The two major frontier techniques that have been used most frequently to estimate inefficiency at the hospital level are DEA and SFA. DEA is a deterministic approach based on the works by Debreu (1951)
and Farrell (1957)
and updated in terms of economic efficiency and productivity by Färe, Grosskopf, and Lovell (1994)
. Coelli et al. (2005)
and Aaronson et al. (2006)
suggest that contextual circumstances should determine which technique is selected when a choice between DEA and SFA has to be made. As our focus is on patient safety in a multi-input, multi-output framework, we opt for a DEA approach that specifically incorporates undesirable outputs. SFA is incapable of doing this.
DEA derives a best practice frontier by solving linear programming problems, which identify those hospitals maximizing outputs given inputs. Individual hospital performance is a proportional measure gauged relative to the frontier that is defined by hospitals deemed to be the best performers. A score of 1.00 indicates that a hospital is operating on the best practice frontier (i.e., one that is efficient). A score >1.00 indicates inefficiency with the difference between the actual score and 1.00 measuring the amount all outputs could be increased, holding inputs constant. In some instances, a hospital may be unique to all others in the sample, in which case it would receive an efficiency score of 1.00 because it lies on its own frontier. Therefore, we reiterate that efficiency is a product of the distribution of hospitals and that this performance measure is a relative measure of productivity and not an absolute measure of efficiency in the engineering sense.
Many benefits of DEA have been cited in the literature. Coelli et al. (2005)
provide a comprehensive discussion, and we provide a brief summary in Appendix A
. Two benefits of DEA that are particularly germane to our research are as follows. First, hospitals located in the interior of the frontier are strictly inefficient. This permits additional analyses exploring factors that separate best practice performers from less efficient producers. Second, we can decompose the DEA total efficiency measure into its various sources of inefficiency as shown in the following equation:
Total efficiency is measured under assumptions of constant returns to scale (CRS) (i.e., productivity does not increase with size) and strong disposability of outputs (SDO) (i.e., all outputs are considered desirable). Pure technical efficiency, variable returns to scale (VRS), measures only the input–output correspondence absent any scale or congestion effects. CRS/VRS measures scale efficiency attributed to hospital size. Congestion is derived by assessing productivity under assumptions of strong and weak disposability of outputs (WDO) (i.e., some outputs may be undesirable). In the approach taken here, we follow Balk's (1998)
assertion that no complete production study can ignore the output of undesirables, particularly from a social point of view. Earlier studies have used congestion analysis to account for pollution (Färe, Grosskopf, and Lovell 1994
), hospital uncompensated care (Valdmanis, Kumanarayake, and Lertiendumrong 2004
; Ferrier, Rosko, and Valdmanis 2006
), and hospital mortality rates (Clement et al. 2008
Total efficiency is a multiplicative total of all sources of inefficiency as seen in equation (1)
. (Interested readers should see Färe, Grosskopf, and Lovell 
for more information regarding this decomposition and the relevant economic proofs.) Here, we focus on quality congestion as it impacts the total production of hospital care. While an input minimization approach has been typically used in health care DEA studies, our quality-congestion analysis requires an output orientation because of the possible association of a high volume of patients served with the occurrence of more patient safety events. In other words, we need to measure how patient safety may be compromised if too many outputs are being produced, holding inputs fixed.
Under the SDO assumption, expansion of all outputs is desirable and reducing one output leads to the possible increase of another output. In contrast, the assumption of WDO treats expansion of some outputs as undesirable. The WDO technology is illustrated in . Two production frontiers (i.e., maximum combinations of outputs that can be produced with fixed levels of inputs), represented by curves CD and EF, are shown in the figure. The distance between points B and A on the two output-based production frontiers illustrates congestion—good output cannot be expanded without increasing the “bad” output. Unlike the typical output substitution in productivity studies (i.e., where one good is substituted or traded off for the production of another good), the good and bad outputs move in the same direction as seen by the line. If congestion did not occur, the higher production frontier, CD, would represent greater adjusted output (i.e., the downward adjustment would be 0).
