Appendix – KCE report 270 Clustering pathology groups on hospital stay similarity
1 Prospective payment items
1.1 Clinical biology
For clinical biology, independent of actually used clinical biology, two sets of RIZIV – INAMI nomenclature codes can be declared:
- fee per service, theoretically representing 25% of the reimbursement
- lumps sums per admission and per day or per prescription, theoretically representing 75% of the reimbursement
Of the Royal Decree “koninklijk besluit van 14 september 1984 tot vaststelling van de nomenclatuur van de geneeskundige verstrekkingen inzake de verplichte verzekering voor geneeskundige verzorging en uitkeringen”, the following are included in the prospective payment:
- article 3, §1, A, II ; §1, B ; §1, C I;
- article 18, §2, B, e) ;
- article 24, §1.
The lump sum nomenclature numbers are:
1.2 Medical imaging
For medical imaging, independent of actually used medical imaging, three sets of RIZIV – INAMI nomenclature codes can be declared:
- fee per service, theoretically representing 73.83% of the reimbursement
- lumps sums per admission and per day or per prescription
- consultancy fee lump sum
Of the Royal Decree “koninklijk besluit van 14 september 1984 tot vaststelling van de nomenclatuur van de geneeskundige verstrekkingen inzake de verplichte verzekering voor geneeskundige verzorging en uitkeringen”, the following are included (defined in “Koninklijk besluit van 10 maart 2009 tot uitvoering van artikel 69, § 1, van de wet betreffende de verplichte verzekering voor geneeskundige verzorging en uitkeringen, gecoördineerd op 14 juli 1994, betreffende de regels voor de berekening van de forfaitaire honoraria inzake medische beeldvorming betaald per opneming voor gehospitaliseerde patiënten”), the following are included in the prospective payment:
- article 17 (except consultancy fee lump sum)
- article 17ter (except consultancy fee lump sum)
The lump sum nomenclature numbers are:
1.3 Pharmaceutical products
For pharmaceutical products for inpatient stays, independent of actually used pharmaceutical products, two sets of nomenclature are declared:
- fee per service, theoretically representing 25% of the reimbursement
- lumps sums per admission, theoretically representing 75% of the reimbursement
Based on the royal decree “koninklijk besluit van 16 mei 2006 tot uitvoering van artikel 37, § 3, van de wet betreffende de verplichte verzekering voor geneeskundige verzorging en uitkeringen, gecoördineerd op 14 juli 1994, met betrekking tot de vaststelling van het aan de ziekenhuizen toegekend forfait voor vergoedbare farmaceutische specialiteiten”, the following lump sums were excluded:
2 Extrapolation effect
2.1 Inpatient
The effect of extrapolating clinical biology, medical imaging, and pharmaceutical products is shown in the figure below. The ‘base’ columns correspond to the data as actually registered in AZV – SHA and ADH – HJA. The ‘extrapolated’ columns correspond to the extrapolation of the fee for service part and the removal of the lump sum parts (the per diem lumps are included for reference, but are excluded in further analysis: they are approximated by the length of stay).
The table below shows the proportion of clinical biology, medical imaging and pharmaceutical products to the total reimbursement for base and extrapolation.
2.2 Day care
The effect of extrapolating clinical biology and medical imaging in day-care stays is shown in the figure below. The ‘base’ columns correspond to the data as actually registered in AZV – SHA and ADH – HJA. The ‘extrapolated’ columns correspond to the extrapolation of the fee for service part and the removal of the lump sum parts.
The table below shows the proportion of clinical biology and medical imaging to the total reimbursement for base and extrapolation.
3 Excluded APR-DRG-SOI
3.1 Particular APR-DRG
APR-DRG’s DDD, MMM, UUU, UAA and XXX are excluded because they are not relevant for inpatient and day-care hospital stay financing.
3.2 Different small size limits
Even using a three year period does not preclude certain APR-DRG-SOIs to have very few stays. The figures below show the impact of excluding APR-DRG-SOIs with different minimum number of stays excluded on number of stays and reimbursement.
3.3 A minimum of six stays
We excluded APR-DRG-SOI with less than 5 stays over a three year period (in any of the two periods used). The table below shows the impact on total reimbursement, number of stays, and length of stay (for inpatient stays).
