ABSTRACT
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 HEALTH ACHIEVEMENT AND THE LONGITUDINAL ANALYSIS OF HEALTH INEQUALITIES
 3 LONGITUDINAL ANALYSIS OF CHANGES IN OTHER RANKDEPENDENT INCOMERELATED HEALTH INEQUALITY INDICES OF HEALTH ATTAINMENT
 4 LONGITUDINAL ANALYSIS OF CHANGES IN INCOMERELATED HEALTH INEQUALITY IN HEALTH SHORTFALLS
 5 EMPIRICAL ILLUSTRATION
 6 CONCLUSIONS
 CONFLICT OF INTEREST
 ACKNOWLEDGEMENTS
 REFERENCES
Changes in rankdependent incomerelated health inequality measures over time may usefully be decomposed into contributions due to changes in health outcomes and changes in individuals' positions in the income distribution. This paper establishes the normative implications of this type of decomposition by embedding it within a broader analysis of changes in the ‘health achievement’ index. We further show that the choice of health inequality measure implies a particular vertical equity judgement, which may be expressed on a common scale in terms of the concentration index of health changes that would be inequality preserving. We illustrate the empirical implications of this choice by reporting results from a longitudinal analysis of changes in incomerelated health inequality in Great Britain using the concentration, the Erreygers and Wagstaff indices of health attainments and the concentration index of health shortfalls. Copyright © 2012 John Wiley & Sons, Ltd.
1 INTRODUCTION
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 HEALTH ACHIEVEMENT AND THE LONGITUDINAL ANALYSIS OF HEALTH INEQUALITIES
 3 LONGITUDINAL ANALYSIS OF CHANGES IN OTHER RANKDEPENDENT INCOMERELATED HEALTH INEQUALITY INDICES OF HEALTH ATTAINMENT
 4 LONGITUDINAL ANALYSIS OF CHANGES IN INCOMERELATED HEALTH INEQUALITY IN HEALTH SHORTFALLS
 5 EMPIRICAL ILLUSTRATION
 6 CONCLUSIONS
 CONFLICT OF INTEREST
 ACKNOWLEDGEMENTS
 REFERENCES
In a recent paper, Allanson et al. (2010; hereafter AGP) considered the characterisation and the measurement of incomerelated health inequality (IRHI) using longitudinal data. In particular, they provided a decomposition of the change in the concentration index (CI) between two periods into the following: (i) an incomerelated health mobility index that reveals whether the pattern of individual health changes favours the initially rich or poor and (ii) a healthrelated income mobility index that captures the effect of income reranking, that is, changes in individuals' positions in the income distribution, on crosssectional IRHI. This paper extends this work in two directions.
First, we explore the normative basis of the AGP decomposition of the change in IRHI by embedding it within a broader analysis of the change in the health of the population. Specifically, we note that the CI is the inequality component of the ‘health achievement’ index of Wagstaff (2002). Accordingly, any change in health achievement can be broken down into parts due to changes in mean health and changes in IRHI, with the AGP mobility indices identifying the extent to which changes in IRHI are driven by changes in health outcomes (i.e. ‘health mobility’) and income reranking (i.e. ‘income mobility’).
Second, we show that the AGP decomposition procedure may also be used to analyse the change in other rankdependent IRHI indices and with IRHI measured with respect to health shortfalls rather than attainments. The choice of health inequality measure can affect the ranking of populations according to IRHI in comparative studies, and this sensitivity will inevitably carry over to mobility indices derived from the decomposition of changes in IRHI measures over time. Specifically, Clarke et al. (2002) showed that the use of relative and absolute indices can produce different rankings and, moreover, that whether health is measured with respect to health attainments or shortfalls also matters in the case of relative indices. The sensitivity of relative measures has led to a search for the class of indices that yield either identical or consistent findings irrespective of the choice between health attainments and shortfalls (Erreygers, 2009; Lambert and Zheng, 2011). However, although health attainments and shortfalls are but ‘two sides of the same coin’ (Clarke et al., 2002), indices of inequality measured with respect to health attainments and shortfalls need not be equivalent to each other. In particular, this paper establishes the nature of the distinct vertical equity judgements1 embodied in relative indices of inequality in health attainments and shortfalls, which shows that each may represent ethically defensible positions in specific contexts.
