Translating Frontiers Into Practice: Taking the Next Steps Toward Improving Hospital Efficiency

Abstract

Frontier techniques, including data envelopment analysis (DEA) and stochastic frontier analysis (SFA), have been used to measure health care provider efficiency in hundreds of published studies. Although these methods have the potential to be useful to decision makers, their utility is limited by both methodological questions concerning their application, as well as some disconnect between the information they provide and the insight sought by decision makers. The articles in this special issue focus on the application of DEA and SFA to hospitals with the hope of making these techniques more accurate and accessible to end users. This introduction to the special issue highlights the importance of measuring the efficiency of health care providers, provides a background on frontier techniques, contains an overview of the articles in the special issue, and suggests a research agenda for DEA and SFA.

Full text

Translating Frontiers into Practice: Taking the Next Steps towards Improving Hospital
Efficiency
Ryan L. Mutter, Ph.D.*
Agency for Healthcare Research and Quality
Center for Delivery, Organization and Markets
540 Gaither Road
Rockville, MD 20850
301-427-1415 (Phone)
301-427-1430 (Fax)
Ryan.Mutter@ahrq.hhs.gov
Michael D. Rosko, Ph.D.
Widener University
School of Business Administration
One University Place
Chester PA, 19013
610-499-4322 (Phone)
610-499-4615 (Fax)
mdrosko@widener.edu
William H. Greene, Ph.D.
New York University
Stern School of Business
44 West Fourth Street, 7-90
New York, NY 10012
212-998-0876 (Phone)
212-995-4218 (Fax)
wgreene@stern.nyu.edu
Paul W. Wilson, Ph.D.
John E. Walker Department of Economics
Clemson University
222 Sirrine Hall
Clemson, SC 29634
864-656-2032 (Phone)
864-656-4192 (Fax)
pww@clemson.edu
*Corresponding author
Keywords: Efficiency, hospital, frontier approaches
Forthcoming, Medical Care Research and Review
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ACKNOWLEDGEMENTS
This paper does not represent the policy of either the Agency for Healthcare Research
and Quality (AHRQ) or the U.S. Department of Health and Human Services (DHHS). The
views expressed herein are those of the authors and no official endorsement by AHRQ or DHHS
is intended or should be inferred.
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ABSTRACT
Frontier techniques, including data envelopment analysis (DEA) and stochastic frontier
analysis (SFA), have been used to measure health care provider efficiency in hundreds of
published studies. Although these methods have the potential to be useful to decision makers,
their utility is limited by both methodological questions concerning their application, as well as
some disconnect between the information they provide and the insight sought by decision
makers. The articles in this special issue focus on the application of DEA and SFA to hospitals
with the hope of making these techniques more accurate and accessible to end users. This
introduction to the special issue highlights the importance of measuring the efficiency of health
care providers, provides a background on frontier techniques, contains an overview of the
articles in the special issue, and suggests a research agenda for DEA and SFA.
3

Cost, access, and quality continue to be important problems confronting health care
delivery in the United States. Total national health expenditures were $1.35 trillion in 2000
(Hartman, Martin, Nuccio, Catlin, et al., 2010) and were estimated to be $2.57 trillion in 2010,
an increase of over 90 percent (Truffer, Keehan, Smith, et al., 2010). Hospital care mirrored this
pattern, starting the period at $416.9 billion and ending at $788.9 billion, which was a growth of
more than 80 percent. Health care expenditures as a percentage of GDP are expected to be 17.3
in 2010 (Truffer, Keehan, Smith, et al., 2010). This is much more than any other industrialized
country. While the U.S. devotes a greater share of its economy to health care than other
developed countries (Anderson, Reinhardt, Hussey, and Petrosyan, 2003), broad health
indicators suggest that it may not all be money well spent. For example, among the 30 countries
that participate in the Organization for Economic Cooperation and Development (OECD), the
U.S. ranks 23rd in life expectancy at birth and 28th in infant mortality rate (Peterson and Burton,
2007). Anderson et al. (2003) indicate that the higher share in GDP allocated to health care in the
U.S. is mostly due to much higher prices and found that most of the broad health services
utilization figures in the U.S. were below the OECD median.
In 2008, there were over 43 million Americans without health insurance coverage
(Heyman, Barnes, and Schiller, 2009). Many more were without adequate health insurance. The
high cost of premiums or premium contributions at work has created a financial barrier to both
health insurance and health care. According to the 2008 Kaiser Survey, about 29 percent of the
uninsured postponed health care because of cost considerations (Kaiser Commission on
Medicaid and the Uninsured, 2009). In contrast, 7 percent of the insured postponed health care.
