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Med Care. Author manuscript; available in PMC 2010 June 1.

Published in final edited form as:

PMCID: PMC2730922

NIHMSID: NIHMS123366

A Review and Comparison of Methods

Benjamin M. Craig, Assistant Member, Health Outcomes & Behavior Program, Moffitt Cancer Center, Tampa, Florida, and Courtesy Associate Professor, Department of Economics, University of South Florida, Tampa, Florida;

There is rising interest in eliciting health state valuations using rankings. Due to their relative simplicity, ordinal measurement methods may offer an attractive practical alternative to cardinal methods, such as time trade-off (TTO) and visual analog scale (VAS). In this paper, we explore alternative models for estimating cardinal health state values from rank responses in a unique multi-country database. We highlight an estimation challenge pertaining to health states just below perfect health (the ‘non-optimal gap’) and propose an analytic solution to ameliorate this problem.

Using rank, a standardized protocol developed by the EuroQol Group, TTO and VAS responses were collected for 43 health states in eight countries: Slovenia, Argentina, Denmark, Japan, Netherlands, Spain, United Kingdom, and United States, yielding a sample of 179,431 state responses from 11,483 subjects. States were described using the EQ-5D system, which allows for three different possible levels on five different dimensions of health. We estimated conditional logit and probit regression models for rank responses. The regressions included 17 health-state attribute variables reflecting specific levels on each dimension and counts of different levels across dimensions. This flexible specification accommodates previously published valuation models, such as models applied in the United Kingdom and United States. In addition to fitting standard conditional logit and probit models, which assume equal variance across health states (homoskedasticity), we examined a heteroskedastic probit model that assumes no variance for the two points anchoring the scale (“optimal health” and “dead”) and relaxes the equal-variance assumption for all other states. Rank-based predictions for the 243 unique states defined by the EQ-5D system were compared to predictions from conventional linear models fitted to TTO and VAS responses.

By construction, the TTO and VAS models assume no variance around the anchoring states of “optimal health” and “dead.” Mimicking this assumption in the probit rank models helps dissolve the ‘non-optimal gap.’ For all other states, variances in TTO and VAS were negatively associated with mean values, which contradict the assumption of homoskedasticity. Estimated health state values from the heteroskedastic probit model for the ranking data were highly correlated with predictions from both TTO and VAS models for the 243 EQ-5D states. Between VAS and rank-based estimates, Lin’s rho, a measure of agreement, was over 0.98 with a mean absolute difference of 0.028. Corresponding measures of agreement between TTO and rank estimates were 0.96 and 0.12, which is similar to the agreement between VAS and TTO.

Rank-based valuation techniques, which offer advantages of flexibility, generalizability, and ease of administration, may be attractive substitutes for TTO and VAS in the measurement of societal values for health outcomes.

The use of ranking methods in eliciting values for health states (1-4) may facilitate data collection where cardinal methods such as time trade-off (TTO) and visual analogue scale (VAS) are not feasible, for example, in populations with limited literacy and numeracy (5). Even in highly-educated populations, the relative flexibility and ease of administration of ranking methods may make this approach an attractive alternative to standard utility measurement techniques. For instance, the EuroQol group is currently exploring the possibility of using ranking to estimate health-state values in a new 5-level version of the widely used EQ-5D classification system (6-8). Unlike standard gamble and time tradeoff techniques, the use of ranking responses has not been formalized within a decision theoretic framework for constructing quality-adjusted life years (QALYs). However, if rank-based estimates are similar to those from common cardinal methods, ranking and other ordinal measurement approaches may offer useful alternatives for deriving weights for QALYs.

A key challenge in estimating health-state values from ranking data is locating the values on the 0 to 1 scale needed for QALYs. The distance between the top anchor for the scale and the next best health state—the ‘non-optimal gap’—presents three methodological problems (3). First, in most measurement protocols, state of “perfect health” or “optimal health” marks the upper bound of a segmented scale, while random utility models—which underlie strategies for analyzing ordinal health-state comparisons—assume a continuous and unbounded scale. Because ranks are not scale-based, they are not subject to the compression of values in cardinal measures caused by proximity to the upper bound. Second, optimal health is logically dominant over all other states, and this dominance is evidently clear to most respondents. Few subjects report ties or rank optimal health below any other health states. This pattern complicates the estimation of values for states near optimal health since models for ranked data rely on differences in orderings to infer distances in cardinal values. Finally, standard models for analyzing ranking data, as the conditional logit regression model, typically assume that variance in health-state values is constant across states, including the states that anchor the scale (“optimal health” and “dead”). If this assumption is violated—for example, because values for mild states vary less than those for severe states—then distances between mild states will be overestimated in models that assume constant variance. Furthermore, allowing variance around anchor state values is inconsistent with TTO and VAS approaches, which take these values to be fixed.

