Malmquist Index

Malmquist Index of Productivity Change in Health Care

The Malmquist index measures how much the productivity of a hospital, practice or health system changed between two periods, judged against the best practice observed among comparable units. Its usual form splits that change into the unit moving closer to or further from the best-practice frontier and the frontier itself moving. Because it is built from frontier distances rather than prices, it suits health services that have no market prices to weight outputs.

Productivity change measured against a moving frontier

A score from data envelopment analysis says how far a hospital sits from the frontier in one year, the level that an efficiency index reports. The Malmquist index compares two such positions in different years.

Thanassoulis, Shiraz and Maniadakis record that Malmquist's 1953 work went unnoticed until Caves, Christensen and Diewert reintroduced it to productivity measurement in 1982. Simar and Wilson explain that Färe, Grosskopf, Lindgren and Roos then merged Farrell's ideas on efficiency with the Caves approach, dropping the assumption of no technical inefficiency, in a 1992 study of 42 Swedish pharmacies. Thanassoulis and colleagues credit the same authors' 1989 working paper on Swedish hospitals with the first empirical use of the index without assuming technical efficiency. Färe, Grosskopf, Norris and Zhang applied the decomposition into technical change and efficiency change to 17 OECD countries in 1994, the version Thanassoulis and colleagues call the classical Malmquist index.

The building block is the distance function, which measures how far a unit lies from the frontier of a given period. In the output-oriented form used by Färe, Grosskopf, Norris and Zhang, it is the smallest factor by which outputs can be divided while staying feasible.

$$D^{s}(x, y) = \inf \left\{ \theta > 0 : (x, y/\theta) \in P^{s} \right\}$$

where $D^{s}(x, y)$ is the output distance function relative to the technology of period $s$, $x$ is the vector of inputs, $y$ is the vector of outputs, $\theta$ is the scaling factor applied to outputs, and $P^{s}$ is the set of all input and output combinations that are feasible in period $s$. Against its own period's frontier the value is at most 1, with 1 meaning the unit is technically efficient. Against another period's frontier, Simar and Wilson point out, it can be below, equal to or above 1.

The index and its two components

The index compares a unit's two positions, first against the earlier frontier and then against the later one. It takes the geometric mean of the two comparisons.

$$M = \left[ \frac{D^{t}(x^{t+1}, y^{t+1})}{D^{t}(x^{t}, y^{t})} \times \frac{D^{t+1}(x^{t+1}, y^{t+1})}{D^{t+1}(x^{t}, y^{t})} \right]^{1/2}$$

where $M$ is the output-oriented Malmquist productivity index for one unit, $t$ and $t+1$ are the two periods, $x^{t}$ and $y^{t}$ are the unit's inputs and outputs in period $t$, $x^{t+1}$ and $y^{t+1}$ are its inputs and outputs in period $t+1$, and each $D$ is the distance function defined above, with the superscript naming the period whose frontier is used.

The same expression can be rearranged into two factors whose product equals $M$.

$$EC = \frac{D^{t+1}(x^{t+1}, y^{t+1})}{D^{t}(x^{t}, y^{t})} \qquad TC = \left[ \frac{D^{t}(x^{t+1}, y^{t+1})}{D^{t+1}(x^{t+1}, y^{t+1})} \times \frac{D^{t}(x^{t}, y^{t})}{D^{t+1}(x^{t}, y^{t})} \right]^{1/2}$$

where $EC$ is efficiency change, the ratio of the unit's own-period efficiency in the later period to that in the earlier one, often called catching up, and $TC$ is technical change, the geometric mean of two measures of how far the frontier moved at the unit's own input and output mix, often called frontier shift. The other symbols are as above, and $M = EC \times TC$.

In this output-oriented form a value above 1 means improvement for the index and each component. The input-oriented version defined by Simar and Wilson runs the other way, with values below 1 showing improvement, so for the pharmacy data they reported reciprocals, as Färe and colleagues had, to make values above 1 denote progress.

Estimating the distance functions from hospital data

Each unit needs four distance functions for each pair of periods. Two are ordinary DEA scores against its own period. The other two compare its data from one year with the frontier built from the other year. The same units must be observed in both periods, with inputs and outputs measured consistently.

