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JMARVolume Ten
1998
Standard Estimation, StandardTightness, and Benchmarking: A
Method With an Application toNursing Services
Rajiv D. BankerUniversity of Texas at Dallas
Hsihui ChangNational Chengchi University
Somnath DasUniversity of Illinois at Chicago
Abstract: This paper presents a new method for estimating standards. Prioraccounting research suggests the need for flexibility in setting standards inorder to allow managers to make trade-offs between efficiency and attaina-bility, and also between inputs with different relative prices to minimize cost.Stochastic Data Envelopment Analysis (SDEA) is adapted to introduce flexi-bility in setting standards and deriving mix and yield variances relative to com-parative data from historical records or from other organizations. This ap-proach to relative performance evaluation and benchmarking has practicalappeal since industry-level data is increasingly being shared, for example, inthe health care sector. Hospital nursing cost data is used to illustrate thisapproach.
Standard setting for performance evaluation and control purposes hasreeeived considerable attention in management aeeounting research. Theprocess of estimating standards, however, has received seant attention.Textbook treatment is typically sketehy and assumes that the productionfunction is known or can be determined using industrial engineering tech-niques such as time and motion studies. Implicit in such methods isthe assumption that the production technology is linear and separable(Christensen and Demski 1995; Darrough 1990). In this paper, we presenta Stochastic Data Envelopment Analysis (SDEA) model to estimate stan-dards from comparative historical or benchmarking data (Banker 1993).^
' Conventional DEA models, tn estimating the production frontier which provides the stan-dard relative to which the performance of all units is measured, require atl variations fromthe frontier to lie only on one side, representing input usage greater than the minimum dueto chance or tnefficiency In operations. By introducing variations on both sides of the fron-tier, such as those caused by random factors attributable to model specification or mea-surement errors, SDEA allows evaluators and standard setters of Individual units in thebenchmark cohort, the opportunity to ex ante specify the likelihood of achieving thestandard.
The authors thank two anonymous reviewers and Mike Shields (the editor)for their commentsand suggestions, and Ms. Viki Ardito, Acting Vice President of Patient Care Services at AltaBates Medical Center in Berkeley, California and Ms. Helen B. Ripple, Associate Director ofMedical Center and Director oJ Nursing at Mqffitt Hospital, University oJ California, San Fran-cisco, Jor access to inforrruxtion on hospital operations and industry practice.
134 Journal of Management Accounting Research, 1998
The SDEA approach allows for the possibility of a nonlinear nonseparablerelation between input eosts and outputs produced, relaxing assumptionsimplicit in industrial engineering methods. We also illustrate how mix andyield variances may be computed relative to such an estimated productiontechnology to enable managers to assess the trade-offs inherent in the useof different inputs.
A question of interest is the tightness of standards. The extant evi-dence suggests that tight standards enhance performance (Chow 1983);however, if the standards are too tight, job stress increases, leading tolower performance (Shields et al. 1998). Recently, organizations have beenbenchmarking their performance relative to comparative data from otherorganizations (Evans et al. 1997). The American Productivity and QualityCenter defines a benchmark as "a measured *best-in-class' achievementrecognized as the standard of excellence for that business process"(Andersen and Pettersen 1996).^ Thus, benchmarking entails comparisonwith the "best in the field." rather than some average target. If used as astandard, such as "best in the field," a benchmark may be perceived to betoo tight, resulting in lower performance. It is important, therefore, to de-velop a procedure to estimate standards based on comparative data in afashion that affords managers the fiexibility to adjust the tightness of thestandards. Additionally, even where fiexible standards can be set usingconventional methods, a simple benchmarking approach that considerseach input separately does not reflect the opportunity for trade-offs amongvarious inputs. The model presented in this paper addresses these bench-marking and standard-setting issues.
Development of useful standards for performance evaluation and con-trol is a problem faced by many organizations. The Tax Equity and FiscalResponsibility Act of 1982 changed the reimbursement of hospital Medi-care costs from a "cost plus" basis to a "comparative" cost basis. Accessto information on relative hospital performance has also increased in re-cent times.^ This has enabled Kroger Company and several other organi-zations to evaluate relative performance and determine "who provides themost cost-effective and highest quality care" [Wall Street Journal 1993).As a result, considerable pressure exists now for hospitals to benchmarktheir performance relative to other hospitals.
Hospital managers we interviewed for this research indicated that ef-ficient use of nursing personnel is an important goal because nursing ser-vices comprise the largest single component of hospital costs. Nursing la-bor costs comprise over half of total hospital costs (American HospitalAssociation 1987). Because of a severe shortage of nurses in the late1980s, salaries of nurses with 20 years experience has more than doubledin the United States over the past ten years (Kunen 1996). In addition tothe importance of controlling nursing costs, comparative data are readilyavailable for the nursing departments of hospitals (Carr 1993), Therefore,
David T. Kearns, chief executive officer of Xerox Corporation, also defines benchmarkingas "the continuous process of measuring products, services, and practices against thetoughest competitors or those companies recognized as industry leaders" (Camp 1989).At present there are eight states that have approved health care data coilection arrange-ments that gather demographic, diagnosis, physician, procedure, payer, fee and other in-formation. Iowa. New York, Pennsyivania and Wisconsin publish both comparative healthcare cost and comparative information collected as a result of mandated disclosures.