To address these problems, we consider alternatives to the conditional logit regression model in analyses of health-state rankings in a large, multi-country dataset. In addition to comparing alternative model specifications for ranking data, we also compare rank-based predictions with predictions from VAS and TTO responses for the 243 unique states defined by the EQ-5D system.

Random utility models (RUMs) provide a theoretical framework for health state valuation studies. The utility of a health state, *j*, for an individual, *i*, is represented by:

$${U}_{ij}={\mu}_{j}+{\epsilon}_{ij}$$

(1)

where the state-specific component, *μ _{j}*, depends only on state attributes, and the error term,

$$\mathrm{Pr}({U}_{ij}>{U}_{ik})=\mathrm{Pr}({\epsilon}_{ij}-{\epsilon}_{ik}<{\mu}_{j}-{\mu}_{k})$$

(2)

Because the probability that state *j* is preferred to state *k* depends on the difference in error terms, additive individual fixed effects are not identifiable. An individual’s rankings would be identical if a constant were added to all of her state-specific utilities. On the other hand, whereas additive individual effects “cancel out” in comparisons of two health states, multiplicative effects may not. Some individuals may respond with greater error than others; at the limit, an individual’s orderings may be virtually uncorrelated with the ordering of the means. In this study, we assume homogeneity in individual-specific multiplicative effects.

In previous research, we built on McFadden’s seminal work on discrete choice (9), and assumed that the randomness term, *ε _{ij}*, comes from a type 1 extreme value distribution (EV-1), Pr(

$$\mathrm{Pr}({U}_{ij}>{U}_{ik})=\frac{{e}^{{\mu}_{j}-{\mu}_{k}}}{1+{e}^{{\mu}_{j}-{\mu}_{k}}}$$

(3)

or, equivalently,

$$\mathrm{ln}\left(\frac{\mathrm{Pr}({U}_{ij}>{U}_{ik})}{1-\mathrm{Pr}({U}_{ij}>{U}_{ik})}\right)={\mu}_{j}-{\mu}_{k}$$

(3a)

The difference between two independent EV-1 errors is logistically distributed, which offers advantages of computational simplicity, parsimony, and robustness of logit estimation (10, 11). Equation 3a shows that the difference between two state-specific components equals the log-odds of choosing one over the other.

Two disadvantages of this formulation include the assumption of constant variance across health states (homoskedasticity) and the slight asymmetry of the EV distribution. Alternative distributions may well provide better characterizations of the variance around health-state values. For example, the randomness term, *ε _{ij}*, may be normally distributed (i.e., following a symmetric bell-curve). A symmetric distribution allows the model to produce equivalent results under a complete reversal of order (“palindromic invariance”). This is intuitively appealing since inferences from a particular rank ordering should not vary depending on whether numbers were assigned to ranks from lowest to highest or highest to lowest. Furthermore, the normal specification implies the familiar probit model (12), which lends itself more easily to a heteroskedastic formulation:

$$\mathrm{Pr}({U}_{ij}>{U}_{ik})=\Phi \left(\frac{{\mu}_{j}-{\mu}_{k}}{\sqrt{{\sigma}_{j}^{2}+{\sigma}_{k}^{2}}}\right)$$

(4)

or, equivalently,

$$\sqrt{{\sigma}_{j}^{2}+{\sigma}_{k}^{2}}{\Phi}^{-1}\left(\mathrm{Pr}({U}_{ij}>{U}_{ik})\right)={\mu}_{j}-{\mu}_{k}$$

(4a)

where ${\sigma}_{j}^{2}$ represent the *j*th state’s variance. We compare this heteroskedastic probit model to the standard conditional logit model.

