Analysis of EQ-5D Profiles

Abstract

Profile data form the cornerstone of analyses of EQ-5D data and, in many cases, are likely to be the primary focus of interest. In this chapter, we provide an overview of methods that can be used to describe the profile data from respondents at a given point in time, and to describe the changes in profiles between different points in time.

The aims of this chapter are

• to demonstrate a variety of analyses that can be performed on the profile data generated from the EQ-5D instruments: EQ-5D-3L, EQ-5D-5L and EQ-5D-Y;

• to explain methods that can be used to describe EQ-5D profile data in cross-sectional (collected at a single point in time) and longitudinal (describing changes over time) designs; and

• to consider the advantages and limitations of each method, and outline in which decision contexts insights from them might be useful.

Profile data form the cornerstone of analyses of EQ-5D data and, in many cases, are likely to be the primary focus of interest. In this chapter, we provide an overview of methods that can be used to describe the profile data from respondents at a given point in time, and to describe the changes in profiles between different points in time.

Even when the ultimate goal of analysis is to generate EQ-5D values and to estimate quality-adjusted life years (QALYs), analysis of profile data provides important insights and should always be the starting point for analysts. For example, summarising EQ-5D patient data simply as values obscures the underlying information about which aspects of their health have been most affected by their condition, or improved by treatment. To know about that, you need to look at the data that respondents have given you: the boxes they ticked on an EQ-5D questionnaire.

The methods presented here need not be treated as alternatives, but rather as complementary. Although they are illustrated using EQ-5D data, and in some cases developed specifically for the analysis of EQ-5D profile data, these same methods could just as readily be applied to other generic or condition specific health status or patient reported outcome (PRO) measures.

It should be noted that we do not cover inferential statistics, either hypothesis testing or estimation, as this book is not intended as a statistical primer and we assume that readers will be able to apply appropriate inference procedures where required. For example, we describe contingency tables, to which measures of association such as a χ2 test could be applied.

2.1 Cross-Sectional Analysis: Describing Health at a Point in Time by Dimension and Level

Exploratory data analysis (EDA) of EQ-5D data, including the use of simple descriptive statistics, is undervalued, and often underreported in papers that contain more complex econometric and psychometric analyses. This is bad practice and wasteful of information, because EDA not only generates information that helps in interpreting more complex analyses, but also generates information about health within populations and about the properties of the EQ-5D which is valuable in itself.

Describing health at the most detailed level possible for the EQ-5D can be done very simply, by reporting the number and percentage of patients reporting each level of problem on each dimension of the EQ-5D. An example of this is shown in Table 2.1, which shows EQ-5D-3L data provided by patients before and after hip surgery, using data from a pilot study for the Patient Reported Outcome Measures (PROMs) programme in the English National Health Service (NHS) (Devlin et al. 2010).

Table 2.1 Frequency of levels by dimension of ‘some problems’, before and after elective hip surgery in the English NHS

This very simple table provides some important information. For example, before hip surgery, 420 of these patients (95.7% of the sample) reported a level 2 problem on mobility, but none reported a level 3 problem. The reason is that Level 3 on the EQ-5D-3L mobility dimension is ‘confined to bed’—and even patients with very poor mobility because of hip problems aren’t confined to bed. That is a problem with the EQ-5D-3L—as has been pointed out previously (Oppe et al. 2011). This issue has been corrected in the EQ-5D-5L (Herdman et al. 2011), where the most severe problem with mobility is ‘unable to walk about’, and is an important advantage of the 5L over the 3L (Janssen et al. 2018).

The information on the types of problems experienced by a sample of patients at any given point in time can be simplified still further by collapsing levels together, to create just two categories: the number and percentage of patients reporting no problems (level 1), and the number reporting any level of problems (levels 2 and 3 for the 3L, and levels 2, 3, 4 and 5 for the 5L). This can also be seen in Table 2.1. For example, before surgery, mobility problems are common in these patients, as might be expected: only 4.3% of these patients had no problems with mobility. Of the 95.7% of patients who reported having at least some problem on mobility before surgery, all reported a level 2, as noted above. However, problems on other dimensions are just as prevalent: 99.8% of patients reported at least some problem with pain and discomfort, and 96.6% at least some problem with usual activities. Over 40% of these patients also reported problems with anxiety and depression—something that might be missed by condition specific instruments focused on mobility and function-related issues specific to hips, such as the Oxford Hip Score.