Delaying health care can lead to complications requiring more expensive treatment, as well as
premature death. Thus, there is evidence in health care of a vicious cycle of high costs, leading to
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poor access, which in turn leads to undesirable outcomes. While the Patient Protection and
Affordable Care Act is estimated to help about 28 million Americans gain new health insurance
coverage by 2016 (McGlynn, Cordova, Wasserman, and Girosi, 2010), the impact of the health
reform legislation on costs and quality is less certain.
While it might be relatively easy to deal with one element of the trifecta of cost, access,
and quality, dealing with all three simultaneously is a much more difficult challenge.
Conceptually, one of the best approaches to addressing all three issues is to improve the
efficiency of health care providers. For example, an increase in efficiency could reduce health
care costs. These savings could be passed on to consumers in the form of reduced health
insurance premiums or health care prices, which could then potentially increase access. Finally,
more efficient processes could lead to better quality. (In the hospital literature, there is some
evidence that more efficient institutions have better outcomes. See, for example, McKay and
Deily [2008].) In the general literature, Deming (1982) called this the “value proposition” (i.e.,
simultaneously decreasing costs and increasing quality).It is a major tenet of total quality
management (TQM).
HOSPITAL INEFFICIENCY MEASUREMENT WITH FRONTIER TECHNIQUES
The above discussion suggests the importance of improving efficiency. But to do so, it is
essential to accurately measure it. Indeed, the following aphorism about the importance of
measurement is frequently attributed to management thought leader Peter Drucker: “If you
cannot measure something you cannot control it. And if you cannot control it you cannot manage
it.” The development of frontier methods, a set of techniques that measure inefficiency as the
distance between a best practice frontier (BPF) and actual performance, has advanced the
practice of efficiency measurement in health care.
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While a variety of frontier techniques exist, data envelopment analysis (DEA) and
stochastic frontier analysis (SFA) are the most frequently applied approaches. Farrell (1957) was
the first to estimate productive efficiently as a distance from a BPF using linear programming
methods. A significant breakthrough occurred when Charnes, Cooper and Rhodes (1978)
generalized Farrell’s single input/output measure to a multiple-input/multiple-output technique,
which they termed DEA. Charnes, Cooper, Lewin, and Seifford (1994) viewed the development
of two-stage analysis as a significant advance in DEA-based research. Combining non-
parametric and parametric methods, researchers in the early 1990s began to explore the factors
that determine inefficiency.
The first published health care application of DEA was by Wilson and Jadlow (1982).
Nunamaker (1983) published the first DEA study of U.S. hospitals. The most recent literature
review article reported that, as of mid-2006, 317 papers on frontier efficiency of health care
organizations had been published. Of these, over 200 used DEA and 57 used SFA. About 25
studies were classified as Malmquist-based (an extension of DEA) productivity studies
(Hollingsworth, 2008).
The history of SFA begins with the near simultaneous publication of articles by Aigner,
Lovell, and Schmidt (1977) and Meeusen and van den Broeck (1977). A major breakthrough was
accomplished by Jondrow, Lovell, Materov, and Schmidt (1982) who derived an estimation of
one-sided residuals, interpreted as inefficiency scores, in a cross-sectional setting. This advance
permitted the estimation of inefficiency for individual units. The ability to obtain producer-
specific estimates of efficiency greatly enhanced the appeal of SFA (Kumbhakar and Lovell,
2000). In the 1990s, panel models were developed that relaxed the assumption of time-invariant
efficiency (Cornwell, Schmidt and Sickles, 1990; Kumbhakar, 1990; Battese and Coelli, 1992).
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Paralleling DEA, SFA studies initially used a two-stage approach to examine the exogenous
factors associated with efficiency. This become outmoded with the development of more
efficient (in a statistical sense) single-stage approaches that incorporate explanatory variables
into the efficiency error component (Kumbhakar, Gosk, and McGuckin, 1991; Battese and
Coelli, 1995).
The first SFA study of a health care organization was published by Wagstaff (1989),
who examined 49 Spanish hospitals. Zuckerman, Hadley, and Iezzoni (1994) published the
first SFA study of U. S. hospitals. Since then at least 27 other U.S studies of hospitals have
been conducted. Their results are synthesized by Rosko and Mutter later in this Special Issue.
