Identifying congestion levels, sources and determinants on intensive care units: the Portuguese case

Abstract

Healthcare systems are facing a resources scarcity so they must be efficiently managed. On the other hand, it is commonly accepted that the higher the consumed resources, the higher the hospital production, although this is not true in practice. Congestion on inputs is an economic concept dealing with such situation and it is defined as the decreasing of outputs due to some resources overuse. This scenario gets worse when inpatients’ high severity requires a strict and effective resources management, as happens in Intensive Care Units (ICU). The present paper employs a set of nonparametric models to evaluate congestion levels, sources and determinants in Portuguese Intensive Care Units. Nonparametric models based on Data Envelopment Analysis are employed to assess both radial and non-radial (in)efficiency levels and sources. The environment adjustment models and bootstrapping are used to correct possible bias, to remove the deterministic nature of nonparametric models and to get a statistical background on results. Considerable inefficiency and congestion levels were identified, as well as the congestion determinants, including the ICU specialty and complexity, the hospital differentiation degree and population demography. Both the costs associated with staff and the length of stay are the main sources of (weak) congestion in ICUs. ICUs management shall make some efforts towards resource allocation to prevent the congestion effect. Those efforts shall, in general, be focused on costs with staff and hospital days, although these congestion sources may vary across hospitals and ICU services, once several congestion determinants were identified.

Appendices

Appendix

1.1 Literature review

Please, see Table 5

Table 5 Literature review on congestion measurement and/or ICU performance evaluation

Service-Mix Index

The inefficiency-corrected SMI for the ICU service (DMU₀) defined by the pair $\left({x}_{i0},{y}_{r0}\right)\in {\mathbb{R}}_{+}^{5+1}$ can be computed as follows:

Use Eq. (19) to obtain the optima ${\left\{{\theta}_0^{*},{p}_1^{-*},\dots, {p}_5^{-*},{p}_1^{+*}\right\}}_{SDH}$ and project (x _i0, y _r0) in the SDH frontier using the transformation $\left({\overset{\vee }{x}}_{i0},{\overset{\vee }{y}}_{10}\right)=\left({\theta}_0^{*}\cdot {x}_{i0}-{p}_i^{-*},{y}_{10}+{p}_1^{+*}\right)\in {\mathbb{R}}_{+}^{5+1}$. This projection removes the technical inefficiency of units. To obtain Ω we follow the strategy proposed by subsection 3.3, considering all environment variables but Z₆ (by obvious reasons).

$$ {\left\{{\theta}_0^{*},{p}_1^{-*},\dots, {p}_5^{-*},{p}_1^{+*}\right\}}_{SDH}=\underset{\theta_0,{\lambda}_1,\dots, {\lambda}_n,{p}_1^{-},\dots, {p}_5^{-},{p}_1^{+}}{ \min}\left\{{\theta}_0-\varepsilon \left({p}_1^{+}+{\displaystyle \sum_{i=1}^5}{p}_i^{-}\right)|\begin{array}{c}\hfill {\displaystyle \sum_{\begin{array}{c}\hfill j=1\hfill \\ {}\hfill j\in {\Omega}^{\hbox{'}}\hfill \end{array}}^n}{\lambda}_j{x}_{ij}+{p}_i^{-}={\theta}_0{x}_{i0}\hfill \\ {}\hfill {\displaystyle \sum_{\begin{array}{c}\hfill j=1\hfill \\ {}\hfill j\in {\Omega}^{\hbox{'}}\hfill \end{array}}^n}{\lambda}_j{y}_{1j}-{p}_1^{+}={y}_{10}\hfill \\ {}\hfill \begin{array}{c}\hfill {\displaystyle \sum_{\begin{array}{c}\hfill j=1\hfill \\ {}\hfill j\in {\Omega}^{\hbox{'}}\hfill \end{array}}^n}{\lambda}_j=1\hfill \\ {}\hfill \begin{array}{c}\hfill {\lambda}_j,{p}_1^{+},{p}_i^{-}\ge 0\hfill \\ {}\hfill j\in {\Omega}^{\hbox{'}};i=1,\dots, 5\hfill \end{array}\hfill \end{array}\hfill \end{array}\right\} $$

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• 1) Re-run the step 1) for all DMUs and obtain the set $\nabla =\left\{\left({\displaystyle \sum_{i=1}^4}{\overset{\vee }{x}}_{ij},{\overset{\vee }{y}}_{1j}\right)=\left({\theta}_0^{*}\cdot {\displaystyle \sum_{i=1}^4}{x}_{ij}-{\displaystyle \sum_{i=1}^4}{p}_i^{-*},{\overset{\vee }{y}}_{1j}\right)\in {\mathbb{R}}_{+}^2,j=1,\dots, n\right\}$. ∇ only contains data from the single output and the first 4 inputs (monetary resources).