In this study, we pool several country-specific data sets provided by members of the EuroQol Group. All studies presented in this secondary analysis (Table 1) are based on the MVH-study protocol. This protocol, first developed and applied in the United Kingdom, has been described in detail elsewhere (13-15). Details about the small differences in the national replication studies that followed the original UK study are also available (3). As we want to focus on the comparison of ordinal and cardinal measures, we only included studies of the EuroQol Group that had both rank and TTO responses based on the original MVH protocol.

In all of the country studies, health states are characterized using the EQ-5D system, which comprises a set of scores on five dimensions (mobility, self-care, usual activities, pain/discomfort, and anxiety/depression) with three possible levels on each. A vector of these five scores may be used as shorthand in identifying specific health states. For instance, the abbreviation 21122 represents a health state with some problems in walking, no problems with self-care, no problems with performing usual activities, moderate pain, and moderate anxiety.

The original MVH-protocol describes a face-to-face interview with several sections. First, respondents are asked to describe their health using the EQ-5D system. Next, respondents order 15 or more cards each describing a health state, including the anchoring states of 11111 and “immediate death”. Respondents are instructed to assume each health state persists for 10 years, followed by death. Following the ranking exercise, the subjects are asked to place each card on the EQ-VAS.

In the TTO exercise that follows, respondents decide whether ten years in the health state is preferred to ‘immediate death.’ If so, a series of trade-offs are presented to determine the number of years in optimal health, *x*, that is equivalent to 10 years in the state presented. If the respondent prefers `immediate death,’ the interviewer presents an alternative series of trade-offs to determine the period (<10 years) in the health state, *y*, followed by the time in optimal health, (10-*y*), that together are equivalent to `immediate death.’ Thus, the TTO exercise produces either an *x* or *y* response for each health state depending on whether the state is regarded as better or worse than death.

In our pooled dataset, respondent data were excluded for a particular method based on any of the following four criteria: (1) only one or two states were valued (other than 11111, “immediate death”, and “unconscious”); (2) all states were given the same value; (3) all states were valued worse than “immediate death”; or (4) the VAS response of “immediate death” was higher than the response for optimal health. In addition, respondents were excluded from the rank sample if they ranked death equivalent to optimal health (1.5%) and from the VAS sample if they reported a difference of less than 3 points on a 101-point interval scale (1.3%). These five criteria resulted in exclusion of 3.7% of the respondents from the rank sample (N=414), 4.5% from the VAS sample (N=470), and 1.5% from the TTO sample (N=173). No further logical consistency criteria were applied.

For use in QALYs, VAS responses are rescaled using anchor state responses.

$${\mathit{VAS}}_{i,j}^{\ast}=\frac{{\mathit{VAS}}_{i,j}-{\mathit{VAS}}_{i,Dead}}{{\mathit{VAS}}_{i,Optimal}-{\mathit{VAS}}_{i,Dead}}$$

(5)

where *VAS _{i,j}* is the raw VAS valuation by respondent

For TTO responses, different transformations are required for states rated as either better than or worse than being dead. For better-than-dead states the transformation is:

$${\mathit{TTO}}_{i,j}^{\ast}={x}_{i,j}\u221510$$

(6)

where *x _{i.j}* is the (better-than-dead) TTO response for state

For worse-than-dead states, various transformations are possible. The alternative that is theoretically consistent with the TTO question transforms responses as follows:

$${\mathit{TTO}}_{i,j}^{\ast}=-(10-{y}_{i,j})\u2215{y}_{i,j}$$

(7)

where *y _{i.j}* is the (worse-than-dead) TTO response for state

Based on Equation (7), negative TTO values can be large compared to positive values for better-than-dead states. In the case of the MVH-protocol, the largest negative value is -19 or -39, depending on whether tradeoffs are recorded in six-month or three-month units (16, 17). When calculating mean values, large negative values could easily dominate the much smaller positive values. Some researchers have suggested that negative values be transformed into the negative one to zero scale. Dolan (13) used the following transformation:

$${\mathit{TTO}}_{i,j}^{\ast}=-(10-{y}_{i,j})\u221510$$

(8)

while Shaw and colleagues (18) divided the negative values obtained by applying Equation (7) by a constant representing the largest possible negative value allowed by that study’s protocol:

$${\mathit{TTO}}_{i,j}^{\ast}=-(10-{y}_{i,j})\u2215(39\times {y}_{i,j})$$

(9)

While both Dolan and Shaw transformations contradict theory, we transformed all worse-than-dead TTO values in this study using Dolan’s method with the interest of comparability to the prevailing practice in previous work.