AHRQ CONFERENCE AND THIS SPECIAL ISSUE
In April 2008, the Agency for Healthcare Research and Quality (AHRQ) released a
commissioned report prepared by the RAND Corporation entitled “Identifying, Categorizing,
and Measuring Health Care Efficiency Measures” (McGlynn, 2008). This report noted that
efficiency measurement techniques developed in the academic literature, namely frontier
techniques, have not been applied in the policy setting. McGlynn (2008) identified these
measures, including SFA and DEA, as among the most promising approaches for measuring
provider efficiency and for suggesting strategies for improving the delivery of health care
services. Consequently, AHRQ hosted an invitational meeting, “Translating Frontiers into
Practice: Taking the Next Steps towards Improving Efficiency,” on August 27 – 28, 2008. This
meeting brought together policy makers, stakeholders, and leading technical experts to discuss
how frontier techniques can be used most effectively to address the problems confronting the
health care system and to identify how the needs of end users should shape the agenda of the
research community.
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AHRQ followed the conference by sponsoring this special issue of Medical Care
Research and Review. The aim of this special issue is to assess the state of the science of using
frontier techniques to measure provider efficiency and to identify steps that can be taken to
increase its accessibility to end users. The papers contained in it review the contributions
made using these techniques, identify where gaps remain, and offer examples of the use of
frontier approaches that hold promise for the future. An overview of the papers in this special
issue follows.
OVERVIEW OF SPECIAL ISSUE ARTICLES
OZCAN AND LUKE
Ozcan and Luke used an extension of DEA, called the Malmquist technique, to examine
the impact of a major restructuring that the Veterans Health Administration (VHA) implemented
in 1995 on productivity change. The VHA decentralized decision making from VHA
headquarters to regions and integrated services and assets regionally to create 21 regional
providers called Veterans Integrated Service Networks (VISNs). The VHA’s restructuring
mirrors efforts in other countries to re-organize health care along regional lines.
The authors calculated a Malmquist Index, a measure of productivity change over time,
which can be decomposed into technical change (i.e., change in productivity associated with an
inter-temporal shift in the best practice production frontier) and efficiency change (i.e., each
unit’s change or “catching up” in measured efficiency relative to each year’s respective
efficiency frontier). Another notable feature of their article was a demonstration of the use of the
Malmquist technique for benchmarking. For example, they found that in 2004 12 VISNs were
off the best-practice production frontier. Ozcan and Luke indicated how much the VISNs would
need to reduce each of the 4 inputs in their model to reach the frontier.
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Their article provides valuable insights into a methodology for analyzing innovations that
are similar to the restructuring that occurred in the VHA. Looking to the future, the Patient
Protection and Affordable Care Act is funding demonstration projects for the creation of
Accountable Care Organizations (ACO) (Health Policy Brief, 2010). Ozcan and Luke’s article
provides a description of a methodology that might be useful for understanding the impact of the
ACOs.
BERNET, MOISES, AND VALDMANIS
Bernet, Moises, and Valdmanis used DEA to measure the social efficiency of hospital
care by incorporating costs from the consumers’ perspective. They did this by adding patient
travel time to the DEA model. This is an important innovation because most evaluations of
hospital cost containment programs consider only hospital expenditures and neglect to consider
the opportunity costs of patients and their families (Folland, Goodman, and Stano, 2007). An
example will highlight the effect of this omission. The Medicare Prospective Payment System
(PPS) created strong incentives for hospitals to discharge their patients earlier. Consequently,
they were less clinically stable when they left the hospital (Rosko and Broyles, 1988).
Recognizing this, many patients were discharged to other post-acute care facilities, such as
skilled nursing facilities or rehabilitation hospitals, while many patients who were discharged
home were given a coordinated home care plan and received visits from home health nurses.
However, it is highly likely that care provided by visiting nurses had to be augmented by
informal care givers, such as family and friends. While the Medicare PPS saved billions of
dollars in inpatient hospital expenditures (Folland et al., 2007), the “true” social saving are
overstated because the increased cost of informal care was not included in the cost calculations.
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Bernet et al. partially overcame this type of specification error by modeling travel time as an
input in the hospital care production process.
Methodologically, they made an interesting adaptation of the Malmquist technique.
Rather than comparing changes over time as is done in the Malmquist approach, they compared
the frontier differences by urgent versus non-urgent condition in an approach similar to that used
by Grosskopf, Margaritis, and Valdmanis (2001). The authors concluded that including patient
travel distance in the model yields a more comprehensive view of social welfare and social costs.