Banker, Chang and Das 135
we illustrate our approach using nursing services. We focus on cost min-imization and control, and consider relatively few inputs and outputs forthe nursing department to keep the illustration simple. However, the ap-proach described in this paper is general and can be applied to otherorganizations.
TTie remainder of the paper is organized as follows. The next sectionreviews prior research on standard costs and variance analysis. Sectionthree describes the proposed method for estimating standards and eval-uating variances. In section four, we illustrate the approach in the contextof nursing services. Finally, section five summarizes the contribution ofthis paper and its conclusion.
RESEARCH ON STANDARD COSTSAND VARIANCE ANALYSIS
Tightness of StandardsMost cost and management accounting textbooks (for example,
Atkinson et al. 1997) prescribe standards that are "efficient and attaina-ble." There is considerable research supporting the use of "efficient" stan-dards. Chow (1983) and Waller and Chow (1985) present experimentalevidence that job standard tightness induces higher effort and conse-quently a higher performance level. Locke and Latham (1984) and Hollen-beck (1987), among others, find that difficult goals often lead to highertask performance than easy goals. However, there is also considerable ev-idence suggesting that "attainability" is important in setting standards. Ina field study. Merchant and Manzoni (1989, 556) find that budget targetsare achieved in about 80-90 percent of the cases they examine and sug-gest "that budget achievability is an important organizational control-system variable." In a study of Japanese automobile engineers. Shields etal. (1998) find that if targets are not attainable, job stress increases andperformance declines.
Thus, two opposite prescriptions result from the research on tightnessof standards. On the one hand, there is a need to set tight standards tomotivate agents to strive for superior performance. On the other hand,setting tight standards can result in lower performance due to job stress.It is important, therefore, that managers have the fiexibility when settingstandards to make trade-offs between tightness and attainability, consis-tent with organizational goals and context.
Cost Variance AnalysisMix variances and yield variances provide valuable information when
there are multiple inputs that can be substituted for each other. Consider,for example, the use of nursing professionals in the production of healthcare services. Registered nurses can be substituted for licensed nurses,or vice versa. Senior hospital managers we interviewed informed us thatmanagers of specific wards often reassign registered nurses and licensednurses from one shift to another within a 24-hour period in order to op-timize resource utilization within their budget constraints. Such substi-tution may be a rational response to changes in input prices or to chang-ing workloads, or it may indeed be attributable to inefficient managementof resources. Textbook analysis of mix and yield variances is based on theassumption that inputs are consumed only in fixed proportions (Wolk and
136 Journal qf Management Accotmting Research. 1998
Hillman 1972; Hasseldine 1967; Petes 1986). Mensah (1982) and Mar-cinko and Petri (1984) consider more general production possibilities byallowing for input substitution and incorrect estimation of input prices,respectively. Barlev and Callen (1986) and Darrough (1990) examine therelation between optimal input standards and input prices, technologyand output level. A limitation of their parametric approaches is that theestimated efficiency measures depend on the assumed production func-tion (Callen 1991). The validity of the assumed production function is anessential determinant of the validity of the standards and the variancesderived from it.
PROPOSED METHOD FOR ESTIMATION OF STANDARDSStandards are typically set based on past experience or an industrial
engineering study. The use of comparative cost data from peer organiza-tions provides an alternative approach to standard setting. For example,in the health care sector, cost data across different hospitals are collectedby and are available from various governmental agencies. Senior hospitalmanagers we interviewed indicated that health care organizations in a re-gion are increasingly sharing cost data on a per patient and per case basis.These data are used for benchmarking the performance of individualunits.
Data Envelopment AnalysisData Enveiopnienl Analysis (DEA) provides a nonparametric represen-
tation ofthe input-output production correspondence (Banker et a!. 1984).It has the advantage of imposing minimal structure on the productionfrontier consistent with observed data (Banker and Maindiratta 1988). Theregularity conditions include monotonocity and convexity. The productionpossibility set is represented by T = {(Y, X)| the outputs Y can be producedfrom the inputs X}, where Y is a vector of outputs and X is a vector ofinputs. Monotonocity implies that if (Yp Xj) e T. Xj > X, and Y2 s Yp then(Y2, X2) E T: that is, less outputs can always be produced from more in-puts. Convexity implies that if (Yp X,) G T and (Y2, X^] £ T, then (KYi+ (1-X)Y2, \X, + {1-\]X^ e T for all 0 < \ < 1; that is, a productionpossibility obtained by interpolating two observed feasible possibilities isalso feasible. The production frontier is the efficient boundary of this pro-duction possibility set. Banker et al. (1989) show this method to be par-ticularly well suited for relative performance evaluation across a cross-section of health care organizations. Callen (1991) provides anintroduction to the DEA method and its applications to management ac-counting. Banker (1993) characterizes the statistical properties of the DEAestimator of an organization's efficiency measured relative to the "bestpractice" frontier gmd proves that the DEA estimator is consistent andmaximizes likelihood. Bogetoft (1994) shows that the DEA efficiency mea-sure is optimal for relative performance evaluation in a multi-agent moralhazard model when the production possibility set is monotone and convex.
