Longitudinal methods to investigate the role of health determinants in the dynamics of income-related health inequality

Abstract

The usual starting point for understanding changes in income-related health inequality (IRHI) over time has been regression-based decomposition procedures for the health concentration index. However the reliance on repeated cross-sectional analysis for this purpose prevents both the appropriate specification of the health function as a dynamic model and the identification of important determinants of the transition processes underlying IRHI changes such as those relating to mortality. This paper overcomes these limitations by developing alternative longitudinal procedures to analyse the role of health determinants in driving changes in IRHI through both morbidity changes and mortality, with our dynamic modelling framework also serving to identify their contribution to long-run or structural IRHI. The approach is illustrated by an empirical analysis of the causes of the increase in IRHI in Great Britain between 1999 and 2004.

1. Introduction

Significant socioeconomic inequalities in health have persisted, or even increased, in many European countries despite considerable improvements in average health (Van Doorslaer and Koolman, 2004; Kunst et al., 2005), leading governments to recognise the need to tackle health inequalities. For example, England committed to reduce socioeconomic inequalities in both infant mortality and life expectancy at birth (Department of Health, 2008). Marmot (2010) argues that such goals cannot be achieved through health policies and health care systems alone but will require action across all the socioeconomic determinants of health because health inequalities are caused by social and economic inequalities in society. It is therefore important to understand how wider changes in socioeconomic conditions impact on health inequalities in order to shape the design of an effective set of public policies to tackle the issue.

There is a considerable body of work in health economics on socioeconomic health inequalities and their determinants (O’Donnell et al., 2008), which focuses primarily on income-related health inequality (IRHI). The main measure of IRHI within this literature is the concentration index (CI), which captures the extent to which good health in any period is concentrated among the rich compared to the poor and is equal to twice the covariance between health and income rank normalised by average health. However the value of this simple bivariate measure is determined not only by the direct relationship between income and health but also by other factors, such as age, gender and lifestyle choices, to the extent that they affect health and are correlated with income. This has led to the development of cross-sectional regression-based decomposition techniques to identify the contribution of inequalities in the individual determinants of health to overall health inequality (see Gravelle, 2003, for an exposition).

This methodology has subsequently been extended to identify the source of changes in IRHI in a population over time by comparing the decompositions from repeated cross-sectional surveys (see, e.g., Wagstaff et al., 2003; Gravelle and Sutton, 2003). In particular, Wagstaff et al. (2003) propose a ‘total differential’ decomposition of the change in the cross-sectional CI between two periods that serves to reveal the contributions of changes not only in the distribution of the determinants of health across income classes but also in the cross-sectional effects of those determinants on health. However, while this type of comparative analysis can be helpful in identifying the determinants of changes in IRHI in a population over time, certain limitations compel a degree of caution. First, as is recognised in Wagstaff et al. (2003), the causal interpretation of the results is problematic because, among other things, the cross-sectional regression model estimates will generally be biased if the health function is dynamic rather than static.¹ There is now considerable evidence of the state-dependence of health (see, e.g. Benzeval and Judge, 2001; Contoyannis et al., 2004), due to the persistence of health conditions, to suggest that dynamic models are more appropriate than static ones.² Second, there are important aspects of the underlying determinants of IRHI changes that cannot be revealed by simply examining changes in cross-sectional relationships and inequalities. In particular, the repeated cross-sectional approach cannot be used to identify the effect of deaths on IRHI (Petrie et al., 2011) and hence the impact of the determinants of mortality on IRHI are also not separately identifiable.

The main aim of this paper is to establish alternative procedures, based on longitudinal or panel data, to investigate the socioeconomic determinants of the health transition process that in part drives changes in cross-sectional IRHI.³ The use of longitudinal data allows the analysis of changes in IRHI to be based on a regression model that captures both the dynamics of morbidity changes and mortality. Our dynamic modelling framework also generates a measure of steady-state or equilibrium health that provides the basis for a complementary analysis of the structural determinants of chronic or persistent IRHI.

Our starting point is the decomposition procedure in Petrie et al. (2011), which shows how the change in IRHI between two periods arises from a combination of changes in health outcomes (i.e. “income-related health mobility”) and changes in individuals’ positions in the income distribution (i.e. “health-related income mobility”). We briefly review this procedure in Section 2 before showing, in the main contribution of the current paper, how it can be expanded upon to explore the contributions from health determinants through the use of regression-based decomposition techniques analogous to those available in the literature for the CI (see Gravelle, 2003). Specifically, we first explain health changes in Section 3 by considering a Two-Part Model which accounts for both morbidity changes and mortality and then show in Section 4 how this may be used to further decompose income-related health mobility into the individual contributions of the various health determinants.⁴ In addition, we demonstrate how our dynamic modelling framework can also be used to analyse the health determinants of chronic or equilibrium IRHI. We employ our procedures in Section 5 to investigate the dynamics of IRHI in Great Britain over the five year period 1999 to 2004 using Quality Adjusted Life Years (QALYs). The final section discusses the contribution of the paper.

2. Review of methods to account for changes in IRHI using longitudinal data

Petrie et al. (2011) propose a decomposition procedure to identify the contributions of health changes and income re-ranking to changes in IRHI between two periods. This section briefly outlines their procedure in order to provide the basis for our subsequent analysis.

Assigning the dead a health status of zero⁵, Petrie et al. (2011) propose the following decomposition of the change in CI from an initial period s to a final period f:

$${\widetilde{CI}}_{ff}^h-C{I}_{ss}^h=({\widetilde{CI}}_{ff}^h-C{I}_{fs}^h)+(C{I}_{fs}^h-C{I}_{ss}^h)=M^R-M^H$$ (1)

where a tilde above a measure indicates that it is defined only over the sub-population in the initial period who survive to the final period $\left\{{\Omega }_s\cap {\Omega }_f\right\}$ rather than the entire population in the initial period Ω_s.⁶ Thus the CI in the initial period $C{I}_{ss}^h=2\mathrm{cov}(h_{is},R_{is})/{\overline{h}}_s$ is defined over Ω_s and is twice the covariance between individuals’ health h_is and fractional income rank R_is normalised by average health ${\overline{h}}_s$, whereas the CI in the final period ${\widetilde{CI}}_{ff}^h=2\widetilde{\mathrm{cov}}(h_{if},{\tilde{R}}_{if})/{\tilde{\overline{h}}}_f$ is analogously defined but only over the sub-set of individuals who survive to the final period $\left\{{\Omega }_s\cap {\Omega }_f\right\}$. Finally, $C{I}_{fs}^h=2\mathrm{cov}(h_{if},R_{is})/{\overline{h}}_f$ is defined over Ω_s as the CI of final period health h_if with respect to initial period income ranks R_is.⁷

The index M^R in (1) provides a measure of health-related income mobility, which captures the effect on IRHI changes of income reranking not only due to the re-shuffling of survivors within the income distribution, but also due to the loss of individuals from the population as a result of death (see Petrie et al., 2011, for further discussion). M^H is interpreted as a measure of income-related health mobility, which captures the effect on IRHI changes of differences in relative health changes between individuals with different levels of initial income. For the rest of the paper we focus on explaining the determinants of income-related health mobility rather than health-related income mobility.

