Signature
E_h = sum_(g=1)^G [n_hg * L_ref_g]; R_h = O_h / E_h
| Inputs | Definition | Unit |
|---|---|---|
n_hg | Number of discharges of hospital h in case-mix group g, one value per group | count |
L_ref_g | Reference ALOS of group g, from a stated reference population on the same stay definition | days |
O_h | Bed-days actually used by the discharges of hospital h in the groups included | bed-days |
E_h | Bed-days hospital h would use if each group stayed for the reference ALOS | bed-days |
|---|---|---|
R_h | Observed bed-days divided by expected bed-days | ratio, 1 meaning stays as expected |
GNumber of case-mix groups, equal to the length of the n_hg and L_ref_g lists (count)
Function
Average length of stay calculation and bed-day costing function
Maps the bed-days and discharges of a period to the average length of stay (ALOS), and relates length of stay to bed capacity, payment trim points, a case-mix adjusted comparison and the cost of an admission or of a change in stay. The records follow the notation of the Average Length of Stay article, where L_bar is the average stay in days and N the number of discharges.
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Implementations
Excel
Case-mix adjusted stay ratio with SUMPRODUCT
With the hospital's discharges per group in a range named GroupDischarges and the reference ALOS in a range of the same size named RefALOS, Excel returns the ratio of observed to expected bed-days.
=ObservedBedDays/SUMPRODUCT(GroupDischarges,RefALOS)
Assumptions
Grouping captures differences in expected stay
Patients in the same group are expected to stay equally long wherever treated, so the remaining difference reflects the hospital. The adjustment is only as good as the grouping system.
Same scope for observed and expected bed-days
O_h and E_h cover the same groups and exclusions. The Australian relative stay index, for example, excludes among others patients who died or were transferred within 2 days of admission and stays over 120 days.
Worked examples
Case-mix adjusted stay ratio for illustrative Hospital A
Hospital A has 300 hip replacement and 100 stroke discharges, with reference ALOS of 5 and 15 days. Expected bed-days are 3,000 and observed are 3,050, a ratio of about 1.017, stays about 1.7% longer than expected.
n_hg = [300, 100]; L_ref_g = [5, 15]; G = 2; O_h = 3050; E_h = 3000; R_h = 1.0167
Case-mix adjusted stay ratio for illustrative Hospital B
Hospital B has 100 hip replacement and 300 stroke discharges. Expected bed-days are 5,000 and observed are 4,800, a ratio of 0.96, stays 4% shorter than expected.
n_hg = [100, 300]; L_ref_g = [5, 15]; G = 2; O_h = 4800; E_h = 5000; R_h = 0.96
Common errors
Ranking hospitals on crude average length of stay
On crude ALOS, Hospital B (12.0 days) looks far less efficient than Hospital A (7.6 days). After adjustment B's ratio of 0.96 is below A's 1.017, so the ranking reverses: B's higher crude figure reflects its larger share of stroke patients.
Sources
Australian relative stay index definition
Productivity Commission. Report on Government Services 2026, Section 12: Public hospitals, key terms. Canberra: Productivity Commission; 2026. Relative stay index: actual patient days for acute care separations in selected AR-DRGs divided by expected patient days adjusted for casemix, with the stated exclusions.
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