• 2) Compute the ratio for the j-th unit ${\boldsymbol{\xi}}_{\boldsymbol{j}}=\left({\displaystyle \sum_{\boldsymbol{i}=1}^4}{\overset{\vee }{\boldsymbol{x}}}_{\boldsymbol{i}\boldsymbol{j}}\right)/{\overset{\vee }{\boldsymbol{y}}}_{1\boldsymbol{j}},\boldsymbol{j}=1,\dots, \boldsymbol{n}$, which represents the efficient unitary cost of such DMU _j .

• 3) Compute $\mathbf{\mathcal{B}}={\displaystyle \prod_{\boldsymbol{j}=1}^{\boldsymbol{n}}}\left({{\boldsymbol{\xi}}_{\boldsymbol{j}}}^{1/\boldsymbol{n}}\right)$, which represents the unitary costs’ national average (baseline).

• 4) The SMI for the DMU₀ is then SMI₀ = ξ₀/$\boldsymbol{\mathcal{B}}$, where ${\boldsymbol{\xi}}_0=\left({\displaystyle \sum_{\boldsymbol{i}=1}^4}{\overset{\vee }{\boldsymbol{x}}}_{\boldsymbol{i}0}\right)/{\overset{\vee }{\boldsymbol{y}}}_{10}$

Gini’s Specialization Index

The GSI _k for the hospital k is computed as follows, [45, 46]:

• 1) Let ℒ be the number of Disease Related Groups (DRG);

• 2) Sort DRGs by discharges treated, in ascending order;

• 3) Let ${\mathbf{\mathcal{D}}}_{\boldsymbol{w}}^{\boldsymbol{k}}$ be the number of the w-th DRG group discharges;

• 4) Let ${\boldsymbol{q}}_{\boldsymbol{i}}^{\boldsymbol{k}},\boldsymbol{i}=1,\dots, \mathbf{\mathcal{L}}-1$, be the ratio of total discharges treated by the first i DRGs, i.e., ${\boldsymbol{q}}_{\boldsymbol{i}}^{\boldsymbol{k}}={\varSigma}_{\ell =1}^i{\mathbf{\mathcal{D}}}_{\ell}^{\boldsymbol{k}}/{\displaystyle \sum_{\boldsymbol{w}=1}^{\mathbf{\mathcal{L}}-1}}{\mathbf{\mathcal{D}}}_{\boldsymbol{w}}^{\boldsymbol{k}}$

• 5) Compute GSI _k  ∈ [0; 1] using Eq. (20).

$$ GS{I}_k=\left\{{\displaystyle \sum_{i=1}^{L-1}}\left(\frac{i}{L}-{q}_i^k\right)/{\displaystyle \sum_{i=1}^{L-1}}\left(\frac{i}{L}\right)\right\}=\left\{{\displaystyle \sum_{i=1}^{L-1}}\left(\frac{i\cdot {\displaystyle {\sum}_{w=1}^{L-1}}{D}_w^k-L\cdot {\displaystyle {\sum}_{l=1}^i}{D}_l^k}{L\cdot {\displaystyle {\sum}_{w=1}^{L-1}}{D}_w^k}\right)/{\displaystyle \sum_{i=1}^{L-1}}\left(\frac{i}{L}\right)\right\} $$

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Bootstrapping

Based on Simar and Wilson [47] and Daraio and Simar [38], the output-oriented bootstrap algorithm is as follows:

• 1) Compute the n output-oriented DEA efficiency scores, under the strong or the weak disposability assumption, Eqs. (6) and (7), respectively; for the sake of generality, let’s suppose we obtain the set of efficiency scores, Φ = {θ _j , j = 1,  … , n}, with a standard ${\sigma}_{\varPhi^{"}}$ deviation and an interquartile range ${r}_{\varPhi^{"}}$.

• 2) Reflect Φ and obtain the 2n-length set Φ '  = {2 − θ ₁, 2 − θ ₂,  ... , 2 − θ _n , θ ₁, θ ₂,  ... , θ _n }.

• 3) Consider only those p DMUs such that θ _b  > 1 , b = 1 ,  .  .  .  , p < n; from Φ^′, create the 2p-length set Φ^″ = {2 − θ ₁, 2 − θ ₂,  … , 2 − θ _p , θ ₁, θ ₂,  … , θ _p } ⊂ Φ^′;Φ ' ' has a standard deviation ${\sigma}_{\varPhi^{{\prime\prime} }}$ and an interquartile range r _Φ ' '.