VAS and TTO approaches assume the anchor values do not vary. On the other hand, rank-based models based on homoskedastic specifications (e.g. conditional logit) assume the same nonzero variance for these anchor states as for all other states. As a result, the latter models should produce higher values for optimal health and lower values for “immediate death” compared to TTO and VAS models. This difference may explain the presence of the non-optimal gap found by Craig, Busschbach and Salomon (3). On the other hand, heteroskedastic models can restrict the variability of anchor state values, and produce results more similar to TTO and VAS-based estimations.

We estimate values from VAS and TTO using ordinary least squares regression. Because the purpose of this paper is prediction and not proof of statistical significance, the inclusion of random effects—which would produce more conservative standard errors around estimated coefficients—is not necessary.

Under the EV-1 error distribution assumption and no ties in rank, the likelihood for an observed set of rank responses is:

$$L=\prod _{i\in N}\prod _{j\in {K}_{i}}\frac{\mathrm{exp}({\mu}_{j}-{\mu}_{c})}{\sum _{k\in {K}_{i}}{s}_{ijk}\mathrm{exp}({\mu}_{k}-{\mu}_{c})}$$

(10)

where *j* does not equal *k* and *s _{ijk}* is an indicator of the

For identification, the model requires one state-specific component, *μ _{c}*, be constrained to zero. This constraint is equivalent to dividing the numerator and denominator by exp(

As for the other QALY scale anchor, the coefficient for immediate death has an important role but an ungainly premise. If the coefficient is unity, disutility is EV-1 distributed on an inverted QALY scale. Otherwise, disutility is distributed in proportion to EV-1 on the inverted scale. In theory, the coefficient represents log odds that “immediate death” has greater disutility than optimal health. Intuitively, this log odd should equal infinity; however, such an infinite proportionality would violate the QALY scale. In practice, the coefficient on immediate death is identified primarily by the proportions of respondents rating relatively severe states as being worse than dead, rather than by direct comparisons between optimal health and death. For this reason, previous analyses using conditional logit models for empirical ranking datasets have estimated finite coefficients for immediate death and transformed all model coefficients so the estimated coefficient for death equals 1 on the disutility scale (2-4). We adopt this same approach in the present study.

In contrast to the conditional logit, the exploded probit estimation expands ranks into a series of pair wise comparisons. Attributes for states *j* and *k* are included in the likelihood of each pair:

$$L=\prod _{i\in N}\prod _{j\in {K}_{i}}\prod _{\underset{k<j}{\overset{k\in {K}_{i}}{}}}\Phi \left(\frac{-{1}^{{s}_{ijk}}({\mu}_{k}-{\mu}_{j})}{\sqrt{{\sigma}_{j}^{2}+{\sigma}_{k}^{2}}}\right)$$

(11)

where *s _{ijk}* is an indicator of the

Three probit specifications are examined. In the first specification, all state-specific variances are assumed equal, including anchor variances. This homoskedastic specification is similar to the conditional logit model, which has the same number of parameters. In the second specification, we constrained the variances of the QALY scale anchors to be zero. As a result, all pairwise comparisons between optimal health and “immediate death” are logically ranked, contributing no information to the estimation. Aside from anchor-specific heteroskedasticity, we constrained the non-anchor variances to be equal. The third specification relaxes the assumption of equal variances in non-anchor states. Likelihood ratio tests are used to compare these three alternative specifications. We also compare results from the second and third probit models to TTO and VAS predictions, because they share a common scale with fixed anchors.

The probit and logit likelihood functions in Equations (10) and (11) were shown in the absence of ties in rank. To handle ties, we adopt the approach suggested by Efron (19), which replaces each observed tie with two synthetic observations reflecting two possible unambiguous orderings of the given pair of states, each assigned 50% weight in the likelihood.