To consider the normative implications of the choice of IRHI measure, we build on the observation that different measures embody different IRHI equivalence criteria reflecting alternative vertical equity judgements.2 An IRHI equivalence criterion specifies how, given the joint distribution of health and income, an additional amount of health should be distributed so as to leave IRHI unchanged. For example, relative and absolute indices are invariant to equiproportionate and uniform changes in health, respectively. But, in contrast to absolute indices, relative indices imply different inequality equivalence criteria depending on whether health is measured with respect to attainments or shortfalls because an equiproportionate change in attainments will not generally constitute an equiproportionate change in shortfalls and vice versa. In this paper, we characterise the IRHI equivalence criteria on a common scale by expressing them in terms of the CI of health changes that would be inequality preserving or distributionally neutral. For each health inequality measure, this provides a comparable benchmark against which to quantify the degree to which the actual distribution of health changes deviates from distributional neutrality and therefore results in a change in measured health inequality.3
In keeping with the income inequality literature (see, e.g. Lambert, 2001), we label the vertical equity judgements implied by the IRHI equivalence criteria of absolute and relative measures of inequality in health attainments as ‘leftist’ and ‘rightist’, respectively. These labels were introduced by Kolm (1976) on the basis that a uniform increase in income levels is commonly perceived as more egalitarian than an equiproportionate rise, although the identification of these positions with the political spectrum is problematic in that an equiproportionate reduction in incomes (e.g. due to the imposition of a tax) is generally viewed as more progressive than a uniform decrease (Atkinson, 1983). Our usage of the labels reflects the fact that health attainment is a ‘good’ like income, and it is similarly apparent in this context that whether ‘leftist’ views would be considered more egalitarian than ‘rightist’ views will depend on whether one is evaluating positive or negative health changes—an egalitarian might well prefer uniform health improvements but equiproportionate health depreciation.
We follow Zheng (2007) in further identifying ‘intermediate’ IRHI measures that increase as a result of equiproportionate improvements in health across income classes but fall in response to uniform improvements, ‘extreme rightist’ measures that fall as a result of equiproportionate health improvements and ‘extreme leftist’ measures that increase in response to uniform improvements. We proceed to demonstrate that relative measures of inequality in health shortfalls reflect an ‘extreme leftist’ view, in contrast to the ‘rightist’ view embodied in relative measures of inequality in health attainment.
2 HEALTH ACHIEVEMENT AND THE LONGITUDINAL ANALYSIS OF HEALTH INEQUALITIES
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 HEALTH ACHIEVEMENT AND THE LONGITUDINAL ANALYSIS OF HEALTH INEQUALITIES
 3 LONGITUDINAL ANALYSIS OF CHANGES IN OTHER RANKDEPENDENT INCOMERELATED HEALTH INEQUALITY INDICES OF HEALTH ATTAINMENT
 4 LONGITUDINAL ANALYSIS OF CHANGES IN INCOMERELATED HEALTH INEQUALITY IN HEALTH SHORTFALLS
 5 EMPIRICAL ILLUSTRATION
 6 CONCLUSIONS
 CONFLICT OF INTEREST
 ACKNOWLEDGEMENTS
 REFERENCES
We draw upon the literature on the ethical foundations of health inequality measurement to elucidate the normative basis of the mobility indices resulting from the longitudinal analysis of IRHI. In particular, we show how the decomposition of changes in IRHI can be embedded within a broader analysis of changes in population health as measured by the ‘health achievement’ index of Wagstaff (2002).