M^H will be zero if relative health changes are unrelated to initial income rank and will be positive if health changes are equalising, which will be the case if the poor either enjoy a larger share of total health gains or suffer a smaller share of total health losses compared to their initial share of health. M^H in turn depends on the progressivity and scale of health changes:

$$M^H=Pq=\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta h}\right)\left(\frac{{\overline{\Delta h}}_f}{{\overline{h}}_f}\right)$$ (2)

where $C{I}_{fs}^{\Delta h}=2\mathrm{cov}(\Delta h_{if},R_{is})/{\overline{\Delta h}}_f$ is the CI of health changes Δh_if = h_if − h_is ranked by initial income, and ${\overline{\Delta h}}_f$ is the average health change between the two periods. Progressivity is captured by the disproportionality index $P=\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta h}\right)$, which will be positive if the poor experience a larger share of total net health changes than their initial share of health. For any given P, the gross redistributive effect M^H is proportional to the scale of net health changes $q=\left({\overline{\Delta h}}_f/{\overline{h}}_f\right)$. Note that positive values of P imply that health changes will be equalising for net health improvements and disequalising for net health deteriorations.

M^H is defined over Ω_s and therefore captures the effect on the CI of relative health changes due to both morbidity changes and mortality. To distinguish ex-post the impact of these two distinct health outcomes on income-related health mobility these health change types are denoted by the superscripts MB and MT, respectively, with their separate contributions to M^H identified as:

$$M^H=Pq=P^{MB}{q}^{MB}+P^{MT}{q}^{MT}=\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta MB}\right)\left(\frac{{\overline{\Delta h}}_f^{MB}}{{\overline{h}}_f}\right)+\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta MT}\right)\left(\frac{{\overline{\Delta h}}_f^{MT}}{{\overline{h}}_f}\right)$$ (3)

where ${\overline{\Delta h}}_f={\overline{\Delta h}}_f^{MB}+{\overline{\Delta h}}_f^{MT}$, with $\Delta h_{if}^{MB}$ set equal to the morbidity change for survivors and zero for those that die, and $\Delta h_{if}^{MT}$ defined as the loss of health due to mortality for those who die and zero for survivors; and $C{I}_{fs}^{\Delta MB}$ and $C{I}_{fs}^{\Delta MT}$ are the CIs of health changes due to morbidity changes and mortality, respectively.

Finally, we note that if there are no deaths then (1) yields the Allanson et al. (2010) decomposition:

$$\begin{array}{l}{\widetilde{CI}}_{ff}^h-{\widetilde{CI}}_{ss}^h=\left({\widetilde{CI}}_{ff}^h-{\widetilde{CI}}_{fs}^h\right)+\left({\widetilde{CI}}_{fs}^h-{\widetilde{CI}}_{ss}^h\right)\\ ={\tilde{M}}^R-\left({\widetilde{CI}}_{ss}^h-{\widetilde{CI}}_{fs}^{\Delta h}\right)\frac{{\widetilde{\overline{\Delta h}}}_f}{{\tilde{\overline{h}}}_f}={\tilde{M}}^R-\tilde{P}\tilde{q}={\tilde{M}}^R-{\tilde{M}}^H\end{array}$$ (4)

where not just ${\widetilde{CI}}_{ff}^h$ but also all other statistics have been defined over $\left\{{\Omega }_s\cap {\Omega }_f\right\}$, which will be identical to Ω_s in this special case. More generally, (4) will prove useful in providing the basis for the analysis of the impact of morbidity changes on IRHI within the initial period sub-population that survive until the final period.⁸

In the following two sections we expand upon the Petrie et al. (2011) decomposition analysis by developing a regression-based decomposition procedure to identify the socioeconomic determinants of income-related health mobility. This is useful since M^H is determined not only by the direct relationship between health changes and initial income but also by other factors that affect health mobility and are correlated with initial income.

3. Modelling the determinants of mortality and morbidity changes

To identify the socioeconomic determinants of M^H, we first need a model which explains health changes due to both mortality and morbidity changes. We use a Two-Part Model (TPM; see Leung and Yu (1996) or Puhani (2000) for a discussion) for this purpose:

$$S_{i,t+1}^{*}={\gamma }_0+\sum _{k=1}^K{\gamma }_k{x}_{kit}+{\gamma }_h{h}_{it}+{\vartheta }_{i,t+1};{\vartheta }_{i,t+1}\sim N\left(\mathrm{0,1}\right);$$ (5a)

$$\Delta h_{it+1}=\left\{\begin{array}{cc}\hfill -h_{it};\hfill & \hfill \text{if}{S}_{i,t+1}^{*}<0\hfill \\ \hfill f(\Delta x_{ki,t+1},x_{ki,t},h_{it});\hfill & \hfill \text{if}{S}_{i,t+1}^{*}\ge 0\hfill \end{array}\right)\forall i\in \left\{{\Omega }_1\cap {\Omega }_t\right\}\text{for}t=\mathrm{1,...}T-1\begin{array}{c}\hfill (5\text{b})\hfill \\ \hfill (5\text{c})\hfill \end{array}$$

which is defined over a T-period panel that is only unbalanced due to death and hence, in period t, comprises the subset of individuals $\left\{{\Omega }_1\cap {\Omega }_t\right\}$ remaining alive through to that period. The first part of the TPM, (5a), determines the probability of survival until period t + 1 as a function of health h_it in the preceding period and a set of mortality risk factors x_kit (k = 1,…,.K). This is assumed to take the form of a standard probit model, with the link between the value of the latent index variable $S_{i,t+1}^{*}$ and observable survival status S_i,t+1 following the rule that S_i,t+1 = 1 (alive in period t + 1) if $S_{i,t+1}^{*}\ge 0$ and S_i,t+1 = 0 (dead in period t + 1) otherwise. For those who die, the change in health Δh_it+1 is equal to −h_it in (5b) since their health status in period t + 1 is zero by definition. For those who survive, Δh_it+1 is given by the dynamic health function f(Δx_ki,t+1, x_ki,t, h_it) in (5c), where it is plausible to assume that the set of morbidity determinants is the same as the survival function risk factors.⁹

To model the dynamic health changes of the surviving population we follow Hauck and Rice (2004) and Contoyannis et al. (2004), among others, in assuming the existence of a stable dynamic health function. We specify a similar but less restrictive health function that takes the form of a first-order autoregressive distributed lag model with fixed effects, which allows for both current and lagged effects of health determinants x_k (k = 1,…K), health persistence and individual heterogeneity:¹⁰

$$h_{i,t+1}={\alpha }_0+\sum _{k=1}^K{\delta }_k{x}_{ki,t+1}+\sum _{k=1}^K{\alpha }_k{x}_{kit}+(1-\lambda )h_{it}+{\eta }_{i,t+1};\forall i\in \left\{{\Omega }_1\cap {\Omega }_{t+1}\right\};t=1,\dots ,T-1$$ (6)

which is defined over an unbalanced panel that in period t comprises the subset of the original population who survive until at least the following period; and where η_i,t+1 = λμ_i + ɛ_i,t+1 is an error term, composed of a fixed individual effect λμ_i and a period specific disturbance η_i,t+1. Eq. (6) may also usefully be expressed in the form of an Error Correction Model (ECM) of the change in health:

$$\Delta h_{i,t+1}=\sum _{k=1}^K{\delta }_k\Delta x_{ki,t+1}+\lambda \left(h_{it}^{*}-h_{it}\right)+{\epsilon }_{i,t+1}$$ (7)

where

$$h_{it}^{*}={\beta }_0+\sum _{k=1}^K{\beta }_k{x}_{kit}+{\mu }_i$$ (8)

may be interpreted as a long-run steady-state or equilibrium health function with parameters ${\beta }_0={\alpha }_0/\lambda$ and ${\beta }_k=\left({\alpha }_k+{\delta }_k\right)/\lambda$, such that $\left(h_{it}^{*}-h_{it}\right)$ corresponds to the ‘equilibrium error’ in period t and $\lambda \left(0\le \lambda \le 1\right)$ determines the rate of adjustment towards equilibrium. Hence, the change in health depends on the effects of contemporaneous changes in the determinants, the initial extent of any disequilibrium in health and the size of the idiosyncratic health shock. Eq. (6) collapses to the static model $h_{i,t+1}=h_{i,t+1}^{*}+{\epsilon }_{i,t+1}$ if there are no lagged effects of health determinants and full adjustment/no persistence in health (i.e. if the α_k s (k = 1,…,K) all equal zero and λ = 1).

