An Information-theoretical Approach to Classify Hospitals with

Respect to Their Diagnostic Diversity using Shannon’s Entropy

Thomas Ostermann

and Reinhard Schuster

Institute of Integrative Medicine, Witten/Herdecke University, 58313 Herdecke, Germany

Institute of Mathematics, University of Lübeck, 23562 Lübeck, Germany

Keywords: Entropy, Diagnostic Diversity, Hospital Comparison, Classification.

Abstract: In Germany hospital comparisons are part of health status reporting. This article presents the application of

Shannon’s entropy measure for hospital comparisons using reported diagnostic data. We used Shannon’s

entropy to measure the diagnostic diversity of a hospital department by means of reported ICD–9–codes.

Entropy values were compared both with respect to the hospital status (i.e. primary, secondary, tertiary or

specialized hospital) and specialisations (e.g. surgery, gynaecology). There were relevant differences in

entropy values between the different types of hospitals. Primary hospitals differed from specialized

hospitals (0.535 ± 0.09 vs. 0.504 ± 0.07). Furthermore, specialized departments like obstetrics or

ophthalmology did generate lower entropy values than area-spanning departments like paediatrics or general

internal medicine, having significantly higher values. In conclusion, we showed how entropy can be used as

a measure for classifying hospitals. Besides of hospital comparisons, this approach can be implemented in

all fields of health services research for measuring variability in nominal or ordinal data. The use of entropy

as a measure for health services research and classification algorithms should be encouraged to learn more

about this measure, which unreasonably has fallen into oblivion in health services research.

1 INTRODUCTION

With emerging costs in the health care sector,

hospitals more and more compete for limited

financial ressources. Hence, not only the comparison

of hospitals with respect to their outcome, but also

with respect to structural qualities plays an important

role for health care providers, physicians and of

course for the the patients (Betzler and Haun, 1998;

Aiken et al., 1998). In Germany however, hospital

comparisons are part of health status reporting and

thus have been put on a legal basis by the Federal

Gouvernment. Despite this fact, some

methodological problems are still unsolved

(Wegscheider, 2004; Schulz et al., 2004).

According to Frick et al. (2003) one of the main

problems is the handling of differences observed in

outcome and performance of hospitals, which are

mainly produced by two sources. On the one hand,

they emerge from the infrastructural properties of

the hospitals themselves; on the other hand they

emerge from the patients population treated in the

hospitals and their diagnostic diversity and severity.

One methodological approach of solving this

problem is to make a statistical adjustment for these

differences and then to perform a comparison on all

hospitals. Another approach is to compare only

those institutions, which have similar structural

premises and join them into groups by means of

cluster analysis.

This however requires the development of a

measure, which maps these characteristics onto a

numerical value. In the work of Gerste (1996) such a

measure was pragmatically calculated from the

relative differences of the ICD-9-codes as can be

seen in Figure 1.

Figure 1: Hospital group-profiles “Surgery” adapted from

Gerste (1996).

325

Ostermann T. and Schuster R..

An Information-theoretical Approach to Classify Hospitals with Respect to Their Diagnostic Diversity using Shannon’s Entropy .

DOI: 10.5220/0005197103250329

In Proceedings of the International Conference on Health Informatics (HEALTHINF-2015), pages 325-329

ISBN: 978-989-758-068-0

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

Most of these approaches for grouping hospitals by

means of diagnostic data are empirically motivated

and e.g. ask for the number n of diagnoses needed to

account for, say 80% of the patient volume. Only a

few systematic and intersubjetively valid approaches

have been proposed for this problem so far (Izsak,

1994).

One method based on the concept of Shannon’s

entropy was proposed quite early by Elayat et al.

(1978) to cluster hospitals in homogenous groups

and has been adapted by Farley (1989) for the

analysis of case-mix specialization and the

concentration of diagnoses in hospitals quite early.

Although information-theoretical approaches are

commonly accepted and applied as a measure for

diversity in other fields of science (Nayak, 1985;

Ricotta and Avena, 2003), it did not become an

accepted method in Health Services Research so far.

For the special case of diagnostic diversity, a

MEDLINE search only found one result dealing

with the relevance of diagnostic diversity and patient

volumes for quality and lenght of stay in pediatric

intensive care units (Ruttimann et al., 2000).

This article is based on a very short conceptual

paper from Erben (2000) and presents the

application of Shannon’s entropy measure for the

calcualtion of diagnostic diversity on a broarder

dataset from hospitals and discusses the results with

respect to other hospital performance measures.