Our analysis focuses on a single transition between an initial period s and some final period f (f > s). Let h_{it} and R_{it} denote, respectively, the health attainment and the fractional income rank of individual i (i = 1, …, n) in period t (t = s, f ), where the health measure lies in the bounded interval a ≤ h_{it} ≤ b with a ≥ 0 by assumption. Following Wagstaff (2002),4 the level of health achievement in any period is defined as the weighted average health of all individuals, where the weights are determined by individuals' ranks in the income distribution. Specifically, let H_{tt} denote ‘health achievement’ evaluated on the basis of health outcomes in period t and fractional income ranks in period t,
 (1)
where the rankdependent weights are given by
 (2)
is the average health attainment and is the extended CI (cf. Yitzhaki, 1983; Wagstaff, 2002). H_{tt} may be interpreted as a Yaaritype (1988) social evaluation function defined over the health domain, where the ‘distributional judgement’ parameter v allows for the calibration of the poverty focus of the evaluation (EssamaNssah, 2005) by controlling the rate at which the weights decrease from poorest to richest (Wagstaff, 2002). Throughout the remainder of this paper, we implicitly assume that v = 2, which yields the (ordinary) CI CI_{tt}.5 H_{tt} cannot be considered as a full social welfare function as it does not allow for the direct effect of the level of income on welfare.6
CI_{tt} provides a measure of relative IRHI, which typically will be positive because of the positive association between income and health attainment. Within the framework provided by Equation ((1)), CI_{tt} may be interpreted as a ‘cost of inequality’ index in the sense of Atkinson (1970), providing a measure of the proportion of average health attainment that could be sacrificed with no loss of ‘health achievement’ if the remainder were to be distributed equally. Bleichrodt and van Doorslaer (2006) have shown that the CI implies social preferences over health distributions must satisfy inter alia the principle of incomerelated health transfers whereby a positive health transfer from a richer to a poorer individual does not lead to a reduction in ‘health achievement’, holding the income rankings of the two individuals constant. If health attainment is an increasing function of income then such transfers will on average be from healthier to unhealthier individuals, but this might not be so in particular cases. Bleichrodt and van Doorslaer (2006, p. 955) concluded that the principle will ‘be more acceptable the stronger the correlation between health and […] income’.7
2.1 Allanson–Gerdtham–Petrie analysis of changes in relative inequality in health attainments
The incomerelated health mobility index in Equation ((4)) captures the effect of health changes on relative IRHI, which is determined by the relationship between relative health changes and initial income ranks,
 (5)
where CI_{f − s,s} is the CI of health attainment changes ranked by initial period income. Hence, AGP evaluated health mobility on the basis of the social weights associated with individuals' initial income ranks, which gives greater weight to the health prospects of those who start with lower income. Following Dardanoni (1993), this asymmetric treatment of individuals can be justified on the grounds that the initially poor are disadvantaged to the extent that they face a worse lottery of future health possibilities than those who are better off. will be positive if the prospect of health mobility is deemed socially desirable in the sense that the resulting distribution of health is less unequal than the initial health distribution, when both are judged on the basis of the social weights from the initial period.
Progressivity is captured by the Kakwanitype (1977) progressivity index P^{CI} = (CI_{ss} − CI_{f − s,s}). P^{CI} will be positive (negative) if the poorest individuals either enjoy a larger (smaller) share of total net health gains or suffer a larger (smaller) share of total net health losses compared with their initial share of health attainment and equals 0 if relative health changes are independent of income. So, for example, positive values of P^{CI} imply that health changes will be equalising for net health improvements and disequalising for net health deteriorations. For any given P^{CI}, the gross impact of health mobility on IRHI is proportional to the scale of health changes . P^{CI} can provide a useful measure of the performance of health improvement programmes in targeting the poor: A given reduction in IRHI can be achieved either by a smallscale but highly targeted intervention or by a largerscale but broader health programme.
3 LONGITUDINAL ANALYSIS OF CHANGES IN OTHER RANKDEPENDENT INCOMERELATED HEALTH INEQUALITY INDICES OF HEALTH ATTAINMENT
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 HEALTH ACHIEVEMENT AND THE LONGITUDINAL ANALYSIS OF HEALTH INEQUALITIES
 3 LONGITUDINAL ANALYSIS OF CHANGES IN OTHER RANKDEPENDENT INCOMERELATED HEALTH INEQUALITY INDICES OF HEALTH ATTAINMENT
 4 LONGITUDINAL ANALYSIS OF CHANGES IN INCOMERELATED HEALTH INEQUALITY IN HEALTH SHORTFALLS
 5 EMPIRICAL ILLUSTRATION
 6 CONCLUSIONS
 CONFLICT OF INTEREST
 ACKNOWLEDGEMENTS
 REFERENCES
Allanson et al. (2010) considered the decomposition of the change in the CI of health attainments, but the same approach may also be applied to a number of alternative rankdependent IRHI indices that have appeared in the literature: the generalised concentration index (GC), relative and slope inequality indices (RII and SII respectively), and the Wagstaff (2005) index (WI) and Erreygers (2009) index (EI) that ‘normalise’ or ‘correct’ the CI to take account of the bounded nature of the health measure. Table 1 provides definitions for all the IRHI indices considered in the study with respect to health attainments, together with the corresponding sets of mobility indices.