Examining the profile data by each dimension and level in this manner is a good starting point to understanding the nature of the health problems reported in the data you have collected. However, there are limitations to this way of reporting the data. Because the focus is on the frequency of observations in each level within each dimension, it doesn’t tell us how these problems combine in the people reporting them. For example, are the people who report a level 3 on Anxiety and Depression also the same people who report a level 3 on Usual Activities? For this reason, it is also important to examine the way that observed levels of problems on each dimension combine into EQ-5D profiles, which is covered in Sect. 2.3.

2.2 Longitudinal Analysis: Describing Changes in Health Between Two Time Points by Dimension and Level

In addition to describing health states at any one point in time, if you have collected EQ-5D profile data at more than one time point, you are likely also to be interested in describing the changes between them—for example, before and after surgery, or between various time points in a clinical trial, compared to baseline. This too can be done at the level of the EQ-5D dimensions, as is also shown in Table 2.1.

‘Eyeballing’ the differences in numbers and percentages of patients in each of the levels tells us about the nature of the changes in health that resulted from surgery. For example, the results in Table 2.1 show there were quite striking improvements in patients’ Anxiety and Depression, Self-care and Pain and Discomfort—not just Mobility. And because of the issue with level 3 Mobility noted above, whereby the worst level of problem these patients were likely to report on mobility was level 2, the only improvements to mobility that were possible as a result of hip replacement surgery were from ‘some’ to ‘no’ problems. This issue with the use of the EQ-5D-3L to measure health outcomes from hip surgery would not have been apparent if these patients’ data had been analysed just in terms of EQ-5D values.

It can however be difficult readily to get an overall picture of improvements, even for these relatively simple EQ-5D-3L data. As with the analysis of cross-sectional data, this does not summarise the extent of improvement across dimensions. As noted in Sect. 2.1, one way of handling this is to collapse the levels into just two categories: no problems and some problems. The shift between these two categories provides a simpler way of capturing change. The change in health between time points, reported in this manner, provides a way of summarising the overall extent to which patients go from any level of problem to no problem within each dimension. This may be useful in some contexts, but it has some limitations as an indicator of improvement because of the loss of information caused by aggregation of levels. It doesn’t capture improvements other than shifts to no problem, so other improvements that may be of value to patients, for example from extreme to moderate problems, are not captured. That means, that if applied to the EQ-5D-5L, the advantages of its more refined descriptive system will be lost.

2.3 Cross-Sectional Analysis: Describing Health at a Point in Time Using Profiles

While describing the number and percentage of observed levels within each dimension (as in Table 2.1) gives very useful information dimension-by-dimension, it does not tell you anything about the way these problems are combined in the health states reported by patients.

One of the most simple and instructive things you can do with an EQ-5D profile data set is to report the cumulative frequency of these profiles. This will reveal the extent to which your observations are evenly distributed over many profiles, or instead concentrated on a relatively small number of health profiles.

The results can sometimes be quite surprising. For example, in Table 2.2 we show the cumulative frequency of self-reported EQ-5D-3L profiles reported by 7294 respondents in the 2012 Health Survey for England. In this example, the great majority of respondents self-reported their health using only a small number of profiles. The top three most frequently reported profiles represented almost three quarters of the respondents.

