Intertemporal analysis of organizational productivity in residential aged care networks: scenario analyses for setting policy targets

Abstract

With an increasing ageing population, there is a growing concern about how the elderly would be looked after. The primary purpose of this paper is to develop scenario analysis using simulated data where various criteria are incorporated into modeling policy targets, and apply an intertemporal productivity analysis to observe inefficiencies as reform unfolds. The study demonstrates how dynamic network data envelopment analysis (DN-DEA) can be used to evaluate the changing productivity of residential aged care (RAC) networks over time. Results indicate that it takes 9 years for 90 % of the RAC networks to have 85 % or more of the total beds in high-level care, and an optimal bed capacity is reached by the end of year 7. Number of beds and registered nurses employed are the main sources of inefficiency. The common core inefficient cohort identified with the paper’s method represents a sub-group of RAC networks more deserving of closer managerial attention because of their constantly inefficient operations over time.

Appendices

Appendices

1.1 Appendix A

1.1.1 An introduction to traditional data envelopment analysis

This brief introduction is for the benefit of those who may not be familiar with traditional DEA (TDEA); network DEA and dynamic network DEA are extensions of TDEA. TDEA is a non-parametric efficient frontier technique that calculates a comparative ratio of weighted outputs to weighted inputs for the modeled production in each organizational unit—often reported as an estimate of relative technical efficiency.

TDEA operates under the condition of Pareto optimality where a decision-making unit (DMU), or in the current study, an RAC network, is not efficient if an output can be raised without raising any of the inputs and without lowering any other output. Similarly, a DMU is not efficient if an input can be decreased without decreasing any of the outputs and without increasing any other input (Charnes, Cooper and Rhodes [29]). We also note that there are many different ways a DMU and its divisions or sub-units can be conceptualized and such divisions do not have to be physical units. For example, a DMU could be a particular demographic such as 65–75 year-olds in a suburb, and divisions could be males and females in this age group or various income brackets before health care is investigated.

TDEA generates information on whether performance can be improved relative to observed benchmark behavior in a peer group, rather than relative to measures of central tendency. The relative efficiency estimate (or score) is often written as a number between 0 and 1, and a DMU with an estimate of less than 1 is labeled inefficient. Those units on the efficient frontier (benchmark units) then determine the potential improvements for the inefficient units that lie off the frontier.

Key advantages of DEA include the property that no functional structure is imposed on the data in estimating relative efficiency. In other words, DEA does not pre-suppose a particular production technology found in all the DMUs in a sample. The significance of this approach is that an organization’s efficiency can be estimated based on other observed performance in that sample by benchmarking peers that are better at managing their inputs and outputs. Similarly, variable weights are endogenously determined rather than relying on subjective weights. Another advantage of DEA is its ability to process multiple inputs and multiple outputs in arriving at a single estimate for a DMU—something which cannot be achieved with ratio analysis because individual ratios are most unlikely to be independent. Gelade and Gilbert ([30], p. 497) state that, “… the way in which performance measures are combined in DEA rests on strong theoretical foundations of production economics, allowing overall efficiency to be computed in circumstances in which additive scales cannot be legitimately constructed.”

In summary, DEA captures the interactions among multiple inputs and multiple outputs that can be traced to operational processes, and locates relative weaknesses of an inefficient organization against its best performing peers. The reader is encouraged to refer to Cooper, Seiford and Zhu [31] for a more in-depth treatment of DEA. A paper that bridges theory and practice by demystifying DEA mathematics is Bougnol, Dula, Lins and da Silva [32].

1.2 Appendix B

1.2.1 Background to data generation

In order to operationalize data simulation, and thus, the illustration of DN-DEA in residential aged care, a number of assumptions are made regarding the organization of RAC, the two care levels therein, and the various interacting variables of service delivery. For example, in profiling metropolitan residential aged care, full occupancy is assumed at any given point in time, which is often what is seen in real life. In fact, the statistics compiled on RAC by the Australian Institute of Health and Welfare [33] indicate an occupancy rate of 93 % as at 30 June 2009. Furthermore, all else the same, the length of stay for a resident at a given care level is assumed to be prolonged when more staff resources are available. Similar to Laine et al. [11], Schnelle, Simmons, Harrington, Cadogan, Garcia and Bates-Jensen [34] also conclude that higher quality of care in nursing homes is associated with higher staffing levels.

Once again all else the same, ALOS would drop when ARCS rises because of a higher level of care needed, which may trigger relocation to another care level. Australian Institute of Health and Welfare ([33], p.79) reports that approximately 10 % of the low-care residents were reclassified as high-care in 2008–09. This figure is used to determine the proportion of residents to be annually transferred from LLC to HLC. Nevertheless, such relocation is unlikely to involve a change of address since the ‘ageing in place’ policy introduced by the Aged Care Act of 1997. According to the Australian Institute of Health and Welfare [33], only 2 % of permanent residents who moved from low-level to high-level care in 2008–09 were actually relocated to another facility; the report indicates ALOS at the time of separation to be about 3 years.