The derivation of the incomerelated health and healthrelated income mobility indices, M_{H} and M_{R}, respectively, is straightforward for these other IRHI indices.9 However, the appropriate definition of the progressivity and scale indices, P and q, respectively, is less obvious: We define P such that differences between alternative progressivity indices can be interpreted as reflecting differences between the IRHI equivalence criteria implied by the underlying health inequality measures, with q defined conformably such that M_{H} = Pq.
To this end, we characterise the IRHI equivalence criterion implied by each IRHI index in terms of the CI of health changes that would be inequality preserving or distributionally neutral, where each such benchmark may be expressed as some multiple θ of the CI of initial health CI_{ss}. To quantify the degree to which the actual distribution of health changes deviates from distributional neutrality, we then define P in each case as the difference between the benchmark function θCI_{ss} and the actual CI of health changes CI_{f − s,s}.
P is a dimensionless measure that provides a generalisation of the progressivity index P^{CI} in Equation ((5)). For all IRHI measures that have the same IRHI equivalence criterion, P will be identical, reflecting the common vertical equity judgement, whereas the scale indices q will differ because of the alternative normalisations. More generally, the ratio of P values between any two IRHI measures will be equal to the ratio of incomerelated health mobilities if these are expressed as proportions of the corresponding final health inequalities.10
3.1 Incomerelated health inequality measures embodying a ‘rightist’ inequality equivalence criterion
The decomposition results for the CI were discussed in the previous section. In this case, θ = 1, and the benchmark function θCI_{ss} = CI_{ss} implies that an equiproportionate change in health will be deemed to be inequality preserving. The RII is commonly used instead of CI to measure relative inequality and is similarly invariant to equiproportionate changes. Thus, P^{RII} = P^{CI}, reflecting the common ‘rightist’ vertical equity judgement.
Wagstaff et al. (1991) noted that the RII is equal to the CI divided by twice the variance of the relative rank variable, where var(R) = (n − 1)^{2}/(12n^{2}) is a constant determined by the sample size n (Milanovic, 1997). Thus, the two measures and the resultant mobility and scale indices simply differ by a multiplicative factor, with the RII values being approximately six times the corresponding CI values in large samples.
3.2 Incomerelated health inequality measures embodying a ‘leftist’ inequality equivalence criterion
The GC is an absolute measure of inequality that is invariant to uniform health changes, so θ = 0, and the benchmark function is simply given as θCI_{ss} = 0. may thus be expressed as the product of the progressivity index P^{GC} = −CI_{f − s,s}, which provides an alternative ‘leftist’ indicator of targeting performance to P^{CI}, and a scale factor . The belief that the priority of healthcare policy should be to ‘heal the sick’, who are disproportionately poor, may be taken to imply that healthcare improvements should be equalising not just in relative but also in absolute terms,11 that is, that P^{GC} should be positive for beneficial health interventions.12 Conversely, a uniform loss in health may be seen as less egalitarian than an equiproportionate one given that the poor typically start with worse health.
The SII and EI are similarly invariant to uniform health changes. Thus, P^{GC} = P^{SII} = P^{EI}, reflecting the common ‘leftist’ vertical equity judgement. The SII provides an absolute IRHI index equal to the RII multiplied by mean health, so the set of SII and GC mobility and scale indices differ by the same multiplicative factor as the corresponding RII and CI indices. The EI provides a dimensionless IRHI index that is equal to (4/(b − a))GC, so the EI mobility and scale indices are equal to (4/(b − a)) times the corresponding GC indices.
