Interval Scaling

Interval Scaling of Health Outcome Scores and Utility Values

Interval scaling is the property that lets differences between scores be compared, so that a gain of 10 points means the same amount of health wherever on the scale it occurs, even though the zero and the size of one point were fixed by convention. Health economists rely on it whenever they average quality of life scores, compare change scores between trial arms or take the difference between two utilities in a QALY gain. Stevens placed the interval level between the ordinal and ratio levels in his 1946 classification, and the four levels side by side are set out under measurement scale type. For an analyst the practical question is which statements about health data survive a change of origin and unit, including statements about QALY gains.

Equal differences with a conventional zero

Stevens defined each level of measurement by the operations needed to build it and by the transformations that leave it intact. He wrote that an interval scale can be erected only if there is an operation for determining equality of intervals, and that its zero point is a matter of convention or convenience, since the form of the scale stays the same when a constant is added. On his examples, the Centigrade and Fahrenheit scales, it is meaningless to say that one value is twice another. The permissible transformations form the general linear group.

$$x' = a + b x, \quad b > 0$$

where $x$ is a value on the original scale, $x'$ is the same value after rescaling, $a$ is a constant that moves the origin and $b$ is a positive constant that changes the unit.

What the transformation preserves is the ratio of two differences.

$$\frac{x'_1 - x'_2}{x'_3 - x'_4} = \frac{b(x_1 - x_2)}{b(x_3 - x_4)} = \frac{x_1 - x_2}{x_3 - x_4}$$

where the subscripts label four observations, the constant $a$ cancels in each difference and $b$ cancels in the ratio. A statement that the gain from treatment A is twice the gain from treatment B keeps its truth under any permissible rescaling, while a statement that one patient's score is twice another's does not.

Which statements about health data survive rescaling

Stevens listed the mean, the standard deviation and the product-moment correlation as statistics that stay meaningful on an interval scale. He noted that a case at the mean stays at the mean only under transformations as restricted as the linear group, while the median keeps its place under any order-preserving change. The test for a statement about a health outcome is whether its truth changes when $a$ and $b$ change.

The standardised mean difference and the mean difference both behave well on an interval scale, which suits them to continuous patient-reported outcome data. Percentage change from baseline carries ratio meaning that an interval scale does not support.

Where interval claims for health measures come from

Stevens observed that most psychological measurement aspires to create interval scales and sometimes succeeds, and health measurement is in the same position. An interval claim rests on a model of how the numbers were produced.

Roudijk, Donders and Stalmeier state that utilities for expected utility calculations should have interval-scale properties, so a standard gamble value takes its interval meaning from the axioms of expected utility and loses it to the extent that respondents depart from them. The time trade-off leans on the QALY model holding for each respondent.

Salomon showed that a conditional logit model fitted to rankings of EQ-5D states produces valuations on an interval scale, so differences can be compared, with an origin and unit fixed arbitrarily by the model's identifying assumptions. Ranks are invariant under positive affine transformations, so anchors must be chosen, and Salomon set full health at 1 and compared rescalings that put dead, ranked alongside the states, or the worst EQ-5D state at 0. Devlin and colleagues describe discrete choice experiment data as giving non-anchored relative values, so a value set built from them needs anchoring information such as time trade-off responses.

Rating scales are the contested case. The scoping review by Åström and colleagues records claims that rating scales do not possess interval scale properties and that the visual analogue scale cannot measure preferences on a cardinal scale.

Summed questionnaire scores and Rasch measurement

Many quality of life questionnaires add item responses into a total and then treat the total as interval data. Grimby, Tennant and Tesio argue that such cumulative scores are bound to an artificial range, which distorts intervals towards the margins. A raw move near either end may understate the real change, while the same move near the centre may overstate it, so the meaning of a raw gain depends on the starting point. They treat this as separate from a ceiling effect.

