Geometric Mean LOS

Concept

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Theoretically, Geometric Mean Length of Stay (Geometric Mean LOS) is a statistical measure that summarises the central tendency of hospital length of stay using the geometric mean rather than the arithmetic mean. It is particularly appropriate for positively skewed length-of-stay distributions, where a small number of patients with exceptionally long admissions can substantially inflate the arithmetic mean. The measure is founded on the theory of geometric averages and log-transformed data and is widely used in hospital reimbursement systems and health services research.

Mathematically, the Geometric Mean LOS is calculated as the nth root of the product of all individual lengths of stay or, equivalently, as the exponential of the arithmetic mean of the natural logarithms of the observed lengths of stay. This approach reduces the influence of extreme values while preserving the multiplicative characteristics of skewed healthcare utilisation data.

In practice, Geometric Mean LOS is used in Diagnosis-Related Group (DRG) payment systems, hospital benchmarking and health economic analyses. It provides a stable estimate of expected resource utilisation for patient groups, supports reimbursement calculations and facilitates comparisons of hospital efficiency across providers and healthcare systems.

Purpose

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Used to summarise typical hospital length of stay for skewed data, support DRG reimbursement, benchmark healthcare providers and inform health economic analyses of inpatient resource utilisation.

Mathematical Formulae

Primary Formula

Geometric Mean LOS = (x? ? x? ? ? ? x?)^(1/n)

Equivalent logarithmic form

Geometric Mean LOS = exp[(�ln(x?)) � n]

where:

• x? = individual length of stay
• n = number of observations

Supporting Formulae

Arithmetic Mean LOS

Mean LOS = �x? � n

Length of Stay

LOS = Discharge Date ? Admission Date

Related Mathematical Methods

• Geometric mean
• Logarithmic transformation
• Arithmetic mean
• Median
• Length of stay analysis
• Diagnosis-Related Group analysis

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Example

Five patients have hospital stays of:

2, 3, 4, 5 and 20 days.

Geometric Mean LOS

= (2 ? 3 ? 4 ? 5 ? 20)^(1/5)

= 1,200^(1/5)

� 4.13 days

Although the arithmetic mean is 6.8 days, the Geometric Mean LOS of 4.13 days better reflects the typical stay because it is less influenced by the single prolonged admission.

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Excel Implementation

Function	Example Formula	Health Economics Application
GEOMEAN	=GEOMEAN(A2:A101)	Calculate the Geometric Mean LOS.
AVERAGE	=AVERAGE(A2:A101)	Compare arithmetic and geometric mean LOS.
MEDIAN	=MEDIAN(A2:A101)	Compare geometric mean with the median LOS.
LN	=LN(A2)	Calculate logarithms for manual geometric mean calculations.
EXP	=EXP(AVERAGE(B2:B101))	Calculate the geometric mean from log-transformed LOS values.

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VBA (Optional)

Automate calculation of Geometric Mean LOS by DRG, hospital, specialty and reporting period while generating reimbursement reports, benchmarking dashboards and healthcare utilisation analyses.

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Sources

• Centers for Medicare & Medicaid Services. MS-DRG Definitions Manual.
• Fetter RB, Shin Y, Freeman JL, Averill RF, Thompson JD. Case Mix Definition by Diagnosis-Related Groups.
• OECD. Health at a Glance.
• Hosmer DW, Lemeshow S, Sturdivant RX. Applied Logistic Regression.
• Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.