ECMs are applicable to stationary, as well as co-integrated non-stationary, series where changes in exogenous factors have differing short and long-term effects or there is persistence of shocks (see Castle et al., 2010; De Boef and Keele, 2008; Hendry, 2006; Banerjee et al., 1990; Wickens and Breusch, 1988). For example, in the current context, when a regular smoker quits there are likely to be both short-term as well as long-term benefits for health: In the ECM, the short-term effects are captured by the impact on health from the contemporaneous change in cigarette consumption, while the long-term effects are captured through the impact of the equilibrium error such that the individual's health slowly improves back to that of a non-smoker. Similarly, it might take the individual a number of periods to fully recover from the effects of an exogenous health shock, such as a car accident, with the degree of persistence inversely related to the rate of adjustment towards equilibrium λ. More generally, current health is determined not only by current health shocks and levels of health determinants but also by the health shocks and levels of health determinants experienced over the entire life course of an individual (see e.g. Galobardes et al., 2007).

The ECM, thus, provides a parsimonious representation of the complex lagged response to health from changes in health determinants and the persistence of health shocks. For our analytical purposes, the main attraction of this representation is the clear distinction between the short-run dynamics and the implied long-run health relationship (see Wickens and Breusch, 1988). In particular, it is possible using the ECM to identify both the short-term impact on IRHI due to contemporaneous changes in health determinants and also how these factors contribute to chronic or persistent IRHI.¹¹

4. Analysing the determinants of income-related health mobility

This section shows how the Two-Part Model may be used to analyse the role of health determinants in driving changes in IRHI through both morbidity changes and mortality, and to identify their contribution to long-run or structural IRHI. We first consider the impact of morbidity changes on IRHI changes within the sub-population in the initial period that survive until the final period, as this serves to clarify links with existing procedures for the decomposition of the CI.

4.1. An analysis of the determinants of morbidity changes conditional on survival

The ECM model of morbidity changes (7) may be used to analyse the socioeconomic determinants of income-related health mobility conditional upon survival. Specifically, if we consider consecutive periods such that f = s + 1 (s = 1, ..., T − 1) then ${\tilde{M}}^H$ in the sub-population $\left\{{\Omega }_1\cap {\Omega }_f\right\}$¹² may be decomposed to yield:

$${\tilde{M}}^H=\tilde{P}\tilde{q}\left\{\sum _{k=1}^K\left({\widetilde{CI}}_{ss}^h-{\widetilde{CI}}_{fs}^{\Delta k}\right)\frac{{\hat{\delta }}_k{\widetilde{\overline{\Delta x}}}_{kf}}{{\widetilde{\overline{\Delta h}}}_f}+\left({\widetilde{CI}}_{ss}^h-{\widetilde{CI}}_{ss}^{EqE}\right)\frac{\hat{\lambda }(\widetilde{\overline{h_c^{*}-h_s}})}{{\widetilde{\overline{\Delta h}}}_f}+\left({\widetilde{CI}}_{ss}^h-{\widetilde{CI}}_{fs}^{\hat{e}}\right)\frac{{\tilde{\overline{e}}}_f}{{\widetilde{\overline{\Delta h}}}_f}\right\}\left(\frac{{\widetilde{\overline{\Delta h}}}_f}{{\tilde{\overline{h}}}_f}\right)$$ (9)

where ${\widetilde{\overline{\Delta x}}}_{kf}$ is the average change in morbidity determinant k, with ${\widetilde{CI}}_{fs}^{\Delta k}$ being the corresponding CI ranked by initial income; the ${\hat{\delta }}_k$ s, and $\hat{\lambda }$ are estimates of the corresponding parameters of the dynamic health function (7); $\left(\tilde{\overline{{\hat{h}}_s^{*}-h_s}}\right)$ is the mean predicted equilibrium error in period s, with ${\widetilde{CI}}_{ss}^{EqE}=2\widetilde{\mathrm{cov}}\left(\left({\hat{h}}_{is}^{*}-h_{is},{\tilde{R}}_{is}\right)\right)/(\tilde{\overline{{\hat{h}}_s^{*}-h_s}})$ the corresponding CI ranked by initial income; and ${\tilde{\overline{e}}}_f$ is the mean regression residual $e_{if}={\hat{\epsilon }}_{if}$, with ${\widetilde{CI}}_{fs}^e$ the corresponding CI ranked by initial income.¹³ Hence (9) may be written as:

$$\begin{array}{l}{\tilde{M}}^H=\tilde{P}\tilde{q}=\sum _{k=1}^K\left({\widetilde{CI}}_{ss}^h-{\widetilde{CI}}_{fs}^{\Delta k}\right)\frac{{\hat{\delta }}_k{\widetilde{\overline{\Delta x}}}_{kf}}{{\tilde{\overline{h}}}_f}+\left({\widetilde{CI}}_{ss}^h-{\widetilde{CI}}_{ss}^{EqE}\right)\frac{\hat{\lambda }(\tilde{\overline{{\hat{h}}_s^{*}-h_s}})}{{\tilde{\overline{h}}}_f}+\left({\widetilde{CI}}_{ss}^h-{\widetilde{CI}}_{fs}^e\right)\frac{{\tilde{\overline{e}}}_f}{{\tilde{\overline{h}}}_f}\\ =\sum _{k=1}^K{\tilde{P}}_{\Delta k}{\tilde{q}}_{\Delta k}+{\tilde{P}}_{EqE}{\tilde{q}}_{EqE}+{\tilde{P}}_e{\tilde{q}}_e\end{array}$$ (10)

where ${\tilde{M}}^H$ can be viewed as the sum of contributions due to changes in the K health determinants, the predicted disequilibrium error and contemporaneous health shocks in (7). Each term in (10) is expressed in terms of the scale and progressivity of the health changes due to that element, with this further decomposition revealing how the average level of health changes and their distribution across initial income ranks respectively impact on income-related health mobility. For example, a positive scale index ${\tilde{q}}_{\Delta k}$ implies a positive average health impact due to the changes in the kth health determinant and if the poor enjoy a larger share of these health gains than their initial share of health then the progressivity index ${\tilde{P}}_{\Delta k}$ will also be positive giving rise to a positive impact on ${\tilde{M}}^H$ and thus a reduction in IRHI.

