benchmark hospitals and to study the correlates of hospital cost inefficiency. The regression error is split into a symmetric noise term and a one-sided inefficiency term, and only assumptions about their distributions separate the two. Those assumptions and the case-mix controls decide how much excess cost is labelled inefficiency, so the scores are signals for investigation, not measured waste.

Why a hospital frontier needs a noise term

Production theory defines the frontier as the most output obtainable from given inputs, and technical efficiency as a unit's shortfall from that frontier. Aigner, Lovell and Schmidt noted that earlier parametric frontiers, fitted by mathematical programming, were extremely sensitive to outliers and gave estimates with no known statistical properties. Their alternative made the frontier itself stochastic, so a producer hit by drought would count as unlucky on their measure, not as inefficient.

Hospital data have obvious sources of noise. Jacobs, studying NHS trusts for the Centre for Health Economics, cites unexpected spending on repairs and temporarily high unobservable disease severity. A deterministic frontier reads every such shock, and every coding error, as inefficiency.

The composed error model

The model adds two error terms to a regression of logged output on logged inputs. For a production frontier and hospital $i$ the model is written as follows.

$$\ln y_i = \mathbf{x}_i \boldsymbol{\beta} + v_i - u_i, \qquad v_i \sim N(0, \sigma_v^2), \qquad u_i \ge 0$$

where $y_i$ is output, $\mathbf{x}_i$ is a row vector of logged inputs and other regressors, $\boldsymbol{\beta}$ is the vector of frontier parameters, $v_i$ is symmetric noise with variance $\sigma_v^2$, and $u_i$ is non-negative inefficiency, independent of $v_i$. Aigner, Lovell and Schmidt concentrated on a half-normal inefficiency term and also considered an exponential one, and the Stata manual for its frontier command adds a truncated normal option.

The model collapses to a deterministic frontier when the noise variance is zero and to an ordinary stochastic production function when the inefficiency variance is zero. Aigner, Lovell and Schmidt argued that efficiency should be measured against the noisy frontier $\mathbf{x}_i \boldsymbol{\beta} + v_i$, not against $\mathbf{x}_i \boldsymbol{\beta}$ alone, which in the log model gives a simple ratio.

$$TE_i = \frac{y_i}{\exp(\mathbf{x}_i \boldsymbol{\beta} + v_i)} = \exp(-u_i)$$

where $TE_i$ is the technical efficiency of hospital $i$, between 0 and 1. A cost frontier reverses the sign because inefficiency raises spending, and the Stata manual writes both cases as $v_i - s u_i$, with $s = 1$ for production and $s = -1$ for a cost function.

Estimating the frontier by maximum likelihood

The parameters are fitted by maximum likelihood estimation from the density of a normal plus a half-normal variable. The Stata manual gives the log-likelihood for this model.

$$\ln L = \sum_{i=1}^{N} \left\{ \frac{1}{2} \ln\left(\frac{2}{\pi}\right) - \ln \sigma_S + \ln \Phi\left(-\frac{s \epsilon_i \lambda}{\sigma_S}\right) - \frac{\epsilon_i^2}{2 \sigma_S^2} \right\}$$

where $N$ is the number of hospitals, $\epsilon_i = \ln y_i - \mathbf{x}_i \boldsymbol{\beta}$ is the composed residual, $\sigma_S = (\sigma_u^2 + \sigma_v^2)^{1/2}$, $\lambda = \sigma_u / \sigma_v$, $\Phi$ is the standard normal distribution function and $s$ is the sign defined above. Software also reports $\gamma = \sigma_u^2 / \sigma_S^2$.

Aigner, Lovell and Schmidt read $\lambda$ as an indicator of the relative variability of the two error sources, with values near zero meaning that noise dominates. They warned that $\sigma_u^2$ is not the variance of $u$, which is $(\pi - 2)\sigma_u^2 / \pi$, so $\gamma$ is not literally the inefficiency share of residual variance. The mean of the half-normal term is $\sigma_u \sqrt{2/\pi}$.

Inefficiency skews ordinary least squares residuals negatively in a production function and positively in a cost function, and the Stata manual describes Coelli's third-moment test built on that skewness. A test of no inefficiency lies on the boundary of the parameter space, so the standard likelihood ratio test is not valid and a one-sided version is used.

Recovering inefficiency for each hospital

Each hospital's residual still mixes $v_i$ and $u_i$. Hospital studies by Mutter and colleagues and by Rosko and colleagues follow Jondrow and colleagues and predict inefficiency from its distribution conditional on the residual, given in the Stata manual as follows.

$$\hat{u}_i = E(u_i \mid \epsilon_i) = \mu_{*i} + \sigma_* \frac{\phi(\mu_{*i} / \sigma_*)}{\Phi(\mu_{*i} / \sigma_*)}, \qquad \mu_{*i} = \frac{-s \epsilon_i \sigma_u^2}{\sigma_S^2}, \qquad \sigma_* = \frac{\sigma_u \sigma_v}{\sigma_S}$$

where $\hat{u}_i$ is predicted inefficiency, $\phi$ is the standard normal density, and $\mu_{*i}$ and $\sigma_*$ are the parameters of the normal distribution that, truncated at zero, gives the conditional distribution. The manual also gives the estimator $E\{\exp(-s u_i) \mid \epsilon_i\}$, which for a cost frontier is a ratio of observed to frontier cost of 1 or more.

The prediction shrinks each residual towards the average inefficiency and inherits every distributional assumption of the model.

