Nominal Scale

Nominal Scale in Health Economic Data and Value Set Models

In Stevens's 1946 classification of measurement scale types, the nominal level is the weakest of the four, with no order, no distance and no zero. Health economic datasets are full of such variables, including trial arm, sex, region, provider, diagnosis group and the identity of each dimension of a health status instrument. Codes like these can be written as digits, but adding, averaging or ranking them gives results that change when the codes are swapped, so regression models bring them in as sets of indicator variables. The same device sits inside the value sets that turn EQ-5D profiles into utilities, where each level on each dimension gets its own indicator and its own estimated decrement. Nominal in this sense refers to the level of measurement, not to a nominal cost, which in costing means a cost stated in the prices of the year it was incurred.

What a nominal code records

A nominal scale preserves only identity. Two observations either share a category or they do not, and nothing else about the numbers used as labels carries information. Any one-to-one relabelling is therefore permissible, so calling the control arm 0 and the intervention arm 1, or the reverse, or A and B, leaves every legitimate conclusion unchanged.

The summaries that survive relabelling are those built on counting. The number and proportion of patients in each category, the most common category, and cross-classifications such as a contingency table of trial arm against hospital admission are all meaningful. Tests of association for such tables, and measures of agreement on a categorical judgement such as the kappa statistic, work on frequencies rather than on the codes themselves. A mean or median of the codes has no meaning, because a different labelling would give a different answer with the same data.

Binary variables are the simplest nominal case. Alive or dead, admitted or not and responder or non-responder each split a population into two categories, and a 0 and 1 coding becomes useful in a model only because the mean of a 0 and 1 indicator equals the proportion coded 1, which is a count-based statistic.

Nominal labels inside health state descriptions

Health state classification systems mix nominal and ordinal information. The five dimensions of the EQ-5D are nominal categories, since no dimension ranks above another and their order on the form is a convention. Within each dimension the response levels are ordered by severity, which makes them ordinal. A profile code such as 21111 strings the levels together in the fixed order of the dimensions, so each digit is a level and its position names the dimension.

The profile code as a whole works as a label for one of the possible health states. Devlin, Parkin and Janssen describe the conventional level labels as a categorical description rather than a number. The England EQ-5D-5L value set reported by Devlin and colleagues in 2018 makes the point concrete, since the profiles 11211 and 21111 are different states with level 2 on different dimensions, and both receive 0.950, the highest value in the set below full health, which is 1 for 11111. Treating 21111 as a five-digit number larger than 11211 would say nothing about health.

Indicator coding in value set regressions

Value set studies estimate how much each problem on each dimension lowers the value of a health state below full health. The usual specification treats every level above no problems on every dimension as its own category, coded by an indicator that is 1 when the profile has that level on that dimension and 0 otherwise. The model is

$$U_{s} = 1-\sum_{d=1}^{5} \sum_{l=2}^{5} \beta_{dl} \, D_{sdl}$$

where $U_{s}$ is the value of health state $s$, $d$ indexes the five dimensions, $l$ indexes the levels above no problems, $\beta_{dl}$ is the estimated decrement for level $l$ on dimension $d$, and $D_{sdl}$ is 1 when state $s$ has level $l$ on dimension $d$ and 0 otherwise.

Devlin and colleagues describe the least restrictive model for the England EQ-5D-5L value set as having a parameter for each step away from no problems on each dimension, 20 parameters in all from four levels on five dimensions, with each decrement subtracted from a constant of 1. That is the least restrictive specification rather than the final one, since the published England value set came from a model with restrictions on its parameters. The German EQ-5D-5L value set of Ludwig, Graf von der Schulenburg and Greiner reports its coefficients as the decrement from level 1 to the respective level, coded as regular dummies.

Indicator coding treats the ordered levels as if they were nominal categories, so the model does not assume that the step from level 2 to level 3 is the same size as the step from level 3 to level 4. The order then reappears in the estimates, and a value set whose decrements do not rise with severity signals a problem with the data or the model. A model that instead entered the level number as one continuous variable per dimension would impose equal steps, which is one of the assumptions behind the level sum score.

Worked example with illustrative provider codes

The numbers in this example are illustrative. A costing study records the treating hospital for 300 patients, 100 at each of three hospitals. Mean cost per patient is GBP 1,200 at hospital A, GBP 2,100 at hospital B and GBP 1,500 at hospital C, so the overall mean is GBP 1,600.

Step 1 (hospital entered as a number). The data file codes the hospitals 1, 2 and 3. A linear regression of cost on that code, with equal numbers in each group, has slope equal to half the difference between the means of the groups coded 3 and 1. The slope is half of 1,500 minus 1,200, which is 150, so the fitted means are GBP 1,450, GBP 1,600 and GBP 1,750.