The interpretation of ${\tilde{P}}_{EqE}$ and ${\tilde{q}}_{EqE}$ are similar in terms of the impacts of health changes due to the process of adjustment towards the equilibrium levels of health implied by individuals’ initial conditions, where this process may generally be expected to have a negative impact on ${\tilde{M}}^H$ and hence exacerbate IRHI. To see this point, note that the contribution of the equilibrium error to ${\tilde{M}}^H$ can be expressed as

$${\tilde{P}}_{EqE}{\tilde{q}}_{EqE}=\left({\widetilde{CI}}_{ss}^h-\frac{{\widetilde{\overline{\hat{h}}}}_s^{*}{\widetilde{CI}}_{ss}^{{\hat{h}}^{*}}-\tilde{\overline{h}}{\widetilde{CI}}_{ss}^h}{(\tilde{\overline{{\hat{h}}_s^{*}-h_s}})}\right)\frac{\hat{\lambda }(\tilde{\overline{{\hat{h}}_s^{*}-h_s}})}{{\tilde{\overline{h}}}_f}=\left({\widetilde{CI}}_{ss}^h-{\widetilde{CI}}_{ss}^{{\hat{h}}^{*}}\right)\frac{\hat{\lambda }{\widetilde{\overline{\hat{h}}}}_s^{*}}{{\tilde{\overline{h}}}_f}$$ (11)

where ${\widetilde{\overline{\hat{h}}}}_s^{*}$ and ${\widetilde{CI}}_{ss}^{{\hat{h}}^{*}}$ are the mean and CI respectively of implied equilibrium health in the initial period. Interpreting ${\widetilde{CI}}_{ss}^{{\hat{h}}^{*}}$ as a measure of chronic IRHI then one would expect $\left({\widetilde{CI}}_{ss}^h-{\widetilde{CI}}_{ss}^{{\hat{h}}^{*}}\right)<0$ in the light of the empirical evidence that IRHI is worse in the long-run than in the short-run (see e.g. Jones and Lopez Nicolas, 2004; Allanson et al., 2010).¹⁴ Thus, ${\tilde{P}}_{EqE}{\tilde{q}}_{EqE}$ will generally be negative since $\hat{\lambda }{\widetilde{\overline{\hat{h}}}}_s^{*}/{\tilde{\overline{h}}}_f$ will be positive if, as will usually be the case, average health is positive and 0 ≤ λ ≤ 1.

4.2. Incorporating the determinants of mortality

Extending the regression decomposition analysis to incorporate those who die is more problematic as the inherent non-linearity of the TPM (5) does not lend itself to being used in a simple decomposition of M^H into its determinants. To address this problem we adopt a hierarchical decomposition procedure in which we first break down M^H into elements due to health changes resulting from expected morbidity changes, expected mortality and health shocks, and then further decompose the first two of these elements to determine the separate contributions of the health determinants.

For this purpose, we note that individual health changes between any two consecutive periods, s and f (s = 1, ..., T − 1), may be written from (5) and (7) as

$$\begin{array}{l}\Delta h_{if}=E\left(\Delta h_{if}\right)+{\upsilon }_{if}\\ =E\left(\Delta h_{if}^{MB}\right)+E\left(\Delta h_{if}^{MT}\right)+{\upsilon }_{if}\\ =\left\{\text{Prob}\left(S_{if}=1\right)E\left(\Delta h_{if}|S_{if}=1\right)\right\}+\left\{\text{Prob}\left(S_{if}=0\right)\left(0-h_{is}\right)\right\}+{\upsilon }_{if}\\ =\left\{\Phi \left(z_{is}\right)\left(\sum _{k=1}^K{\delta }_k\Delta x_{kif}+\lambda \left(h_{is}^{*}-h_{is}\right)\right)\right\}-\left\{\left(1-\Phi \left(z_{is}\right)\right)h_{is}\right\}+{\upsilon }_{if};\forall i\in \left\{{\Omega }_1\cap {\Omega }_s\right\}\end{array}$$ (12)

where $z_{is}={\gamma }_0+{\sum }_{k=1}^K{\gamma }_k{x}_{kis}+{\gamma }_h{h}_{is}$, and $\Phi \left(\cdot \right)$ denotes the cumulative density function of the standard normal distribution. Hence, the health change of an individual from the original population who has survived until period s will be the sum of the expected health change $E\left(\Delta h_{if}\right)$ and an error term υ_if that captures the effect of health shocks.¹⁵ The decomposition of the former into morbidity-related and mortality-related components, $E\left(\Delta h_{if}^{MB}\right)$ and $E\left(\Delta h_{if}^{MT}\right)$ respectively, parallels that underlying (3).¹⁶ Thus the first stage of the decomposition procedure straightforwardly yields:

$$\begin{array}{l}{M}^H=\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta h}\right)q=\left(C{I}_{ss}^{\Delta h}-\frac{2}{{\overline{\Delta h}}_f}\mathrm{cov}\left(\Delta m{b}_{if}+\Delta m{t}_{if}+u_{if},R_{is}\right)\right)\frac{{\overline{\Delta h}}_f}{{\overline{h}}_f}\\ =\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta mb}\right)\frac{{\overline{\Delta mb}}_f}{{\overline{h}}_f}+\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta mt}\right)\frac{{\overline{\Delta mt}}_f}{{\overline{h}}_f}+\left(C{I}_{ss}^h-C{I}_{fs}^u\right)\frac{{\overline{u}}_f}{{\overline{h}}_f}\\ =P^{E(MB)}{q}^{E(MB)}+P^{E(MT)}{q}^{E(MT)}+P^u{q}^u=Pq;\forall i\in \left\{{\Omega }_1\cap {\Omega }_s\right\}\end{array}$$ (13)

where $\Delta m{b}_{if}\equiv {\widehat{\Delta h}}_{if}^{MB}$, $\Delta m{t}_{if}\equiv {\widehat{\Delta h}}_{if}^{MT}$ and $u_{if}\equiv {\hat{\upsilon }}_{if}$ are the sample counterparts of $E\left(\Delta h_{if}^{MB}\right)$, $E\left(\Delta h_{if}^{MT}\right)$ and υ_if respectively, with corresponding means, ${\overline{\Delta mb}}_f$, ${\overline{\Delta mt}}_f$ and ${\overline{u}}_f$, and CIs, $C{I}_{fs}^{\Delta mb}$, $C{I}_{fs}^{\Delta mt}$ and $C{I}_{fs}^u$, defined over all those from the original population who survive until the initial period s.¹⁷ We note that (13) provides estimators of the progressivity and scale indices in (3), given that $C{I}_{fs}^{\Delta mb}$ and $C{I}_{fs}^{\Delta mt}$ only depend on expected morbidity-related and mortality-related health changes conditional on income rank (see Duclos et al., 2003).

In the second stage we make use of the Taylor-series expansions of $E\left(\Delta h_{if}^{MB}\right)$ and $E\left(\Delta h_{if}^{MT}\right)$ to obtain the following linear approximations:

$$E\left(\Delta h_{if}^{MB}\right)=\left\{\sum _{k=1}^K{\rho }_{kis}\Delta x_{kif}+{\rho }_{EqE,is}\left(h_{is}^{*}-h_{is}\right)\right\}+\left\{{\pi }_{0is}+\sum _{k=1}^K{\pi }_{kis}{x}_{kis}+{\pi }_{his}{h}_{is}\right\}+{\varpi }_{if};$$ (14a)

$$E\left(\Delta h_{if}^{MT}\right)={\tau }_{0is}+\sum _{k=1}^K{\tau }_{kis}{x}_{kis}+{\tau }_{his}{h}_{is}+{\omega }_{if};$$ (14b)

where the ρs, πs and τs are the parameters from the linearisation, for which definitions are given in Appendix A, and ϖ_if and ω_if are approximation errors. The terms in the first bracket of (14a) capture the influence on expected morbidity changes of the determinants of morbidity change conditional on survival $E\left(\Delta h_{if}|S_{if}=1\right)$, while those in the second bracket may be interpreted as selection effects that capture the influence on expected morbidity changes of the determinants of survival $\text{Prob}\left(S_{if}=1\right)$.

