Testing the Expert Based Weights Used in the UK’s Index of Multiple Deprivation (IMD) Against Three Preference-Based Methods

The Index of Multiple Deprivation (IMD), used widely in England, is an important tool for social need and inequality identification. It summarises deprivation across seven dimensions (income, employment, health, education, housing and services, environment, and crime) to measure an area’s multidimensional deprivation. The IMD aggregates the dimensions that are differentially weighted using expert judgement. In this paper, we test how close these weights are to society’s preferences about the relative importance of each dimension to overall deprivation. There is not agreement in the literature on how to do this. This paper, therefore, develops and compares three empirical methods for estimating preference-based weights. We find the weights are similar across the methods, and between our empirical methods and the current IMD, but our findings suggest a change to two of the weights.


Introduction
Deprivation is multidimensional; low income and other material and social disadvantages affect an individual's well-being (Atkinson 2003;Stiglitz et al. 2009). In the UK, the Index of Multiple Deprivation is a multidimensional index used to measure deprivation in small geographically-defined areas. The IMD is used extensively by national and local government to identify pockets of high deprivation and to direct poverty alleviation policies, to classify local authority districts into those eligible for additional funding and used within formulae that determine funding for health care, policing and housing across England.
The IMD includes seven dimensions of deprivation: Income, Employment, Health and Disability, Education, Skills and Training, Barriers to Housing and Services, Living Environment, and Crime. The IMD aggregates these dimensions into one summary deprivation measure, in which the dimensions are differentially weighted. In multidimensional indices those dimensions with higher weights impact on total deprivation more, and increased achievement in one dimension can compensate for decreased achievement in another. 1 The weight given to a dimension is a judgement about the dimension's importance in the aggregate.
A variety of methods are used to estimate dimension importance for indices. These include expert-based, correlation-based, and preference-based weights (see Decancq and Lugo 2013; OPHI 2012 for reviews). Expert-based weights are based on experts' opinions about each dimension's importance to the overall experience of deprivation. Many multidimensional deprivation indices use expert-based weights and most of these weight all dimensions equally. 2 Expert-based weights have been criticised because experts may not accurately represent the population being assessed by the index, which raises concerns about paternalism. The IMD is an example of an index with expert-based weights that differ across dimensions. (Noble et al. 2000(Noble et al. , 2004Smith et al. 2015). The IMD weights were applied based on theoretical and normative considerations about the dimensions' importance to the experience of deprivation. The reliability of the expert-based dimension weights in the IMD has been questioned (Deas et al. 2003).
Correlation-based weights are based on data about deprivation and the correlation between the different dimensions in the population. Correlation-based weights summarise data and do not reflect preferences. An extensive literature calculates weights based on the correlation between dimension deprivations in the population using principal component analysis (PCA) or factor analysis (FA) (Ram 1982;Noorbakhsh 1998). Both, PCA and FA assume one single, latent variable (or construct) exists to be measured and that this is best measured using a set of variables (corresponding to the index's dimensions). The weight assigned to each dimension reflects the accuracy with which the variable measures the latent factor. A limitation of this method is that many multidimensional indices do not aim to improve a single (latent) construct's 1 In addition to weights, the choice of indicators, their transformed distributions and the aggregation function will also lead to implicit dimension weighting. See Decancq and Lugo (2013) for a discussion of these issues. 2 Equal weights may be explicit and normative because each dimension is believed to be equally important. Often, however, equal weighting is implicit because researchers want to avoid the contentious task of setting weights (OPHI 2012). Examples of multidimensional indices with equal weights include the Human Development Index (HDI) (UNDP 1990), the Human Poverty Indices (UNDP 1999), the Commitment to development index (Birdsall and Roodman 2003), the Multidimensional Poverty Index (Alkire and Santos 2010) and the New Zealand Index of Socioeconomic Deprivation (NZiDep) (Salmond et al. 2006). measurement, but to summarise several constructs into a single measure of aggregate deprivation. When a multidimensional index measures multiple, independent, latent constructs, PCA and FA can offer no guidance on dimension weights.