Stochastic Data Envelopment AnatysisDEA uses the "best practice frontier" as the standard relative to which
performance is measured. It allows for deviations only on one side of thefrontier—the side that represents less output generation or more input
Banker, Chang and Das 137
consumption than the best practice due to unfavorable random conditionsor due to inefficiency in operations. That is, DEA yields best practice tar-gets which, although "efficient," may be almost impossible to "attain" bya given organization being evaluated relative to a benchmark group. Sto-chastic Data Envelopment Analysis (SDEA) extends DEA by allowing forrandom effects such as measurement errors or uncontrollable factors thatmay cause deviations on either side of the efficient production frontierSpecifically, the SDEA model adds a parameter that weights deviationsbeyond the best practice frontier relative to those on its inefficient side.''The fiexibility of the SDEA approach to choose how much relative weightto place on deviations on the two sides of the estimated production func-tion, allows managers to make the trade-off between tightness and attain-ability of the benchmark.
The basic objective is to determine the production frontier that rep-resents an efficient and attainable standard relative to which an organi-zational unit can measure its performance. The SDEA model is repre-sented as a linear programming problem minimizing the weighted sum ofdeviations inside and outside the production frontier that are estimatedfrom the observed comparative data for j = 1 n organizations. The fron-tier is computed based on available observations without imposing anyother assumptions about the specific underlying functional form of theinput-output correspondence.^
n
Minimize 2 7Uj + (1 - 7)Vj (1)
subject to:K 1
S 2 Xj, - X^,) - (Vj - V^) + (Uj - U^) > (Xji - X^,)| y j y 2 j j,&— 1 1 — 2
Vj.m = 1 nWji,, Zjp Uj, Vj > 0
where yj is the quantity of output k for organization j . Xj, is the quantityof input i for organization j . ^ The Vj represent deviations outside the fron-tier. These result from favorable random effects, such as model specifi-cation and measurement errors, net of deviations in the other directiondue to inefficiency or unfavorable random conditions. Similarly, the u, rep-resent net deviations inside the frontier. These may result from randomeffects or inefficiencies or both.^ The n(n - 1) constraints for j . m = 1 n.
'' An alternative is to use the parametric stochastic frontier estimation technique to arrive atbenchmark cost standards (Aigner et al. 1977). This parametric approach, however, im-poses assumptions different from basic regularity conditions, and estimated functions oftenviolate those conditions (Caves and Christensen i980). Furthermore, it does not provideflexibility in specifying the degree of attainability. See Dopuch and Gupta (i997) for anapplication of tliis approach to public school expenditures in Missouri.
^ If additional a priori information is available about the input-output correspondence thenthe estimation of the model can be modified appropriately to reflect such information(Banker 1992).
^ Please refer to appendix A for a glossary of all symbols used in this paper. Observe that theconstraint for j = m is obviously redundant as both sides are identically equal to zero.
' This assumption is consistent with an assumption common in stochastic production fron-tier estimation (Aigner et al. 1977; Forsund et al. 1980) that inefficiency distribution is one-sided while other random factors causing deviations in observed output are symmetricallydistributed about zero.
138 Journal qf Management Accounttng Research. 1998
j T m, in the linear program (1) ensure that after we adjust the observedXjj* = Xji - Uj + Vj, j = 1 n, the resultant n points (Yj, X*j), where Yj- (yii,---.yjk) and X*j = (x*ji, X]2,...,x,,), lie on the boundary of a monotoneand convex production possibility set. The objective of the programmingproblem is to find points (Yj, X*j) that minimize the weighted sum of de-viations Uj and Vj, while satisfying the constraints to ensure that the es-timated production set is monotone and convex."
We require that 0 < -y i. This is the parameter specified ex ante torepresent the relative weight on deviations on the two sides of the frontierand reflects the ex ante likelihood of attaining the targets implied by theestimated standards. Observations j with optimal Uj* = Vj* = 0 are on theestimated production frontier and are represented by the set denoted Rl-v).The value of y has a useful interpretation: 100 y represents the ex anteexpected percentage of observations outside the estimated productionfrontier (Koenker and Bassett 1978; Bassett and Koenker 1982). Specifi-cally, for -y = 0.5, a median function is estimated with equal probability ofan observation lying Inside or outside the estimated production frontier.In this case, all deviations from the frontier may be attributed to an as-sumed symmetric random error and none to inefficiency in production.^
For SDEA to be properly implemented, careful attention must be givento the definition and measurement of inputs and outputs, the specificationof the appropriate functional relationships between inputs and outputsand robustness of results to infiuential observations. " Computationally,it is easier to solve the dual formulation as it has fewer constraints whenthe number of observations n > K + I + 3. This dual linear program isgiven in appendix B.
Previous applications of SDEA have focused on assessing the robust-ness of inferences about factors influencing productivity. Banker et al.(1987) assess changes in input usage due to a regime shift caused byimplementing a gain-sharing contract among workers. Banker etal. (1991)identify variables that impact the productivity of software maintenanceprojects. The objective behind the use of SDEA in these studies was toensure that their results were not sensitive to the possibility of measure-ment error in the data. In contrast, the primary contribution of this paperis in adapting SDEA to provide managers the flexibility in setting the tight-ness of standards for relative performance evaluation.