Hence P^E(MB)q^E(MB) and P^E(MT)q^E(MT) in (13) can be decomposed to reveal the progressivity and scale indices of the various health determinants of expected morbidity changes and mortality:

$$\begin{array}{c}\hfill P^{E(MB)}{q}^{E(MB)}=\left\{\sum _{k=1}^K\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta m{b}^{\Delta k}}\right)\frac{{\overline{\Delta mb}}_f^{\Delta k}}{{\overline{h}}_f}+\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta m{b}^{EqE}}\right)\frac{{\overline{\Delta mb}}_f^{EqE}}{{\overline{h}}_f}\right\}\hfill \\ \hfill +\left\{\sum _{k=1}^K\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta m{b}^0}\right)\frac{{\overline{\Delta mb}}_f^0}{{\overline{h}}_f}+\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta m{b}^k}\right)\frac{{\overline{\Delta mb}}_f^k}{{\overline{h}}_f}+\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta m{b}^h}\right)\frac{{\overline{\Delta mb}}_f^h}{{\overline{h}}_f}\right\}\hfill \\ \hfill +\left(C{I}_{ss}^h-C{I}_{fs}^v\right)\frac{{\overline{v}}_f}{{\overline{h}}_f}\hfill \\ \hfill =\left\{\sum _{k=1}^K{P}_{\Delta k}^{E(MB)}{q}_{\Delta k}^{E(MB)}+P_{EqE}^{E(MB)}{q}_{EqE}^{E(MB)}\right\}\hfill \\ \hfill +\left\{P_0^{E(MB)}{q}_0^{E(MB)}+\sum _{k=1}^K{P}_k^{E(MB)}{q}_k^{E(MB)}+P_h^{E(MB)}{q}_h^{E(MB)}\right\}+P_v^{E(MB)}{q}_v^{E(MB)}\hfill \\ \hfill =\left\{\sum _{k=1}^K{P}_{\Delta k}^{E(MB)}{q}_{\Delta k}^{E(MB)}+P_{EqE}^{E(MB)}{q}_{EqE}^{E(MB)}\right\}+\left\{P_z^{E(MB)}{q}_z^{E(MB)}\right\}+P_v^{E(MB)}{q}_v^{E(MB)};\hfill \end{array}$$ (15a)

$$P^{E(MT)}{q}^{E(MT)}=\left\{\sum _{k=1}^K\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta m{t}^0}\right)\frac{{\overline{\Delta mt}}_f^0}{{\overline{h}}_f}+\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta m{t}^k}\right)\frac{{\overline{\Delta mt}}_f^k}{{\overline{h}}_f}+\left(C{I}_{ss}^h-C{I}_{fs}^{\Delta m{t}^h}\right)\frac{{\overline{\Delta mt}}_f^h}{{\overline{h}}_f}\right\}+\left(C{I}_{ss}^h-C{I}_{fs}^w\right)\frac{{\overline{w}}_f}{{\overline{h}}_f}=\left\{P_0^{E(MT)}{q}_0^{E(MT)}+\sum _{k=1}^K{P}_k^{E(MT)}{q}_k^{E(MT)}+P_h^{E(MT)}{q}_h^{E(MT)}\right\}+P_w^{E(MT)}{q}_w^{E(MT)}$$ (15b)

where $\Delta m{b}_{if}^{\Delta k}\equiv {\hat{\rho }}_{kis}\Delta x_{kif}$, $\Delta m{b}_{if}^{EqE}\equiv {\hat{\rho }}_{EqE,is}\left(h_{is}^{*}-h_{is}\right)$, $\Delta m{b}_{if}^0\equiv {\hat{\pi }}_{0is}$, $\Delta m{b}_{if}^k\equiv {\hat{\pi }}_{kis}{x}_{kis}$, $\Delta m{b}_{if}^h\equiv {\hat{\pi }}_{kis}{h}_{is}$, $\Delta m{t}_{if}^0\equiv {\hat{\tau }}_{0is}$, $\Delta m{t}_{if}^k\equiv {\hat{\tau }}_{kis}{x}_{kis}$ and $\Delta m{t}_{if}^h\equiv {\hat{\tau }}_{his}{h}_{is}$ are the sample counterparts of the corresponding expressions in (14), with means and CIs defined in terms of the morbidity-related and mortality-related health change effects of the various health determinants, rather than the determinants themselves, because the linearisation parameter values (which capture the marginal effects) vary across individuals in the Taylor series approximations;¹⁸ and $v_{if}$ and w_if are the approximation residuals, which will in general not equal zero on average. In the final line of (15a), we combine terms to yield $P_z^{E(MB)}{q}_z^{E(MB)}$, which captures the overall influence on M^H of the expected morbidity changes due to the combined effect of the selection terms in (14a), with $q_z^{E(MB)}$ equal to the sum of the individual scale indices of the selection terms and $P_z^{E(MB)}$ to the weighted sum of the progressivity indices with weights in proportion to the individual scale indices.¹⁹

4.3. Analysing the determinants of structural IRHI

The contribution of the equilibrium error $P_{EqE}^{E(MB)}{q}_{EqE}^{E(MB)}$ in (15a) could be further broken down to identify the ‘apparent’ contribution of each equilibrium health determinant to M^H through the adjustment process, but this is misleading inasmuch as the causes of the disequilibrium in the initial period are unknown.²⁰ Instead, it is more meaningful to simply analyse the determinants of structural or equilibrium IRHI in the initial period. Following Gravelle (2003), the CI of equilibrium health in the sub-set of the original population that has survived until period s (s = 1, T − 1) may be decomposed to yield:

$$\begin{array}{l}C{I}_{ss}^{{\hat{h}}^{*}}=\frac{2}{{\overline{\hat{h}}}_s^{*}}\left\{\sum _{k=1}^K{\hat{\beta }}_k\text{Cov}\left(x_{kis},R_{is}\right)+\text{Cov}\left({\hat{\mu }}_i,R_{is}\right)\right\}\\ =\sum _{k=1}^K\frac{{\hat{\beta }}_k{\overline{x}}_{ks}}{{\overline{\hat{h}}}_s^{*}}C{I}_{ss}^k+\frac{G{C}_{fs}^{\hat{\mu }}}{{\overline{\hat{h}}}_s^{*}}=\sum _{k=1}^K{\hat{\eta }}_{ks}C{I}_{ss}^k+\frac{G{C}_{fs}^{\hat{\mu }}}{{\overline{\hat{h}}}_s^{*}};\forall i\in \left\{{\Omega }_1\cap {\Omega }_s\right\};\end{array}$$ (16)

where $C{I}_{ss}^k$ is the CI of x_ks ranked by initial income and ${\hat{\eta }}_{ks}$ is the corresponding elasticity of health evaluated at sample means, which is equal to the share of equilibrium health attributable to that determinant; and $G{C}_{fs}^{\hat{\mu }}$ is the generalised concentration index of the estimated fixed effects ${\hat{\mu }}_i$ ranked by initial income. Hence the contribution of each health determinant to equilibrium IRHI is simply the product of the health elasticity and the CI of that determinant with respect to initial income rank.