Consider the following thought experiment: In a hypothetical country, in time period t, citizens' wellbeing is measured by their housing quality, health, and mobile phone ownership. At time t everyone who is in poor health lives in poor housing and does not own a mobile phone and everyone in good health lives in good housing and owns a mobile phone. There is perfect correlation between these three variables. PCA or FA would generate equal weights. From a normative perspective, however, we believe that not owning a mobile phone is not as deleterious for wellbeing as living in poor housing or being in poor health. Now the country's government improves the housing of all citizens. In time period t + 1, 90% of individuals previously living in poor housing now live in good housing, but nothing else has changed. The correlation between housing and health, and housing and mobile phone ownership is now lower. Therefore, correlation based weights will change and housing will receive a smaller weight even though the normative importance of housing to wellbeing has not changed.
Preference-based weights are based on individuals' preferences and can be either inferred from the relationship between individual wellbeing and deprivation in dimensions or directly elicited from individuals using surveys. Fleurbaey et al. (2009), Haiksen-DeNew andSinning (2010), and Schokkaert (2007) derive weights based on the relationship between individuals' (subjective) well being and their experience of deprivation across a set of dimensions. Adler and Dolan (2008), Fusco et al. (2013), Bellani (2013) and Benjamin et al. (2014) derive weights from a sample of individuals' stated preferences about the importance of achievements in each dimension for wellbeing. This paper applies and compares three empirical methods to estimate preferencebased weights for the IMD. The paper is based on research reported in the working paper Dibben et al. (2007). The methods we apply to obtain preference-based weights differ in how directly preferences are elicited. In the first empirical method, we estimate weights based on the relationship between individuals' self-reported social exclusion and their achievements in the IMD dimensions. In doing so, we observe how achievements act through the individual's and society's preferences to affect their experience of social exclusion within the society to which they belong. In the second empirical method, we estimate weights using a stated preference survey and directly ask members of the general public to state the most deprived individuals from a set of multidimensionally deprived individuals. In the third empirical method, we estimate weights based on how much money the government spends alleviating deprivation across the IMD dimensions such that the weights are proportional to the relative government spending. We argue individuals' preferences influence government spending through the democratic process.
In this paper we detail the methods used to elicit preference-based weights, the assumptions underlying these methods and the challenges faced when applying each method. Each method takes a slightly different, but related, conceptual approach and this enables us to assess the stability of preference-based weights across the elicitation methods. If we find that weights differ across methods, our results can prompt discussion and a decision based on empirical evidence. If we find weights are the same across methods we provide strong support for a set of weights.

3 2 The Index of Multiple Deprivation in England
The IMD is a multidimensional deprivation index used to measure deprivation in England at the super output area level. 3 The IMD combines seven deprivation dimensions: Income, Employment, Health and Disability, Education, Skills and Training, Barriers to Housing and Services, Living Environment, and Crime. Deprivation in each dimension is measured by a set of indicator variables and their respective thresholds below which an area is considered deprived (Table 1, column 1). For example, five indicators and thresholds are used to measure Income deprivation, and each indicator counts the proportion of an area's population who are deprived for that indicator. Dimension indicators are combined or aggregated to obtain a score for that domain. The aggregation method varies across the dimensions. For instance, the indicators within the Income Deprivation dimension are believed to measure a single underlying construct-income deprivation-and as such are combined using FA.
Standardised dimension scores are aggregated following Eq. (1) to provide a multidimensional deprivation index score for a super output area: x j denotes deprivation in dimension j = 1,…,q and overall deprivation is summarised by X = (x 1 ,…,x q ). An area's deprivation is the weighted mean of the (transformed) deprivations I j (x j ). The dimensions, x j , are measured in different units, thus a transformation function or standardisation is required giving, I j (x j ). The dimension weights are non-negative (w j ≥ 0). The index is increasing in deprivation and can be used to assess if one geographically defined area is worse or better off (more or less deprived) than another.
In the five IMDs since 2000 (IMD 2000, IMD 2004, IMD 2007, IMD 2010and IMD 2015, expert-based dimension weights have been used. The weights are unchanged since IMD 2004 and take account of theoretical and normative considerations based on existing literature and the quality of dimension indicator data 4 (Noble et al. 2000(Noble et al. , 2004. The existing literature suggests that having a low income and being dislocated from the labour market are key determinants of other deprivations, and therefore these dimensions should carry greater weight Thus, the Employment and Income dimensions were given weights of 22.5%; Health and Disability, and Education Training and Skills dimensions 13.5%; and Barriers to Housing and Services, Living Environment and Crime dimensions 9.3%.