Intuitively, for the one-input one-output case, observe that the constraints require that (yj- y^)/(Xj* - x^*) ^ 1/W|. where x,* = Xj * u, + v, and x^* = x^ - Un, + v^. That Is. theslope of the chord joining the points j and m is greater than the slope 11 / Wj) of the produc-tion frontier at tlie point. This property characterizes the boundary of a monotone andconvex set. The general result for the multiple-input multiple-output case follows from asimilar supporting hyperplane theorem for convex sets.For any data set. \iy s l / n then Vj* = 0 for allj and all estimated deviations are attributedto inefficiency or unfavorable random conditions as in the usual DEA model. This is amaintained assumption based on the best a priori information that is available. For ex-ample, if we knew a priori, that there is no random influence, then we could model this bysetting -y ^ l /n . Alternatively, if it were knoun a priori that all the units in a sample areefficient but they are subject to symmetric random shocks, then we could model this bysetting 'I' = 0.5. and attribute equal weights to deviations on both sides. Of course, as inany econometric analysis, if the maintained assumptions are not valid then the propertiesof the estimators are distorted.
° See Cooper et al. (1996) for a detailed discussion of some limitations of the DEA approach.
Banker, Chang and Das . / 139
Mix and Yield Variance RatiosIn order to construct ratio measures of yield and mix variances based
on the production frontier estimated using SDEA, we proceed as follows.The lowest cost C* (Y ,, P ) to produce the outputs Y . given the input pricesPp for the organization indexed "o," based on the benchmark set R(7) isestimated using the following linear program (Banker and Maindiratta1988):
I
C* (Y^.PJ = Minimize 2 Po. x, (2)r , • • •. 1 = 1
subject to:
k = 1 K
\ j . X, > 0
This linear program identifies the cost minimizing inputs x, that canproduce the outputs Y for the organization indexed "o." The input pos-sibilities are determined with reference to the observations j G R(7) thatlie on the estimated production frontier. Since the production set is mon-otone and convex, we search over the set of all convex combinations:l(SjeR( i ^Y,, SjgR( |\jXj)|2jeR(. , Xj = 1] of the observation in R(7) that produceas great an output as Y , . The objective function in linear program (2)ensures that the Input cost to produce this output is minimized, given theinput prices po,.
The total cost variance ratio (TCVR ,) for the organization indexed "o"is then measured as:
where 2{=i PoiX , is the actual cost for organization "o" and x,* are the costminimizing inputs x, solving the linear program (2). This ratio is analyzedinto two components: yield variance ratio (YVR ,) and mix variance ratio(MVR ,). The yield variance ratio is estimated using the Banker et al. (1984)model relative to the benchmark set R(7) as follows:
= Minimize h , (3)
subject to:
— Yok
This linear program identifies the smallest quantities of the inputs Xj thatare sufficient to produce the outputs Y,,. keeping the input proportions(mix) the same as the observed proportions in X ,. As in the previous linear
140 Journal qf Management Accounting Research. 1998
program, input possibilities are determined with reference to the obser-vations j G R(7). The constraints are similar to those in the previous pro-gram as we search over all convex combinations of the observations inR(7). The objective function in linear program (3) ensures that the maxi-mum proportionate reduction possible in the inputs is identified.
The mix variance ratio can now be obtained as MVR = TCVR^ / YVR ,.Depending on the value of 7. we have different values for the mix, yieldand total cost variance ratios. We use figure 1 to illustrate these measures.The two axes represent the two inputs (X and X ) used in production. LetRR' represent the relative price (isocost) line and let C^ be the actual ob-served cost for point A. representing a given output level Y with corre-sponding inputs usage (X^ . X ), respectively. Let PESP' be a productionisoquant such that it is the lower bound of the set of all input combina-tions that produce at least output level Y. Then the coordinates at the pointE represent the cost minimizing input combinations (X^E. Xf ) with a re-sulting overall optimal cost of CE- The total cost variance ratio is definedas OT/OA. which is the ratio of optimal cost (Cg) to actual cost (CJ:
OT _ OT^ OSOA " OS OA*
The component of the total variance given by OS / OA. which measureshow far the firm is inside the isoquant, is used to measure the yield var-iance ratio. The term OT/OS refiects the higher cost due to sub-optimalmix of inputs and is used to measure the (input) mix variance ratio. Thus,the total cost variance ratio Is represented as the product of the yield var-iance ratio (OS/OA) and mix variance ratio (OT/OS).
ILLUSTRATIVE APPLICATION TO NURSING SERVICESWe focus on the cost minimization and control problem in the nursing
department of a hospital. For parsimony and expositional convenience inthis illustration, we assume that in the hospital production function, theproductivity of nursing inputs is separable from other inputs such as lab-oratory, dietary services, patient monitoring equipment and physician in-puts. However, within the nursing department, we specificcilly allow forsubstitutions among different types of nursing inputs. Indeed, one of thebenefits of the DEA and the SDEA models over traditional approaches tostandard setting is the ability to allow for input (and output) substituta-bility or separability based on available a priori information (Banker 1992).