2 MATERIAL AND METHODS

Shannon’s entropy is based on a system of mutually

exclusive and exhaustive events A

, A

,…, A

and a

set of probabilities p

:= p(A

), p

:= p(A

),…, p

p(A

). Then, the entropy is given by

(,, log

nkk

pp pp









(1)

where 0 log 0 = 0 is assumed. The largest value of E

is given for the equal distribution of the events A

with

nkp

,,2,1;



, which is easy to proof.

Thus, to standardize E on the interval [0, 1], it has to

be divided by

)log(

max

E 

In the following, we use the term entropy for this

kind of standardised entropy value. Expect from a

scaling factor  which depends on which logarithm

is used, there is only one such function E, which

quantifies the content of information in the above

defined way.

To illustrate our approach, we will give the

original example from Erben (2000): In his work he

analysed the L4-hospital diagnosis statistics and

thus, the events A

are given by aggregated three-

digit-ICD-9 codes (e.g. 820 = “Fracture of the neck

of the femur”).

Suppose now, we have two hospitals A and B.

Hospital A is highly specialized in orthopaedic

surgery. In this case, 75% of all cases are covered by

one two three-digit-ICD-9 codes and four three-

digit-ICD-9 codes include about 90% of all cases

treated in this hospital. In total, 71 three-digit-ICD-9

cover the complete spectrum of this hospital. This

leads to a value of E=0,368.

Hospital B is an orthopaedic clinic and

additionally serves as a university clinic. In this

case, 17 three-digit-ICD-9 codes are needed to cover

75% of all cases. In total 113 three-digit-ICD-9

codes are needed to cover the complete range of

cases. This leads to an entropy-value of E = 0,729.

With a low value for the specialized clinic A and a

value double that size for the university clinic B,

entropy thus might serve as a good indicator for

diagnostic diversity.

To see if the results of this example are

generalizable to a greater variety of hospital

departments, we analysed the complete spectrum of

hospital departments in Schleswig- Holstein. For

further analysis these departments are subdivided

with respect to their area (e.g. internal medicine,

surgery, gynaecology) but also with respect to the

hospital status they are affiliated with. This status is

defined as follows:

Primary hospitals include at least the fields of

Internal Medicine and Surgery, and according to

requirements may include the fields of Obstetrics

and/or Gynaecology, Otorhinolaryngology and

Ophthalmology and, in special cases Urology and

Orthopaedics.

Secondary hospitals: Additionally to the fields

covered by primary hospitals, secondary hospitals

include paediatrics, neurology and dental surgery.

Tertiary hospitals: The range of services of

tertiary hospitals significantly goes far beyond those

of secondary and primary hospitals. This includes

the provision of a highly differentiated range of

technical equipment e.g. medical devices like a

positron emission tomograph.

Specialized hospitals offer the best medical

therapy and care for a limited range of diagnoses

including the referral of complicated cases.

Just like in the original work of Erben (2000), the

statistics are based on the data set of the aggregated

three-digit ICD-9-codes from the L4-hospital

statistics of 1998. We excluded hospitals with

incomplete diagnostic data, which led to a sample of

HEALTHINF2015-InternationalConferenceonHealthInformatics

326

977 hospitals and hospital departments that were

included in our analysis.

Descriptive statistics and the calculation of the

entropy were performed with SPSS for Windows

Version 20.

3 RESULTS

Figure 2 shows the distribution of entropy values for

all 977 institutions. The distribution is quite

symmetric with a minimum entropy value of 0.12, a

maximum of 0.95 and a median of 0.516, which is

near the mean of 0.520.

Figure 2: Entropy-distribution of all hospitals and

departments (N=977).

Figures 3 and 4 show the subgroup analysis of

entropy with respect to the hospital classification

and the medical fields of the hospitals.

Primary Hospitals Secondary

Hospitals

Tertiary Hospitals Special Hospitals

0,00

0,20

0,40

0,60

0,80

1,00

Entropy

Figure 3: Boxplot of Entropy-values for all hospitals and

departments subdivided for hospital status.

With median-values from 0.50 to 0.52 and a similar

interquartile ranges (IQR), primary, secondary and

tertiary hospitals do not extremely differ in their

entropy-values. Nevertheless there is a difference

with respect to the range. As can also be seen from

the distribution of entropy parameters in the Table 1,

there is an evidence of outliers in the group of

primary to tertiary hospitals, which are marked in

the boxplot-figure with stars and circles. This effect

is not observed in the group of specialized hospitals.

Sonst.