5. Empirical analysis

We employ the decomposition procedures to analyse the health determinants of IRHI changes in Great Britain between 1999 and 2004, treated as a single five year transition period as in Petrie et al. (2011). Our empirical analysis employs data from the British Household Panel Survey (BHPS; University of Essex, Institute for Social and Economic Research, 2007), which is a nationally representative longitudinal survey of private households in Great Britain. Specifically, we use the data from 1999 and 2004 to construct an unbalanced panel consisting of observations on the sub-set of individuals in the BHPS for whom full information on health, income and a range of other socioeconomic variables was available in both 1999 and 2004 or for whom full information was available in 1999 and the individual was known to have died by 2004. The analysis is restricted to these two waves by the availability of data to construct our preferred health measure, which is defined in terms of Quality Adjusted Life Years (QALYs) and derived from the responses to the SF-36 questionnaire using the SF-6D preference-based algorithm (Brazier et al., 2002).

To prevent outlier incomes exerting undue influence, we follow common practice when using the BHPS (see e.g. Jones and Lopez Nicolas, 2004; Jenkins and Van Kerm, 2011) by excluding individuals from the panel if their equivalised income fell in the top 1% or bottom 1% of the distribution in either wave. Sample weights were used throughout the analysis with these being given by a set of adjusted BHPS cross-sectional weights for 1999, where the adjustments were made using inverse probability weights (see Wooldridge, 2002) to allow both for missing data in either 1999 or 2004 and for non-mortality related sample attrition between 1999 and 2004 (see Petrie et al. (2011) for further discussion). Standard errors for all inequality and mobility measures were generated using a bootstrap procedure in which re-sampling was carried out at the cluster (Primary Sampling Unit) rather than individual level within each major stratum, reflecting the sample design.²¹

5.1. Definition of variables

The trimmed sample comprises observations on 9677 individuals of whom 599 had died by 2004, representing 6.3% of the weighted sample. Table 1 provides definitions and descriptive statistics for all the variables used in the empirical analysis. The health variable is bounded in the unit interval with full health corresponding to a value of one, the lowest possible health utility of anyone alive being equal to 0.301, and with death assigned a QALY of zero. The average QALY score fell among those who survived until 2004, with this morbidity-related decline being reinforced by health utility losses due to mortality.

Table 1.

Variable definitions and summary statistics.

Variable	Attribute	Mean	Std. Dev	Min	Max
HEALTH99	Health 1999	0.800	0.128	0.301	1
ΔHEALTH	Change in health	−0.052	0.202	−1	0.495
LNINCOME99	Logarithm of income 1999	2.893	0.616	0.322	4.403
SMOKING99	Cigarettes smoked per day 1999	3.494	7.340	0	60
AGE99	Age 1999	47.823	18.729	16	96
AGESQ99	Age squared 1999	2637.779	1900.315	256	9216
MALE	Gender (Male = 1)	0.477	0.499	0	1
NONWHITE	Race (Non White = 1)	0.161	0.367	0	1
ADVEDUC	Advanced qualifications 1999	0.350	0.477	0	1
STDEDONLY	Standard qualifications 1999	0.312	0.463	0	1
SURVIVAL	Survival status (Alive in 2004 = 1)	0.937	0.243	0	1
Survivors only
HEALTH99	Health 1999	0.809	0.120	0.301	1
ΔHEALTH	Change in health	−0.010	0.118	−0.699	0.495
LNINCOME99	Logarithm of income 1999	2.917	0.615	0.322	4.403
ΔLNINCOME	Income growth	0.127	0.735	−11.043	3.439
SMOKING99	Cigarettes smoked per day 1999	3.525	7.370	0	60
ΔSMOKING	Change in smoking	−0.440	4.760	−40	30
AGE99	Age 1999	46.056	17.731	16	93
AGESQ99	Age squared 1999	2435.548	1737.122	256	8649
MALE	Gender (Male = 1)	0.476	0.499	0	1
NONWHITE	Race (Non White = 1)	0.169	0.375	0	1
ADVEDUC	Advanced qualifications 1999 (At least A Levels)	0.366	0.482	0	1
STDEDONLY	Standard qualifications 1999 (GCSEs/O levels/CSEs only)	0.324	0.468	0	1

Source: Authors’ calculations from BHPS data.

The income variable was defined as the natural logarithm of annual household income, equivalised using the McClements scale (Taylor, 1995) to take account of household composition and deflated by the CPI to take account of inflation. Other determinants of health and survival included in the analysis were the number of cigarettes usually smoked each day, age in years, the square of age, gender, ethnicity and highest level of educational attainment. Changes in log income and smoking were also included in the specification of the dynamic health function to capture the possible short-run effects of changes in these variables on health. Of those alive in both years, average incomes rose by 13.5% from an average of £18494, while smoking fell by 0.44 cigarettes per day from an average of 3.53 per day. Both average incomes and cigarette consumption were higher in 1999 among those who survived until 2004 compared to those who did not.²²

5.2. Accounting for changes in IRHI using longitudinal data

Table 2 shows that cross-sectional IRHI was 0.02095 in 1999 and 0.02523 in 2004.²³ The remainder of the table presents results from the Petrie et al. (2011) decomposition of the increase in IRHI of 0.00492 between the two years. This reveals three main points of interest, where all the reported measures are significantly different from zero at conventional significance levels. First, the negative value (−0.02506) for the income-related health mobility index M^H indicates that health changes among the initial population had a disequalising effect. Average health depreciation was 0.05170 with the progressivity index P value (0.36285) implying that relative health losses were concentrated among the worse-off. Second, 86.4% of the income-related health mobility was due to selective mortality, while 13.6% was due to selective morbidity changes of the survivors: mortality accounted for 82.2% of the net loss in health over the period and these losses were more concentrated among the poor than net morbidity losses. Third, the negative value for the health-related income mobility index M^R (−0.02078) indicates that health inequality among the population alive in 2004 was reduced by the combined effect of the dead no longer contributing to the calculation of IRHI and of income re-ranking among the survivors.²⁴

Table 2.

Decomposition of IRHI changes.

Average health 1999 $({\overline{h}}_s)$	0.80016***
	0.00190
Average health change $({\overline{\Delta h}}_f)$	−0.05170***
	0.00286
Conc. index of health changes $(C{I}_{fs}^{\Delta h})$	−0.34191***
	0.03064
Concentration index 1999 $(C{I}_{ss}^h)$	0.02095***
	0.00129
Concentration index 2004 $\left({\widetilde{CI}}_{ff}^h\right)$	0.02523***
	0.00128
Change in inequality $\left({\widetilde{CI}}_{ff}^h-C{I}_{ss}^h\right)$	0.00429***
	0.00133
Income-related health mobility (MH)	−0.02506***
	0.00231
Due to:Morbidity changes (PMBqMB)	−0.00342***
	0.00106
Mortality (PMTqMT)	−0.02164***
	0.00200
Progressivity index (P)	0.36285***
	0.03038
Due to: Morbidity changes (PMB)	0.28790***
	0.10211
Mortality (PMT)	0.38101***
	0.02731
Scale index (q)	−0.06907***
	0.00410
Due to: Morbidity changes (qMB)	−0.01227***
	0.00205
Mortality (qMT)	−0.05680***
	0.00326
Health-related income mobility (MR)	−0.02078***
	0.00246

Source: Authors’ calculations based on Eqs. (1)–(3). Bootstrapped standard errors in italics based on 2000 replications. Statistical significance at 1%, 5% and 10% levels are denoted by ***, ** and *, respectively.

5.3. Two-Part Model of health changes

The further decomposition to analyse the determinants of the income-related health mobility index M^H in Table 4 is based on the estimates of the Two-Part Model.

Table 4.

Decomposition of the income-related health mobility index.