Specification of Inpnts and OntpntsThe focus of much of prior empirical research on performance evalu-
ation in the health care sector has been on examining the aggregate effi-ciency of entire hospitals rather than of individual services or depart-ments. With the exception of Byrnes and Valdmanis (1989) and Morey etal. (1990), most prior studies have evaluated the overall productive effi-ciency without examining the mix and yield components. Also, most priorresearch has used parametric functional forms to estimate hospital pro-duction functions. In contrast. Banker et al. (1986). Banker et al. (1989),Capettini et al. (1985), Grosskopf and Valdmsinis (1987) and others haveused Data Envelopment Analysis (DEA). •
Banker, Chang and Das 141
FIGURB 1ninstration of Total Cost Variance Ratio
RR' Relative Price LinePESP' Production Frontier
Yield Variance Ratio
Mix Variance Ratio
= OS/OA
= OT/OS
Total Cost Variance Ratio = OT / OA= (OS/OA)*(OT/OS)
Most hospitals employ different types of nurses and nursing assis-tants, such as registered nurses, licensed nurses, apprentice nurses andunlicensed assistants, each of whom has a different wage rate and skillset. A hospital is interested in the lowest cost combination of the various
142 Journal qf Management Accounting Research. 1998
categories of nurses that provides the required level of service at the de-sired levels of quality. Quality considerations preclude the use of appren-tice and unlicensed assistants for regular nursing tasks (Kunen 1996).Moreover, a registered nurse cannot be completely replaced by a licensednurse. Licensed nurses cannot administer intravenous injections, makepatient assessments and perform other higher skill tasks that only reg-istered nurses can. However, to the extent licensed nurses are substitut-able for registered nurses, the hospital manager may be able to lowernursing department costs by employing lower wage nurses.
Prior research studies on hospital cost have used several measures ofhospital output. For example. Tatchcll (1983) used the case mix approach.More recently, diagnostic related groups (DRGs). a variation of the casemix approach is used in some studies. However, there are more than 400DRGs. which makes it difficult to perform statistical analysis with limitedobservations. A more detailed review and discussion of the measurementof hospital outputs is provided in Tatchell (1983) and Banker et al. (1989).In this paper, we illustrate the SDEA approach using total weighted in-patient days as a measure of output, where the weights indicate the apriori assessment by hospital professionals of different expected loads ofdemand for nursing services depending on the severity of individual com-ponents of the case mix.
Each hospital functions within its own externally determined environ-ment and. to some extent, certain factors affecting the efficiency of thehospital may be uncontrollable by the manager. When hospitals in a ref-erence group do not all have the same set of external constraints, suchfactors need to be incorporated Into the benchmark cost variance analysis.For example, the proportion of elderly patients (over 65 years) served by ahospital leads to lower operating efficiency, but it is often among the fac-tors which are beyond the control of the hospital manager (Pauly andDrake 1970). Therefore, we measure hospital output with two separatevariables, one measuring older patient days and another measuring theremaining patient days. Thus, our model comprises two inputs: registerednurse hours (RNH) and licensed nurse hours (LNH) and two outputs: el-derly patients days (EPD) and other patient days (OPD).
The objective of this paper is to illustrate the SDEA approach and notto measure the performance of the sample hospitals or to present policyinferences about the sample. Hence, we do not use a broader range ofinputs and outputs. For example, one could use DRGs as outputs, butthat merely complicates the setting without providing any additional in-tuition about the benefits or limitations of the proposed method. Moreover,we have introduced one form of patient mix control with the output mea-sure for older patient days, to illustrate how the method deals with mul-tiple outputs. The proposed method thus allows the possibility of incor-porating a diverse set of inputs and outputs as well as more controlvariables.
Description of DataWe illustrate our approach using Medicare audited data from 66 gov-
ernment-supported hospitals located in one state. Teaching and charitable
Banker. Chang and Das 143
hospitals are excluded from the analysis. Data related to Inpatient-days,wage rates and nursing-hours for registered and licensed nurses are avail-able from audited cost reports. Table 1 provides descriptive statistics forthis data set. For example, the average wage rate for registered nurses is67 percent higher than that for licensed nurses.
Empirical ObservationsInitially, the estimates of mix and yield variance ratios for the entire
sample are examined. Table 2 presents the distribution of the ratio mea-sures for yield (panel A), mix (panel B) anri total (pane! C) variances, cor-responding to different levels of 7. As the value of 7 increases from 0.01to 0.5, the yield variance ratios and total variance ratios increase whilethe mix variance ratios decrease. This increase in the yield variance ratiois expected because increasing the value of 7 increases the penalty weighton deviations attributed to yield inefficiency, which is minimized in theSDEA program.
The correlation matrix for yield variance ratios in panel A of table 3shows that these scores are very highly correlated for each of the six dif-ferent values of 7 and suggests that the relative value of a hospital's yieldvariance ratio is robust to the choice of the parameter 7. Similarly, the mixvariance ratios are also very highly correlated across riifferent values of 7,as shown in panel B of table 4. Although not reported here, similar cor-relations are observed for the total cost variance ratios as well. We notethat while the relative values of variance ratios are robust to the choice of7, the targets or estimated standards themselves are different.