The nuances identified by their methodology have the potential to yield interesting insights. For
example, while their results suggest that public hospitals are less efficient than their private
counterparts, when travel costs are included, the efficiency gap narrows. The authors suggest that
this indicates that public hospitals provide an increased value-added by being closer to the
population in need. In future applications, the methodology proposed by Bernet et al. could be
valuable in the assessment of the social efficiency impact of hospital closures, mergers, and
relocations.
MURPHY, ROSENMAN, MCPHERSON, AND FRIESNER
Many of the studies in the frontier literature yield a single measure of inefficiency for a
decision-making unit (DMU). However, Murphy, Rosenman, McPherson, and Friesner describe
an approach that generates efficiency measures within the organization at the level of the hospital
cost center. Their technique used a combination of DEA, spreadsheet modeling, and regression,
and it took into account the sequential nature of hospital production rather than assuming, as
many frontier studies do, that production takes place jointly. They assessed the extent of shared
inefficiency across eight cost centers that encompass many of hospitals’ activities, and they
identified the cost centers within the hospitals that contribute the most to the shared inefficiency.
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Murphy et al. applied their approach to a panel of mid- to large-size nonprofit hospitals in
the State of Washington using Washington State Department of Health data from 2003 – 2007.
They found evidence of considerable shared inefficiency across hospital cost centers. Their
results suggest that three cost centers affect the total performance of the firm. They are plant, the
emergency department, and an “other” category, which consists of housekeeping, laundry,
nutrition, and central services.
Hospital managers and policymakers have sometimes struggled with the results of
frontier studies because they do not know what to do in response to them. Understanding the
relative performance of an institution can be useful, but decision makers want to know what they
need to do to improve efficiency. The approach described by Murphy et al. offers insight into
the performance of institutions (i.e., inefficiency is shared) and holds promise to help managers
and policymakers design interventions to improve the efficiency of operations (e.g., focus on
centers that are the greatest contributors to the overall inefficiency of the firm).
Although the methodology of Murphy et al. has some technically appealing aspects to it,
it is also a fairly novel approach. As such, continued work is needed to maximize its potential.
The technique does not have explicit statistical foundations, and Murphy et al. operate without a
conceptual framework concerning the mechanism through which inefficiency among hospital
cost centers is shared. Addressing these limitations through future research could add to this
approach, which has the potential to help bridge the gap between frontier analysis and improved
practice.
ROSKO AND MUTTER
In the final article in this issue, Rosko and Mutter reviewed 27 studies where SFA has
been applied in the analysis of U.S. hospitals. They used X-Inefficiency Theory as the
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framework for much of their discussion. X-Inefficiency Theory posits that environmental
pressures, including internal and external factors, can influence the level of effort (and hence
efficiency) given by firm managers. Rosko and Mutter reviewed where SFA has been used to
examine the impact of these factors on hospital cost-inefficiency.
The internal factors examined in the literature include ownership type, system
membership, and specialty hospital status. Although inconsistent results were found for
ownership type, the literature provided support that other internal characteristics, including
system membership and the provision of general medical and surgical services, were
associated with lower cost-inefficiency. There were some important nuances related to these
findings, which are discussed in the review article.
Public payment policy, critical access hospital (CAH) status, market unemployment
rate (a proxy for uncompensated care), health maintenance organization (HMO) penetration,
and hospital competition were the external factors analyzed in the literature. In general,
higher Medicare and Medicaid share of inpatients, a higher unemployment rate, and greater
HMO penetration were associated with lower hospital cost-inefficiency. Conversion to CAH
status was associated with greater cost-inefficiency, and cost-inefficiency increased, on
average, with the number of years an institution participated in the CAH program. The
findings with respect to hospital competition were mixed. Overall, these results supported
the contention of X-Inefficiency Theory that external pressures can shock organizations into
better performance.
Rosko and Mutter reviewed other results from the hospital SFA literature of potential
interest to policymakers and practitioners, including factors associated with changes in
hospital inefficiency over time; the relationship of SFA-derived measures of inefficiency
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with other measures of hospital performance, including outcome measures of quality; the
association between hospital inefficiency and hospital expansion and closure; and the
application of SFA to hospital benchmarking and budgeting.