3.4 Excluded small size APR-DRG-SOI
4 Partitioning around medoids
4.1 Algorithm
The Partitioning Around Medoids (PAM) algorithm partitions n objects into k groups (clusters). A medoid is an actual data point that is considered representative for a cluster. The input to the algorithm is a matrix of dissimilarities. A dissimilarity is a measure how ‘far’ individual objects are from each other. In this analysis, the Euclidean distance is used as dissimilarity measure:
\[ d_{ij} = \sqrt{\sum_{k = 1}^p (x_{ik} - x_{jk})^2} \]
with \(x_{ik}\) being the value of the \(k^{th}\) dimension of the p-dimensions of object i and \(x_{jk}\) being the value of the \(k_{th}\) dimension of the p-dimensions of object j .
The aim of the algorithm is to minimise the overall dissimilarity between the medoid and the other members of a cluster, and that for each cluster. The function that needs to be minimised (the objective function) is
\[ f(x) = \sum_{i = 1}^n \sum_{j = 1}^n d_{ij}z_{ij}\]
with
- \(\sum_{i = 1}^n z_{ij} = 1\) for each j where \(j \in \lbrace 1, \ldots, n\rbrace\)
- \(z_{ij} \leq y_{i}\) where i and j in \(\lbrace 1, \ldots, n\rbrace\)
- \(0 \leq z_{ij} \leq 1\)
- \(0 \leq y_i \leq 1\)
- \(\sum_{i = 1}^n y_i = k\) (number of clusters)
Multiplying the Euclidean distance \(d_{ij}\) by \(z_{ij}\) ensures that the function is minimised only for the elements in the particular cluster.
The algorithm works as follows:
- Build phase – search initial medoids:
- The objective function is defined as follows:
- Calculate the distances between a possible medoid and all other data points.
- Sum these distances.
- Do this for k possible medoids.
- The objective function is the sum of k distance sums.
- Find the k data points that decrease the objective function as much as possible.
- Assign all remaining data points to one of k clusters based on the closest medoid.
- The objective function is defined as follows:
- Swap phase – exchange data points and medoid per cluster:
- For each cluster, search the data point that when used as medoid, decreases the objective function from step 1a the most.
- If such a data point exists, go to step 1c, using the new data point as medoid of its cluster. If no data points can be found that decrease the objective function from step 1a, stop the algorithm.
4.2 Heuristic Identification of Noisy Variables (HINoV)
The HINoV algorithm works as follows:
Cluster the data using only the available variables. Do this for each of the j available variables.
Calculate the adjusted Rand index for the clusters formed from all distinct pairs of the available variables.
For each variable, sum all adjusted Rand indices in which the variable is used in step two. The size of the summed adjusted Rand indices indicates how well a variable contributes in determining the clusters.
Plot all summed adjusted Rand indices: a sharp drop in the curve indicates which variables can be considered ‘noisy’ because they do not contribute much in determining the clusters curve (see e.g. the green arrow in the example below). These can be excluded (e.g. variables D to F in the example below).
Perform the actual cluster analysis with the remaining variables.
5 Validation plots
5.1 Description
Aside from stripes plots, another validation graph combines two plots: a dissimilarity plot (Hahsler M, Hornik K. Dissimilarity Plots: A Visual Exploration Tool for Partitional Clustering. Journal of Computational and Graphical Statistics. 2011;20(2):335-54) and a Silhouette Width plot (Rousseeuw PJ. Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. Journal of Computational and Applied Mathematics. 1987;20:53-65).
A dissimilarity plot is constructed using a complex algorithm not further explained here (see previous reference of Hahsler 2011 for a detailed explanation). It can be combined with a Silhouette Width plot. The silhouette width for a data point corresponds to how close the data point is on average to the other data points in the same cluster compared to on average all other data points in other clusters . The figures below show examples of dissimilarity and silhouette width plots.
5.2 Results
The figures below strongly suggest that the clusters are homogeneous as the lower part of the square has differences in luminosity between the clusters. There is also no evidence that more clusters than three are needed: the pattern of the middle squares in the figures below follow the pattern shown in Figure 5 (
) and not that of possible subclusters (
). But similar to the stripes plots, the clusters are not well separated: some of the APR-DRG-SOIs from other clusters are equally darkly coloured as the own APR-DRG-SOI. This effect is stronger for inpatient stays than for day-care stays. For example, for the inpatient stays in the 2009-2011 period, in the first ‘row’ of the plot showing the medium cluster, the ‘columns’ of low and high show dark coloured points while for well separated clusters, these points should be coloured lighter indicating a larger distance from the medium cluster. The ‘striped’ nature of the silhouette plots point to the same lack of separation.