The same report identifies roughly three-quarters of permanent admissions as aged 80 years or older. In Australia, resident dependency levels are determined by the Aged Care Funding Instrument (http://www.health.gov.au/acfi). According to Australian Institute of Health and Welfare ([33], p.83), about three-quarters of permanent residents were appraised as high-care as at 30 June 2009. This ratio guides the initial data generation for number of beds.

Interactions among the five inputs lead to the primary output of length of stay, as well as the undesirable carry-overs such as hospitalization and mortality. Assuming a performance evaluation period of 1 year, the relationships put forward below depict interactions among the variables and divisions outlined in Fig. 1 in the associated paper. Various assumed data ranges and weights, while arbitrary for illustrative purposes because we are not privy to real-world data where we could observe distributions, follow reasonable expectations and can be easily changed for further what-if analyses. For example, to reflect the increased focus by policy makers on high-level care beds within an RAC network, the divisional weights LLC (0.20) and HLC (0.80) are allocated for the purpose of computing overall efficiency estimates. Essentially, an algorithm is modeled where the number of beds primarily determines RN-FTE, which in turn drives OC-FTE; the average resident classification score for a division determines the average length of service for registered nurses; the average length of stay is affected by ARCS and RN-ALS in opposite directions; and, the annual number of hospitalizations are linked to the number of beds and ARCS.

We allow a small range of variation in determining staffing levels in order to reflect differences in managerial decision-making and regional differences in the availability of labor. Thus, initially the number of beds is randomly generated in the range of 50–100 for an LLC division, followed by randomly generated numbers of registered nurses in a narrower range. For each bed, we allocate 0.10–0.12 of an RN-FTE (i.e., a minimum of 5 RN-FTE for 50 residents); similarly, for each RN-FTE, 1.5–2.5 other caregivers are allocated, where once again, random number generation proceeds in the assumed ranges. For an HLC division, the number of beds range between 150 and 300. For each bed, 0.12–0.15 of an RN-FTE is allocated, and for each RN-FTE 2–3 other caregivers are allocated, thus reflecting the more intensive care at this level.

The average length of service (RN-ALS) of registered nurses—considered to be the key caregivers—is linked to the average resident classification score (ARCS) that captures the overall health of residents in a division. That is, as the health of residents deteriorates and ARCS rises, we assume that registered nurses with greater experience would be recruited to better meet the more complex care needed. Residential classification scores (on a scale of 1–10) are allowed to randomly vary where the distribution for LLC is skewed towards the lower end, and a corresponding RN-ALS of 0.5–1.0 year is allowed for every point in the ARCS. Similarly, the residential classification scores for an HLC division are skewed towards the higher end.

In determining the average length of stay of residents (years) in a division, we begin with random number generation between these ranges: LLC (2–5); HLC (1–4). ALOS is then adjusted to reflect ARCS. That is, for every point in ARCS above 1, ALOS is reduced by 1/10th of a year in recognition of the non-discretionary element in managing residents’ health. We adjust ALOS for the second time by raising it by 1/10th of a year for every year of RN-ALS above 2 years in recognition of the discretionary element in residents’ health which can be shaped by the quality of care. Here, the focus is specifically on the experience of registered nurses rather than just physical staffing levels.

Finally, the undesirable outputs from divisions (carry-overs) are modeled. Random data generations adhere to the following ranges: for each bed in an LLC division, the annual number of hospitalizations or ANH equals (0.01–0.08); this number is then multiplied with ARCS. For an HLC division, the annual number of hospitalizations varies in the range of 0.06–0.10, followed by factoring in ARCS. The average severity of hospitalizations (ASH) is considered generally unpredictable and modeling adheres to random number generation, but a distributional skew towards 1 for an LLC and towards 10 for an HLC division is included. Progressively higher ranges in the form of (5–15) for an LLC division, and (10–30) for an HLC division, are assumed for MR or mortality rate (%)—allowed to vary randomly.

The following algorithm summarizes the steps in generating initial data for 526 RAC networks. Data generation incorporates the variable relationships shown in Fig. 1. The tilde symbol (∼) is used to indicate how the values for a given variable are distributed. All figures below are at the DMU level unless stated otherwise.

Generate RAC networks and number of high- and low-care beds:

• The number of residents is approximately based on Table A5.2 found in the Australian Institute of Health and Welfare [33]. In this report, the overall ratio of high-care to low-care beds is about 3:1, which we use to set the parameters for generating the number of beds. Thus,

  • Number of HLC beds (input):

    • ∼ U[150,300], integers only.