Since the usefulness of a study’s findings are determined, in part, by the
appropriateness of the methods employed, Rosko and Mutter also reviewed some of the
recent advances in controlling for output heterogeneity. Unmeasured variations in output (i.e.,
type or quality of product) might result in overestimates of cost-inefficiency. For example, if it
costs more to treat more severely ill patients, and no variables for severity are included in the
SFA model, then hospitals that tend to treat these more expensive types of patients will have
higher cost inefficiency estimates. This problem is compounded by the likelihood that certain
types of hospitals (e.g., tertiary care hospitals) tend to attract patients with more complicated
problems. The controls for hospital quality and patient burden of illness highlighted in the
review have the potential to yield more accurate SFA estimates of hospital efficiency.
FUTURE DIRECTIONS
Much of the research presented in this special issue consists of innovative applications of
frontier approaches that are suggestive of future directions that can be taken that could improve
the estimates generated by these techniques and that could make their results more actionable by
decision makers. To build upon the directions indicated in these papers, the following section
presents a research agenda for DEA and SFA.
A RESEARCH AGENDA FOR DEA
While numerous references to DEA models can be found in the academic literature, DEA
is but one of several estimators one might use in a non-parametric model of production. A non-
parametric model of production consists of a set of minimal assumptions about the process that
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generates observed data on input and output quantities. Typically, these assumptions describe a
production set consisting of all feasible combinations of input and output quantities; the
production set is usually (but not always) assumed to be convex. A non-parametric model must
also include assumptions about the probability process that generates observed data; for DEA
estimators to be statistically consistent, one must assume that the joint probability density of
inputs and outputs is zero everywhere outside the production set and strictly positive over at least
some region of the production set. Other assumptions on the smoothness of the boundary (or
frontier) of the production set may be made, but in a non-parametric model, no assumptions
about the specific form of the frontier or the joint density of inputs and outputs are made. See
Simar and Wilson (2000b; 2008) and Kneip, Simar, and Wilson (2008) for specific assumptions
and additional details.
The DEA efficiency estimator provides an estimate of distance from a fixed point to the
boundary of the production set along some path chosen a priori by the researcher. This amounts
to measuring distance (typically, in the case of DEA estimators, by solving a linear program)
from the fixed point of interest to a DEA estimate of the frontier. DEA estimates of either the
frontier or efficiency involve uncertainty, creating a need for statistical inference; both the true
frontier and hence the true level of efficiency are unobservable.
Although DEA estimators have been used since Farrell (1957), researchers have only
recently derived the estimator’s statistical properties. Under certain assumptions the DEA
frontier estimator is a consistent, maximum likelihood estimator (Banker, 1993), with rates of
convergence given by Korostelev, Simar, and Tsybakov (1995). Consistency and convergence
rates of DEA efficiency estimators have been established by Kneip, Park, and Simar (1998). See
Simar and Wilson (2000b) for a survey of the statistical properties of DEA estimators.
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It is important to note that consistency, while perhaps the most fundamental property of
an estimator, is at the same time a very weak property. In particular, DEA estimators suffer from
the curse of dimensionality in the sense that their convergence rates decrease with increasing
numbers of dimensions (i.e., numbers of inputs and outputs). Consequently, for a given sample
size, estimation error increases exponentially with dimensionality. In non-parametric regression,
various approaches to dimension reduction have been developed to deal with this problem, but to
date it is not clear how such methods might be used with DEA estimators. Given that many
applications of DEA estimators involve relatively small sample sizes, more research is needed in
this area.
While DEA estimators have been widely used, inference about the underlying model
structure or the efficiencies that are estimated is often ignored even today. In the past, inference
was problematic since properties of DEA estimators were unknown and consistent bootstrap
methods were not available, but this is no longer true today. Simar and Wilson (1998; 2000a)
proposed bootstrap methods for inference about efficiency based on DEA estimators in a
multivariate framework, and Simar and Wilson (2001a; 2001b) proposed bootstrap methods for
testing hypotheses about the structure of the underlying nonparametric model of production, but
consistency of these procedures has not been established. Banker (1993; 1996) proposed tests of
model structure based on ad-hoc distributional assumptions, but simulation results obtained by
Kittelsen (1999) and Simar and Wilson (2001a) show that these tests perform poorly in terms of
both size and power.