GGY

IVM

URO

ZKH

CUC

CHI

IMD

KIH

NUR

ORT

PSY

GGF

HNO

AUG

GBH

Medical Field

0,00 0,20 0,40 0,60 0,80 1,00

Entropy

OTHER

Figure 4: Boxplot of Entropy-values for all hospitals and

departments subdivided into medical fields (CHI=

Surgery; IMD= Internal Medicine; HNO=

Otorhinolaryngology; GGF= Gynaecology and Obstetrics;

URO= Urology; PSY= Psychiatry; ORT= Orthopaedics;

IVM= Intensive Care; KIH= Paediatric Intensive Care;

NUR= Nuclear Medicine and Radiology; ZKH= Dental

Surgery; GGY= General Gynaecology; GBH= Obstetrics;

AUG= Ophthalmology; CUC= Traumatology).

Table 1: Statistical parameters for the distribution of

entropy subdivided by hospital classification.

Primary Secondary Tertiary Specialized

Mean  SD

Median  IQR

Minimum

Maximum

Range

0.535



0.09

0.506  0.06

0.33

0.79

0.46

0.510



0.07

0.505  0.06

0.29

0.95

0.66

0.505  0,07

0.510  0,06

0.12

0.95

0.83

0.504



0.07

0.503  0.12

0.40

0.69

0.29

Whilst the subgroup analysis of entropy by

means of hospital categories did not yield to

promising conclusions, the subdivision with respect

to medical fields does generate some interesting

findings. On first sight, highly specialized

departments like obstetrics (0.44) or ophthalmology

(0.46) generate lower entropy values than area-

spanning departments like Nuclear Medicine and

AnInformation-theoreticalApproachtoClassifyHospitalswithRespecttoTheirDiagnosticDiversityusingShannon's

Entropy

327

Radiology (0.52) or general gynaecology (0.56),

which have significantly higher values (p<0.05).

4 CONCLUSIONS

This article examines the application of Shannon’s

entropy measure to hospital diagnostic admission

data. Although Shannon's entropy has been widely

used as a measure for diversity in various scientific

fields, it has only sparsely been applied for

classification purposes in health services research.

Based on the frequencies of three-digit-ICD-9

codes at hospital admission, we showed the

possibilities of Shannon’s measure of entropy as a

possibility for analysing nominal scaled data of

health status reporting of hospitals. Based on the L4-

statistics, we exemplified how entropy can be used

for clustering hospitals by using their routine

diagnostic data.

Although the ICD-9 data from our example dates

back almost 10 years, our approach can easily be

adapted to hospital data based on ICD-10 or DRGs.

Especially in DRG-data a clearer differentiation of

specialized hospitals might be possible.

With increasingly limited financial resources in

the health care sector, hospitals as well as networks

of general practitioners are seeking for markers

which distinguish them from competitors in their

fields (Sabatino et al., 1992). The assessment of

diversity therefore might be one promising

approach, which especially in the life sciences is

already a highly important issue. From the level of

molecular biology i.e., the diversity of gene

expressions is actually discussed, whereas on the

level of evolutionary biology the diversity of species

in the animal and plant kingdom is of relevance. In

both situations entropy has been applied as a

measure to assess the diversity or complexity (Pueyo

et al., 2007).

Using entropy as a diversity marker can basically

be implemented in all fields of health services

research, where categorical data emerges. One actual

example is the application of entropy as a measure to

assess the diversity of medical devices in large

inventories of medical equipment (Brindle et al.,

2008). For diagnostic or therapeutic data, entropy

might be useful e.g. for measuring the variability of

diagnostic data.

One particular application might be the

comparison of suspected diagnosis at referral of

hospitalized patients versus the proven admission

diagnosis at intake. Another example is the use of

entropy for the analysis of clinical pathways.

Especially in integrated care, the question arises,

whether entropy is created by transferring patients

from a primary hospital to a specialized clinic or

vice versa. This might lead to a sequential

calculation of entropy by dividing diagnostic data

with respect to the pathways patients were admitted.

Hence, one might wonder why outcomes

research has not used this measure e.g. for the

diversity classification in health outcomes.

Especially for the task of measuring variability in

nominal or ordinal data parameters common

parameters like standard deviation or the variation

coefficient are not applicable and thus, entropy can

be used for such purposes.

Although our examples are quite

comprehensible, one has to be aware that entropy is

just a marker for variety and does not measure a

difference in the distribution of categorical data. For

example, groups one and four in Fig. 1 of the

hospital group-profiles “Surgery” from Gerste

(1996) show a similar distribution, where group one

can be created by shifting group four to the right.

Although their profile is completely different, their

entropy, if it had been calculated, would be quite

similar. Hence, entropy only classifies with respect

to similar structures (e.g. centroids in special

diagnostic groups), but does not give a clue whether

these structures are similar to each other in respect

of contents.

Thus, according to Jost (2006) it is important to

distinguish between entropy and true diversities

when interpreting such indices as it is not obvious on

what basis these indices were computed.

