Improvement Potential in Danish Elderly Care

Transcription

1 Health Care Management Science 7, , Kluwer Academic Publishers. Manufactured in The Netherlands. Improvement Potential in Danish Elderly Care JENS LETH HOUGAARD Institute of Economics, University of Copenhagen, Denmark DORTE KRONBORG Department of Statistics, Copenhagen Business School, Denmark CHRISTIAN OVERGÅRD The Danish National Institute of Social Research, Herluf Trolles Gade 11, DK-1052 Copenhagen K, Denmark Abstract. This paper examines Danish elderly care resource utilization among municipalities. The available production data are aggregated, which means that it is virtually impossible to allocate costs to specific services. We show that Multi-directional Efficiency Analysis (MEA) provides a much more subtle performance picture than Data Envelopment Analysis (DEA) because we are able to assess the input specific relative improvement potentials. The empirical results show considerable improvement potential for all inputs. The largest relative potential for improvement is found for administrative staff. Keywords: elderly care, efficiency, multi-directional efficiency analysis, data envelopment analysis 1. Introduction The supply of elderly care is one of Danish municipalities maor activities. About 120,000 employees are in daily contact with more than 200,000 users. The services consist of housing offers (old age housing, nursing homes, etc.), various forms of home care in terms of practical help as well as personal care and activation programs including physiotherapy and occupational therapy. 1 In 2000 total net operation expenditures on elderly care was 23.1 billion DKK (approx. 3.1 billion Dollars) corresponding to roughly 20% of total budgets. This amount will increase drastically in the near future because of an increasing proportion of older people in the population. The costs of elderly care are financed primarily by taxes. The service level is politically determined by the local authorities but is subect to a general regulation stating that the municipalities are obliged to deliver certain fundamental services to the elderly (no minimum standards required explicitly) in terms of the above mentioned activities. In view of the increasing proportion of elderly it is extremely important for the economic viability of the municipalities that resources are used efficiently in elderly care. In order to ensure efficient use of resources it is necessary to obtain knowledge of best performance standards in the production of elderly care. DEA has been one of the most popular tools of efficiency assessment as well as implicit benchmarking. See, e.g., Cooper et al. [5] for a recent survey of the DEA method and its applications. Corresponding author. 1 These activities are the basis for the selection of representative outputs in section 4. There have been a few previous attempts to investigate elderly care using DEA. Some studies have focused on nursing homes by analysing American data (Nyman and Bricker [20] and Rosko et al. [21]) and Dutch data (Kooreman [15]). Scandinavian studies focus more broadly on the entire service provided for the elderly by the local municipalities as a natural consequence of the way that elderly care is organized in Scandinavia: The Danish Ministry of Finance [17] uses DEA to analyse a sample of 273 Danish municipalities. 40% of the municipalities had an improvement potential of over 25%. 2 Another report was conducted by the Danish Ministry of Social Affairs [18] using a reduced sample of 136 municipalities. A large variation in efficiency was found and the improvement potential amounted to an average of 35%. 3 Data on Norwegian municipalities were analysed in Edvardsen et al. [8], Erlandsen et al. [9], Erlandsen [6] and Erlandsen and Førsund [7]. In short, their results indicate a similarly large improvement potential of at least 25% on average. In particular, [9] found no connection between client satisfaction and efficiency in the elderly care, and [8] found that the 25% least efficient municipalities have significantly lower ratios between residents and homecare recipients than the benchmark units. 2 Some of the efficient municipalities were interviewed in an attempt to explain the differences in efficiency. The following factors turned out to influence the level of efficiency: Decentralisation with effective superior leadership, relatively low number of reported sick days among the staff, low level of employee turnover, homogeneous effective inspections of home help, re-inspections in light of changes in the elderly persons needs and explicit IT-policies. 3 To analyse the productivity development a Malmquist index was computed for the time period It turned out that the average productivity of the municipalities was nearly unchanged throughout this period.