Kneip et al. (2008) and Park, Joeng, and Simar (2010) derived the limiting distributions
of DEA efficiency estimators under variable returns to scale and constant returns to scale
(respectively), with arbitrary numbers of inputs and outputs. These distributions contain several
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unknown quantities, and are not useful in a practical sense for inference. Kneip et al. (2008) also
proposed two bootstrap procedures for inference about efficiency and proved consistency of both
methods. The first approach uses sub-sampling, where bootstrap samples of size m < n are drawn
(independently, with replacement) from the empirical distribution of the original n sample
observations. Simulation results provided by Kneip et al. (2008) indicate that in finite-sample
scenarios, coverages of confidence intervals for efficiency estimated by bootstrap sub-sampling
are quite sensitive to the choice of the sub-sample size m. In a more recent paper, Simar and
Wilson (2009) provide a data-based method for optimizing the choice of m; Monte Carlo
evidence indicates that the method works well in terms of coverages of estimated confidence
intervals. Simar and Wilson (2009) also show how subsampling methods can be used for
hypothesis testing in the DEA context and give examples including testing of returns to scale
(i.e., constant versus variable returns to scale). This should be especially useful in hospital
studies, where there is some debate about whether returns to scale are constant. Moreover, it is
easy to adapt the methods described by Simar and Wilson (2009) to other testing situations (e.g.,
testing whether for-profit hospitals differ in terms of efficiency from non-profit hospitals).
The second, full-sample bootstrap procedure described by Kneip et al. (2008) requires for
consistency not only smoothing of the distribution of the observations as proposed in Simar and
Wilson (1998; 2000a) but also smoothing of the initial DEA estimate of the frontier itself,
resulting in a formidable computational burden. Kneip, Simar, and Wilson (2009) developed a
computationally efficient full-sample bootstrap method for inference; while the method works
well for estimating confidence intervals for efficiency scores, it is not suitable for estimating
critical values for test statistics. For the latter, the sub-sampling methods of Simar and Wilson
(2009) appear to be more promising for most applied researchers.
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When DEA estimators are used to estimate Malmquist indices, infeasibilities often arise
when (estimated) frontiers from two time periods (or two groups) cross each other. Using the
hyperbolic efficiency estimator introduced by Färe, Grosskopf, and Lovell (1985) would avoid
this problem, but until very recently, this estimator has only been used while imposing constant
returns to scale in order to avoid computational difficulties. Wilson (2010) provides a simple
numerical procedure for computing hyperbolic DEA estimates and derives asymptotic properties
for the hyperbolic DEA estimator.
DEA estimators are known to be quite sensitive to outliers; apart from the curse of
dimensionality, this may be the most serious drawback to using DEA estimators, particularly
with hospital data in which outliers are common. One approach is to use outlier detection
methods to find the outliers. An alternative approach is to measure efficiency relative to an
estimated partial frontier instead of an estimate of the full frontier as with DEA.
Cazals, Florens, and Simar (2002) developed the notion of an order-m frontier. In this
approach, efficiency is measured in the input direction by comparing a given hospital’s input
usage to an estimate of the expected minimum input usage among m hospitals producing at least
as much of each output as the given hospital of interest. In the output orientation, efficiency is
measured by comparing a hospital’s output usage to an estimate of the expected maximum
output among m hospitals using at least as much of each input as the hospital of interest. This
approach has been shown to be robust with respect to outliers; in addition, the usual, parametric
rate of convergence is obtained, avoiding the curse of dimensionality, although the variance of
the estimator is affected by the number of inputs and outputs. The order-m efficiency estimator is
asymptotically normally distributed, provided m is not too large. Wilson (2010) extended the
order-m idea to the hyperbolic orientation.
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Daouia (2003), Aragon, Daouia, and Thomas-Agnan (2005) and Daouia and Simar
(2007) developed input- and output-oriented conditional order-α estimators and derived their
asymptotic properties. These results were subsequently extended to an unconditional, hyperbolic
order-α estimator by Wheelock and Wilson (2008). In each case, efficiency is measured relative
to a quantile of the joint distribution of inputs and outputs lying close to the full frontier, as
opposed to the full frontier itself. This has advantages similar to those of the order-m approach
(i.e., the estimators attain the parametric rate of convergence while avoiding the curse of
dimensionality, are robust with respect to outliers, and are asysmptotically normally distributed).
Only a few applications of the order-m and order-α estimators have been attempted at the
time of this writing, and most of these have been in banking (e.g., Wheelock and Wilson, 2008;
2009). Given that hospital data often contain outliers, and the sensitivity of DEA estimators to
outliers, these methods seem promising for researchers in health care. Moreover, the methods
have been included in the FEAR package for use with R. (See Wilson [2008] for details.) This
makes them more accessible for applied researchers.
Regardless of whether DEA, partial frontier, or even stochastic frontier estimators are
used, one must decide which variables to include in the model and what their role should be.