Individual's Experience of Social Exclusion
An individual's experience of social exclusion may reflect the allied but less experiential state of multiple deprivation. We measure an individual's experience of social exclusion and achievements in dimensions of the IMD using data from the millennium poverty The dimension was described as: "…the total amount of money that a household has each week for each adult living in this household. This is the money available to cover housing costs, bills, grocery shopping etc. In the following situations, people will be described as living in a household where income is:" The dimension was described as: "…the person described is in paid employment or not. In the following situations the people will either be: Employed-either employed, retired, or looking after home/family The dimension was described as: "…health is measured by whether the person has a longterm illness or disability, which limits their daily activities or the work they can do. In the following situations the people described will either have: Limits on their daily activities and work due to long term illness Respondent reported air pollution as problem in area.
We measure social exclusion using responses to the following question:

Have there been times in the past year when you've felt isolated and cut off from society or depressed, because of a lack of money?
This question fits with Townsend's conceptualization of deprivation as not only a state, but also a process that excludes people from social norms with consequences for the wellbeing of that person (Townsend 1979). We therefore use feeling isolated and cut off from society as a proxy for the individual's experience of multidimensional deprivation.
We develop a set of regressors that represent achievements in each IMD dimension. For each dimension, we create a variable coded as 1 if the individual is 'dimension deprived' and otherwise coded as 0. To do this, we match dimension indicator variables from the IMD 2004 with PSE variables as summarised in Table 1, columns 1 and 2. If exact equivalents for a dimension indicator were not available in the PSE then variables of most relevance to the dimension were used. Each PSE variable was used to create binary outcome for an individual: either above or below the IMD threshold (Table 1). These binary variables were combined into dimension indicator variables. Consistent with a union measure of deprivation, an individual was considered to be deprived in a dimension if they were below the IMD-equivalent threshold in any of the PSE variables for that dimension. The number of individuals considered deprived for each dimension are presented in Table 2.
We estimate the effect of being deprived in a dimension on the experience of social exclusion using a logistic regression model in which we estimate the probability that individual, i, experiences social exclusion (Pr(ESE i = 1) as a function of experiencing deprivation in the IMD dimensions (Greene 2011). We follow the specification of the IMD and specify a linear additive relationship between the dimensions as in Eq. (1): Subjective measures, such as the PSE social exclusion measure, can be affected by idiosyncratic individual differences and individual differences that lie within the 'private sphere' (for example, religious belief) that should not be considered in a deprivation measure (Schokkaert (2007). The error term ε i in Eq. (2) captures idiosyncratic differences (2) Pr ESE i = 1 = β inc INC i + β emp EMP i + β hea HEA i + β bhs BHS i + β le LE i + β crime CRIME i + ε i (2) to control for their influence on social exclusion. We estimate Eq. (2) with and without these controls. We use marginal effects to calculate the impact of moving from being not deprived in a dimension to being deprived in a dimension on the probability of experiencing social exclusion. We calculate weights (scaled to sum to 1) for the IMD by dividing each marginal effect by the sum of all the marginal effects. These weights describe the relative importance of each dimension on underlying deprivation (social exclusion).
Based on the responses to the social exclusion question in the PSE, 240 individuals experienced social exclusion and 1330 did not. Table 3, column 2 reports the relationship between experiencing social exclusion and the dimension variables. All but one of these variables were statistically significantly related to probability of an individual experiencing social exclusion. Being deprived in the Barriers to Housing and Services dimension was not statistically significantly related to social exclusion. The ranking of the IMD dimensions from the regression-based weights is: Income, Health and Disability, Employment, Education, Skills and Training, Living Environment, Crime and Barriers to Housing and Services. Figure 1 reports PSE weights based on the rescaled marginal effects.
We test the robustness of the estimated weights in two ways. We test robustness to the choice of proxy by re-estimating the weights using individuals' feeling depressed as a proxy for the experience of deprivation. We test robustness to the inclusion of additional control variables. The weights derived from this alternative proxy and/or with the control variables are broadly similar and are available from the authors on request.

General Population Stated Preference Survey
We use a survey-based stated preference method, a discrete choice experiment (DCE) to find out which dimensions society considers to be worse than others, and how much worse 5 in order to assess how society judges individuals experiencing deprivation in one or multiple dimensions. We assume that deprivation states can be described by the dimensions, and that the relative importance of dimensions can be inferred from responses to a survey in which respondents judge if one multidimensional deprivation state is worse than another.