To verify that the variance ratios are independent of the output mix,we present in table 4 the correlation between the variance ratios and theproportion of elderly patient days serviced by a hospital. For each of thesix values of 7, both Pearson anri Spearman correlations are not signifi-cant at conventional levels of significance. This suggests that once we con-trol for the differences in the levels of the two outputs in estimating therelation between inputs and outputs, the resulting variance ratios do notdepend on the differences in output mix across units.
Our earlier empirical finding of a decrease in the mix variance ratiowith an increase in the value of 7 (table 2) does not follow from any a prioritheoretical analysis. To explore this further, we examine the relation be-tween the level of 7 and the ratio of the optimal relative inputs to the actualrelative inputs. These results reported in table 5 indicate that the ratio isincreasing in the value of 7. i.e., when we place relatively less penalty ondeviations outside the production frontier, the optimal ratio of RNs to LNs(given their wage rates) increases relative to the actual observed ratio. Thisimplies that the relative productivity of RNs is estimated to be higher forhigher values of 7. Therefore, hospitals should use more of the RNs thanthe LNs for higher values of 7 than for lower values. The sensitivity of theproduction frontier to the values of 7 is illustrated in figure 2. We fix theoutput levels at their median values and then plot six production func-tions, corresponding to the six values of 7. As the value of 7 increases, theproduction isoquant moves outward (away from the origin) as expected,because increasing the value of 7 implies that more of the deviations are
144 Journal qf Management Accounting Research, 1998
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Banker, Chang and Das 145
y
Panel0.010.10.20.30.40.5
Panel0.010.10.20.30.40.5
Panel0.010.10.20.30.40.5
TABLE 2Descriptive Statistics of Variance Ratio
MeanStd.Dev.
A: Yield Variance Ratio Estimates0.8900.9070.9320.9400.9841.014
B: Mix Variance0.9510.9430.9260.9240.9040.895
0.1040.1130.1550.1490.1780.190
Ratio Estimates0.0520.0620.0890.0840.1020.103
2 5 %
0.8210.8290.8380.8510.8900.928
0.91510.9020.8710.8670.8390.828
C: Total Cost Variance Ratio Estimates0.8470.8540.8560.8630.8780.894
0.1150.1130.1130.1120.U20.111
0.7780.7820.7830.7870.7990.810
Estimates
Qoartiles50%
0.9020.9080.9580.9641.0001.000
0.9700.9600.9570.9530.9450.929
0.8410.8570.8590.8590.8750.891
7 5 %
1.0001.0001.0001.0001.0021.055
0.9910.9940.9930.9890.9810.972
0.9640.9650.9680.9770.9890.998
attributed to random measurement errors, rather than the inefficiency ofthe particular unit. Consequently, the estimated production frontier rep-resents more attainable benchmark targets for higher values of 7. Inter-estingly, for our sample data set. the estimated production frontier is alsosteeper for higher values of the parameter 7. indicating the relativelyhigher productivity of RNs to LNs for higher values of -y.
Illustration in the Context of a Speelflc HospitalWe now illustrate how our approach was implemented in a specific
hospital (hereafter referred to as ABC Hospital) included in our sample of66 hospitals. Table 6 presents the actual data and the variance ratio es-timates for ABC hospital. Similar to our observations for the entire sample,the yield variance ratio increases and the mix variance ratio usually de-creases as we increase the value of 7. Our interest is in determining thebenchmark cost against which the ABC hospital should measure perform-ance. For the six values of 7, the benchmark nursing cost ranges from$1.51 million to $1.62 million, which was considerably lower than the ac-tual cost of $1.82 million. The increase in the benchmark cost with highervalues of 7 means that the target is more attainable.
An important decision for the ABC hospital manager is to choose thevalue of 7 to determine the appropriate level of attainability for benchmarkcosts. A choice of a large value for 7 implies that the managers expect a
146 Journal qf Manc^ement Accounting Research, 1998
TABLE 3Correlation Matrix for Variance Ratio Estimates
0.01 0.1 0.2 0.3 0.4 0.5
PANEL0.01
0.1
0.2
0.3
0.4
0.5
PANEL0.01
0.1
0.2
0.3
0.4
0.5
A: Yield Variance Ratio1.0000
[0.0000]0.9330
(0.0001)0.7139
(0.0001)0.7177
(0.0001)0.6216
(0.0001)0.5785(0.0001)
0.9768(0.0001)1.0000
(0.0000)0.9076(0.0001)0.9109(0.0001)0.8125(0.0001)0.7610(0.0001)
(YVR)
0.8942(0.0001)0.9497(0.0001)l.OOOO
(0.0000)0.9949(0.0001)0.9274(0.0001)0.8753(0.0001)
B: Mix Variance Ratio (MVR)1.0000
(0.0000)0.8902
(0.0001)0.7616
(0.0001)0.7365
(0.0001)0.5844(0.0001)0.4581(0.0001)
0.9123(0.0001)l.OOOO
(0.0000)0.9391
(0.0001)0.9370
(0.0001)0.7668(0.0001)0.6668(0.0001)
Spearman correlations are abovethe diagonal. Siginificance levels
0.8591(0.0001)0.9704(0.0001)1.0000
(0.0000)0.9853(0.0001)0.8822(0.0001)0.8016
(0.0001)
0.8631(0.0001)0.9361(0.0001)0,9843(0.0001)1.0000
(0.0000)0.9300(0.0001)0.8816(0.0001)
0.7888(0.0001)0.9418(0.0001)0.9689(0.0001)l.OOOO
(O.OOOO)0.8941(0.0001)0.8206(0.0001)
the diagonal and Pearson> are in parentheses.