They state that Rasch modelling of ordinal data, given certain conditions and within a probabilistic framework, allows raw scores to be transformed into interval scale measures. They also hold that the Rasch model is the only item response model that complies with the axioms of fundamental measurement, and they acknowledge that the model's demands on data have led others to adopt wider item response models. For a questionnaire of $n$ yes or no items of equal difficulty set at zero, the Rasch estimate for a raw score $r$ is

$$\theta = \ln\left(\frac{r}{n - r}\right)$$

where $\theta$ is the person's location on the latent trait in logits, $r$ is the number of items answered positively and $n$ is the number of items, with $r$ strictly between 0 and $n$. The logit stretches the scale at both ends, and real instruments with items of differing difficulty need conversion tables from fitted models.

Grimby and colleagues add that a minimal clinically important difference along an ordinal scale should be managed cautiously, because the same numeric increment may mean a different change depending on the baseline value.

Interval properties in QALY arithmetic

A health state utility value enters the QALY calculation as a weight on time. The question is what happens to the QALY gain when every utility is rescaled linearly.

$$Q = \sum_{j} t_j u_j$$

where $Q$ is the QALY total for one treatment, $t_j$ is the time in years spent in health state $j$ and $u_j$ is the utility of that state. Applying $u'_j = a + b u_j$ to every utility gives

$$Q' = \sum_{j} t_j (a + b u_j) = aT + bQ$$

where $T$ is the total time alive, the sum of the $t_j$. For the difference between two treatments,

$$\Delta Q' = b \, \Delta Q + a \, \Delta T$$

where $\Delta Q$ is the QALY gain on the original scale and $\Delta T$ is the difference in time alive.

When both treatments keep patients alive for the same time, $\Delta T$ is zero and the gain changes only by the unit $b$, so treatment rankings and ratios of QALY gains are unaffected. A treatment that improves quality of life within a fixed survival period can be ranked on interval-scaled utilities, although a cost per QALY set against a threshold still needs the unit fixed by anchoring dead at 0 and full health at 1. When survival differs, the term $a \Delta T$ shifts the gain. With $b$ equal to 1, placing dead at 0.1 instead of 0 adds 0.1 QALYs to the gain of a treatment that adds one year of life, through the relabelling alone.

Roudijk and colleagues, and Prieto and Sacristán, differ on whether utilities with dead at 0 have the ratio properties the QALY model needs, as set out under ratio scale.

Worked example with a rescaled rating scale and a summed score

All numbers here are illustrative. A rating line runs from 0, the worst imaginable health, to 100, and a panel's mean ratings place dead at 20, a moderate state M at 60, a mild state S at 80 and full health at 100. The analyst wants values with dead at 0 and full health at 1.

Step 1 (the linear rescaling). The rescaling that sends 20 to 0 and 100 to 1 is

$$u = \frac{r - r_{\text{dead}}}{r_{\text{full}} - r_{\text{dead}}}$$

where $u$ is the rescaled value, $r$ is the mean rating of a state, $r_{\text{dead}}$ is 20 and $r_{\text{full}}$ is 100. In the form $a + bx$, the unit is $b = 1/80 = 0.0125$ and the origin shift is $a = -20/80 = -0.25$.

Step 2 (rescaled values). State M sits 40 points above dead, giving 40/80 = 0.5, and state S sits 60 points above dead, giving 60/80 = 0.75.

Step 3 (a ratio of differences). On the rating line the gap between S and M is 20 points and the gap between M and dead is 40 points, a ratio of 20/40 = 0.5. On the rescaled values the gaps are 0.25 and 0.5, a ratio of 0.25/0.5 = 0.5. The comparison of gaps survives.

Step 4 (a ratio of values). On the rating line S is 80/60 = 1.33 times M, while on the rescaled values S is 0.75/0.5 = 1.5 times M. The improvement between M and S is one third on the raw ratings and one half on the rescaled values.

Step 5 (a summed score). A separate questionnaire has 20 yes or no items of equal difficulty. Patient X moves from a raw score of 8 to 12 and patient Y from 14 to 18, both gaining 4 raw points. With the Rasch formula, patient X moves from ln(8/12), or minus 0.405, to ln(12/8), or 0.405 logits, a gain of 0.81. Patient Y moves from ln(14/6), or 0.85, to ln(18/2), or 2.20 logits, a gain of 1.35.