In the survey we define multidimensional deprivation states wherein each state refers to a hypothetical person's circumstances. 6 The dimensions included are based on the IMD dimensions and indicators (Table 1, column 3). A hypothetical person's circumstances in each dimension could be deprived or not deprived based on the IMD thresholds for the dimension's indicators. In our study there are 128 multidimensional deprivation states (2 7 ). We match these states with their mirror image to create 128 pairs of hypothetical states that describe two people who experience multidimensional deprivation. A mirror image of a state is created as follows, if one state is deprived in the income dimension then its mirror image is not, if one state is not deprived in the employment dimension then its mirror image is, and so on. An example of a pair of multidimensional deprivation states is presented in Fig. 2 Fig. 1 Comparison of IMD weights and empirical weights by method 5 DCEs are based on Lancaster's theory of value (Lancaster 1966) and can be used to elicit the relative importance of different product characteristics in the demand for a good or a service. DCEs have been applied in transportation research, and in environmental and health economics to elicit preferences for nonmarket goods (Kanninen 2007). 6 The hypothetical people are all adults: we take this perspective to avoid confounding respondents' weights for the dimensions with the deprived individuals' characteristics.
We ask survey respondents to report which of the two individuals in a pair of deprivation states most needs additional government support. By asking which person needs additional government support, we incorporate the purpose of the IMD: the distribution of government funding. The respondents' choices reveal information about the trade-offs they make between deprivation on the different dimensions when deciding who needs additional government support. We developed questionnaires that explained to respondents each dimension's meaning, and the two states an individual could be in. In the questionnaire, the dimensions and indicators were explained in way that was consistent with the hypothetical person perspective presented in the choice tasks. The deprivation thresholds were chosen both to match those in IMD 2004 and to be meaningful and understandable to the general public. The 128 pairs of states are too many to ask one respondent to assess. The pairs were randomly divided into eight groups of 16 pairs and eight versions of the questionnaire developed. After respondents assessed 16 pairs, they completed questions about their socioeconomic characteristics.
The questionnaire was sent to a random sample of 1000 households in England drawn from the Royal Mail's small user postcode address file in August 2006. 7 One week after the initial mailing a postcard was sent to the whole sample, to thank respondents and remind non-respondents to respond. A second questionnaire was sent to non-respondents three weeks later. The second mailing contained a revised covering letter urging those who had not yet responded to do so and another copy of the questionnaire.  Fig. 2 Example of a discrete choice experiment pair 7 Included alongside the questionnaire was a covering letter explaining the use of the IMD and the relevance of this study and a prepaid return envelope was also included.
From the questionnaire responses, we observe which of the two hypothetical persons a respondent states should be given more government support. Thus, we have a binary dependent variable. We assume that respondents select the person they believe is most deprived and analyse responses within the framework of random utility theory. We assume respondents perfectly discriminate between the two states and know the relative importance they give to each dimension when deciding who is most deprived, but that we, the analyst, cannot observe all the factors that influence respondents' choices (McFadden 1973). We estimate the effect of being deprived in a dimension on respondent's choice using a logistic regression model in which we estimate the probability that respondent i states that individual j is most deprived (Pr(D = 1)), as a function of the observable, deprivation dimensions as in Eq. (1), and an additive random (unobservable) component, ε j (Greene 2011). We follow the specification of the IMD and specify a linear additive relationship between the dimensions as in Eq. (1): The random component ε j represents inter-individual differences in state j's assessed deprivation due to heterogeneity in respondents' preferences, measurement errors and/or (3) Pr D j = 1 = β inc INC j + β emp EMP j + β hea HEA j + β bhs BHS j + β le LE j + β crime CRIME j + ε j the functional form specification (Manski 1977). Each respondent makes 16 choices, therefore we have 16 observations per respondent and estimate a random effects logit model. 8 We estimate marginal effects and calculate the weight for each dimension by transforming the marginal effects onto a 0 to 1 scale. 251 individuals returned the general population survey (response rate = 25.1%). The socioeconomic characteristics of the respondents are summarised in Table 4. Respondents are not representative of the population in England as at census 2001. Respondents under represent people under the age of 60 years and under represent people with no or 'O' level (or equivalent) educational qualifications. Responses are weighted by age and education, based on population proportions in the census 2001 to correct for the sample composition. Table 5 reports the marginal effects of the survey responses, for the unweighted and weighted samples. Overall, most dimensions are statistically significant determinants of respondents stating that a hypothetical individual should receive more government support. Weighting the responses to correct for sample representativeness has a small impact on the results: each dimension's weight changes slightly but the dimensions' relative importance do not change. The ranking of the IMD dimensions from the survey-based weights is: Income, Living Environment, Health and Disability, Education, Skills and Training, Barriers to Housing and Services, Crime and Employment. Table 5, column 5 and Fig. 1 report the DCE weights for the IMD based on the rescaled marginal effects estimated for the weighted sample.