Non-parametric statistical methods are available for testing whether a particular variable acts as
an input or output (Simar and Wilson, 2001b) or how environmental variables might play a role
(Daraio, Simar, and Wilson, 2010). In health-care research, it seems important to control for
“quality,” but it is less clear how quality enters the production process. The available testing
methods may be useful here. While most would agree that quality is an important variable, there
is less agreement on how quality should be measured. Necessarily, anything that is to be
measured must be clearly defined, but unfortunately, discussions of health-care quality often
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proceed without a clear definition of what is being measured. Part of the difficulty surrounding
notions of quality relates to data availability; presumably, quality is related to health outcomes,
but data on patients’ overall level of health are often sketchy at best. With regard to quality, more
research is needed on all sides, including theoretical econometricians and statisticians, data
collectors, health-care practitioners, and empirical researchers.
A RESEARCH AGENDA FOR SFA
The stochastic frontier model builds on a production/cost function view of the process of
hospital care. The starting point of the analysis is a decision on the level of aggregation that is of
interest, possibly the hospital as a whole or some functional unit within it, such as the emergency
department. This is both a matter of appropriate model selection and a question of the interests
of the stakeholder engaged in the research.
Although the analysis can theoretically take place with respect to production or costs, in
the hospital care setting, it is the latter that is usually of greater interest. Indeed, that this is the
primary dimension on which DEA and SFA will differ – DEA will provide results about the
process of production while SFA will focus immediately on the cost consequences of production.
One of the important avenues of research will be to aggregate and to reconcile these two sources
of information.
A natural place to begin the search for best practice is total cost. A simple comparison of
“average” costs of hospitals immediately hits several obstacles. Costs differ greatly across
hospitals for many reasons having little to do with relative success. Hospitals differ on
dimensions that will substantively affect costs, such as the setting – urban/rural, local labor
market conditions – and case mix, whether mainly emergency/outpatient or inpatient, and with
respect to the latter, which types of care are mostly provided and whether the hospital is involved
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in teaching as well as treatment, or only the latter. This presents two challenges to the analyst:
(1) determining what to use as the output (i.e., the activity level of the hospital [patient days,
discharges, etc.]) and (2) to distinguish clearly the aspects of the cost consequences of these
activities that are the target of the study from the features of the hospital and its activity mix that
are not related to efficiency.
The overall purpose of frontier modeling, as discussed at several other points in this
issue, is to compare hospitals (their costs), not directly with respect to their costs, but rather on
the level of costs compared to what they could be. Thus, there are two degrees of comparison: in
absolute terms efficiency as measured with respect to how well a given hospital does compared
to how well it could do, and, second, in relative terms with respect to how well hospitals do
compared to each other.
The stochastic frontier is one of two commonly used methods that are specifically
designed to measure costs from the first viewpoint above, then to compare firms (hospitals) on
the second basis as a corollary. The methods differ on specific assumptions about the form of
the technology and how deviations from the frontier (best practice) arise and are ultimately
measured. The stochastic frontier model assumes that best practice differs across hospitals both
systematically and randomly, in ways that are not always measurable. The technique seeks to
measure efficiency against a hospital specific benchmark that accommodates differences such as
those noted above (e.g., case mix). A comparison of hospitals then measures them in terms of a
frontier that recognizes the differences that substantially affect costs. Superficially, the approach
makes sense. A hospital that treats an inherently expensive case mix of patients is not inefficient
compared to one that treats a less expensive case mix because of the case mix per se, though it
may be so (or not) for other reasons. Because there is vast heterogeneity in hospital types and in
20

the settings (markets) in which they operate, an important element of the ongoing research on
hospital cost efficiency relates to the appropriate ways to handle these differences. Where these
elements can change over time, such as in local labor markets, repeated observations of hospitals
that can reveal dynamic effects becomes yet another feature in the ongoing research.
What follows is a consideration of some specific dimensions of the ongoing research
about hospital costs. There are two large issues that impinge on the type of analysis that can be
done and which, ultimately, will color what can be expected of a cost study however carefully
done: ownership type and the impact of “quality” on hospital costs.