3.3 Incomerelated health inequality measure embodying a ‘variable’ inequality equivalence criterion
Finally, we note that the WI does not embody some fixed vertical equity judgement, as is the case with rightist and leftist measures, because implies that the benchmark function depends upon the scale of the average health change as well as the initial distribution of health.13 Note first that if , which will be the case if average health in each period is equal to the midpoint of the range,14 then WI embodies a ‘leftist’ vertical equity judgement because will be equal to 0. For low average levels of both initial and final health, , so will be positive, tending to as average initial health tends to a if a = 0 and to positive infinity if a > 0. Thus, in this case, WI will provide an ‘intermediate’ measure if a = 0 and tend to an ‘extreme rightist’ one otherwise. Conversely, for high initial and final average health , so WI will provide an ‘extreme leftist’ measure with tending to negative infinity as average initial health tends to b.15
4 LONGITUDINAL ANALYSIS OF CHANGES IN INCOMERELATED HEALTH INEQUALITY IN HEALTH SHORTFALLS
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 HEALTH ACHIEVEMENT AND THE LONGITUDINAL ANALYSIS OF HEALTH INEQUALITIES
 3 LONGITUDINAL ANALYSIS OF CHANGES IN OTHER RANKDEPENDENT INCOMERELATED HEALTH INEQUALITY INDICES OF HEALTH ATTAINMENT
 4 LONGITUDINAL ANALYSIS OF CHANGES IN INCOMERELATED HEALTH INEQUALITY IN HEALTH SHORTFALLS
 5 EMPIRICAL ILLUSTRATION
 6 CONCLUSIONS
 CONFLICT OF INTEREST
 ACKNOWLEDGEMENTS
 REFERENCES
Thus far, we have focused on incomerelated inequalities in health attainment, but the preceding analysis may also be applied to changes in incomerelated inequalities in health shortfalls or morbidity. For this purpose, let u = (b − h) be some measure of health shortfalls,16 with corresponding bounds 0 ≤ u ≤ (b − a). We first show that the conclusions drawn from the longitudinal analysis of IRHI will be sensitive to the choice between attainments and shortfalls if IRHI is evaluated in relative terms because the two approaches embody fundamentally different vertical equity judgements. We subsequently note that this is not an issue if the IRHI index satisfies the ‘mirror’ condition—that health inequalities measured with respect to attainments and shortfalls are of equal size but of opposite sign—which is the case for all the other IRHI indices considered in this paper (Erreygers, 2009).
Hence, the change in ‘health achievement’ in Equation ((4)) can be rewritten as
 (7)
where is the CI of final period health shortfalls ranked by initial income. Thus, reductions in both average health shortfalls and (the scale of) relative IRHI in shortfalls will improve overall wellbeing, with the AGP decomposition of serving to identify the sources of the latter as before.
The healthrelated income mobility index will be negative if, as expected, is positive, implying that those who move up the income ranking will tend to be less unhealthy in the final period than those who move down. Thus, reranking will generally exacerbate inequalities in health shortfalls as well as in health attainment.
In contrast to relative IRHI indices, the results of the decomposition analysis will be insensitive to the choice between health attainments and shortfalls for any IRHI index that satisfies the ‘mirror’ condition. Specifically, let JJ (JJ = GC, SII, EI, WI) be one such index, then by definition, , with and , and . Thus, the signs of the mobility and scale indices change, but the sign of the progressivity index does not as the vertical equity judgement is the same irrespective of the choice of health measure.
5 EMPIRICAL ILLUSTRATION
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 HEALTH ACHIEVEMENT AND THE LONGITUDINAL ANALYSIS OF HEALTH INEQUALITIES
 3 LONGITUDINAL ANALYSIS OF CHANGES IN OTHER RANKDEPENDENT INCOMERELATED HEALTH INEQUALITY INDICES OF HEALTH ATTAINMENT
 4 LONGITUDINAL ANALYSIS OF CHANGES IN INCOMERELATED HEALTH INEQUALITY IN HEALTH SHORTFALLS
 5 EMPIRICAL ILLUSTRATION
 6 CONCLUSIONS
 CONFLICT OF INTEREST
 ACKNOWLEDGEMENTS
 REFERENCES
In the preceding sections, we have shown how to decompose changes in IRHI for a number of commonly used measures and thereby revealed that between them they embody only four distinct vertical equity judgements. We illustrate the empirical implications of these four vertical equity judgements by reporting results from a study of the dynamics of income and health in Great Britain for the CI, EI and WI of health attainments and for the CI of health shortfalls. Results for the other IRHI measures considered in the paper can readily be derived using the mathematical relationships between the various measures set out earlier.