Worked example with two hospitals on a cost frontier

The figures are illustrative. A half-normal cost frontier for acute hospitals gives $\sigma_u = 0.20$ and $\sigma_v = 0.10$. Hospital A spends GBP 67.5 million against a frontier cost $\exp(\mathbf{x}_A \boldsymbol{\beta})$ of GBP 50 million, so $\epsilon_A = \ln(67.5/50) = 0.30$, and hospital B spends GBP 52.6 million against the same frontier cost, so $\epsilon_B = \ln(52.6/50) = 0.05$.

$$\sigma_S^2 = 0.20^2 + 0.10^2 = 0.05, \qquad \sigma_S = 0.2236, \qquad \lambda = \frac{0.20}{0.10} = 2, \qquad \gamma = \frac{0.04}{0.05} = 0.8, \qquad \sigma_* = \frac{0.20 \times 0.10}{0.2236} = 0.0894$$

where the symbols are as defined above. With $s = -1$, $\mu_{*i} = 0.8 \epsilon_i$. For A, $\mu_{*A} = 0.24$ and $\mu_{*A} / \sigma_* = 2.683$, where $\phi = 0.0109$ and $\Phi = 0.9964$, a ratio of 0.0109.

$$\hat{u}_A = 0.24 + 0.0894 \times 0.0109 = 0.2410$$

where $\hat{u}_A$ is A's predicted inefficiency. For B, $\mu_{*B} = 0.04$ and $\mu_{*B} / \sigma_* = 0.447$, where $\phi = 0.3610$ and $\Phi = 0.6726$, a ratio of 0.5367.

$$\hat{u}_B = 0.04 + 0.0894 \times 0.5367 = 0.0880$$

where $\hat{u}_B$ is B's predicted inefficiency. As frontier cost over observed cost, A scores $\exp(-0.2410) = 0.786$ and B scores $\exp(-0.0880) = 0.916$.

A deterministic reading puts the two hospitals 0.210 apart. The stochastic model pulls both towards the mean inefficiency of $\sigma_u \sqrt{2/\pi} = 0.20 \times 0.798 = 0.160$ and narrows the gap to 0.130. About a fifth of A's residual is attributed to noise, while B is credited with some inefficiency because noise may have flattered its costs. The ranking holds, but the amount of shrinkage rests entirely on the half-normal assumption and the estimated $\lambda$.

Cost frontiers and what the inefficiency term contains

Hollingsworth points out that a production frontier needs all outputs aggregated into one measure, which he calls questionable in health care, so researchers often estimate cost frontiers, since costs sum in money. A typical hospital model regresses logged total cost on outputs such as admissions and outpatient visits, input prices and product descriptors such as a case-mix index, dividing through by the wage to impose linear homogeneity in input prices. Mutter and colleagues used a translog form. Including case mix and producer characteristics also lets hypotheses about the correlates of inefficiency be tested statistically.

A cost frontier changes what $u_i$ means. Jacobs explains that the term reflects technical inefficiency only if allocative efficiency is assumed, and otherwise combines technical and allocative failures that the method cannot separate, although the two call for different policy responses. Rosko and colleagues chose SFA precisely because cost efficiency is the broader measure.

Evidence from hospital applications

Rosko and colleagues date the first health care application to Wagstaff's 1989 study of 49 Spanish hospitals. Rosko and Mutter counted at least 27 SFA studies of United States hospitals by 2011, and an earlier review of 20 found SFA results relatively insensitive to several model variations.

Mutter, Rosko and Wong studied 1,290 urban acute hospitals in 20 states in 2001. Adding comorbidity measures lowered mean estimated inefficiency from 0.174 in their first model to 0.143 in the first model that included them, and they concluded that patient burden of illness had been masquerading as inefficiency.

In Jacobs's NHS study, stochastic cost frontiers with a half-normal inefficiency term replicated five DEA specifications. Mean efficiency lay between 0.831 and 0.876, scores were spread much more narrowly than with DEA, and they correlated at around 0.7 with the Department of Health's regression cost indices. Jacobs concluded that the data held a large amount of random noise and that savings from bringing up poorer performing trusts would be very modest.

Choosing between SFA and data envelopment analysis

Jacobs describes a trade-off. SFA allows for noise but imposes a functional form and strong assumptions about the inefficiency term, DEA needs no functional form but counts all deviation as inefficiency, and each rests on assumptions that cannot be tested. Rosko and colleagues report that the hospital literature has established no preference for either and that the choice should follow the aim of the analysis and the data. Hollingsworth concludes that a mix of DEA and SFA specifications often makes most sense, in the hope of consistent results.

Panel data allow fixed effects for each hospital, whose standard errors Jacobs suggests could show how far a trust differs from best practice. The WHO's 2001 comparison of 191 health systems used country fixed effects and, as Cylus and Pearson describe, ranked Oman first. Gravelle and colleagues judged the approach inappropriate, partly because 50 countries had one year of data, and in a between-country model Oman fell to 169th.

Reading the scores and common misreadings

An SFA score is relative to the frontier estimated from the sample, and Cylus and Pearson note that frontier methods assume all units share the same production possibilities. Jacobs concluded that scores should not be read as accurate point estimates, that several specifications should give ranges that act as signals, and that funding decisions should not rest entirely on them. Hollingsworth adds that inefficiency may be under or overestimated if the normality assumption fails, and that functional form is another source of error.

Three misreadings recur. The inefficiency term is treated as proven waste, when it also holds unmeasured case mix or quality. A cost frontier score is called technical efficiency, when it may include allocative inefficiency. Scores from different samples or models are compared directly, although each frontier belongs to its own data. SFA compares producers such as hospitals or health systems and is unrelated to the cost-effectiveness frontier used for treatments in health technology assessment. Change over time is tracked with a Malmquist index, and single-number summaries are covered under efficiency index.

Sources

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