Hospitals differ discretely between three ownership types: government, for profit (FP)
and not for profit (NFP). This discussion will focus on differences between FP and NFP
institutions. It is distinctly possible (though not inevitable), that cost structures and outcomes
will differ substantively across these two types. In a statistical setting, it might seem appropriate
to perform separate analyses of these two broad types of hospitals. However, this approach
might turn up what appears to be differences in efficiency aspects of hospital costs that are
actually related to differences between FP and NFP types that are not what they appear
superficially to be. This is known as a “sample selection” problem. Unfortunately (for the
analyst), the differences between FP and NFP may well be real, but the conventional analytical
methods are inappropriate for uncovering those differences. A second related, though possibly
of lesser consequence, dimension of hospital environment is whether the hospital is a member of
network or a multi-hospital system. Compounding this is the existence of a variety of different
types of system structures (Bazzoli, Shortell, Dubbs, Chan, and Kralovec, 1999). Zuckerman,
Hadley, and Iezzoni (1994) provide some empirical evidence and wrestle with the issue of using
pooled versus group-specific frontiers. (A disadvantage of group-specific frontiers is that
21

inefficiency cannot be compared across groups.) This topic could benefit from further
consideration by the field.
A significant component of the costs of hospital patient care is “quality,” broadly defined.
Fundamentally, quality is not directly measurable – it is impossible to agree on an objective
measure – but it is obviously a substantive element of the cost structure. It is an important
question at the outset of the discussion whether quality is something that hospitals produce along
with other outputs, as measured by inpatient days or outpatient discharges, or whether quality is
an input into the process, along with materials, labor (nurses, doctors, technicians, etc.) and
capital (beds, etc.) Properly accounting for quality of care might reasonably be viewed as the
signature open end in contemporary research on hospital costs. Regardless of how quality is
measured and interpreted, its presence in the cost structure of the hospital implies that cost
efficiency always embodies this hidden feature of production. For example, the most natural
outcome will be that unaccounted for investment in quality will show up in a cost model as if it
were inefficiency, a distinctly unappealing result if one seeks a cost basis for measuring
(in)efficiency. Recent and ongoing research in econometric methodology has considered this
problem of how to accommodate unmeasured heterogeneity, such as quality, in hospital costs.
One aspect of this study is how best to make use of observable proxies for quality such as
investments in technology (i.e., structures), institution of broad procedure regimes (i.e.,
processes), and in-hospital mortality (i.e., outcomes) (Romano and Mutter, 2004).
Finally, there are many fine, but still substantive, technical points about the model
building itself that the researcher will want to consider. The search for the right model is
ultimately the search for a model specification that will properly isolate inefficiency from the
production process itself and both observed (e.g., at least to some extent, case mix) and
22

unobserved (e.g., quality) heterogeneity from the measurement of inefficiency. One such feature
of the model is the way that it handles the inevitable variation across hospitals in the technical
parameters of the model equations. These are econometric issues including, for example,
“heteroscedasticity,” and “parameter heterogeneity,” that impinge on the ability of the model to
produce appropriate estimates of hospital efficiency measures. One of the largely unanswered
(satisfactorily) questions in the broad field of efficiency analysis is the extent to which departures
from purity in the structural features of the cost model are propagated in the implied estimates of
quantities of interest, particularly measures of efficiency. In short, can the imperfect model be
trusted to provide useful results in spite of its imperfections? The answer to this question is a
matter of degree, not of type.
The preceding suggests an agenda for the researcher interested in formulating a modeling
framework for measuring efficiency in hospital care. There remains a detail that, ultimately, is
of crucial importance. That is, packaging the results of the stochastic frontier modeling in a way
that is useful for the policy maker or manager. Stochastic frontier results come in the form of
quantitative efficiency measures, for example, a hospital (and possibly time) specific measure of
the percentage difference between what is achieved and what appears to be theoretically
possible. There is a question of the statistical uncertainty in the estimates. It may be appropriate,
for purposes of overall policy analysis, to suggest a credible range of values for this quantity
rather than a single number. If so, the methodology remains incomplete on how such a range
should be constructed. Finally, in the search for best practice, it may evolve that ranks of
hospitals, not hospital specific estimates are the measures of interest. This implicitly suggests
that the best practice is determined in the field, not in the abstract. Turning the analysis in this
23

direction will then call upon the researcher to continue the study to pinpoint more precisely the
features of those hospitals that provide the standards for best practice.
CONCLUSION
The application of DEA and SFA in hundreds of published studies over the past several
decades indicates the need for techniques to measure efficiency in the health care sector. It also
suggests the usefulness of frontier methods. Nevertheless, these approaches are not without
drawbacks. Some important methodological questions remain unanswered, and the accessibility
of the insights from these methods to end users needs to be increased. The articles in this special
issue are intended to make contributions in both of these areas, and the research agendas for
DEA and SFA are intended to highlight where particular gaps in understanding remain and to
suggest where future work might be particularly useful.
24

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