0.7060(0.0001)0.7958(0.0001)0.9068(0.0001)0.9098(0.0001)1.0000
(0.0000)0.9852(0.0001)
0.6282(0.0001)0.7331(0.0001)0.7958(0.0001)0,8274(0.0001)1.0000
(0.0000)0.9742(0.0001)
correlations
0.6530(0.0001)0.7413
(0.0001)0.8608
(0.0001)0.8704(0.0001)0.9666(0.0001)1.0000
(0.0000)
0.4668(0.0001)0.5744(0.0001)0.6495(0.0001)0.7043(0.0001)0.9216(0.0001)1.0000
(0.0000)
are below
large proportion of the nofse In the output metrics to be purely randomand unrelated to the worker's effort. This results in setting a loose stan-dard, one that Is easily attainable by the worker. The choiee of the param-eter 7 is therefore critical in the implementation of the SDEA method, sincedepending upon the choice of the parameter, managers can set either atight or a slack standard. Tbe value of 7 is chosen by the supervisingmanager, possibly in consultation with the subordinate worker being eval-uated, as in participative budgeting.
How did the ABC hospital managers choose the value of the 7 param-eter? As part of the initial implementation of this method, managers ex-perimented with different values of the parameter to assess, based on pastexperience, what may be a good initial value for the parameter. Nursingdepartment managers carefully evaluated factors such as task uncertaintyand complexity that are likely to introduce random variations in output,independent of managerial effort level. Based on this evaluation, the ABChospital chose the value of 7 = 0.3 implying a 30 percent chance, on av-erage, of attaining the benchmark standard if operations are efficient.
Banker. Chang and Das 147
TABLE 4Correlation Matrix for Proportion of Elderly Patient Days
vs. Variance Ratio Estimates(Significance levels are in parentheses)
Correlation With YVR Correlation With MVR
y
0.01
0.1
0.2
0.3
0.4
0.5
YVRMVR
Spearman
-0.1086(0.3855)
-0.0861(0.6767)
-0.0523(0.4920)
-0.0659(0.5989)0.1150(0.3578)0.1632(0.1903)
denotes estimate of yielddenotes estimate of mix
Pearson
-0.0432(0.7302)0.0192
(0.8764)0.U58
(0.3547)0.1061
(0.3963)0.2214
(0.0740)0.2279
(0.0657)
variance ratio.variance ratio.
Spearman
-0.1380(0.2691)
-0.0725(0.5627)
-0.1369(0.2730)
-0.0709(0.5711)
-0.2324(0.0605)
-0.2609(0.0343)
Pearson
-0.0294(0.8416)
-0.0329(0.7944)
-0.1213(0.3319)
-0.0713(0.5692)
-0.1693(0.1742)
-0.1840(0.1392)
The weight parameter is crucial in determining the benchmark, sincethe likelihood of achieving the target increases with this weight parameter.A manager has ex ante knowledge of factors that make her hospital's nurs-ing services more or less inefficient relative to the referent group of hos-pitals. For example, if a hospital manager believes that relative to its ref-erent hospitals, the average years of experience of nurses at her hospitalis lower and if nursing productivity increases with experience, then usingthe "unadjusted" frontier benchmark could create unattainable targets forthe hospital's nurses. Thus, the manager may set a relatively high level of7-
For each value of 7 in linear program (1) we first identify the set R(7)of referent observations that lie on the production frontier 0).Note that this set may include some observations that are not on the fron-tier for a lower value of 7. We next estimate the optimal cost based on theinput-output data for the benchmark set R(7). Finally, we compute the
TABLESRatio of Optimal Relative Inputs to Actual Relative Inputs
(Relative Inputs = Registered Nursing Hours/Licensed Nursing Hours)
y
0.010.10.20.30.40.5
Mean
1.1101.1751.1811.2131.2971.392
Std. Dev.
0.4270.4590.4650.4960.5370.584
2 5 %
0.8550.9100.9130.9091.0001.000
Quartiles
5 0 %
1.0001.0001.0061.0521.1461.227
7 5 %
1.3511.4181.4191.4621.6011.707
148 Journal qf Management Accounting Research, 1998
FIGURE 2Production Frontiers for Different Values of y
(Elderly FaUent Days = 9,500, Other Fatient Days = 16,561]
180
•3160 -
120 h
.2 100
80
—
A \
' .1
D
E
•
*
\ F
\\\\\
idJBABCDEF
! -X.
o.oro.rOJ0.30.40.3
25 30 35 40 45Registered Nurse Hours (in Thousands)
50 53
TABLESSDEA Estimates for the ABC Hospital
Fanel A: Actual Data for the ABC Hospital
Registered nursing hoursHourly wage rate of RNLicensed nursing hoursHourly wage rate of LNTotal nursing costElderly patient daysOther patient days
43,884 hours$17.94/hour113,333 hours89.12 / hour$1,821,3519,597 days7,005 days
Panel B: Variance Ratio Estimates
YVR MVR TCVR ONC ROFT
0,010,10.20.30.40,5
0.83870.84880.85190.87250.89250.9440
0.99170.99850.99060.97930.89250.9440
0.83180.84750.84870.85440.86120.8911
$1,515,000$1,543,629$1,545,787$1,556,233$1,568,496$1,623,006
0.84991.09261.09611.U231.13151.2165
'YVR = yield variance ratio.MVR = mix variance ratio.