2 226 J.L. HOUGAARD ET AL. We argue, from a methodological point of view, that DEA efficiency is not well-suited to assess the utilization of resources in elderly care when performance data are highly aggregated since DEA represents a way to aggregate performance information into a single efficiency score. This conceals a lot of relevant information with respect to the utilization of individual production factors. We therefore suggest using the potential improvements approach introduced in Bogetoft and Hougaard [3], see also Asmild et al. [1] and Bogetoft and Hougaard [4], where efficiency is measured in proportion to improvement potential rather than in proportion to current activities as in DEA. As such, we take for granted that it is meaningful to perform non-parametric efficiency analyses on production data for elderly care (for general criticism related to health care see, e.g., Newhouse [19]) and focus on how the potential improvements approach perform relative to DEA. We obtain a much more subtle performance picture by using the potential improvements approach since it becomes possible to disaggregate the efficiency score which allows us to asses the relative improvement potentials for each input or output category separately. In the present case of Danish elderly care it is shown that DEA scores conceal significant differences in the utilization of the different categories of staff and operational expenditures. DEA tends to seriously underestimate the relative improvement potential in administration and service staff while there seems to be overall agreement with respect to nursing staff and operational expenditures. The results show that there are considerable relative improvement potentials for all input categories, especially for the administration staff, which has a maximum of 44% on average. Looking at the cost side, however, efficient utilization of the nursing staff (which amounts to 85% of the total staff) provides by far the largest absolute saving potential. The administrative staff only amounts to 3% of the total staff. Since elderly care primarily consists of labour intensive service production, reductions in staff may lead to quality reductions. For the present study no reliable quality indicators were available and consequently our results do not take into account the qualitative differences between municipalities. The specific size of the improvement potentials is therefore to be interpreted with considerable caution. The interesting result is the overall performance pattern and the fact that it directly provides a useful starting point for the local authorities in their aim of improving the performance of elderly care. 4 Since the data are highly aggregated in the sense that several different production processes; nursing homes, home care, etc., are covered, it is very difficult to explain the variation in the relative improvement potentials even when the disaggregated results are applied. However, we are able to obtain some useful indications: A proportional odds model based on the relative improvement potential reveals a weak tendency towards better resource utilization of the nursing staff in rural 4 In fact, many municipalities realize the necessity of reducing administration expenses not only in elderly care. municipalities. Moreover, there is also a weak tendency of economies of scale since there is increased efficiency in operational expenditures with an increasing number of elderly. The paper is organized as follows: Section 2, provides a further motivation for our analysis. In section 3 the new methodology is presented and compared to the DEA methodology. The data and the model are introduced in section 4 and the results are presented in section 5. These results are further analysed in section 6 where we try to explain some of the variation in resource utilization. Section 7 closes with final remarks. 2. Motivation As mentioned in section 1, Danish elderly care is mainly provided by the 275 municipalities. On the output side each municipality is responsible for a wide variety of integrated care services primarily financed by taxes. The input side mainly consists of staff and other production costs. With the current data registration it is virtually impossible to allocate costs or staff to specific services. The aggregated data publicly available from Statistics Denmark conceal a lot of information with respect to cost utilization of the various care services. This is unfortunate because this information would prove useful for detailed efficiency assessments. Thus, with the currently available data set the challenge is to find a method for efficiency assessment that involves as little information aggregation as possible. However, in the long run it would be preferable to change the entire structure of the data registration process. Using DEA to analyse efficiency on a highly aggregated data set only adds to the problem mentioned above since the individual DEA efficiency scores aggregate performance information into a single number. Hence, there is a need for disaggregating the efficiency assessments where efficiency is measured in each input and/or output dimension separately while maintaining the computational convenience of DEA and the simple interpretation of the efficiency results. Disaggregating the efficiency assessments will provide politicians and management responsible for the daily operations of elderly care with a much more detailed performance picture, which will improve their possibilities for optimal resource allocation. Theoretically, however, this requires an explicit benchmark selection procedure. In DEA a benchmark selection is implicitly obtained for a given production unit by making proportional changes relative to the unit s current production, see figure 1. The relative performance measured in each dimension is identical for all dimensions. Empirically, there are often huge differences between the efficiency of the separate dimensions, which the DEA methodology conceals. Consider figure 1: Production unit x uses two inputs to produce one output under input requirement set L (i.e., the set of input combinations that yield a given level of output). It is evident that for production unit x there is considerable improvement potential for input 2 whereas the improvement potential is rather limited for input 1.