	Progressivity, P	Scale, q	Mobility, MH	Share
Income-related health mobility	0.36285***	−0.06907***	−0.02506***	100.0%
	0.03038	0.00410	0.00231
Resulting from health changes due to:
Expected morbidity changes	0.23853**	−0.01219***	−0.00291**	11.6%
	0.10446	0.00206	0.00113
Due to:
ΔLNINCOME	1.11026***	0.00112***	0.00124***	−4.9%
	0.11261	0.00036	0.00037
ΔSMOKING	−0.13404	0.00010	−0.00001	0.1%
	0.09091	0.00015	0.00002
Equilibrium error	0.31376***	−0.01146***	−0.00359***	14.3%
	0.10168	0.00178	0.00105
Combined selection	0.27703***	−0.00195***	−0.00054***	2.2%
	0.10416	0.00033	0.00018
Approx. error	−0.39082	0.00000	0.00000	0.0%
	0.25487	0.00001	0.00000
Expected mortality	0.36967***	−0.05713***	−0.02112***	84.3%
	0.02807	0.00334	0.00206
Due to:
HEALTH99	0.08482***	−0.07083***	−0.00601***	24.0%
	0.00823	0.01655	0.00163
LNINCOME99	−0.00450	0.01806	−0.00008	0.3%
	0.01364	0.01678	0.00036
SMOKING99	0.14770***	−0.00469***	−0.00069***	2.8%
	0.02236	0.00139	0.00021
AGE99	0.17361***	0.14460**	0.02510**	−100.2%
	0.01386	0.06641	0.01166
AGESQ99	0.22695***	−0.19432***	−0.04410***	176.0%
	0.01423	0.03409	0.00820
MALE	0.05458***	−0.01486***	−0.00081***	3.2%
	0.01394	0.00292	0.00026
NONWHITE	−0.01568	0.00112	−0.00002	0.1%
	0.02509	0.00130	0.00005
ADVEDUC	−0.18551***	0.00522**	−0.00097**	3.9%
	0.01791	0.00217	0.00043
STDEDONLY	0.03335*	0.00324*	0.00011	−0.4%
	0.01863	0.00190	0.00009
Intercept	0.11475***	0.05533*	0.00635	−25.3%
	0.01353	0.03706	0.00426
Approx. error	−0.42125*	0.00000	0.00000	0.0%
	0.24637	0.00026	0.00007
Residual health shocks	−4.13427	0.00025*	−0.00103**	4.1%
	93.92231	0.00014	0.00044

Source: Authors’ calculations based on Eqs. (13), (15a) and (15b). Bootstrapped standard errors in italics based on 2000 replications. Statistical significance at 1%, 5% and 10% levels are denoted by ***, ** and *, respectively.

5.3.1. Estimation methods

We estimate the ECM (7) by ordinary least squares (OLS)²⁵ and the probit model by maximum likelihood. Consideration of individual heterogeneity in the ECM is problematic as we only have a single transition and individual-specific effects are therefore not identifiable.²⁶ However, we did experiment on a reduced sample with a Mundlak (1978) type parameterisation of the unobserved individual effect as a function of within-individual 3-year averages of the time-varying levels of smoking, income and self-assessed health over the period 1996–98, and found that this made little difference to the conclusions.²⁷ Another estimation issue with the ECM is the potential endogeneity of the contemporaneous changes in smoking and income given that health shocks may influence both income and smoking status (see e.g. Kapteyn et al., 2008; García-Gómez, 2011). Others have suggested that conditioning on initial health may limit the bias due to reverse causation (see e.g. Contoyannis et al., 2004; Van Ourti et al., 2009). We further explored this issue by excluding individuals from the sample whose economic activity status changed to long-term sick in 2004, as their change in income was mostly likely driven by a change in health, and found that this had a relatively small effect on the results.²⁸ Finally, we note that heterogeneity and endogeneity may also arise in the estimation of the probit model (see Balia and Jones, 2008) but consideration of these econometric issues lies beyond the scope of the current study.

5.3.2. Empirical results

Column [1] in Table 3 reports the results from the estimation of the probit survival model (5a) over the entire sample of individuals alive in 1999, where the dependent variable is 2004 survival status. All other things equal, the survival chances of those who were in better health in 1999 were, as would be expected, significantly better than those in worse health. Higher levels of income and education improved survival chances, though the effect for income was not significant, while smoking had a significant negative effect. The quadratic in age implies that those who were in their early twenties in 1999 had the highest probability of survival, with survival chances declining with age at an increasing rate thereafter. Finally, men were less likely to survive than women, while the effect of ethnicity was insignificant.

Table 3.

Two-Part Model and the implied Equilibrium Health Function.

	[1] Probit survival model	[2] Error Correction Model conditional on survival	[3] Equilibrium health function
Left hand side variable	SURVIVAL	ΔHEALTH	$h_s^{*}$
Explanatory variables	Coeff.	Coeff.	Coeff.
	Std error	Std error	Std error
ΔLNINCOME	–	0.00819***	–
		0.00237
ΔSMOKING	–	−0.00022	–
		0.00034
HEALTH99	2.03316***	−0.51034***	–
	0.23017	0.01340
LNINCOME99	0.07344	0.01623***	0.03179***
	0.06229	0.00274	0.00539
SMOKING99	−0.01485***	−0.00092***	−0.00181***
	0.00413	0.00021	0.00040
AGE99	0.02964**	0.00050	0.00098
	0.01315	0.00044	0.00080
AGESQ99	−0.00064***	−0.00002***	−0.00004***
	0.00011	0.00000	0.00001
MALE	−0.33276***	0.01456***	0.02854***
	0.06746	0.00270	0.00519
NONWHITE	0.11852	−0.00494	−0.00967
	0.14809	0.00391	0.00680
ADVEDUC	0.21339**	0.01048***	0.02054**
	0.09378	0.00397	0.00817
STDEDONLY	0.14268	0.01389***	0.02722***
	0.09265	0.00391	0.00811
Constant	0.63080	0.36988***	0.72478***
	0.42130	0.01552	0.02447
Sample size	9677	9078
Pseudo R2/R2	0.3656	0.2617
Wald χ2(9)	650.76***	–
F(11,398)	–	139.44***

Source: Authors’ calculations based on Eqs. (5a), (7) and (8), with the coefficient estimates in column [3] derived by manipulation of the estimation results in column [2]. Robust standard errors in columns [1] and [2] allow for the sample design. Bootstrapped standard errors in column [3] are based on 2000 replications. Statistical significance at 1%, 5% and 10% levels are denoted by ***, ** and *, respectively.

Column [2] reports the results for the dynamic health function conditional upon survival (7) with the dependent variable being the change in morbidity between 1999 and 2004. The first two regression coefficients show the short-run impact of changes in income and smoking on health.²⁹ Thus increases in income led to contemporaneous improvements in health, consistent with other evidence that short run movements in individual health are related to transitory income shocks (see e.g. Benzeval and Judge, 2001). Conversely, increases in smoking are estimated to have had a contemporaneous negative impact on health, though this effect was small and not significant.