TCVR = total cost variance ratio.ONC = optimal nursing costs.
ROPT = ratio of optimal relative inputs (RN:LN) to actual relative Inputs.
Banker, Chang and Das 149
total cost variance ratio for ABC hospital (denoted by Index "o") as a ratioof the estimated optimal cost to the cost for that observation (indexed "o").
The benchmark cost corresponding to 7 = 0.3 was Si.556,233, whichwhen divided by the actual cost of $1,821,351 for the ABC hospital yieldsthe total cost variance ratio (TCVR) of 0.8544 in panel B of table 6. Thisbenchmark cost reflects the trade-off between efficiency and attainabilityin the sense that it is neither the tightest benchmark cost of $1,515,000(=$1,821,351*0.8318) corresponding to a value of 7 = 0.01, nor is it arelatively loose standard of $1,623,006 (=$1,821,351*0.8911) corre-sponding to the value of 7 = 0.5. The total variance in nursing cost isassessed as an unfavorable variance of $265,118. of which $232,222(=$1.821.351*(1 -0.8725)) is due to the unfavorable yield variance and thebalance of $32,896 is attributable to an unfavorable mix variance.
Alternatively, instead of pre-specifying 7. we could iteratively find avalue of 7 that puts the subject hospital Just on the production frontier. Alower value of 7 indicates better performance. Having ex post identified thevalue of 7 for which the subject hospital is at the production frontier, themanager of the hospital nursing unit can then seek to explain whether thevalue of 7 is consistent with the extent of noncontrollable factors that thesubordinates faced in performing their tasks. Although implementable,this approach is computationally much more demanding because of itsiterative nature.
Because of the relatively large unfavorable yield variance, the nursingdepartment at ABC hospital appointed a task force to explore what factorswere responsible for the hospital's low nursing productivity relative to thebenchmark group of hospitals. The task force inquired about staffing,training and administrative practices at the hospitals identified in the op-timal referent group (Xj > 0) by linear program (3). They identified short-comings in planning and scheduling and assignment of nurses to differentnursing tasks as the potential sources of inefficiency and recommendedthe acquisition of new software for planning and scheduling. The taskforce also investigated how allocations and assignments were made be-tween registered and licensed nurses to specific wards and patients, whichmay have contributed to the unfavorable mix variance. However, the mixvariance was not considered sufficiently large to necessitate an immediatechange in procedures.
CONCLUSIONDeveloping appropriate standards for performance evaluation Is a
common but important problem faced by many organizations. While re-gression approaches provide average standards under an assumed pro-duction function, the conventional DEA method uses the best practicefrontier as the standard. However, such a benchmark representing thevery best practice may be eonsidered too tight or unachievable, leading tolower morale and performance (Elnathan et al. 1996). This paper dem-onstrates how these limitations can be overcome with a Stochastic DataEnvelopment Analysis.
We illustrate the implementation of our approach using a sample of 66hospitals. We show how a total cost variance ratio may be decomposedinto its mix and yield components, and how managerially useful inferencesmay be drawn for purposes of cost control. Thus, the primary contribution
150 Journal qf Management Accounting Research, 1998
of this paper is in its adaptation of the SDEA approach to provide man-agers the flexibility to set attainable standards while benchmarkingagainst best practice organizations, allowing for trade-offs between sub-stitutable Inputs and outputs.
I APPENDIX AGlossary of Symbols Used in the Text
7 - relative weight on the upward and downward deviations (manage-rial choice parameter).
Uj = deviation of organization J inside the production frontierVj = deviation of organization J outside the production frontier.
Wjk = implicit weight on output k when evaluating the efficiency of orga-nization j . •
Ymk = quantity of output k for organization m.Yjk - quantity of output k for organization j .Zj, = implicit weight on input i when evaluating organization j ,
x^^ - quantity of input i for organization m.Xji = quantity of input i for organization J.
u^ = deviation of organization m inside the production frontier,v^ = deviation of organization m outside the production frontier,
x^, = quantity of input 1 for organization m.Xj, = quantity of input 1 for organization].Oj = weight on referent organization m when evaluating organization J.e jj = weight on referent organization j when evaluating organization m.Pol = price of input i for organization o.hg = radial efficiency of organization o.Xp, = quantity of input i for organization o.Yok = quantity of output k for organization o. *'X, = quantity of input i.\j = relative weight on organization J when evaluating organization o.
APPENDIX BThe dual formulation to the linear program in equation (1) is repre-
sented as follows:
Maximize Sji^ Sm=i.m*j %J^i ' ^i)
subject to:
n
S VVjk - ymk) ^ 0 Vj = 1 n; k - 1 KI m=I
n
- S V S i - x ^ i l ^ O VJ = 1 n; i = 2 1
1 - 7 Vj = 1 n
Banker. Chang and Das • 151
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