3 IMPROVEMENT POTENTIAL IN DANISH ELDERLY CARE 227 potentials found in the subsequent analysis can be considered as long run potentials. 5 In the present paper the potential improvements approach is used to estimate the improvement potential in each input dimension, i.e. different types of staff and costs, for each local municipality since we are primarily interested in the utilization of the different types of staff. The results are not the usual DEA efficiency scores but rather vectors of input specific scores relative to the selected benchmark unit. These input (or output) specific scores relative to the benchmark, S PI, are referred to as MEA-scores. 3. Multi-directional Efficiency Analysis (MEA) Figure 1. Illustration of a case where production plan x uses two inputs to produce one output with input requirement set L. Thus, it seems that selecting a benchmark for x, this benchmark must use relatively less input 2 than in the current production of x. However, if the benchmark is selected proportionally to the current production x,asindea,wefindavery different result, see S F in figure 1. One way to solve this problem, while maintaining the computational convenience of DEA, is to replace the use of the Farrell index with an approach that takes the actual improvement potentials into account as in [1,3,4]. It is suggested that for each individual production unit, benchmarks are selected in proportion to the specific improvement potentials, ( x1 x1, x 2 x ) 2, x 1 x 2 in the two dimensional case instead of proportionally to the current production, x, as done implicitly in DEA. That is, to select the production plan S PI as benchmark for x in figure 1. In [3] it is demonstrated that the axiomatic foundation of such a benchmark selection procedure is much more attractive than the implicit benchmark selection procedure underlying the Farrell index used in DEA. We implicitly assume that the production units involved are free to alter their current activities by allowing for nonproportional scaling of production factors. Hence, we assume that a change in input mix do not meet any limitations or bounds. This is particularly relevant for the present study since we consider a production process that is an aggregate of (at least) three separate production processes. The improvement potentials indicated by non-proportional scaling could be difficult to achieve in the short run since they might require structural changes in the production of elderly care for some municipalities. Now, using DEA (i.e., proportional scaling) such problems are avoided since downsizing leaves the original production activity unchanged. Hence, the improvement The efficiency analysis naturally becomes input oriented because we are primarily concerned about the utilization of the different types of staff. The result of an input oriented MEA approach directly indicates how performance could be improved by reorganizing the use of staff. Alternatively, we could have maximized municipality outputs given their budgets (or actual total costs which are available in the present data set) with the obvious changes in the programs equations (1) and (2). The programs would then provide a nonproportional version of the cost indirect approach along the lines of Färe et al. [12]. To provide some intuition consider figure 1, where x is the input vector of a given production unit and L is the input requirement set (for example, determined by convex envelopment of the data set as in DEA). In the standard version of DEA, Farrell s (input) efficiency index E F (x) = min{ψ R: ψx L}, for x L, is applied to determine the efficiency score of each production unit. Thus, in DEA the production plan S F is selected as an implicit benchmark for production unit x. Using the potential improvements approach the production plan, S PI, is selected in the following way: First the ideal reference plan for x is found as x (x),where x i (x) = min{ x i R: (x 1,..., x i,...,x m ) L }, for all i. That is, the ideal reference plan for x is found as the full input reduction potential in each separate input dimension. The ideal reference point x is generally infeasible. Now, the benchmark, S PI, is found as the largest possible reduction of x in the direction of x : For the general case of m inputs, reductions are made in proportion to the input specific excesses given by (x i0 x i (x 0)) i=1,...,m. See [3,4] for further theoretical details and results. If x is efficient then x = x (x) and consequently all input specific excesses are equal to zero. Therefore, x = S PI and all input specific MEA scores become zero. Let N be the number of municipalities and let municipality N use m inputs x i, i = 1,...,m, to produce s outputs 5 It goes without saying that even proportional downsizing tends to meet serious obstacles in real life adaptation to benchmark performance standards and the process rarely happens over night.

4 228 J.L. HOUGAARD ET AL. y r, r = 1,...,s. The mathematical programs involved in the calculation of the relative input specific MEA-scores for a given municipality (x 0,y 0 ) can now be determined as follows: First, the ideal reference plan for (x 0,y 0 ) is found by solving the m linear programs (one for each input dimension): min θ i, s.t. λ x i θ i, λ x ( i) x ( i)0, i = 1,...,i 1,i+ 1,...,m, λ y r y r0, λ = 1, λ 0, r = 1,...,s,, where the condition λ = 1 is used if the technology is assumed to exhibit variable returns to scale (without this condition, constant returns to scale is assumed). Letting (λ,θi ) solve the above programs for i = 1,...,m, the ideal reference plan for (x 0,y 0 ) is given by (θ1,...,θ m ).If(θ 1,...,θ m ) = x 0 we know that municipality (x 0,y 0 ) is (input) efficient and consequently all relative input specific MEA scores are 0. Assuming that (θ1,...,θ m ) x 0, consider the following linear programming problem: max β s.t. λ x i x i0 β ( x i0 θi0), i = 1,...,m, λ y r y r0, λ = 1, λ 0, r = 1,...,s,, where the condition λ = 1 is used if the technology is assumed to exhibit variable returns to scale (without this condition, constant returns to scale is assumed). The solution (λ,β ) to this program can be used to determine the benchmark selection: S PI (x 0 ) = x 0 β ( x 0 θ ), and the vector of relative input specific MEA-scores: ( xi0 Si PI ) ( (x 0 ) β (x i0 θi = ) ) x i0 i=1,...,m x i0, i=1,...,m with respect to municipality (x 0,y 0 ). As such, the relative input specific MEA scores indicate the relative improvement potential for municipality (x 0,y 0 ) in each input dimension. The relative input specific MEA scores are constrained between 0 and 1. (1) (2) Applying DEA scores for comparison, the relative improvement potential is found as 1 ψ where ψ solves the input oriented envelopment problem: min ψ s.t. λ x i ψx i0, i = 1,...,m, λ y r y r0, λ = 1, λ 0, r = 1,...,s,, under