Comparison of Unadjusted and Adjusted Production Frontiers
Once the quality-congestion measure is derived, we can adjust outputs to account for poor quality. This is accomplished by dividing the outputs by the quality-congestion measure, thereby discounting unadjusted output to form quality-adjusted output. If high-output hospitals have no quality congestion, we can assert that a volume–outcome relationship exists.
High-volume hospitals have been shown to have lower mortality rates than low-volume hospitals for certain technology-intensive, complicated procedures. A considerable body of literature supports the statistical association between high procedure volume and better outcomes. See Dudley et al. (2000)
for a review. Staff expertize developed as a result of learning by doing has been hypothesized as an explanation for this relationship, although the reasons for the association are still not entirely understood (Elixhauser, Steiner, and Fraser 2003
Despite the benefits of DEA identified above, there are some drawbacks. DEA has been criticized for its deterministic nature, which assumes no measurement error (Newhouse 1994
). But the stochastic nature of demand may lead hospital decision makers to overestimate resource needs. Depending on one's perspective, this might overstate inefficiency. For example, assuming fixed resources and random patterns of admissions (Grannemann, Brown, and Pauly 1986
), unoccupied beds might represent reservation quality (Joskow 1980
) rather than slack resources. Conversely, operations that include poor forecasting and inflexible staffing systems could be accurately characterized as inefficient. The reality is probably a mix of the two situations, so the inefficiency estimates might be better viewed as an upper bound as there is no compensation for periods of excessive utilization.
DEA has also been criticized for its inability to capture quality differences (Newhouse 1994
). However, with our data we can capture more quality differences than previous DEA studies, and this should partially address the second criticism.
A recent DEA-based study by Clement et al. (2008)
empirically assessed how poor quality outcomes detract from overall hospital productivity. This was accomplished by applying the WDO technology to a sample of hospitals to measure the production of undesirable outputs, such as in-hospital mortality. We follow the approach used in Clement et al. (2008)
but expand upon it by adjusting outputs (specifically the subset of the Agency for Healthcare Research and Quality [AHRQ] Patient Safety Indicators [PSIs] that are nurse-sensitive outcomes) to define undesirable outcomes rather than the AHRQ Inpatient Quality Indicators (IQIs) used by Clement et al. (2008)
. We also expand on the work by Clement et al. (2008)
by comparing the slack (i.e., excess inputs) values for inputs between the model where all outputs are considered desirable and the slack values for inputs in the case when outputs are adjusted to account for relatively poor quality. Below, we discuss the insights we can gain by making these comparisons.
Slack is derived using the input-based Cooper, Seiford, and Zhu (2000)
Using the results derived from this second-step analysis, we measure the differences in the slack values for inputs obtained for the high-quality hospitals as compared to the medium- and low-quality hospitals. For example, we can analyze the over- or under-utilization of nurses, for example, by hospital quality status. If the difference between slacks is positive, it would suggest that the hospital is employing excess input that leads to inefficiency. If the difference is negative, it implies that inputs need to be increased to improve quality of care. In fact, the difference indicates the amount of inputs that need to be increased.
Unlike congestion, slack does not impede total production, but may either represent a quality input or excessive inputs leading to inefficiency. (It should be noted that while the congestion measure is multiplicative, the slack measure is additive.)
The identification of slack values can also direct management's attention to areas where inefficiency exists and adjust accordingly to optimize both production and quality of care (Sherman 1984
). Therefore, this analysis can show managers how much their hospital needs to increase an input (to increase quality) or decrease an input (to reduce inefficiency) as compared with their hospital's peers.
We also assess quality differentials and slack by organizational factors (ownership, teaching status, resource expenditures, payer-mix, and system membership) and market characteristics (health maintenance organization [HMO] penetration and hospital competition). We conclude by analyzing quality and slack in order to develop a direct link between production performance and input slack.