The empirical analysis employs data from waves 9 and 14 of the British Household Panel Survey (University of Essex, ISER, 2007), which is a nationally representative longitudinal survey of private households in Great Britain, to construct a balanced panel consisting of observations on the subset of individuals in the survey who were alive in 1999 and 2004 and for whom full information on health, income and a range of other socioeconomic variables was available in both years. The resultant sample comprises observations on 9258 individuals in Great Britain. Sample weights were used throughout the analysis, with these being given by a set of adjusted British Household Panel Survey crosssectional weights for 1999, where the adjustments were made using inverse probability weights (Wooldridge, 2002) to allow both for missing data in either year and for nonmortalityrelated sample attrition between the years (see Petrie et al., 2011, for further discussion). Standard errors were generated for all estimates using a bootstrap procedure that took into account the sample design, with resampling carried out at the cluster level (primary sampling unit) rather than at the individual level within each major strata.18
We employ a measure of health attainment that is expressed in terms of QALYs and derived from responses to the SF36 questionnaire using the SF6D preferencebased algorithm (Brazier et al., 2002). This measure is bounded in the unit interval with full health corresponding to a value of 1 and death assigned a score of 0,19 so our measure of health shortfalls is simply equal to one less the measure of health attainment. We rank individuals on the basis of annual household income, equivalised using the McClements scale (Taylor, 1995) to take account of household composition.
Table 2 shows that average health declined over the 5year period, which is to be expected given the balanced nature of the panel. Health attainments were concentrated among the rich and health shortfalls among the poor in 1999, as is indicated by the signs of the corresponding measures of health inequality. The benchmark function value for each measure gives the CI of health changes that would preserve IRHI. For the two relative measures, CI and CI ^{u}, the benchmark values are identical to the corresponding levels of initial inequality, given that equiproportionate health changes are inequality preserving in these cases. But note that such health changes will be concentrated among the rich in the case of the ‘rightist’ CI and among the poor for the ‘extreme leftist’ CI ^{u}. The benchmark value for the ‘leftist’ EI is equal to 0 by definition, given that uniform health changes would be inequality preserving. Finally, given the high levels of average health in both 1999 and 2004, the WI benchmark value implies that health changes would have to be concentrated among the poor to be inequality preserving, making WI an ‘extreme leftist’ measure but less so than CI ^{u}.
Table 2. Decomposition of changes in incomerelated health inequalities   Health attainments   Health shortfalls 


Average initial health outcome   0.8091 (0.0017)   0.1909 (0.0017) 
Average health outcome change   −0.0096 (0.0016)   0.0096 (0.0016) 
Concentration index of health changes   −0.3179 (0.1162)   −0.3179 (0.1162) 
 Concentration index CI  Erreygers index EI  Wagstaff index WI  Concentration index CI^{u} 
Initial inequality  0.0167 (0.0011)  0.0541 (0.0036)  0.0876 (0.0058)  −0.0709 (0.0047) 
Benchmark function value  0.0167 (0.0011)  0 (−)  −0.0533 (0.0035)  −0.0709 (0.0047) 
Change in inequality  0.0084 (0.0012)  0.0262 (0.0039)  0.0377 (0.0062)  −0.0293 (0.0050) 
Incomerelated health mobility  −0.0040 (0.0011)  −0.0122 (0.0036)  −0.0158 (0.0057)  0.0118 (0.0046) 
Progressivity index  0.3347 (0.1158)  0.3179 (0.1162)  0.2646 (0.1177)  0.2471 (0.1183) 
Scale factor  −0.0119 (0.0020)  −0.0382 (0.0064)  −0.0596 (0.0097)  0.0506 (0.0086) 
Healthrelated income mobility  0.0044 (0.0010)  0.0141 (0.0031)  0.0220 (0.0048)  −0.0176 (0.0038) 
Final inequality  0.0251 (0.0013)  0.0803 (0.0039)  0.1253 (0.0059)  −0.1002 (0.0046) 
Health inequality rose over the 5year period according to all the IRHI measures considered in the study, which is evident from the larger magnitudes of all the reported inequality values in 2004. The subsequent decomposition results show how much of the rise in IRHI in each case was driven by changes in health outcomes and how much by income reranking.