The choices that drive a DEA score drive the index too. Outputs need adjusting for case mix, since a shift towards sicker patients can look like falling productivity. Maniadakis, Hollingsworth and Thanassoulis computed indices of both productivity and quality change for NHS acute hospitals and found similar productivity changes when service quality was included, but of smaller magnitude.

Returns to scale and the decomposition debate

Simar and Wilson show that the index measures productivity change only if distances are measured against a constant returns to scale frontier, the conical hull of the production set. With a variable returns frontier they give a one-input example in which a unit whose ratio of output to input does not change is wrongly shown as improving.

Efficiency change can be split further into pure efficiency change and scale efficiency change, an enhanced decomposition that Färe and colleagues described. Splitting technical change under variable returns is disputed. Ray and Desli called the Färe decomposition internally inconsistent and proposed another, and Simar and Wilson argued that the Ray and Desli scale component confounds movement of units with changes in the shape of the technology. Health studies are therefore safer computing the headline index against a constant returns frontier and treating finer splits as diagnostics with stated assumptions.

Worked example with three hospitals over two years

All figures in this example are illustrative. Three acute hospitals each use one input, whole-time equivalent (WTE) clinical staff, to produce case-mix-adjusted inpatient episodes. Staffing is unchanged between the years and the frontier has constant returns to scale, so each distance function is the hospital's episodes per WTE divided by the best value on the relevant frontier.

1. Locate the frontiers. Hospital B is best in both years, so the frontier is 12.0 episodes per WTE in year 1 and 13.5 in year 2.

2. Compute hospital C's four distance functions. Against its own years, C scores 8.0 / 12.0 = 0.6667 in year 1 and 10.0 / 13.5 = 0.7407 in year 2. Its year 2 data score 10.0 / 12.0 = 0.8333 on the year 1 frontier, and its year 1 data score 8.0 / 13.5 = 0.5926 on the year 2 frontier.

3. Efficiency change. The catch-up term compares the two own-year scores.

$$EC_{C} = \frac{0.7407}{0.6667} = 1.111$$

where $EC_{C}$ is hospital C's efficiency change, 0.7407 is its year 2 score against the year 2 frontier and 0.6667 is its year 1 score against the year 1 frontier.

4. Technical change. The frontier shift term uses all four scores.

$$TC_{C} = \left[ \frac{0.8333}{0.7407} \times \frac{0.6667}{0.5926} \right]^{1/2} = \left[ 1.125 \times 1.125 \right]^{1/2} = 1.125$$

where $TC_{C}$ is hospital C's technical change, 0.8333 and 0.5926 are its cross-period scores, and 0.7407 and 0.6667 are its own-period scores.

5. Productivity change. The index is the product of the two components, 1.111 × 1.125 = 1.250. As a check, with one input and one output under constant returns the index equals the ratio of the two productivity levels, 10.0 / 8.0 = 1.25.

6. Hospitals A and B. Hospital A's own-year scores are 10.0 / 12.0 = 0.8333 and 11.0 / 13.5 = 0.8148, so its efficiency change is 0.8148 / 0.8333 = 0.978. Its technical change is also 1.125, and its index is 0.978 × 1.125 = 1.100, matching 11.0 / 10.0. Hospital B is on the frontier in both years, so its efficiency change is 1, and its year 2 data score 13.5 / 12.0 = 1.125 against the year 1 frontier, a cross-period value above 1. Its index is 1.125.

7. Interpretation. Hospital C's productivity rose by 25%, made up multiplicatively of a 12.5% frontier shift and an 11.1% catch-up. Hospital A's productivity rose by 10%, yet it fell behind best practice because the frontier moved further. All three share the same frontier shift only because a one-input, one-output constant returns frontier is a straight line through the origin. With several inputs and outputs, technical change is assessed at each hospital's own mix and differs between them.

Reading the index in NHS and health system studies

Studying the first five years of the early 1990s NHS internal market reforms, Maniadakis, Hollingsworth and Thanassoulis found a slowdown in the first year, then progress and a net gain overall. Technical change dominated, since hospitals were already largely relatively efficient, and small efficiency regress in later years did not bear out the claim that the reforms would raise hospital efficiency. Quality followed different trends, which the authors suggest may have been the price of the productivity gains.