In all of the regression models for VAS, TTO, and ranking data described above, the state-specific component, *μ*, is modeled as a function of 18 indicator variables: two level indicators for each of the five dimensions, six indicators for the seven possible combinations of any 1s, 2s, or 3s, a count of 2s squared, and a count of 3s squared. In addition to these variables, the rank-based regressions include an indicator for “immediate death,” so predicted values can be anchored on the QALY scale after estimation. An indicator for optimal health is included in the rank-based models, but its coefficient is constrained (as described in the above sections). Both the United States and United Kingdom models published previously are nested within this specification (13, 18).

First, we examined variance specifications from the conditional logit and exploded probit models. If health-state values were distributed EV-1, the estimated coefficient for “immediate death” should be 1; however, the estimated coefficient is 10.28 (95% CI 10.17, 10.39), which clearly rejects the hypothesis that (dis)utility on the QALY scale is distributed EV-1. Using the probit specification, we considered whether utility is distributed standard normal (i.e., whether the variance equals 1). In our first probit model, estimated variance is 0.0161 (95% CI 0.0160, 0.0163), clear evidence against standard normality. These findings confirm that health state values on the QALY scale have smaller variances than those described by EV-1 and standard normal distributions.

Comparing the first two probit specifications (i.e., variable anchors and constant anchors), use of constant anchors increased the maximum predicted value (0.687 to 0.750) and reduced the minimum predicted value (-0.054 to -0.083) compared to variable anchors. The revised scale with constant anchors is comparable to those used by the cardinal measures. For example, the gap between optimal health and the mild health states, found in previous work (3), is likely attributable, in part, to the choice of a scale with variable anchors (i.e., conditional logit). In addition to anchor states having no variance, non-anchor states may have different variances. These findings on anchor-related compression argue against using the conditional logit model with rescaling based directly on observed rankings of optimal health and death, and in favor of the more flexible exploded probit.

Comparing the second and third probit specifications, a likelihood ratio test rejects homoskedasticity in non-anchor states at a significance level of 0.05 (4046.75 χ^{2}(49); p-value<0.001). While we can statistically reject homoskedasticity in non-anchor states, the difference in predicted values between the second and third specifications is small (i.e., mean absolute difference = 0.004). Only the predicted value of ‘pits’ (33333) changes by more than 0.01, and its value decreases from -0.083 to -0.119.

To better understand state-specific heteroskedasticity, we estimated the state-specific means and variances of the adjusted TTO and VAS responses for the 49 non-anchor states represented in the dataset for comparison to the heteroskedastic probit estimates (Figure 1). The patterns from cardinal measures suggest boundary effects in mild states, as exhibited in the negative correlation between means and standard deviations. The estimates from the heteroskedastic probit do not indicate such a relationship. There may be little predictive advantage from allowing state-specific variances in the probit model.

Table 2 describes the coefficients of the linear TTO and VAS models as well as the coefficients from the second and third probit models. Table 3 lists correlations and measures of agreement between ordinal and cardinal values. Using the coefficients from the heteroskedastic probit, Figure 2 further illustrates these relationships.

All predicted values are highly correlated, with Pearson’s rho and Spearman’s rho well above 0.95. The correlation between the rank and VAS predictions is around 0.99. Drawing a 45 degree line (0,0) to (1,1) in Figure 2, we observe VAS values and rank-based values strongly agree. Lin’s rho, a measure of precision and accuracy, is over 0.98 (Table 3). The mean difference between TTO and all other predictions is substantial, over 0.11 on the QALY scale; however, the mean absolute difference between VAS and rank-based predictions is small, around 0.028. The difference between TTO and other predictions is greatest in poorer health states (Figure 2).

This paper introduces a novel method for analyzing health state ranking data and produces an international value set based on eight countries, including five in Europe, one in Asia, one in South America, and one in North America. Key questions for future research concern sample sizes and design components needed for rank-based valuation of health states. The demand for such methodological work is increasing, because of the recognition that TTO exercises may be impractical for developing countries and arbitrary rescaling of TTO negative responses needs to be addressed. Furthermore, new descriptive systems (e.g., a 5-level EQ-5D) are being developed that require valuation studies to inform country-specific health policy making. Cardinal valuation techniques are limited to trade offs in thermometer units (VAS) or in intervals of time (TTO) or risk (standard gamble). Rank based techniques are more flexible and can incorporate a nearly limitless range of tradeoffs.