Table 2.2 Prevalence of the 10 most frequently observed self-reported health states and frequency of reporting of the worst possible health state in EQ-5D-3L

In contrast, Table 2.3 shows the cumulative frequency of profiles reported by 996 respondents from the general public in the EQ-5D-5L value set study for England for their self-reported health on the EQ-5D-5L. This shows, in comparison to Table 2.2, a larger number of unique health states observed in this data set, and the observations are less concentrated on a small number of states. A large proportion of observations are accounted for by profile 11111 (no problems on any dimension) in both data sets, which is not surprising given that both samples comprise members of the general public, many of whom would not regard themselves as ill. But in general, this ‘ceiling effect’ is somewhat less in the EQ-5D-5L data (Devlin et al. 2018). Obviously, the states observed and their cumulative frequency will differ from data set to data set, but in general the EQ-5D-5L yields less concentrated data, reflecting the advantages of the larger number of response options.

Table 2.3 Prevalence of the 10 most frequently observed self-reported health states and frequency of reporting of the worst possible health states in EQ-5D-5L

Understanding these patterns of observations in your data is important for three reasons:

• (i) The way self-reported health problems are combined may be useful, as a complement to clinical information, for understanding and planning for patients’ treatment needs.

• (ii) The combination of problems into health profiles determines the distribution of EQ-5D values data. For example, Parkin et al. (2016) show that the clustering of observations on particular EQ-5D-3L profiles contributes to the unusual ‘two group’ distribution that is often seen in EQ-5D-3L values data.

• (iii) The characteristics of the distribution of problems at baseline may have important implications for the potential for and nature of health improvements that can be observed at later time points.

Looking at the cumulative frequency is a simple and effective way of getting an insight into the distribution of health profiles in a data set. However, a limitation is that it does not provide a summary statistic that allows us readily to (a) describe how good or bad the health states are, or (b) the extent to which the observations cluster on just a few health states, or are evenly spread out over the available heath states described by the descriptive system. Having a summary statistic to characterise the degree to which there is clustering or dispersion of observed health states is useful, especially if one wanted to compare this characteristic, for example to find out whether there are changes in the distribution of profile data from a group of patients observed at different time points, or between EQ-5D profile data from patients with different conditions.

2.4 Longitudinal Analysis: Describing Changes in Health Between Two Time Points Using Profiles

Descriptive analyses of profile data such as Table 2.1 can be very useful, but they contain a lot of information and sometimes an overall summary is required. One way of summarising profile data is to generate a single number for each profile using weights, for example using value sets. However, as noted in Chap. 1, this introduces possible problems of information loss and bias. The good news is that there are ways of summarising changes in EQ-5D health status without using value sets, just using the data that respondents have given you.

2.4.1 The Paretian Classification of Health Change (PCHC)

Devlin et al. (2010) introduced a way of summarising changes in profile data called the Paretian Classification of Health Change (PCHC). The approach is based on the principles of a Pareto improvement in Welfare Economics, drawing an analogy with the challenge of summing up changes in utility of different individuals, where utility can be measured only in ordinal terms. The idea is simple: an EQ-5D health state is deemed to be ‘better’ than another if it is better on at least one dimension and is no worse on any other dimension. And an EQ-5D health state is deemed to be ‘worse’ than another if it is worse in at least one dimension and is no better in any other dimension. Using that principle to compare a person’s EQ-5D health states between any two time-points, there are only four possibilities:

• (i) Their health state is better

• (ii) Their heath state is worse

• (iii) Their health state is the same

• (iv) The changes in health are ‘mixed’: better in at least one dimension, but worse in at least one other.

Applying this to the English NHS PROMs pilot hip replacement data, we found that under 5% had no change, 82% had improved health, under 5% had worse health, and under 10% had a ‘mixed’ change (Devlin et al. 2010). In other words, this simple analysis provides a very clear summary of what is happening to patients’ health because of hip surgery—without relying on value sets. It also highlighted important differences in the benefits from hip surgery, compared with the other types of elective surgery analysed in the English NHS PROMs pilot, shown in Tables 2.4 and 2.5. Looking at Table 2.4, hip replacement operations were by far the best in terms of success in reducing the number of patients who had problems, with knee replacement operations a close second. Hernia and varicose vein repairs were much less successful, and cataract removals had a very low success rate, with more patients getting worse than improving—although the last of these should be interpreted carefully because the EQ-5D may not be capturing the kind of benefits that cataract operations provide. The numbers of patients who worsened or had no change show the same pattern.