Moreover a variety of entropy measures does

exist summarized in the generalized diversity indices

 proposed by Patil andTaillie (1982)

)(







with





















1)ln(

)(







ifp

As a special case (=1)  does also include

Shannon’s entropy measure.

A detailed mathematical analysis of Leinster and

Cobbold (2012) found that Shannon’s entropy might

be more sensitive to rare events, while others like

Simpson’s measure of diversity (=2) is not that

much influenced by such events. This property still

makes Shannon’s entropy to be one of the most

reliable diversity indices.

HEALTHINF2015-InternationalConferenceonHealthInformatics

328

Nevertheless different sensitivities to the occurence

of rare events have to be taken into account very

carefully, when deriving conclusions from entropy

measures (Ricotta and Seidl, 2006).

In our case of diagnostic diversity of hospitals

we therefore believe that Shannon’s entropy is a

proper choice.

In conclusion the use of entropy as a measure for

health services research and classification algorithms

based on entropy have to be encouraged to learn

more about this measure, which unreasonably has

fallen into oblivion in health services research.

REFERENCES

Aiken, L.H., Sloane, D.M., Sochalski. J., 1998. Hospital

organisation and outcomes. Quality in Health Care

7:222–226.

Betzler, M., Haun, P., 1998. Hospital Comparison – Status

quo and prospects. Chirurg 69:1300-1304.

Brindle, G.W., Gibson, C.J., 2008. Entropy as a measure

of diversity in an inventory of medical devices. Med

Eng Phys.;30(3):399-401.

Elayat, H.A., Murphy, B.B., Prabhakar, N.D., 1978.

Entropy in the hierarchical cluster analysis of

hospitals. Health Serv Res 13: 395-403.

Erben, C.M., 2000. The concept of entropy as a possibility

for gathering mass data for nominal scaled data in

health status reporting. Stud Health Technol Inform

2000; 77:118-9.

Farley, D.E., 1989. Case-Mix specialization and the

concentration of diagnoses in hospitals using

information theory. J Health Economics 8: 185-207.

Frick, U., Binder, H., Barta, W., Cording, C., 2003. Fair

Hospital Comparisons - Does the method contracted in

Germany enable unbiased results? Gesundheitswesen

65: 8-18.

Gerste, B., 1996. Bildung von Krankenhausgruppen auf

Fallmix-Basis. In Arnold M, Paffrath D (Eds.):

Krankenhaus-Report ’96.Stuttgart: Gustav-Fischer:

115-126.

Izsak, J., 1994. Applying the jackknife method to

significance tests on diagnsotic diversity. Methods Inf

Med 33: 214-9.

Jost, L., 2006. Entropy and diversity. Oikos, 113(2), 363-

375.

Leinster, T., Cobbold C.A., 2012. Measuring diversity: the

importance of species similarity. Ecology, 93(3), 477-

489.

Nayak, T.K., 1985. On Diversity Measures Based on

Entropy Functions. Commun. Statist.- Theory Meth.

14: 203-215.

Patil, G.P., Taillie, C., 1982. Diversity as a concept and

its measurement. J. Am. Stat. Assoc. 77, 548–567.

Pueyo, S., He, F., Zillio, T., 2007. The maximum entropy

formalism and the idiosyncratic theory of biodiversity.

Ecol Lett.10(11):1017-28.

Ricotta, C., 2002. Measuring taxonomic diversity with

parametric information functions. Community Ecology

3(1): 95-99.

Ricotta, C. Avena G.C., 2003. An information-theoretical

measure of taxonomic diversity. Acta Biotheor 51: 35-

41.

Ricotta, C., Szeidl, L., 2006. Towards a unifying approach

to diversity measures: bridging the gap between the

Shannon entropy and Rao's quadratic index.

Theoretical population biology 70(3): 237-243.

Ruttimann, U.E., Patel, K.M., Pollack, M.M., 2000.

Relevance of diagnostic diversity and patient volumes

for quality and leght of stay in pediatric intensive care

units. Pediatr Crit Care Med 1: 133-9.

Sabatino, F., Koska, M.T., Burke, M., Hudson, T., 1992.

Clinical quality initiatives: the search for meaningful-

and accurate-measures. Hospitals 66(5):26-32, 34, 36-

40.

Schulz, H., Barghaan, D., Watzke, B., Koch, U., Harfst,

T., 2004. Clinical Comparison as an instrument for

Quality Assurance in the treatment of patients with

mental disorders: The Significance of Risk

Adjustment. ZaeFQ 98: 663-672.

Wegscheider, K., 2004. Methodological requirements for

center comparisons (profiling) in health care. ZaeFQ

98: 647-654.

AnInformation-theoreticalApproachtoClassifyHospitalswithRespecttoTheirDiagnosticDiversityusingShannon's

Entropy

329