• 4) Compute a bandwidth $d\approx \left(1.06\cdot {\sigma}_{\boldsymbol{\Phi}}\cdot \min \left\{{\sigma}_{{\boldsymbol{\Phi}}^{\mathbf{{\prime\prime}}}},\kern0.5em \frac{r_{{\boldsymbol{\Phi}}^{\mathbf{{\prime\prime}}}}}{1.34}\right\}{(2p)}^{4/5}\right)/\left(\boldsymbol{n}\cdot {\sigma}_{{\boldsymbol{\Phi}}^{\mathbf{{\prime\prime}}}}\right)$.

• 5) Randomly (with reposition) draw a n-length sample from Φ' (step 2)) and obtain the set ${\varPhi}^{\star }=\left\{\tilde{\theta_j^{\star }},\mathrm{j}=1,\dots, n\right\}$, with a standard deviation σ ⋆ and an arithmetic mean m ⋆.

• 6) Use a perturbation χ _j  = d ⋅ ζ _j , where ${\boldsymbol{\zeta}}_{\boldsymbol{j}}\sim \mathbf{\mathcal{N}}\left(\boldsymbol{\mu} =0,\boldsymbol{\sigma} =1\right)$, to obtain the set

  $$ {\varPhi}^{\star \star }=\left\{\tilde{\theta_j^{\star \star }}=\frac{\tilde{\theta_j^{\star }}+{\chi}_j-{m}^{\star }}{\sqrt{1+{\left(\frac{d}{\&^{\star }}\right)}^2}}+{m}^{\star };\mathrm{s}.t.{\chi}_j=d\cdot {\zeta}_j;{\zeta}_j\sim \mathbf{\mathcal{N}}\left(\mu =0,\sigma =1\right);\mathrm{j}=1,\dots, n;\tilde{\theta_j^{\star }}\in {\varPhi}^{\star}\right\} $$

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• 7) Reflect those n units from Φ ^⋆⋆, as follows:

  $$ {\theta}_j^{\star \star }=\left\{\begin{array}{cc}\hfill 2-\tilde{\theta_j^{\star \star }}\hfill & \hfill if\ \tilde{\theta_j^{\star \star }}<1\hfill \\ {}\hfill \tilde{\theta_j^{\star \star }}\hfill & \hfill otherwise\hfill \end{array}\right. $$

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• 8) Create the set ${\Im}^{\star \star }=\left\{\left({x}_{ij}^{\star },{y}_{rj}^{\star}\right)\in {\mathbb{R}}_{+}^{m+s}:\kern0.5em {x}_{ij}^{\star }={x}_{ij}\cap {y}_{rj}^{\star }={y}_{rj}\frac{\theta_j}{\theta_j^{\star \star }},\mathrm{j}=1,\dots, n\right\}$, and re-run Eqs. (6) and (7) to project units in the new frontier and to obtain the bootstrap-based efficiency scores, under the strong or the weak disposability assumption, resp.

• 9) Repeat steps 5)-8) B times, where B is large, say B ~ 1,000 iterations.

• 10) Let m _Bj and σ _Bj be the arithmetic mean and the standard deviation of those B bootstrap-based efficiency score for unit j. Bias is then bias _j  ≈ m _Bj  − θ _j and the bias-corrected DEA efficiency score is $\hat{\theta_j}={\theta}_j-bia{s}_j\approx 2\cdot {\theta}_j-{m}_{Bj}$. Still, this bias correction shall not be performed if |bias _j | ≤ σ _Bj /4.

Some additional graphics and tables

Please check Figs. 2, 3, 4 and 5, as well as Tables 6, 7 and 8.

Table 6 Pearson’s correlation coefficients for the non-categorical environment variables, Z₂ up to Z₁₀

Table 7 Degree of Scale Economies. Values are presented in the form “average ± std. deviation”

Fig. 2

Nadaraya-Watson regression of Gamma functions, Γ ₀(z ₇) and Γ _weak , 0(z ₇), against the Gini’s Specialization Index, z ₇. τ is the Kendall’s correlation coefz ₉ficient, * (resp. **) indicates significance (resp. Lack of significance) at 5 %

Fig. 3

Nadaraya-Watson regression of Gamma functions, Γ ₀(z ₈) and Γ _weak , 0(z ₈), against the Population Density, z ₈. τ is the Kendall’s correlation coefficient, * (resp. **) indicates significance (resp. Lack of significance) at 5 %

Fig. 4

Nadaraya-Watson regression of Gamma functions, Γ ₀(z ₉) and Γ _weak , 0(z ₉), against the Wealth Index, z ₉. τ is the Kendall’s correlation coefficient, * indicates significance at 5 %

Fig. 5

Nadaraya-Watson regression of Gamma functions, Γ ₀(z ₁₀) and Γ _weak , 0(z ₁₀), against the Aging Index, z ₁₀. τ is the Kendall’s correlation coefficient, * indicates significance at 5 %

Table 8 Congestion sources (10 % trimmed mean of Marginal Products for each input, given a single output (InpD))