Table 2.4 Changes in health for five surgical procedures according to the PCHC

Table 2.5 Changes in health state for three conditions according to the PCHC, taking account of those with no problems

One problem with this analysis is that ‘No change’ is confounded when patients record no problems according to any of the dimensions before treatment, because they are, according to the EQ-5D, healthy patients whose only alternative would be for their condition to worsen as a result of treatment. Recording no problems at all is rare for patients who have conditions serious enough to require a joint replacement but may occur for conditions whose need for treatment may not be fully captured by their EQ-5D profile. Table 2.5 shows for the three conditions to which this applies the PCHC taking into account those with no problems before surgery. In each case, this shows a slightly better performance than suggested by Table 2.4.

The advantage of the PCHC is that it provides a high-level summary of the nature of changes in health reported by patients, without the need to introduce any external scoring system or preference weighting.

The limitations of the PCHC are:

• (i) It focuses on whether there is improvement or worsening in self-reported health, and does not account for the magnitude of those changes. It does not differentiate between small improvements and big improvements (e.g., both a shift from level 5 to level 4, and a shift from level 5 to level 1, are counted as improvements).

• (ii) It takes no account of whether the changes occur in dimensions that matter a lot to people or in dimensions that may be considered less important.

• (iii) The PCHC will not be informative in cases where mixed changes dominate the changes in health self-reported by patients.

The PCHC can be extended to give information about the composition of differences between profiles according to how dimensions and levels differ. These are illustrated using newer data on hip replacement patients in the English NHS PROMs programme that was instituted following the pilot study referred to earlier, using simple graphs. They also show how data can be compared at different time periods. This could be adapted to compare, for example, patients in different populations.

First, Fig. 2.1 shows the PCHC for three years in graphical form.

Fig. 2.1

The PCHC for hip replacement patients in the English NHS, 2009–12

Figure 2.2 shows which dimensions were improved for those patients whose PCHC category was ‘Improved’

Fig. 2.2

Percentage of hip replacement patients who improved overall, by the dimensions in which they improved, English NHS 2009–2012

This shows that improvements were spread over all dimensions, but were most frequently found in Pain and Discomfort, followed by Usual Activities and Mobility, with Self-care and Anxiety and Depression improving for less than 50% of those who improved overall.

Figure 2.3 shows which dimensions were worsened for those patients whose PCHC category was ‘Worsened’.

Fig. 2.3

Percentage of hip replacement patients whose health worsened overall, by the dimensions in which they worsened, English NHS 2009–12

This shows the opposite pattern to improvements. Worsening health was spread over all dimensions, but was most frequently found in Anxiety and Depression and Self-Care followed by Usual Activities, with Pain and Discomfort and Mobility getting worse for less than 20% of those whose health was worse overall.

Figure 2.4 shows a comparison of PCHC ‘Mixed’ patients, which is more complicated because it involves both worsening and improving in different dimensions.

Fig. 2.4

Percentage of hip replacement patients who had a mixed change overall, by the dimensions in which they improved and worsened, English NHA 2009–12

For the EQ-5D-3L, it is possible to show every possible change in every dimension. Each dimension can change in one of three ways—no change, improvement or worsening—each of which has three possible specific level changes, resulting in 9 categories for each dimension. Table 2.6 shows how these can be displayed for the hip replacement data.

Table 2.6 Changes in levels in each dimension for hip patients, NHS PROMs, 2009–10, percentages of total and of type of change

This shows that the dominant change for Mobility, Usual Activities and Pain and Discomfort is an improvement from level 2 to level 1, but for Self-care and Anxiety and Depression it is no change from ‘no problems.’ Within change categories, it is notable that in each dimension improvements are dominated by a change from level 2 to level 1; that improvements from level 3 to level 1 and worsening from 1 to 3 are rare, reflecting the rarity of level 3 observations in the data set; and worsening from 2 to 3 is the most common amongst those who worsened overall in Usual Activities and Pain and Discomfort.