Government Spending
Government spending, arguably, reflects society's assessment of the relative importance of factors influencing their own lives, and those of their fellow citizens through the electoral system. During elections political parties put before the electorate manifestos detailing different options about the manner and degree to which revenues are raised and how the state's resources will be spent. For instance, before the 1997 election the Labour party emphasised education's importance. This, therefore, provided a mandate for the Labour party, after winning the election, to put their policies into action, and increase government spending on the education sector (Department for Education and Skills 2004). Based on the assumption that the political system allows the population's preferences to influence government policy and through this the amount of money spent on various social policies, we derive weights by calculating the proportion of government spending allocated to each IMD dimension. We assume that government spending associated with each IMD dimension represents the value to society of keeping individuals out of a particular deprivation state. Government spend is reviewed for financial year 2003-2004 for each major central government department and local government. "Appendix A of ESM" shows how departmental budgets are allocated to IMD dimensions. The total spending attributed to each dimension is added together and a percentage of total spend calculated for each dimension. This percentage indicates the emphasis given by local and national government to each IMD dimension, and translates to each dimension's weight given within the overall index. We assume that the national debate acted out within the democratic process affects systems of government and that spending decision are not based on precise accounting processes but rather on a broader debate about the importance of providing social goods to reduce deprivation in specific areas of society. The differential cost of satisfying the same level of need in different dimensions is not accounted for in the wider debate, although may be important in the particular functioning of government. Table 6 reports the total government spend attributed to each IMD dimension, as detailed in "Appendix A (ESM)". Health and Disability and Income Deprivation are given the greatest share of resources and Employment is given the lowest share. The percentage of government spend attributed to each IMD dimension represents the weight that should  Table 6 and Fig. 1 report the government spend-based weights for the IMD.

Discussion
All three empirical methods produce similar weights (Fig. 2), and suggest a close correspondence between what is important to individuals who experience social exclusion, what people say is important when judging hypothetical others and how governments allocate spending to alleviate deprivation. The weights represent a plausible weight range for the IMD within which, for most dimensions, the existing expert elicited weights sit. The weight range is fairly narrow for some dimensions: the Income weights range from 21.60 (PSE) to 25.39 (government spend) and the Education, skills and training weights range from 11.44 (DCE) to 13.02 (PSE). The narrow range indicates a 'consensus' about these dimensions' importance. There is a wider range of weights for other dimensions, however: the Living Environment weights range from 8.16 (government spend) to 24.02 (DCE) and Employment weights range from 3.76 (DCE) to 17.38 (PSE). In these cases, expert opinion is needed to understand why differences arise across the methods, how the methods affect the estimated weights, and to select an appropriate weight. The benefit of the method outlined in this paper is that this sensitivity is identified and the search for an appropriate response prompted.
All three empirical preference-based methods suggest that Employment should be given less weight and that Health and Disability should being given a higher weight than they currently receive in the IMD. The existing weights are 'outliers'. The low Employment weight derived from the DCE implies that respondents do not view unemployment as a significant problem for individuals 'over and above' deprivation in the other IMD dimensions. The low weight from the government spending implies that government does not spend a lot on alleviating unemployment. The PSE weights give a lower weight to Employment than the IMD, but still suggest that employment has a substantial influence on a person's feeling of social exclusion (even after controlling for income deprivation).