    • 526 RAC networks are generated. The total number of generated HLC residents is 117,940 which closely approximates the figure of 117,884 from Table A5.2 (Australian Institute of Health and Welfare [33]).

  • Number of LLC beds (input):

    • ∼ U[50,100], integers only.

    • This produces an LLC population of 39,804.

Select networks to be grown and apply growth rates:

• A compound growth rate ∼ U[0.05,0.10] is applied to high-level care beds in those networks that have a proportion of high-level care beds below 85 % of the total beds. The number of beds is rounded to an integer in each year. This gives rise to the variables of hlbeds, llbeds for each network-year.

  • In each year, the growth in hlbeds simultaneously reduces llbeds without dropping lower than a minimum number of 50 llbeds for a given network.

The variables below are generated for each network-year:

• Number of Registered Nurses (input):

  • hlrn ∼ U[0.12,0.15] * hlbeds

  • llrn ∼ U[0.1,0.12] * llbeds

• Number of Other Caregivers (input):

  • hloc ∼ U[2, 3] * hlrn

  • lloc ∼ U[1.5, 2.5] * llrn

• Average Resident Classification Score (input):

  • Underlying HLC RCS ∼ Triangular distribution with minimum 0.5, maximum 10.5, mode 8.57, rounded to integers that result in values 1–10.

  • Underlying LLC RCS ∼ Triangular distribution with minimum 0.5, maximum 10.5, mode 1.43, rounded to integers that result in values 1–10.

  • The underlying RCS’s are fully randomized across network-years. The average is then taken for each network-year to form hlarcs and llarcs.

• Average Length of Service of Registered Nurses (input):

  • hlals ∼ U[0.5, 1] * hlarcs

  • llals ∼ U[0.5, 1] * llarcs

• Average Length of Stay (output):

  • hlalos ∼ U[1, 4] − 0.1 * (hlarcs − 1) + 0.1 * (hlals - 2)

  • llalos ∼ U[2, 5] − 0.1 * (llarcs − 1) + 0.1 * (llals - 2)

• Number of Hospitalizations (undesirable carry-over):

  • hlh ∼ ceil(U[0.06, 0.10] * hlbeds * hlarcs)

  • llh ∼ ceil(U[0.01, 0.08] * llbeds * llarcs)

• Average Severity of Hospitalizations (undesirable carry-over):

  • Underlying HLC severity of hospitalization ∼ Triangular distribution with minimum 0.5, maximum 10.5, mode 8, rounded to integers that result in values 1–10.

  • Underlying LLC severity of hospitalization ∼ Triangular distribution with minimum 0.5, maximum 10.5, mode 3, rounded to integers that result in values 1–10.

  • The underlying hospitalization severities are fully randomized across network-years. The average is then taken for each network-year to form hlash and llash.

• Mortality Rate (%) (undesirable carry-over):

  • hlmort ∼ U[10, 30]

  • llmort ∼ U[5, 15]

Number of Residents Transferred from low-care to high-care (intermediate product): Permanent residents in low-care are ranked in descending order on their residential classification scores (RCS) and the top 10 % is winsorised. This top 10 % represents the total number of residents transferred from low-care to high-care.

1.3 Appendix C

1.3.1 Dynamic network range-adjusted measure of efficiency (DN-RAM)

$$ {\varGamma}_o^{*}=\underset{\lambda_k^t,{s}_{ok}^{t-},{s}_{ok}^{t+},{s}_{o\left(k,h\right)}^t,{s}_{ok}^{\left(t,t+1\right)}}{ \min }1-{\displaystyle {\sum}_{t=1}^T{W}_t{\displaystyle {\sum}_{k=1}^K\frac{w_k}{m_k+{i}_k+{b}_k}\left[{\displaystyle {\sum}_{m=1}^{m_k}\frac{s_{mok}^{t-}}{R_{mk}^{t-}}+{\displaystyle {\sum}_{l=1}^{i_k}\frac{s_{lo\left(k,h\right)}^t}{R_{l\left(k,h\right)}^t}+{\displaystyle {\sum}_{n=1}^{b_k}\frac{s_{nok}^{\left(t,t+1\right)}}{R_{nk}^{\left(t,t+1\right)}}}}}\right]}} $$

(C.1)

subject to

$$ \begin{array}{ll}{x}_{ok}^t={X}_k^t{\lambda}_k^t+{s}_{ok}^{t-}\hfill & \left(k=1,\dots, K;t=1,\dots, T\right)\hfill \end{array} $$

(C.1.a)

$$ \begin{array}{ll}{y}_{ok}^t={Y}_k^t{\lambda}_k^t+{s}_{ok}^{t+}\hfill & \left(k=1,\dots, K;t=1,\dots, T\right)\hfill \end{array} $$