The strength of agreement between VAS and probit predictions ameliorates a major shortcoming of previous rank-based techniques, relating to the location of estimates on the 0 to 1 QALY scale, and the large gap that can arise between optimal health and the next best state when relying on rankings of these states to estimate distance. The use of constant anchors in the probit model brings the analysis of ranking data in line with approaches for analyzing VAS or TTO data, and shrinks the “non-optimal gap” considerably. We acknowledge, however, that some concerns persist about relying solely on ranking data to define distance between the best and second-best states. Inferences about this distance—whether modeling rankings with constant anchors or not—relies on either inversions or ties in the ordering of the two best states, yet optimal health logically dominates all other states. For distances between pairs of non-anchor states, inversions or ties can result from three different phenomena: heterogeneity across individuals in values for the two states, differences between the two states that fall below a minimal threshold for detection, or errors. For pairs of states that only include the top anchor, the latter two apply. Thus, treating the two types of orderings as if they arise from the same measurement process may not be appropriate. Some component of the non-optimal gap may remain even when using constant anchors. Nevertheless, the empirical finding in this paper that the use of constant anchors produces a close alignment between rank-based and VAS-based estimates indicates that this modeling strategy, in the absence of exogenous information to help define the scale, marks a clear advance over previous methods.

The heteroskedastic probit estimation provides evidence that statistically rejects the assumption of state-specific homoskedasticity; however, the relationship between mean and variance among the health state distributions is not clearly defined. It is important to recognize that the basic structure of estimation models for ranking data makes it difficult to identify the systematic relationship between means and variances of health states that appears in TTO and VAS responses (Figure 1). This is because the ranking of health states will be relatively insensitive to many types of monotonic transformation of the underlying scale. As a simple illustration, imagine marking points along an elastic band representing cardinal valuations for a series of states. Pulling on one end of the band will expand the distances between states near this end, without changing the overall ranking of states. Similarly, compression or expansion of variances in health states that relate systematically to their locations on the scale will largely preserve rank orderings in population-level datasets, which means simultaneous identification of distances on a latent cardinal scale and variances in distributions of individual values on this scale are difficult.

A larger issue is the randomness in rankings that may result from experimental design factors, societal heterogeneity, or individual-specific randomness in addition to inter-state covariance. For example, some respondents may answer with greater measurement error than others due to illiteracy, numeracy, or poor salience. Previous work on logical inconsistencies shows that exclusion of individuals who did not adequately complete a measurement task can significantly change estimates (22). Furthermore, some subjects may better understand a given descriptive system with the evaluation of each state or get bored as the exercise progresses, producing order-dependent variance. Either through practice, education or interest, measurement-related error can cause states to appear more similar, resulting in compression of health state values. By controlling for logical consistency and order of evaluation directly within the estimator, future estimators may reduce compression bias.

In addition to measurement error, some subjects may be more consistent in their preferences than others. Individual-specific heteroskedasticity may be incorporated in rank models, but would likely be difficult to distinguish from measurement error; one possible remedy would be to include individual covariates reflecting cognitive function in the variance regression. Besides individual-specific randomness, levels of agreement about the values of particular health states may vary across different groups of people. While mean differences across countries in TTO and VAS responses have been considered previously (5, 23), we are not aware of studies that have considered cross-national differences in variance. The heteroskedastic probit described here could easily accommodate such analysis..

TTO has been the dominant choice-based method for health state valuation and the only valuation technique that accounts for variation in time. It has been criticized because of its greater cognitive burden, its constant proportionality assumption (24), and its rescaling of “worse than dead” responses to a unit interval (16). Previous research has also shown that subjects provide more logically inconsistent responses using TTO than rank, likely due to its complexity (22). Our results support the use of a simpler valuation exercise. We provide a novel estimation technique built from empirically tested assumptions and a strong theoretical foundation in random utility. The prediction results strongly agree with those of VAS and are representative of an international sample of eight countries. More research is needed to design valuation studies that best utilize these rank-based innovations.

Benjamin M. Craig, Assistant Member, Health Outcomes & Behavior Program, Moffitt Cancer Center, Tampa, Florida, and Courtesy Associate Professor, Department of Economics, University of South Florida, Tampa, Florida.

Jan J. V. Busschbach, Professor, Department of Medical Psychology and Psychotherapy Erasmus MC.

Joshua A. Salomon, Associate Professor of International Health, Department of Population and International Health, Harvard University.

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