Interpretation. The linear rescaling keeps comparisons of gaps and changes ratios and percentages, so only gap comparisons carry over from the raw ratings, and the rescaling cannot show that the original ratings had equal intervals. In the summed score, equal raw gains correspond to measured gains that differ by a factor of 1.35/0.81 = 1.67, with the larger gain near the top of the scale.

Assumptions behind an interval claim

An interval claim for a set of health scores is a claim about how they were produced. Each assumption below can fail in practice.

1. Equal units across the range. A point near the top must mean the same change as a point in the middle, which raw summed scores may not deliver. Grimby and colleagues call this invariance of comparisons, a given difference meaning the same interval at whatever level of the variable.
2. The same unit across people and time. A given score difference must mean the same amount of health for every patient and at every assessment.
3. The elicitation model holds. Standard gamble values need expected utility, and ranking or choice values need their random utility model.
4. Consistent anchors. Values from rankings or choices carry no origin or unit until anchors are added.
5. Linear rescaling. Stevens noted that if a scale still serves its purpose when its values are squared or cubed, it is not even an interval scale.

Boundaries with ordinal, ratio and cardinal scaling

Interval scaling sits between two neighbouring properties. Ordinal scaling keeps only order, so any increasing transformation is allowed and means of ordinal codes can reverse under recoding. A ratio scale adds a true zero and allows only multiplication by a positive constant, which makes statements such as twice as costly meaningful for costs and life years.

Salomon's cardinal values from rankings are interval-scaled valuations. Interval scaling is the formal property, cardinal scaling is the claim that a valuation method delivers it, and health state valuation is the process that produces the values.

Limitations and common misreadings

Most errors read more into a set of numbers than the method behind them supports. Five recur in health outcome work.

1. Decimals taken as proof of interval data. A mean of ordinal codes has decimals and still lacks equal intervals.
2. Percentage improvement on an interval scale. The result depends on where zero sits, as Step 4 shows.
3. A rescaled rating scale read as a utility scale. Linear rescaling keeps whatever interval meaning the ratings had and adds none.
4. One minimal important difference across the whole range of a summed score. The same raw change can stand for different amounts of change at different baselines.
5. Interval utilities assumed enough for every QALY comparison. They suffice when time alive is equal between arms, and the position of dead matters once survival differs.

Sources

• Åström M, Thet Lwin ZM, Teni FS, Burström K, Berg J. Use of the visual analogue scale for health state valuation: a scoping review. Quality of Life Research. 2023;32(10):2719-2729. https://doi.org/10.1007/s11136-023-03411-3
• Devlin NJ, Shah KK, Feng Y, Mulhern B, van Hout B. Valuing health-related quality of life: an EQ-5D-5L value set for England. Health Economics. 2018;27(1):7-22. https://doi.org/10.1002/hec.3564
• Grimby G, Tennant A, Tesio L. The use of raw scores from ordinal scales: time to end malpractice? Journal of Rehabilitation Medicine. 2012;44(2):97-98. https://doi.org/10.2340/16501977-0938
• Prieto L, Sacristán JA. Problems and solutions in calculating quality-adjusted life years (QALYs). Health and Quality of Life Outcomes. 2003;1:80. https://doi.org/10.1186/1477-7525-1-80
• Roudijk B, Donders ART, Stalmeier PFM. Setting dead at zero: applying scale properties to the QALY model. Medical Decision Making. 2018;38(6):627-634. https://doi.org/10.1177/0272989X18765184
• Salomon JA. Reconsidering the use of rankings in the valuation of health states: a model for estimating cardinal values from ordinal data. Population Health Metrics. 2003;1:12. https://doi.org/10.1186/1478-7954-1-12
• Stevens SS. On the theory of scales of measurement. Science. 1946;103(2684):677-680. https://doi.org/10.1126/science.103.2684.677