Unfortunately, it is much more difficult to display the same analysis for the 5L version, as there are 25 possible categories for each dimension.

2.4.2 The Probability of Superiority

Buchholz et al. (2015) introduced a nonparametric effect size measure, the probability of superiority (PS), to analyse paired samples of EQ-5D profile data in the context of assessing changes in health in terms of improvement or deterioration. This measure was initially recommended by Grissom and Kim (2012). For each dimension, the number of patients with positive changes is divided by the total number of matched pairs (i.e. the number of respondents scoring EQ-5D at both time-points). To account for patients with no changes, that is ‘ties’, half the number of ties is added to the numerator. PS is therefore the probability that within a randomly sampled pair of dependent scores, the score obtained at follow-up will be smaller than the score obtained at baseline. It ranges from 0 to 1 and is

• <0.5 if more patients deteriorate than improve,

• = 0.5 if the same number of patients improve and deteriorate or do not change and

• >0.5 if more patients improve than deteriorate.

This is a further, useful way of examining the nature of change in EQ-5D data. A limitation is that it focuses on changes at the dimension level, rather than on how this combines at the patient level.

2.4.3 Health Profile Grid (HPG)

A further way of summarising changes in health in an EQ-5D data set is the Health Profile Grid (HPG), also introduced by Devlin et al. (2010). The HPG relies on profiles being ordered from best to worst. This can be done using a value set, a scoring system based on equally weighted dimensions and levels, or a scoring system based on the EQ VAS predicted from the profile (see Chap. 4).

The HPG plots the profiles between any two points in time. The example shown in Fig. 2.5, again taken from the English NHS PROMs pilot, shows profiles before and six months after hip replacement surgery. The rank ordering is determined by the EQ-5D-3L values according to the value set for the United Kingdom (Dolan et al. 1997). The PCHC category for each profile change is also shown.

Fig. 2.5

Health profile grid for hip operations, English NHS

The location of each point shows improvement and worsening according to the profiles’ rank order. The 45° line represents ‘no change’; the further above the line, the greater the improvement in health; below the line means health has worsened. The pattern of observations in the HPG in Fig. 2.5 suggests that most patients experience benefit from hip replacement surgery, as the observations lie predominantly above the 45° line. There is a spread of health profiles from less to more severe before surgery, but a much narrower distribution after surgery, concentrated in the least severe profiles, with some outliers. The PCHC category adds to this by identifying cases where overall improvement and worsening of the patients’ ‘before’ and ‘after’ profiles according to their rank are ‘Mixed Change’, that is they include both improvements in at least one dimension and worsening in at least one other. In these data, every mixed change case included only one dimension which changed in the opposite direction to the overall change according to the profiles’ rank.

By contrast, the HPG shown in Fig. 2.6, for the English NHS PROMs pilot cataract surgery data, shows a much more mixed picture of improvements and worsening. The immediately obvious observation is that similar numbers improved and worsened. However, another feature is that most of those with the worst health profiles before surgery improved and most of those with the worst profiles after surgery had amongst the least severe health profiles before surgery. Unlike the clear-cut conclusions that may be drawn from the hip HPG, such a pattern suggests further investigation is required into the impact of cataract operations on patients’ health-related quality of life (HRQoL).

Fig. 2.6

Health profile grid for cataract operations, English NHS

Presenting the profiles in this manner can suggest clusters of patients, characterised by the nature of their profiles at time point 1, and the direction and magnitude of the change between the time-points. However, it is important not to rely on visual inspection alone to identify clusters, because some of the gaps that are apparent simply identify EQ-5D health profiles that are very infrequently observed, for example states having no problems in four dimensions and the worst state in the other. It is essential to test for these formally using statistical cluster analysis techniques. An example, with clusters identified using a k-means procedure, is shown in Fig. 2.7.