the assumption of variable returns to scale. In the efficiency literature there are alternative approaches in obtaining non-proportional scaling generalized by the directional distance function approach (see, e.g., Färe and Grosskopf [10]), which is closely related to MEA. In MEA each production unit is scaled in individual directions given by (x (x) x) as obtained in equations (1). For this specific choice of direction we rest on the firm axiomatic foundation provided in [3] and [14]. In particular, non-proportional scaling can alternatively be obtained implicitly using the Russell index, which generalises the Farrell index used in DEA, see, e.g., Färe and Lovell [11]. However, the implicit benchmark selection is unfortunately not unique and suffers from many of the same problems as the implicit selection of the Farrell index (see [3]). Moreover, the index value is the average of the input specific efficiency scores relative to the selected (efficient) benchmark and as such contains practically no managerial information. But clearly, the input specific efficiency scores considered separately can be interpreted along the lines of the MEA scores (but relates to a different benchmark selection procedure). 4. Data and the model The analysis applies official data from Statistics Denmark from The complete database covers a population of 275 Danish municipalities. We use a reduced sample of 266 municipalities, because seven municipalities have registered no administration staff and two municipalities have registered no amount of assigned home care hours per week. To model the service of elderly care we use the following inputs and outputs: Inputs: (1) administration staff, (2) nursing staff, (3) service staff, 6 (4) operational expenditures. Inputs 1 to 3 are measured in full time obs. Input 4 is measured in 1000 s of Danish Kroner (DKK). Salaries and payments between relevant counties and the state are excluded from operational expenditures. 6 E.g., Janitors, cleaning and kitchen help. Staffs that clean in private homes are included in nursing staff. (3)

5 IMPROVEMENT POTENTIAL IN DANISH ELDERLY CARE 229 Table 1 Descriptive statistic. Variable Min Max Mean Median Standard Total deviation sum Input Administration staff a Nursing staff a Service staff a Operational expenditures (in 1000 DKK) Output Amount of assigned hours of home care per week Number of residents in nursing homes Number of clients in day treatment centre and day nurseries a Measured in full-time obs. Outputs: (1) amount of assigned hours of home care per week, 7 (2) number of residents in nursing homes, (3) number of clients in day treatment centres and day nurseries. These outputs represent the maor activities related to elderly care in the municipalities. Obviously, the outputs are proxies for enhancing the quality of life for the elderly. In particular, note that there is no explicit proxy of quality included and consequently quality differences may account for some part of the differences in the subsequent efficiency results. Given the available data, there are no suitable candidates for measures of quality for the general quality of life or for the quality of the specific services involved. Indeed, the very notion of quality in elderly care is hard to define with highly aggregated data. Meaningful quality indicators ought to be connected with specific production processes as witnessed by the many relevant (micro) quality indicators listed by the Centre of Health Systems Research and Analysis in the United States. As expected, the maority of the staff consists of nursing with 79,412 employees (85%), cf. table 1. About 12% of the staff consists of service personnel while the remaining 3% work in administration. All municipalities deliver home care but not all municipalities have residents in nursing homes and clients in day treatment centres and/or day nurseries. However, the maority of the municipalities produce all three outputs and none have restricted their production to homecare. The median is considerably lower than the mean for all variables because the maority of the municipalities are small. Moreover, we focus on three external variables in the analysis of the MEA results (cf. table 2): Share of inhabitants aged 67 and over (67+), share of population in urban 7 In approximately 50% of the municipalities the personal care offered at the nursing homes is also registered as hours of home care (with current data it is impossible to correct for this difference). At first sight this seems problematic since these municipalities may be compared to municipalities without double registration. However, running MEA on a separate subsample of municipalities without double registration shows that the mean relative improvement potentials are of the same magnitude as those of the full sample. Consequently we have chosen to represent the results of the full sample in the subsequent sections. Table 2 Descriptive statistics for the external variables. Variable Min Max Mean Standard error URBAN INCOME areas (URBAN), 8 and average taxable income per household (INCOME) measured in DKK Results We chose to apply a variable returns to scale technology (VRS) to analyse the data described in section 4. Outputs consist of three separate production processes, which include home care, nursing homes and day nurseries. However, as mentioned in section 2, current input data prevents separate analysis since inputs cannot be allocated to these specific output categories. Therefore we construe the production of elderly care as a oint production process. There are no reasons to expect constant returns to scale for nursing homes and day nurseries, due to limitations on building capacity in the short run. Hence, from an intuitive viewpoint the oint production process ought to be characterized by VRS and consequently we shall use this assumption in the sequel. However, there are strong elements of constant returns to scale (primarily in home care) confirmed by an average scale efficiency score of 0.93 (using DEA scores). 10 We focus on MEA results and use DEA results as a source of comparison in the following analysis. In the sample of 266 Danish municipalities, 63 municipalities (24%) were fully efficient. In table 3, both absolute and relative improvement 8 Urban areas are defined as cities with more than 200 inhabitants. 9 The number of inhabitants in the municipalities is highly positively correlated with the percentage of population in urban areas, due to the fact that the Danish population is concentrated in urban areas. The effects of these two variables are inseparable when assessing the influence on the relative improvement potentials (MEA-scores). Therefore only the percentage of population in urban areas is considered. 10 Scale efficiency is defined along the lines of Førsund and Halmarsson [22].