Step 2 (the same hospitals with codes swapped). If hospital B is coded 3 and hospital C is coded 2, nothing about the patients has changed. The slope becomes half of 2,100 minus 1,200, which is 450, and the fitted means become GBP 1,150 for hospital A, GBP 1,600 for hospital C and GBP 2,050 for hospital B.

Step 3 (indicator coding). With hospital A as the reference and one indicator each for B and C, the intercept is 1,200 and the coefficients are 900 for B, which is 2,100 minus 1,200, and 300 for C, which is 1,500 minus 1,200. The fitted means are exactly GBP 1,200, GBP 2,100 and GBP 1,500 whichever hospital is chosen as the reference.

Both numeric codings misstate every hospital's mean and give a slope whose size depends on an arbitrary labelling decision. The indicator model recovers the observed means because it treats the hospitals as unordered categories, which is what they are. In a cost-effectiveness model that applies hospital-specific costs, the numeric coding in step 1 would understate the cost at hospital B by GBP 500, the difference between GBP 2,100 and the fitted GBP 1,600.

Nominal variables in trial and choice data

In trial-based economic evaluation the treatment arm is a nominal variable, and its indicator coefficient in a regression analysis of cost or QALYs is the adjusted incremental cost or effect. Other nominal covariates, such as centre or country, enter in the same way, with one category held as the reference. A subgroup analysis by a nominal factor, such as whether the incremental cost differs by country, adds products of the treatment indicator and the country indicators, and the number of extra parameters grows with the number of categories.

A discrete choice experiment often includes attributes whose levels are unordered, such as the type of provider or the mode of administration of a medicine. The ISPOR good research practices task force report on the statistical analysis of discrete choice experiments, by Hauber and colleagues, describes alternative approaches to coding the data such experiments produce. Whichever coding is used, an unordered attribute needs one parameter per level beyond the reference, and the estimated preference weights for its levels can be compared with each other but not ranked by their codes.

Nominal groupings in costing and payment

Casemix classifications assign each hospital episode to one group. A diagnosis related group code taken as a whole is a nominal label. Some casemix systems build complexity into part of the code, such as a suffix, but the group is identified by the whole code rather than by any arithmetic on it, and two episodes are alike only in the sense of falling in the same group. Each group then carries a price or cost weight, which is a ratio-scale quantity, so a costing study combines a nominal classification with ratio-scale values attached to it.

The same pattern appears when a costing study assigns unit costs by staff grade, ward type or setting of care. The grouping variable is nominal and the unit costs are not, and errors arise when the grouping codes are carried into a model as if they were quantities, as in the hospital coding example.

Boundaries with ordinal and categorical data

Categorical data is the wider term and covers both nominal and ordinal variables. The difference lies in whether the categories have a meaningful order. Severity levels, disease stages and performance status grades have one, so they are ordinal, and analysing them as nominal loses information while still giving valid results. Analysing nominal codes as ordinal or interval invents information that is not in the data.

Some variables change level with the question asked. Cause of death is nominal, while the same deaths grouped by age band are ordinal. A health state in a Markov model is a nominal category for the purpose of tracking where a cohort sits, even though the states usually carry utilities and costs that are interval and ratio scale quantities.

Limitations and common misreadings

Indicator coding is safe but costly in parameters. A variable with many categories, such as a large set of diagnosis groups, needs many indicators, and categories with few observations give imprecise estimates. Value set studies face the same trade-off between flexibility and precision when they choose between the least restrictive model and models with fewer parameters.

1. A numeric code makes a variable quantitative. Hospital, region or dimension codes stay nominal however they are stored.
2. Profile codes can be compared as numbers. A profile code identifies a state, and only a value set attaches a number with preference meaning to it.
3. Indicators for ordered levels throw the order away. The order is not assumed in the model but can be checked in the estimated decrements.
4. A different reference category changes the results. The coefficients change because they become differences from another category, but the fitted means and the fit of the model stay the same.

Sources

• Stevens SS. On the theory of scales of measurement. Science. 1946;103(2684):677-680. https://doi.org/10.1126/science.103.2684.677
• Devlin N, Parkin D, Janssen B. Methods for Analysing and Reporting EQ-5D Data. Cham: Springer; 2020. Chapter 2, Analysis of EQ-5D profiles. https://doi.org/10.1007/978-3-030-47622-9
• 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
• Ludwig K, Graf von der Schulenburg JM, Greiner W. German value set for the EQ-5D-5L. PharmacoEconomics. 2018;36(6):663-674. https://doi.org/10.1007/s40273-018-0615-8
• Hauber AB, González JM, Groothuis-Oudshoorn CGM, et al. Statistical methods for the analysis of discrete choice experiments: a report of the ISPOR Conjoint Analysis Good Research Practices Task Force. Value in Health. 2016;19(4):300-315. https://doi.org/10.1016/j.jval.2016.04.004