Fig. 2.7

Health profile grid showing clusters of changes in health for NHS hip replacement patients, using the k-means procedure

The numbers represent the 6 different clusters of patients identified. Most of the clusters seem to be identified as similar because of the patients similar pre-surgery profiles. Cluster 4 is of more interest, identifiable as the patients with worst health profiles after surgery. Also of interest is the comparison of clusters 2 and 5, with similar, relatively less severe profiles before surgery but with cluster 2 having more severe profiles after surgery. These observations could form the basis of further investigation into whether or not these are real clusters of clinical importance.

It is to possible to improve the appearance of the HPG and reduce the problem of artefactual gaps by including only those health states found within the data. It is also possible to take this further by including only the most frequently found profiles. In many data sets, only a few very common profiles are found, along with many rarer cases, so restricting the analysis to profiles covering, for example, 90% of all observations would be informative.

The advantage of the HPG is that it provides a ready means of displaying and examining the changes in health within a sample of patients. A limitation of the HPG is that it relies upon having a valid and appropriate means of ranking the EQ-5D profiles. The method used to rank the profiles may affect the HPG and the statistical identification of clusters.

2.5 Summarising the Severity of EQ-5D Profiles

It is sometimes useful to summarise the overall ‘severity’ of EQ-5D health states, by means other than generating weighted scores such as values. Because these involve information loss and hidden assumptions about the aggregation of dimensions and levels, they should be used with care.

2.5.1 The Level Sum Score (LSS)

It is possible to summarise a profile by calculating a Level Sum Score (LSS), sometimes misleadingly referred to as the ‘misery score’. This simply adds up the levels on each dimension, treating each level’s conventional label (1, 2 or 3) as if it were a number rather than simply a categorical description.

The best EQ-5D health state involves having no problems on any dimension and is conventionally represented by the label 11111. Treating the level labels as numbers, the best possible score is (1 + 1 + 1 + 1 + 1) = 5. Similarly, the most severe problem on any dimension has the label 3 for the EQ-5D-3L, so the LSS for the worst health state is (3 + 3 + 3 + 3+ 3) = 15. Every other health state on the EQ-5D-3L will have a level sum score between 5 (the best) and 15 (the worst 15), and as these are integer there are 11 possible scores; the larger the score, the worse the health state. For the EQ-5D-5L, the range is between 5 and 25 and there are 21 possible scores.

The LSS has been used as a crude measure of severity to gauge the validity of values obtained in valuation for studies for different health states. Figure 2.8 shows the relationship between the English value set for the EQ-5D-5L and the LSS (Devlin et al. 2018). This shows that, as the LSS increases (states get worse), the values decline.

Fig. 2.8

EQ-5D-5L values (English value set) plotted against the LSS

However, the LSS has some important limitations as a means of summarising health states across dimensions and levels:

• (i) It’s a very crude summary score—for example, the very different EQ-5D-3L profiles 22222, 33211 and 11233 all have the same level sum score (LSS = 10). The Dutch values for these profiles are 0.569, 0.350 and 0.009 respectively (Lamers et al. 2006).

• (ii) Within the LSS scores, the weighted index values derived from profiles have very wide and overlapping ranges.

• (iii) Each score contains a very different number of potential profiles: for example, in the EQ-5D-3L, LSS = 5 and LSS = 15 have just one profile each, but LSS = 10 contains 51 profiles. For the 5L, there are 381 profiles with LSS = 15, but just 5 profiles with a LSS = 6.

• (iv) Giving equal weight to the dimensions and the difference between levels means the LSS is not free from value judgements—it makes a specific assumption about their relative importance (Parkin et al. 2010).

These issues can be seen below, with respect to the EQ-5D-5L. Table 2.7 shows all possible LSSs for the EQ-5D-5L. It also shows descriptive statistics for the English value set for the EQ-5D-5L for all the different LSSs for the EQ-5D-5L. Although the mean and median values relate reasonably well to the order of the LSS, it does show big differences in the standard deviation. Importantly, it shows the overlap between the range of values for the different level summary scores. For example, the range for LSS = 15 includes the mean values of LSS = 12 and LSS = 18 and the lower or upper range respectively of LSS = 10 and LSS = 21. This issue can also be seen in Fig. 2.8. For these reasons, it is wrong to treat the LSS as ordinal.