The same machinery can compare groups instead of periods. Merelie, Amado and dos Santos split 188 Clinical Commissioning Groups in England into two groups of 94 by deprivation and used the group version of the index proposed by Camanho and Dyson to compare them at one point in time. Those in more deprived areas had more resources per head and higher efficiency but were generally less effective.

National NHS productivity figures are built differently. The Centre for Health Economics at York measures growth in NHS outputs, inputs and productivity with a Laspeyres volume chain index, using costs as weights because prices are not available for the vast majority of NHS services, and adjusts outputs for quality. That gives service-wide growth rates but needs cost weights, while the Malmquist index needs a panel of comparable units and no external price or cost weights and attributes each unit's change to catching up or frontier shift.

Where the Malmquist index sits among related measures

The index tracks change in technical efficiency and in the frontier. Hollingsworth notes that stochastic frontier analysis accounts for statistical noise, which DEA does not, so DEA treats every deviation from the frontier as inefficiency, and that carries into an index built on it. The technical index ignores prices, so it cannot show whether the input mix is the cheapest, which the cost Malmquist index of Maniadakis and Thanassoulis addresses where input prices are known. No version shows allocative efficiency in the sense of whether the services produced are those that would do most good.

It is unrelated to the efficiency frontier and cost-effectiveness frontier of health technology assessment, which compare treatments for the same patients by cost and health effect. In benchmarking, the decomposition helps separate a provider's own improvement from a sector-wide shift.

Uncertainty and common misreadings

The distance functions are estimated from a sample, so the index carries sampling error. Simar and Wilson developed a bootstrap procedure for confidence intervals around Malmquist indices and their components. Reanalysing the pharmacy panel, they found only three pharmacies with no significant efficiency changes over the period, where the original study had reported five pharmacies efficient in all periods.

1. Reading the direction wrongly. Some input-oriented definitions, including that of Simar and Wilson, show improvement below 1 unless reciprocals are reported, so the convention needs stating.
2. Treating technical change as new technology. The term records any frontier movement, which can reflect coding or data changes as well as new treatments.
3. Ignoring returns to scale. An index computed against a variable returns frontier can show productivity change where none occurred.
4. Equating activity with health. Episode counts rise with avoidable admissions, and quality can move differently from measured productivity.

Sources

• Castelli A, Nazerai A, Podaneva A, Warmington C. Productivity of the English National Health Service: 2023/24 update. CHE Research Paper 203. York: Centre for Health Economics, University of York; July 2026. https://www.york.ac.uk/media/che/documents/papers/researchpapers/CHE%20RP%20203%20Final%20July%2026%2016.56.pdf
• Färe R, Grosskopf S, Norris M, Zhang Z. Productivity growth, technical progress, and efficiency change in industrialized countries. American Economic Review. 1994;84(1):66-83. https://ideas.repec.org/a/aea/aecrev/v84y1994i1p66-83.html
• Hollingsworth B. Health system efficiency: measurement and policy. In: Cylus J, Papanicolas I, Smith PC, editors. Health System Efficiency: How to Make Measurement Matter for Policy and Management. Health Policy Series 46. Copenhagen: World Health Organization on behalf of the European Observatory on Health Systems and Policies; 2016. p. 99-114. https://iris.who.int/server/api/core/bitstreams/2794c497-bcca-4a89-8161-b65f1b5cb998/content
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• Merelie HB, Amado CAF, dos Santos SP. Measuring the effect of deprivation on primary health care performance using data envelopment analysis and Malmquist indices. Health Care Management Science. 2025;28(3):461-477. https://doi.org/10.1007/s10729-025-09715-9
• Simar L, Wilson PW. Estimating and bootstrapping Malmquist indices. European Journal of Operational Research. 1999;115(3):459-471. https://doi.org/10.1016/s0377-2217(97)00450-5 (text read: author version, http://hdl.handle.net/2078.1/120925)
• Simar L, Wilson PW. Productivity growth in industrialized countries. Discussion paper, Institut de Statistique, Université catholique de Louvain; February 1998. http://hdl.handle.net/2078.1/3943
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