(C.1.b)

$$ \begin{array}{ll}{z}_{o\left(k,h\right)}^t={Z}_{\left(k,h\right)}^t{\lambda}_k^t+{s}_{o\left(k,h\right)}^{t+}\hfill & \left(\forall \left(k,h\right)\kern0.5em input;\forall t\right)\hfill \end{array} $$

(C.1.c)

$$ \begin{array}{ll}{z}_{ok}^{\left(t,t+1\right)}={Z}_k^{\left(t,t+1\right)}{\lambda}_k^t+{s}_{ok}^{\left(t,t+1\right)}\hfill & \left(\forall k;t=1,\dots, T-1\right)\hfill \end{array} $$

(C.1.d)

$$ \begin{array}{ll}{\displaystyle {\sum}_{n=1}^N{\lambda}_{nk}^t=1\left(\forall k,t\right)}\hfill & {\lambda}_{nk}^t\ge 0\left(\forall n,k,t\right)\hfill \end{array} $$

(C.1.e)

$$ \begin{array}{ll}{Z}_{\left(k,h\right)}^t{\lambda}_h^t={Z}_{\left(k,h\right)}^t{\lambda}_k^t\hfill & \left(\forall \left(k,h\right)\kern0.5em input;\forall t\right)\hfill \end{array} $$

(C.1.f)

$$ \begin{array}{ll}{Z}_k^{\left(t,t+1\right)}{\lambda}_k^t={Z}_k^{\left(t,t+1\right)}{\lambda}_k^{t+1}\hfill & \left(\forall k;t=1,\dots, T-1\right)\hfill \end{array} $$

(C.1.g)

$$ {\lambda}_k^t\ge 0,{s}_{ok}^{t-}\ge 0,{s}_{ok}^{t+}\ge 0,{s}_{o\left(k,h\right)}^t\ge 0\left(\forall k,h,t\right),{s}_{ok}^{\left(t,t+1\right)}\ge 0\left(\forall k;t=1,\dots, T-1\right) $$

(C.1.h)

where,

o :

the observed DMU, o = 1,…, N

N :

the number of DMU’s

k, h :

a division (K = number of divisions)

m _k :

number of inputs for division k

r _k :

number of outputs for division k

i _k :

number of input links for division k

b _k :

number of bad carry overs for division k

s ^t − _ok :

input slack for division k, time t

s ^t + _ok :

output slack for division k, time t

s ^t _o(k,h) :

slack for intermediate product link between division k and division h, time t

s ^{(t,t + 1)} _ok :

slack for bad carry over for division k from time t to time (t + 1)

λ ^t _k :

intensity vector for division k, time t

X ^t _k :

is the input matrix for division k, time t where ${X}_k^t=\left({x}_1^t,\dots, {x}_K^t\right)\in {R}^{{\mathrm{m}}_k\times T}$

Y ^t _k :

is the output matrix for division k, time t where ${Y}_k^t=\left({y}_1^t,\dots, {y}_K^t\right)\in {R}^{r_k\times T}$

Z ^t_(k,h) :

intermediate product input link between division k and division h at time t

Z ^{(t,t + 1)} _k :

bad carry over for division k from time t to time (t + 1)

R ^t − _mk :

range of input m, division k, time t = max(x ^t − _mk ) − min(x ^t − _mk )

R ^t _l(k,h) :

range of l ^th, input link variable between division k and h, time t = max(z ^t _l(k,h) ) − min(z ^t _l(k,h) )

R ^{(t,t + 1)} _nk :

range of n ^th, bad carry over variable, division k, time t = max(z ^{(t,t + 1)} _nk ) − min(z ^{(t,t + 1)} _nk )

∑  ^K _k = 1 w _k  = 1, w _k ≥ 0(∀ k):

where w _k is the relative weight of division k determined exogenously

∑  ^T _t = 1 W _t  = 1, W _t  ≥ 0(∀ t):

where W _t is the relative weight of time t determined exogenously

$$ {\varGamma}_o^{t*}=1-{\displaystyle {\sum}_{k=1}^K\frac{w_k}{m_k+{i}_k+{b}_k}\left[{\displaystyle {\sum}_{m=1}^{m_k}\frac{s_{mok}^{t-}}{R_{mk}^{t-}}+{\displaystyle {\sum}_{l=1}^{i_k}\frac{s_{lo\left(k,h\right)}^t}{R_{l\left(k,h\right)}^t}+{\displaystyle {\sum}_{n=1}^{b_k}\frac{s_{nok}^{\left(t,t+1\right)}}{R_{nk}^{\left(t,t+1\right)}}}}}\right]} $$

(C.2)