Incomerelated health mobility had a disequalising effect in this study, irrespective of the choice of health inequality measure, due to the concentration of health losses among the poor being such that the CI of health changes was more negative than any of the benchmark values. Thus, the progressivity index is positive in each case, with the smallest deviation from neutrality given by the ‘extreme leftist’ CI ^{u}, the largest by the ‘rightist’ CI, and with WI and EI in between. Incomerelated health mobility, expressed as a proportion of final health inequality, was 11.5% in the case of CI ^{u} and 15.9% for CI, with the latter value being 35.3% larger than the former where this ratio is also given by the ratio of the corresponding P values.
Although in this example health mobility was disequalising according to all the measures, this need not be the case. If the value of the CI of health changes had fallen between a pair of benchmark values, then one measure would have implied that health mobility was equalising whereas the other would have suggested it was disequalising. For example, a uniform health loss across all income classes, for which CI_{f − s,s} = 0, would have had an equalising effect according to CI ^{u} and WI, been inequality preserving according to EI and been disequalising according to CI. And if the actual CI of health changes had been more positive than any benchmark value, then health losses would have been equalising according to all measures.
In contrast to incomerelated health mobility, healthrelated income mobility will always be either equalising or disequalising irrespective of the choice of health inequality measure. Indeed, healthrelated income mobility, expressed as a proportion of final IRHI, must be the same across all IRHI measures. In this study, those who moved up the income ranking tended to be healthier in 2004 than those who moved down, such that 17.5% of IRHI in 2004 was due to reranking. Thus, income reranking had more of a disequalising effect than health changes for all the IRHI measures. Overall, between 29.0% and 33.4% of IRHI in 2004 had arisen because of the combination of health and income rank changes since 1999.
6 CONCLUSIONS
 Top of page
 ABSTRACT
 1 INTRODUCTION
 2 HEALTH ACHIEVEMENT AND THE LONGITUDINAL ANALYSIS OF HEALTH INEQUALITIES
 3 LONGITUDINAL ANALYSIS OF CHANGES IN OTHER RANKDEPENDENT INCOMERELATED HEALTH INEQUALITY INDICES OF HEALTH ATTAINMENT
 4 LONGITUDINAL ANALYSIS OF CHANGES IN INCOMERELATED HEALTH INEQUALITY IN HEALTH SHORTFALLS
 5 EMPIRICAL ILLUSTRATION
 6 CONCLUSIONS
 CONFLICT OF INTEREST
 ACKNOWLEDGEMENTS
 REFERENCES
This paper elaborates the approach to the longitudinal analysis of IRHI proposed by Allanson et al. (2010). The resultant contribution is twofold. First, we establish the normative basis of the AGP mobility indices by embedding the AGP decomposition of changes in IRHI within a broader analysis of changes in ‘health achievement’. In particular, we show that the incomerelated health mobility index provides a forwardlooking measure in which the redistributive effect of individuals' health changes are evaluated on the basis of their initial income ranks, with this asymmetric treatment justifiable on the grounds that the initially poor are disadvantaged to the extent that they face a worse lottery of future health possibilities than the better off. We further show that income reranking will lead to a loss of ‘health achievement’ to the extent that it exacerbates IRHI.
Second, we demonstrate that the AGP decomposition procedure may be used to analyse changes in rankdependent IRHI measures other than the CI. The choice of health inequality measure is shown to affect the results of the subsequent longitudinal analysis only to the extent that different measures embody alternative inequality equivalence criteria. In particular, relative measures of inequality in health attainments and in health shortfalls embody fundamentally different vertical equity judgements that may affect whether changes in health outcomes are perceived as equalising or disequalising. In contrast, inequality indices that satisfy the ‘mirror’ condition, such as GC, WI and EI, have the practical property that the conclusions of any longitudinal analysis will be invariant to whether IRHI is measured with respect to health attainments or shortfalls. However, the exact choice of health inequality measure should not be guided by analytical convenience but by perceptions of what distribution of health changes between poor and rich would constitute an improvement in IRHI in a particular context, where this may well depend on whether one is evaluating positive or negative health changes. Studies should therefore justify the vertical equity judgements implied by their choice of health inequality measure or present findings for a range of measures to provide policymakers with a more detailed assessment of the nature of the health inequality changes taking place.