6 230 J.L. HOUGAARD ET AL. Table 3 MEA improvement potential for the inefficient municipalities. Variable MEA, variable returns to scale Absolute improvement potential Relative improvement potential Mean Std. dev. a Min Max Total Mean Std. dev. a Min Max Administration staff Nursing staff Service staff Operational expenditures a Std. dev. standard deviation. Table 4 DEA improvement potential for the inefficient municipalities. Variable DEA, variable returns to scale Absolute improvement potential Relative improvement potential Mean Std. dev. a Min Max Total Mean Std. dev. a Min Max Administration staff Nursing staff Service staff Operational expenditures a Std. dev. standard deviation. potentials are summarized for inefficient municipalities. In general, there are large (mean) improvement potentials in absolute and relative terms. Among different categories of staff, the nursing staff has the largest absolute improvement potential (53.55 full time obs on average). This is not surprising since nurses comprise 85% of total staff. The administrative staff has the largest relative improvement potential (44% or 3.28 full time obs on average). Approximately 75% of the inefficient municipalities have their largest relative improvement potential for administration staff. The DEA results for the inefficient municipalities are represented in table 4. The results show large improvement potentials in both absolute and relative terms for all input categories, where nursing staff has the largest mean absolute improvement potential. All relative potentials, are identical per definition (25.9%), see section 3. Comparing DEA and MEA we find that MEA provides a much more subtle performance picture since the relative potentials vary between inputs. In fact, all input categories apart from nursing staff have larger mean relative improvement potential in MEA than in DEA. The large difference for the administrative and service staff is particularly interesting. This difference illustrates the consequences of insisting on proportional reductions found in DEA, thus a lot of information concerning the actual location of the municipalities relative to the production frontier is concealed. We have selected two municipalities with extreme differences between MEA and DEA scores to expand on this aspect. When DEA program (equations (3)) is applied, municipality with reference number 655 has a relative improvement potential of about 20% in all four input dimensions (cf. table 5). Thus, its relative improvement potential is smaller than the mean relative improvement potential of all the inefficient municipalities. However, the MEA analysis gives us a more detailed picture of the comparison between municipality 655 Table 5 Relative improvement potential for municipalities with reference number 655 and 421. Municipality Relative improvement potential reference Administration Nursing Service Operational number staff staff staff expenditures 655 MEA DEA MEA DEA and its benchmark units. Municipality 655 follows the general tendency with large relative improvement potentials for administration and service staff (65 and 51%, respectively) and less relative improvement potential for nursing staff and operational expenditures (15 and 20%, respectively), see table 5. Characterized by a larger relative improvement potential for the administration and the service staff and smaller relative improvement potential for the nursing staff and the operational expenditures, municipality 655 is more extreme compared with the general tendency. Not all municipalities resemble the general tendency. According to DEA, the municipality with reference number 421 is nearly classified as efficient since its relative improvement potential is less then 1%. This is also the case for the administration and the service staff according to the MEA result. However, the MEA results reveal a relative improvement potential of 15 and 11% for the nursing staff and the operational expenditures, respectively. Therefore, municipality 421 can hardlybe characterizedas a best performer despite the misleading DEA-score. The relative MEA and DEA improvement potential for all the municipalities are compared in figure 2 for each of the four inputs. For the administration staff and the service staff, the MEA results reveal a larger relative improvement poten-

7 IMPROVEMENT POTENTIAL IN DANISH ELDERLY CARE 231 Figure 2. Comparing relative MEA and DEA improvement potential for each input variable. tial than the DEA results for most municipalities. For nursing staff there is a tendency of larger DEA improvement potential than MEA improvement potential. The picture is somewhat mixed for the operational expenditures. These tendencies fit well with the results of the mean relative improvement potential in tables 3 and 4. There are no obvious explanations for the emergence of this phenomenon. From a methodological point of view, frontier efficiency implies that for each municipality, then if some inputs have a larger potential in MEA than in DEA other inputs must have a smaller potential. Apart from this observation there is no predetermined pattern. Everything depends on the shape of the production frontier as well as the specific location of the data points. Finally, MEA can also be compared to DEA based on the cost minimizing input combination (for each output level) as common reference point, given a suitable vector of average salaries. Despite the obvious difficulties and variations within groups of staff for each municipality, statistics on average salaries indicate that administration staff yearly average salary is approximately 300,000 DDK while average salaries for nursing and service staff are approximately 240,000 DDK. 11 If we consider the difference between costs of the selected benchmarks of DEA and MEA, respectively, and the minimum cost for each municipality we find that summing over all municipalities the selected MEA benchmarks 11 We assume that operational expenditures have price 1 DDK. We emphasize that these average salaries are not accurate but only intended for the present comparison. have a smaller absolute slack (1.86 billion DDK) with respect to the allocatively efficient combination than DEA benchmarks (1.92 billion DDK). The average difference between MEA benchmarks and allocative efficiency is 8.8% of the total budget (given the price vector above). To conclude, in this particular case reducing inputs according to MEA directions comes closer to allocative efficiency than downsizing accordingtodea The efficient municipalities In this section we consider the set of efficient municipalities with respect to their role and importance as benchmark units. In order to get an idea of the influence of particular benchmarks on the overall efficiency result, it is possible to assess the importance of each benchmark unit by counting the number of municipalities dominated by each benchmark. Despite the fact that some benchmark units dominate several other municipalities, their performance may not be much better than that of the units they dominate. In fact, there may be other benchmark units with a smaller number of dominated units, but much more important as benchmarks in the sense that the municipalities that they dominate are very inefficient relative to this benchmark Municipalities 333 and 661, which are benchmarks for 88 and 20 municipalities illustrate this point. Even though municipal number 333 dominates over four times as many municipalities as municipality 661, the share of the total improvement potential which is caused by the presence of benchmark municipality 333 is smaller for all four input variables than benchmark municipality 661.

8 232 J.L. HOUGAARD ET AL. Table 6 MEA improvement potential for the inefficient municipalities (removing the three most important benchmarks). Variable MEA, variable returns to scale Absolute improvement potential Relative improvement potential Mean Std. dev. a Min Max Total Mean Std. dev. a Min Max Administration staff Nursing staff Service staff Operational expenditures a Std. dev. standard deviation. Figure 3. Peer index for each input variable. An interesting index assesses, for each input dimension, the share of the total improvement potential in that dimension which is caused by the presence of each particular benchmark unit, along the lines of Torgersen et al. [23]. Let K be the set of efficient municipalities where k K. Moreover, let λ k be the weight of municipality with respect to benchmark k in the program equations (2), see section 3. Now, define the potential improvements peer-index of benchmark (efficient municipality) k K for input i as N\K ρi k = λk (x i Si PI) N\K x i Si PI, for i = 1,...,m. The municipalities with the top five peer indexes in each of the four input variables are illustrated in figure 3. The top three peer indexes are all found to belong to the same three municipalities when comparing the peer indexes for each input variable. The top five peer indexes belong to six different municipalities. 13 For all four input variables the share of the 13 It should be noted that the three most important benchmarks are municipalities with double registration (see footnote 7) and this may clearly account for some part of their position. Looking a the sub-sample con- Figure 4. Municipalities grouped according to their number of inhabitants. The bars represent all municipalities while the grey parts only represent the efficient municipalities. top five peer indexes is at least 45%. From a methodological point of view the same benchmarks do not necessarily account for the largest share of the total improvement potential in all dimensions. Thus, it is possible that the three most important benchmarks are outliers. However, removing these three benchmarks from the data set does not imply that the overall performance picture changes radically, see table 6. The absolute and relative potentials are roughly the same and the pattern between the different types of staff also remains the same, see table 3. It is difficult to find common characteristics for the benchmark municipalities. The municipalities with the top three peer index values all have less than 10,000 inhabitants. This is probably due to the small populations found in the maority of Danish municipalities rather than a specific size effect. In figure 4, the over representation of efficient municipalities among the large municipalities (>50,000) is caused by the methodological assumption of VRS (see section 3). 6. Statistical analysis of the variation in MEA-scores Analysis of the influence of external factors on the relative MEA-scores has been categorized into three groups. A proportional odds model (see, e.g., McCullagh and Nelder [16]) is subsequently analyzed. 14 sisting of municipalities without double registration municipalities 741 and 423 are the most important benchmarks accounting for approximately 35% of total slacks in all four inputs. Note that 741 also appears among the top five in the full sample. 14 Cut-points are chosen to ensure a statistically meaningful analysis, i.e. to ensure that too many sparse cells are avoided. One set of cut-points is

