Cluster Analysis

Concept

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Theoretically, Cluster Analysis is an unsupervised multivariate method used to partition observations into groups whose members are more similar to one another than to observations in other groups. It is founded on measures of similarity, dissimilarity and within-group homogeneity rather than a predefined outcome variable. The concept exists to identify latent structure, population segments or patterns within multidimensional data when group membership is not known in advance.

Mathematically, cluster analysis is represented through an optimisation rule, distance measure or probabilistic model that assigns observations to clusters. Different clustering frameworks use different mathematical representations. Partitioning methods such as k-means minimise within-cluster variation, hierarchical methods iteratively combine or separate clusters according to linkage criteria, and model-based methods estimate membership from finite mixture distributions.

In practice, variables are selected, scaled and assessed for suitability before a clustering algorithm is applied. The number and stability of clusters may be evaluated using measures such as the silhouette coefficient, within-cluster sum of squares, gap statistic or information criteria. In health economics, cluster analysis is used to identify patient subgroups, classify healthcare providers, segment patterns of service use and distinguish groups with different costs, outcomes or treatment pathways.

Purpose

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Used to identify internally homogeneous and externally distinct groups within multivariate health data, supporting population segmentation, resource-use analysis, patient profiling and the development of subgroup-specific economic models.

Mathematical Formulae

Primary Formula

There is no universally recognised canonical mathematical formula.

Supporting Formulae

K-means objective function:

J = ????? ????C? ?x? ? ???�

Cluster centroid:

?? = (1 / n?) ? ????C? x?

Euclidean distance:

d(x?, x?) = �[?????(x?? ? x??)�]

Silhouette coefficient for observation i:

s(i) = [b(i) ? a(i)] / max[a(i), b(i)]

where a(i) is the mean distance between observation i and other observations in its assigned cluster, and b(i) is the minimum mean distance between observation i and observations in another cluster.

Total within-cluster sum of squares:

WCSS = ????? ????C? ?x? ? ???�

Related Mathematical Methods

• K-Means Clustering
• Hierarchical Clustering
• Agglomerative Clustering
• Ward?s Method
• Finite Mixture Models
• Latent Class Analysis
• Gaussian Mixture Models
• Principal Component Analysis
• Silhouette Analysis
• Gap Statistic

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Example

A health economist analyses annual healthcare use among 600 patients using four standardised variables: inpatient admissions, outpatient visits, emergency department attendances and total healthcare cost. A k-means analysis with k = 3 identifies the following clusters:

• Cluster 1: low service use and mean annual cost of �1,200
• Cluster 2: high outpatient use and mean annual cost of �4,800
• Cluster 3: frequent emergency and inpatient use with mean annual cost of �15,600

For a patient with standardised values x = (1.2, 0.4, 1.5, 1.3) and a candidate cluster centroid ? = (1.0, 0.5, 1.4, 1.1), the Euclidean distance is:

d(x, ?) = �[(1.2 ? 1.0)� + (0.4 ? 0.5)� + (1.5 ? 1.4)� + (1.3 ? 1.1)�]

d(x, ?) = �(0.04 + 0.01 + 0.01 + 0.04)

d(x, ?) = �0.10 = 0.316

The patient is assigned to the cluster with the smallest distance between the patient?s standardised profile and the corresponding cluster centroid. The resulting segments may be used to estimate subgroup-specific costs, outcomes and intervention effects.

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Excel Implementation

Function	Example Formula	Health Economics Application
STANDARDIZE	=STANDARDIZE(B2,AVERAGE(B$2:B$601),STDEV.S(B$2:B$601))	Standardises healthcare utilisation and cost variables before clustering.
SQRT	=SQRT(SUMXMY2(B2:E2,$H$2:$K$2))	Calculates the Euclidean distance between a patient profile and a cluster centroid.
SUMXMY2	=SUMXMY2(B2:E2,$H$2:$K$2)	Calculates the squared distance used in k-means clustering.
MIN	=MIN(L2:N2)	Identifies the smallest distance across candidate clusters.
MATCH	=MATCH(MIN(L2:N2),L2:N2,0)	Assigns an observation to the nearest cluster.
AVERAGEIF	=AVERAGEIF($O$2:$O$601,H$1,B$2:B$601)	Recalculates a cluster centroid for a selected variable.
SUMPRODUCT	=SUMPRODUCT((O2:O601=H1)*(L2:L601^2))	Calculates the within-cluster sum of squared distances.

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VBA (Optional)

Automate iterative k-means clustering by repeatedly assigning observations to the nearest centroid and recalculating cluster centroids until assignments stabilise.

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Sources

• Everitt BS, Landau S, Leese M, Stahl D. Cluster Analysis.
• Kaufman L, Rousseeuw PJ. Finding Groups in Data: An Introduction to Cluster Analysis.
• MacQueen J. Some Methods for Classification and Analysis of Multivariate Observations.
• Ward JH. Hierarchical Grouping to Optimise an Objective Function.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
• NICE. Health Technology Evaluation Manual.