Table 2.7 Summary statistics for the EQ-5D-5L values (English value set) by all the different LSSs

2.5.2 The Level Frequency Score (LFS)

An alternative, although rarely used, means of summarising profile data is the level frequency score (LFS). The measure was proposed by Oppe and de Charro (2001) and used there to demonstrate the distribution of the EQ-5D-3L profiles in their data on the effects on HRQoL of a helicopter trauma team. The method characterises each health state by the frequency of levels at 1, 2 or 3 (for the EQ-5D-3L) or the frequency of levels at 1, 2, 3, 4 and 5 on the EQ-5D-5L. For example, in the EQ-5D-5L, the full health profile 11111 has 5, 1 s, no level 2, 3, 4 and 5 s, so the LFS is 50000; the worst health profile is 00005; profiles such as 31524 and 53412 would be 11111; 20 profiles such as 13211 have a LFS of 31100.

Oppe and de Charro used the LFS to show the way in which the EQ-5D-3L values data observed in their data (using the UK EQ-5D-3L value set) were distributed over the various EQ-5D-3L profiles (see Table 2.8).

Table 2.8 Number of observations in the LFS according to the UK EQ-5D-3L values

The distribution of EQ-5D-5L profiles by LFS is provided in an Appendix to this chapter.

2.6 Analysing the Informativity of EQ-5D Profile Data

2.6.1 Shannon Indices

Shannon’s indices, originally developed to analyse the information content of strings of text, are widely used in the ecology literature to measure how many species are observed and how evenly animals, or plants are spread over the various categories. It has also been applied widely in assessing distributional characteristics of the EQ-5D (Buchholz et al. 2018), where the categories of interest are EQ-5D profiles and we are interested in a summary measure of how evenly respondents to EQ-5D questionnaires are spread over the profiles defined by the descriptive system. The main application of the Shannon indices has been to compare informational richness and evenness of dimensions, either comparing the EQ-5D-3L with the EQ-5D-5L or to compare similar dimensions between different generic health status instruments (Janssen et al. 2007). It is also possible to apply the Shannon indices to distributions of health profiles.

The Shannon index is defined as:

$$H^{\prime} = - \mathop \sum \limits_{i = 1}^{C} p_{i} \log_{2} p_{i}$$

where H′ represents the absolute amount of informativity captured, C is the total number of possible categories (levels or profiles), and p_i = n_i/N, the proportion of observations in the ith category (i = 1,…, C), where n_i is the observed number of scores (responses) in category i and N is the total sample size. The higher the index H′ is, the more information is captured by the dimension or instrument. In the case of a uniform (rectangular) distribution (i.e., p_i = p* for all i), the optimal amount of information is captured and H′ has reached its maximum (H′max) which equals log₂ C. If the number of categories (C) is increased, H′max increases accordingly, but H′ will only increase if the newly added categories are actually used. The Shannon Evenness index (J′) exclusively reflects the evenness (rectangularity) of a distribution, regardless of the number of categories, and is defined as: J′ = H′/H′max. Variance of the Shannon index can be calculated as described by Janssen et al. (2007) and accordingly standard errors and 95% confidence intervals can be calculated.

The Shannon indices are purely descriptive measures of the informational richness and evenness of a classification system and have no relation to the content, meaning, or clinical relevance of what the instrument aims to measure. Both the Shannon index and the Shannon Evenness index are needed to make a useful interpretation of the measurement scale.

2.6.2 Health State Density Curve (HSDC)

Zamora et al. (2018) introduced a graphical means of depicting the nature of the distribution of EQ-5D profiles, the health state density curve (HSDC). This draws on an analogy with the Lorenz curve in describing an income distribution. The cumulative frequency of health states is compared against the cumulative frequency of the sample or population. A 45° line means that the observed health states are completely evenly spread across the sample: 10% of the sample accounts for 10% of the health states; 50% of the sample accounts for 50% of the health states, and so on.