It was challenging to map the IMD to the methods used. Apportioning government spending separately to Income, Employment and Education, skills and training is complicated. Much government spending serves more than one purpose: for example, to increase a household's income and to incentivise work or to improve population education and to improve their 'employability'. Figures for Employment and Education domains differ from the original working paper. The Employment domain included spend for Education, Skills and Training in the original working paper (Dibben et al. 2007). On reflection, and to avoid double counting, the authors have removed this spend in the analysis presented here. For the Education Domain, 'Cash' value was used in the original paper, whereas we have now used the 'real terms' value. Doing so gives greater clarity as to how the total was arrived at (see "Appendix A of ESM"). DCE respondents were asked to complete the questionnaire from a societal perspective and to state who should receive government support. This question mimics the IMD purpose, but does not ask who is most deprived. DCE respondents may have considered both the individuals' experiences of deprivation and how effective government support would be in alleviating deprivation. It is reasonable to assume that the government can reduce income deprivation, but should government be in the business of providing employment for all? For the PSE data it was a challenge to identify variables included in the data set that measured each of the dimension indicators in the IMD 2004. For the Income dimension, the PSE data included variables that were similar to four out of five of the IMD indicators (no variable measures asylum seeker support). Whereas, for the Education dimension, the PSE data included variables that were similar to only one out of seven IMD indicators. The PSE data has a variable on the adult respondents' educational attainment and does not included data on children's and young people's educational attainment. Benjamin et al. (2014) ask individuals to choose between two alternative lives that differ in 2, 4 or 6 dimensions to elicit weights for a large set of well-being dimensions, and Adler and Dolan (2008) ask individuals to rank alternative multidimensional lives. Both studies apply stated preference methods similar to the one applied here. One concern about stated preference methods is that choices are hypothetical and therefore are unreliable measures of true preferences. Economists apply stated preference methods to value non-market goods, and studies find significant differences between hypothetical and true valuations (Blumenschein et al. 2008;Harrison and Rutström 2008). However, Benjamin et al. (2014) argue that stated preferences reliability is less problematic when "elicited preferences are used […] normatively". The convergence between our weights provides evidence that stated preference methods elicit reliable preferences for deprivation dimensions.
There is circularity in the use of government spending as a proxy for importance of the different dimensions of deprivation. Voters' preferences are reflected in the election's outcome, but government's spending reflects voter's preference and the marginal effectiveness of spending across different policies. Our method implicitly assumes that, at the margin, spending on education and spending on health will have the same effect on reducing education deprivation and health deprivation, respectively.
Our results have three limitations. First, in the DCE, all dimensions have two outcomes either an individual is deprived or not. For the Living Environment dimension this means the person's was living in "decent housing" or "not decent housing". Respondents' may have had an emotional reaction to the word "decent" and this framing effect could explain the high weight given to Living Environment in the DCE weights (Tversky and Kahneman 1981). Second, the data across the three methods are for different years. The government spend is for 2003/4. The DCE was administered in 2006. The PSE data were collected in 1998/99. These data are close in date to 2004, and therefore are comparable to the expert-based weights chosen for the IMD 2004 and used in all subsequent indices (IMD 2007(IMD , 2010(IMD , 2015. Future research could consider if index weights should change over time and the stability of preference-based weights. Given the current interest in measures of wellbeing, social exclusion should be routinely measured by government using the PSE question or a similar question. Such data would provide the opportunity to explore the stability of preference-based weights. Third, the PSE and DCE weights are based on data concerned with an individual's experience of deprivation. The weights from apportioning government spending are based on spending across England and include spending that is not directed at individuals but at areas. The IMD is a measure of area deprivation. It is an open question whether weights would differ if the general population were asked about an area's deprivation rather than individual deprivation (Atkinson 2003).

Conclusion
The IMD is an important tool for social need and inequality identification. Indices assign weights are either explicitly or implicitly to each dimension. These weights are normative judgements about the each dimension's relative importance for overall deprivation. We apply and compare three empirical methods of deriving preference-based weights for the IMD. We compare weights derived from individuals' experience of social exclusion, a survey exploring the trade-offs society makes between different deprivation dimensions and the apportioning government spending on alleviating deprivation. We find a high degree of correspondence between the weights obtained from each method and between the empirical weights and the weights used since IMD 2004. The preference-based weights derived in this study do not consider the robustness of the data available to measure deprivation across the dimensions and this is taken into account in the weights set in the IMD 2004. Nevertheless, a simple swap of the IMD weights for the Employment and Health and Disability achieves a solution very close to that of the average weights across our three methods.