Concept Architecture
Concept
Theoretically, Interval Scaling is a level of measurement in which numerical values represent ordered observations separated by equal intervals, but without a true zero point. It is founded on classical measurement theory and permits meaningful interpretation of differences between values while prohibiting ratio comparisons. Within health economics, the concept underpins many psychometric instruments and statistical analyses in which equal numerical differences are assumed to represent equal differences in the underlying construct.
Mathematically, interval scaling is characterised by invariance under positive linear transformations. Measurements preserve both ordering and equal intervals following transformation, allowing meaningful addition and subtraction but not multiplication or division because the zero point is arbitrary.
In practice, interval scaling is implemented through validated measurement instruments and psychometric methods, including item response theory and Rasch analysis where appropriate. Interval-scaled outcomes are commonly summarised using means, standard deviations and parametric statistical methods, although preference-based utility measures used in health economics are not generally regarded as pure interval scales.
Purpose
Used to represent quantitative measurements in which equal numerical differences correspond to equal differences in the underlying construct, enabling valid statistical analysis and comparison of changes in measured outcomes.
Mathematical Formulae
Primary Formula
x? = a + bx,?b > 0
where:
- x = original interval-scale value
- x? = transformed interval-scale value
- a = additive constant
- b = positive scaling constant
Supporting Formulae
? = x? ? x?
where:
- ? = difference between two interval-scale measurements
Related Mathematical Methods
- Classical measurement theory
- Stevens' theory of measurement scales
- Rasch measurement
- Item response theory
- Linear transformation
Example
A health-related quality of life instrument produces scores ranging from 0 to 100. Patient A improves from 42 to 54, while Patient B improves from 71 to 83. Both patients demonstrate an equivalent improvement of 12 units because interval scaling assumes that equal numerical differences represent equal changes in the underlying construct. It is not valid to conclude that a score of 84 represents twice the measured construct of a score of 42.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| AVERAGE | =AVERAGE(B2:B101) | Calculates the mean interval-scale score across respondents. |
| STDEV.S | =STDEV.S(B2:B101) | Estimates variability of interval-scale measurements. |
| STANDARDIZE | =STANDARDIZE(B2,$E$2,$E$3) | Standardises interval-scale scores for comparative statistical analysis. |
| ABS | =ABS(B2-A2) | Calculates the absolute difference between two interval-scale measurements. |
VBA (Optional)
Automate the calculation of descriptive statistics and score transformations for interval-scaled outcome measures.
Sources
- Stevens SS. On the Theory of Scales of Measurement. Science. 1946;103:677?680.
- DeVellis RF, Thorpe CT. Scale Development: Theory and Applications. Sage.
- Brazier J, Ratcliffe J, Salomon JA, Tsuchiya A. Measuring and Valuing Health Benefits for Economic Evaluation. Oxford University Press.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. Oxford University Press.
Related Concepts (2)
Library
Publications
1
Measuring and Valuing Health Benefits for Economic Evaluation — Brazier, Ratcliffe, Salomon & Tsuchiya, 2nd Edition ed., 2017 (Oxford University Press)
The comprehensive text on the measurement and valuation of health benefits for economic evaluation — defining health, valuation techniques (time trade-off, standard gamble), whose values to use, preference-based measures (EQ-5D, SF-6D), and the construction of QALYs.
BookView source →
Frequently Asked Questions (6)
What is interval scaling?
A measurement scale property in which numerical differences correspond to equal differences in the underlying quantity, allowing meaningful arithmetic operations.
Source: Torrance GW. Measurement of health state utilities for economic appraisal. Journal of Health Economics. 1986;5(1):1-30. doi:10.1016/0167-6296(86)90020-2.
What arithmetic does an interval scale permit?
On an interval scale equal numerical gaps stand for equal amounts of the underlying quantity, so differences between values can be added, subtracted, and averaged meaningfully. What it does not support is statements of ratio, because the zero point is set by convention rather than marking a true absence, so one value cannot be called twice another. Temperature measured in degrees Celsius is the standard illustration. This is why interval-level differences, but not ratios, can be taken from such a scale. Stevens (1946) set out the properties of the interval level.
Source: Stevens 1946
Why does interval scaling matter for health measurement?
Interval scaling matters because economic evaluation performs arithmetic on health values, weighting and summing them to form quality-adjusted life years, which is only valid if equal differences in the values represent equal differences in health. A scale that ranks states but whose intervals are not equal cannot support such calculation. Utilities used in evaluation are therefore required to have at least interval properties, so that differences between them carry consistent meaning.
Source: Torrance 1986
How does interval scaling differ from ordinal scaling?
An ordinal scale conveys only the order of items, showing that one is higher than another but not by how much, whereas an interval scale conveys the size of the differences, so that equal intervals represent equal amounts. Ranking health states is ordinal; assigning values whose differences are equal for equal changes in health is interval. Economic evaluation requires interval-level values, since it combines and weights them, which an ordinal ranking cannot support.
Source: Torrance 1986
How does interval scaling relate to ratio scaling?
An interval scale has meaningful differences but no true zero, so ratios of its values are not meaningful, whereas a ratio scale has a true zero, allowing statements that one value is twice another. Utility scales anchored at death as zero and full health as one are often treated as having ratio properties for the purpose of weighting time, since death provides a meaningful zero. Whether health utilities are interval or ratio affects what calculations are valid.
Source: Torrance 1986
Why is interval-level measurement required for utilities?
Utilities are required to be at least interval-level because quality-adjusted life years weight time by utility and sum across states, operations that assume equal differences in utility represent equal differences in health. If the scale were only ordinal, these calculations would be meaningless, since the size of the intervals would not be defined. Valuation methods are chosen to yield interval or ratio values, and the assumption that they do underlies the arithmetic of economic evaluation.
Source: Torrance 1986
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 1 Sep 2025
Content version: 1.0.0
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- Persistent URI
- https://healtheconomics.wiki/concept/interval-scaling
- Term code
- HE-EE-HU-044
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