6 Cluster members
6.1 Inpatient stays
6.1.1 2009 – 2011
6.1.2 2010 – 2012
6.2 Day-care stays
6.2.1 2009 – 2011
6.2.2 2010 – 2012
7 Cross-border APR-DRG-SOIs
The tables below show the APR-DRG-SOI with negative silhouette widths: these are considered as ‘wrongly’ clustered as they are on average closer to another clusters members than to the members of their own.
7.1 Inpatient stays
7.1.1 2009-2011
7.1.2 2010-2012
7.2 Day-care stays
7.2.1 2009-2011
7.2.2 2010-2012
8 Cluster analysis day-care stays
8.1 Description of the three clusters
8.2 Cluster members
8.2.1 2009 – 2011
8.2.2 2010 – 2012
8.3 Cluster distribution
8.4 Cluster correspondence between 2009-2011 and 2012-2012
8.5 Validation of the three clusters
8.6 Cross-border cases
The two tables below show the APR-DRG-SOI with negative silhouette widths: these are considered as ‘wrongly’ clustered as they are on average closer to another clusters members than to the members of their own.
8.6.1 2009-2011
8.6.2 2010-2012
9 APR-DRG-SOI with low variability pattern
The tables below show an initial assessement per APR-DRG-SOI in the low variability cluster for the pattern of low within and between hospital variability (as shown in the top left panel of the picture below). In an number of cases, the pattern is present for the large majority of the hospitals, except for some hospitals with a low number of stays. This is noted in the remarks columnn.
Also, in APR-DRG-SOIs with a low number of stays, these stays are sometimes concentrated in one or a few hospitals. In that case, no pattern assessment was done and this is noted in the remarks column.
The graphs used for this initial assessment are available in the supplement to this report.
9.1 Inpatient stays 2010 – 2012
9.2 Day-care stays 2010 – 2012
10 Comparison with reference amounts APR-DRG
10.1 Inpatient 2010 – 2012
10.2 Day-care 2010 – 2012
11 Colophon
| Title: | Clustering pathology groups on hospital stay similarity – Appendix |
| Authors: | Stephan Devriese (KCE), Carine Van de Voorde (KCE) |
| Project coordinator: | Nathalie Swartenbroekx (KCE) |
| Reviewers: | Claire Beguin (KCE), Carl Devos (KCE), Sophie Gerkens (KCE), Raf Mertens (KCE), Koen Van den Heede (KCE) |
| External experts and stakeholders: | Benoit Collin (INAMI – RIZIV), Michael Daubie (INAMI – RIZIV), Christiaan Decoster (FOD Volksgezondheid – SPF Santé Publique), Jean Legrand (INAMI – RIZIV), Ingrid Mertens (FOD Volksgezondheid – SPF Santé Publique), Suzanne Van Malderen (FOD Volksgezondheid – SPF Santé Publique) |
| External assessors: | Olivier Thas (Universiteit Gent) |
| External validators: | Guy Durant (UCL), Niel Hens (UHasselt), Wilm Quentin (Technische Universität Berlin, Germany) |
| Acknowledgements: | Stefaan Van de Sande (KCE), Karin Rondia (KCE) |
| Other reported interests: | All consulted experts and stakeholders were selected because of their involvement in the topic of this report. Therefore, by definition, each of them might have a certain degree of conflict of interest to the main topic of this report. |
| Layout: | Ine Verhulst |
| Disclaimer: |
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| Publication date: | 22 June 2016 |
| Domain: | Health Services Research (HSR) |
| MeSH: | Hospital; Health Care Reform; Cluster Analysis; Data Interpretation, Statistical |
| NLM Classification: | WX 157.8 |
| Language: | English |
| Format: | html |
| Legal depot: | D/2016/10.273/63 |
| ISSN: | 2466-6459 |
| Copyright: | KCE reports are published under a “by/nc/nd” Creative Commons Licence http://kce.fgov.be/content/about-copyrights-for-kce-reports. |
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| How to refer to this document? | Devriese S, Van de Voorde C. Clustering pathology groups on hospital stay similarity – Appendix. Health Services Research (HSR) Brussels: Belgian Health Care Knowledge Centre (KCE). 2016. KCE Reports 270S1. D/2016/10.273/63. |
| This document is available on the website of the Belgian Health Care Knowledge Centre. |