9 IMPROVEMENT POTENTIAL IN DANISH ELDERLY CARE 233 Table 7 Results from multiple regression analysis. Variable Administration staff Nursing staff Service staff Operational expenditures χ 2 df p χ 2 df p χ 2 df p χ 2 df p URBAN INCOME Score test MEA-scores: Calculated Wald test statistics (χ 2 ), degrees of freedom (df) and corresponding significance probability (p) for the multiple multi-category ordinal response model for the influence of each of the external variables. The Score test is the test for the proportional odds assumption. The proportional odds model is a multi-category logit model for ordinal responses with H categories, where the relation between the cumulative logits and the external variables, is specified as: P(Y h) log P(Y > h) = logitp(y h) = α h + βx, h = 1,...,H 1, where x denotes the external variables, and β is a vector of parameters describing the potential influence of the external variables. Note that the model is a multiple multi-categorical logit model and the influence of a given external variable is evaluated after correcting for the influence of the remaining variables. Further, note that since the parameter β is independent of the choice of category (h), the model is robust to collapsing or splitting categories. This means that, as long as the proportional odds model assumptions is fulfilled the effect of the external variables is retained when collapsing or splitting categories Results of the proportional odds model Initial analyses of the relationship between potentially influential external variables and the MEA scores reveal that the five maor municipalities; Copenhagen, Frederiksberg, Aarhus, Odense and Aalborg all have an extremely high influence on the assessment of the effects. Therefore, these five municipalities are excluded from the regression analyses. Since the MEA scores are calculated in case of VRS the five maor municipalities becomes efficient by assumption. Under CRS all five municipalities are inefficient. First a score test for the specified structural part of the model was performed. The proportional odds model, gave a satisfactory description of the data sets, see table 7. Furthermore, the results from the multiple regression analysis including all three external variables can be seen in table 7: Only MEA-scores derived from the input variable nursing given by the following intervals: (i) [0, 0.1], (ii) (0.1, 0.4], (iii) (0.4, 1] and is used for MEA-scores with respect to the input variables; administration staff and service staff. The other set is given by the intervals: (i) [0, 0.1], (ii) (0.1, 0.3], (iii) (0.3, 1], and is used for MEA-scores with respect to the input variables nursing staff and operational expenditures as well as for the DEA-scores used for comparison. We have tried with finer intervals of cut-points and found no changes of the overall results. Table 8 Estimated probabilities for each MEA-score category for nursing staff calculated for different values of percentage of population in urban areas. MEA-score URBAN = 60% URBAN = 75% URBAN = 90% [0, 0.1] (0.1, 0.3] (0.3, 1] Table 9 Estimated probabilities for each MEA-score category for operational expenditures calculated for different values of share of inhabitants aged 67 and over. MEA-score 67+ = = =0.20 [0, 0.1] (0.1, 0.3] (0.3, 1] staff seems to be (slightly) affected by the percentage of population living in urban areas (URBAN) having the highest χ 2 value. Excluding the variables INCOME and 67+ from the model for nursing staff since they have no effect, results in a slightly smaller significance probability for URBAN. Marginally, the χ 2 value is corresponding to p = As illustrated in table 8 there is a weak tendency of having a more efficient nursing staff when the variable URBAN decreases. 15 Each of the external variables turns out to have marginal significant effect, when considering the MEA-scores obtained for operational expenditures. 16 Controlling for 67+,URBAN has no significant influence on the scores for operational expenditures (χ 2 = 2.71, p = 0.10). Similarly, no significant effect of INCOME is found (χ 2 = 0.91, p = 0.34). The effect of 67+ is illustrated in table 9. It seems that the estimated proportion of efficient municipalities increases when the share of elderly inhabitants increases. The related MEA scores for administration staff and service staff, successive elimination of external variables does 15 Note, that if the five maor municipalities are included in the analysis this effect vanishes. This is because the five maor municipalities all have very high values of URBAN and are efficient due to the assumption of variable returns to scale. Hence, including these municipalities actually distorts the picture. 16 The calculated values are URBAN: χ 2 = 5.81, p = 0.02, INCOME: χ 2 = 5.68, p = 0.02 and 67+: χ 2 = 6.85, p = 0.01.