A concentrated distribution—that is, where relatively few profiles are reported and are common to a large proportion of the sample—will be show as a curve which lies below the 45° line. The more unevenly distributed the profile data, the further below the diagonal line the HSDC will be. In the extreme, where just one profile is reported by all members of the sample, the HSDC will take a right-angled shape.

Figure 2.9 shows the HSDC for patients from three groups of patients, and overall, from Cambridgeshire NHS in the UK. This shows that for all patients, observed profiles are not evenly distributed, that is a small number of profiles accounts for a relatively large share of the observations. The musculoskeletal patients had the most concentrated data.

Fig. 2.9

HSDC for EQ-5D-5L profiles from Cambridgeshire NHS patients

The HSDC provides a simple means of illustrating this property of a profile data set, in a manner that facilitates comparisons between data sets. It has limitations. As with Lorenz curves, where two curves cross (as is the case with rehabilitation and nursing data shown in Fig. 2.9), there is no unequivocal way of declaring one data set to be more concentrated than another. It also does not tell us which profiles are the most commonly self-reported. Therefore, the HSDC is best seen as a complement to the information from the cumulative frequency of profiles.

2.6.3 Health State Density Index (HSDI) and Other Related Indices

In the analysis of income distribution, the Lorenz curve is often accompanied by the Gini coefficient, which describes the extent of inequality which is apparent as the area between the diagonal line and the curve, divided by the entire area underneath the diagonal. In a similar way, an index can be calculated to summarise the inequality of observed health state profiles. Zamora et al. (2018) introduce a broadly similar summary measure, the Health State Density Index (HSDI). HSDI has a value of 1 where there is total equality, that is where there are the same number of patients in each profile, and HSDI = 0 for total inequality, that is where one profile accounts for all the observations.

The HSDI allows the degree of concentration in self-reported health to be compared both between different sets of patients and between different instruments, for example the 3 and 5 level versions of the EQ-5D. Zamora et al. (2018) use the HSDC to compare the EQ-5D-3L and EQ-5D-5L, their respective HSDIs indicating the advantages of the 5L in differentiating between patients and yielding less concentrated data.

The specific properties of the HSDI may be compared with the Shannon’ indices. Each performs somewhat differently as a measure in capturing specific aspects of the distribution of patients’ data, such as the concentration over the most common states, and the influence of ‘rare’ states. For example, the Shannon index (absolute and relative) is not sensitive to random variations but decreases slowly with “rare health states”. The HSDI decreases slowly with random variations and is strongly affected by infrequently observed health states with large decreases towards zero (total inequality). For more detail see Zamora et al. (2018).

Appendix: Analysis of the LFS for the EQ-5D-5L

For the EQ-5D-5L, the LFS has a total of 102 possible scores, from 00005 (for the worst profile 55555) through to 50000 (no problem on any dimension, state 11111). Like the LSS, a problem with the LFS is that LFSs contain an uneven number of profiles. For example, LFS = 50000 and LFS = 00005 each contain 1 profile, whereas LFS = 11111 (meaning: any health profile containing one level 1, one level 2, one level 3, one level 4 and one level 5) represents 120 different EQ-5D-5L profiles. Table 2.9 is a full list of the possible values for the LFS.

Table 2.9 Distribution of the EQ-5D-5L profiles by LFS

Table 2.10 shows how the LFS could be used to analyse the characteristics of an EQ-5D-5L value set, using data from the English value set (Devlin et al. 2018). It shows the mean and median values for each LFS.

Table 2.10 Summary statistics of EQ-5D-5L values by LFS

The following chart (Fig. 2.10) shows how the EVS and LFS are related. As with the LSS, this gives an indication of the general validity of a value set, in that there are no patterns that indicate that a value set takes perverse or other undesirable characteristics.

Fig. 2.10

EQ-5D-5L values (English value set) plotted against the LFS