10 234 J.L. HOUGAARD ET AL. not change the conclusion, i.e. no significant influences can be found. Analysis of the effects of the external variables on the DEA-scores shows a marginal effect of URBAN (χ 2 = 5.24, p = 0.02). In the model with the variable URBAN included, neither INCOME (χ 2 = 0.29, p = 0.59) nor 67+ (χ 2 = 1.35, p = 0.25) show significant effects. The estimated category probabilities are similar to those obtained from analyzing nursing staff MEA-scores. Therefore, the analysis of the DEA scores indicates that there is a weak tendency of achieving better resource utilization for all inputs in the rural municipalities. However, we can show that this effect only relates to the utilization of nursing staff when using MEA scores Discussion It is well known that efficiency scores obtained from nonparametric MEA and DEA have inherited characteristics, which include dependence between efficiency scores and an unknown distributional form. Consequently, ordinary methods for statistical inference are not directly applicable. Simar and Wilson [22] investigate the size of the problem for standard DEA efficiency scores and states that the most serious problem is that efficiencies are correlated. Also MEA relative excesses are correlated but the correlation structure differs from that for DEA scores. Anyway, asymptotically MEA scores are independent and the model used is sound in that respect. Moreover, the proportional odds model does not rely on any specific distributional form. It is difficult to find common characteristics among benchmarked municipalities because of the limited available data and because of the afore-mentioned problems with statistical inference. However, we find a weak tendency towards better performance with respect to operational expenditures when the number of elderly in each municipality increases, i.e., some degree of economies of scale as should be expected. This tendency cannot be detected using the DEA results since there are no economies of scale with respect to the different categories of staff, which influence the efficiency score. There is a weak tendency towards better utilization of nursing staff among rural municipalities: this only occurs for the nursing staff contrary to the DEA result. The explanation could be related to a higher absence due to sickness among nursing staff in urban areas. Absence could create a more stressful situation for those on the ob resulting in their increased likelihood of sickness, etc. 17 Therefore we should not expect to find the same effect for other input variables. Given the present model it is not surprising that INCOME and 67+ are without effect on the utilization of staff. More detailed micro-econometric studies could be performed if we were able to obtain data on separate activities such as nursing homes, and more specific external variables such as average income for households over 67 years of age. 17 This observation is based on unpublished survey data from the Danish National Institute of Social Research. Finally, some municipalities have organized home care and nursing homes as one unit while others are organized as separate units. We tried to test whether this had an influence on the MEA scores and found no effect. 7. Final remarks The difference between DEA and MEA can be construed as a difference in the level of planning. DEA is usually considered to be a tool for a central planner (principal) who needs to monitor a set of similar sub-units (agents). The radial efficiency scores may play a role in various contractual relations between the central planner and the sub-units, see, e.g., Bogetoft [2]. Proportional budget reductions, such as in the input oriented DEA scores, are typical for public management often referred to as lawn-moving. MEA, on the other hand, focuses on local improvement potentials and seems better suited for the local decentralized management of sub-units. The explicit benchmark selection allows each sub-unit to obtain managerially relevant information on how to reorganize its performance according to peer standards. In the present study the sub-units are Danish municipalities. On a conceptual level, MEA appears to be the most relevant approach because each municipality is free to determine the provision of elderly care. The aspect of the central planner, which is the state, becomes irrelevant. Thereby also the aspects of uniform treatment of sub-units like in the case of lawn-moving, and aspects of monitoring and incentive provision. As in previous efficiency studies of elderly care we find that there is considerable improvement potential with respect to the utilization of staff and operational expenditures (on average between 23 and 44% among the inefficient municipalities). Even though the specific size of these figures should be interpreted with considerable caution because we have not accounted for quality differences. It seems, on the other hand, that such huge differences can hardly be explained by quality differences alone. In particular, the fact that the largest relative improvement potential is found for the administration staff indicates that not all improvements necessarily result in drastic reductions in the quality of services as experienced by the elderly. As in previous efficiency studies of elderly care we also have great difficulties in detecting influential external factors. The problem is caused by the lack of sound data both concerning the activities and relevant external factors. Obtaining data for the separate production processes would make it easier to focus on particular external factors of influence. This is unfortunately not possible with the current data registration. Changing the data registration process remains a long-term proect. Acknowledgements We would like to thank James Weatherall for comments and careful reading of the manuscript and the anonymous referees for many helpful comments.

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