The remainder of the dynamic health function relates to the equilibrium error, where the specification of steady-state or equilibrium health in (8) includes the same set of determinants as the probit survival model. The initial health coefficient provides an estimate of the adjustment parameter $\hat{\lambda }$ equal to 0.51, implying that just over half of the gap between any individual's actual and equilibrium health in 1999 was closed by 2004. Dividing the coefficients on the initial period levels of the health determinants by $\hat{\lambda }$ yields the parameters of the implied equilibrium health function reported in Column [3]. Thus the implied long-run effect on health of a 1% increase in income (0.03179) was nearly four times as large as the contemporaneous effect (0.00819), while the implied long-term effect of smoking an additional cigarette per day (−0.00181) was over eight times the contemporaneous impact (−0.00022). The quadratic in age implies that equilibrium health levels peak before twenty, with health declining at an increasing rate thereafter. Men had significantly higher equilibrium levels of health than women all other things equal. Non-whites had lower equilibrium levels of health though the coefficient was not significantly different from zero. Finally, higher levels of educational attainment are associated with better long-term health than the omitted case of no educational qualifications.

Overall, the Two-Part Model results are credible with expected signs on all coefficients that are significantly different from zero. Parameter estimates from the linearization of the model, which are employed in the hierarchical decomposition of the Petrie et al. (2011) income-related health mobility index, are reported in Appendix A.

5.4. Determinants of income-related health mobility

Table 4 expands upon the Petrie et al. (2011) decomposition analysis by reporting the results from our further hierarchical decomposition of the income-related health mobility index M^H by both morbidity change and mortality determinants using the TPM.

The first-stage decomposition results reveal that health changes due to expected morbidity changes, expected mortality and residual health shocks were all estimated to have had the effect of increasing IRHI over the period, with positive contributions of 11.6%, 84.3% and 4.1% respectively to income-related health mobility. Thus the TPM explains the vast majority of observed income-related health mobility, with the residual term making only a minor contribution.³⁰ Moreover, the estimated progressivity, scale and mobility indices of health changes due to expected morbidity changes and expected mortality closely match the corresponding non-parametric estimates reported in Table 2. Health changes due to expected mortality accounted for over four fifths of overall income-related health mobility, as a result of both the scale of expected health losses due to death and their concentration among the poor.

The second-stage decomposition results suggest that the effects of age on mortality was the principal contributor to income-related health mobility (75.8% = 176.0%–100.2%), with the old in 1999 more likely both to be poor and to die in the following five year period. Health in 1999 also made a disequalising contribution (24.0%) because the marginal effects of initial health on mortality-related health losses were negatively correlated with initial income rank,³¹ such that mortality-related health losses due to initial health were concentrated among the poor even though initial health was not. Similarly, gender contributed to increasing IRHI (3.2%) because the marginal effect of being male tended to be larger for the poor, such that mortality-related health losses due to being male were concentrated among the poor even though men tended to be better off than women on average.³² More obviously, advanced educational attainment also contributed to the disequalising effect of expected deaths (3.9%) because the highly educated were both more likely to be better off and less likely to die. This is also the case for smoking (2.8%) but for the opposite reason—smokers were both less likely to be better off and more likely to die. None of the remaining determinants – income, race and standard qualifications – make significant contributions, consistent with the insignificant effects of these factors on survival. Finally, the intercept term in (14b), which varies across individuals, makes an equalising contribution to income-related health mobility (−25.3%) given that the positive impact of the constant in the survival model varies across individuals with the reduction in the chance of death tending to be larger among the poor due to the non-linearity of the probit function.³³

Expected morbidity changes also made a disequalising contribution to income-related health mobility (11.6%). However, income growth provided a significant equalising contribution to this (−4.9%), with the positive values of q and P arising because income growth improved health and these health gains were concentrated among the poor whereas health in 1999 was concentrated among the rich.³⁴ In contrast, reductions in smoking over the period increased health inequalities, since the distribution of health gains due to the fall in smoking were more concentrated among the rich than the distribution of health in 1999, although the effect was both very small and insignificant. The largest contribution to M^H, of 14.3%, was from the equilibrium error term which, consistent with our expectations based on the discussion of (11), exacerbated IRHI. Specifically, adjustments from observed health in 1999 towards equilibrium health as defined by the socioeconomic determinants in 1999 resulted in a disequalising net health loss as these losses were concentrated among the poor. Finally, the combined effect of survival selection on expected morbidity made only a minor contribution to overall income related health mobility (2.2%) given that the average predicted chance of survival was 93.7%.

5.5. Determinants of implied 1991 equilibrium IRHI

Given the major contribution to income-related health mobility from the adjustment toward equilibrium health (14.3%), it is of interest to further consider the socioeconomic determinants of 1999 equilibrium IRHI per se. The observed and equilibrium CIs in Table 5 show that structural IRHI in 1999 (0.02982) was more severe than would be inferred from the cross-sectional measure (0.02095), which is consistent with the empirical literature on the relationship between long-run and short-run IRHI (see e.g. Jones and Lopez Nicolas, 2004). The decomposition of the equilibrium CI further reveals that just under half (47.2%) of structural IRHI in 1999 was attributable to income differences, with this resulting from income being both concentrated among the rich (by definition), as indicated by the positive CI for the logarithm of initial income (0.11934), and beneficial to health leading to a positive equilibrium health share (0.11799). A further two fifths (39.7% = 50.0–10.3%) of the equilibrium CI was attributable to income-related inequalities in age, with the old more likely to be both poorer and, all other things equal, in worse health. Finally, smoking, education and gender also made significant positive contributions to overall levels of equilibrium IRHI, with smokers more likely to be poor and smoking detrimental to health, while well-qualified individuals and men were more likely to be both richer and have better underlying levels of health.

Table 5.

Decomposition of the implied equilibrium IRHI in 1999.

	Decomposition Analysis
	Conc. Index	Health share	Contribution
	CI	η		Share
IRHI in 1999: $C{I}_{ss}^h$	0.02095***	–	–	–
	0.00129
Equilibrium IRHI: $C{I}_{ss}^{{\hat{h}}^{*}}$	0.02982***	–	0.02982***	100.0%
	0.00240		0.00240
of which due to:
LNINCOME99	0.11934***	0.11799***	0.01408***	47.2%
	0.00165	0.02013	0.00239
SMOKING99	−0.07904***	−0.00810***	0.00064***	2.1%
	0.01647	0.00183	0.00020
AGE99	−0.05111***	0.06015	−0.00307	−10.3%
	0.00351	0.04739	0.00247
AGESQ99	−0.11029***	−0.13519***	0.01491***	50.0%
	0.00631	0.02769	0.00329
MALE	0.05409***	0.01744***	0.00094***	3.2%
	0.00536	0.00313	0.00020
NONWHITE	0.07027***	−0.00199	−0.00014	−0.5%
	0.01858	0.00143	0.00011
ADVEDUC	0.24303***	0.00923**	0.00224**	7.5%
	0.01074	0.00369	0.00090
STDEDONLY	0.02010*	0.01089***	0.00022	0.7%
	0.01110	0.00328	0.00014
constant	0	0.92958***	0	0%
	–	0.03032	–	–

Source: Authors’ calculations based on Eq. (16) using all those present in 1999. Bootstrapped standard errors in italics based on 2000 replications. Statistical significance at 1%, 5% and 10% levels are denoted by ***, ** and *, respectively.

6. Discussion

The traditional approach to understanding the causes of changes in health inequality has been based on repeated cross-sectional analysis using regression decomposition techniques such as those proposed in Wagstaff et al. (2003). This paper develops new regression-based procedures that exploit the additional information contained within longitudinal data to estimate a dynamic health function that captures the persistence of health states and to identify important aspects of the underlying determinants of IRHI changes such as those relating to mortality. Specifically, we employ a Two-Part Model – a probit model of survival together with a dynamic health function conditional upon survival – that provides a unified framework for understanding the role of health determinants in driving changes in IRHI through both morbidity changes and mortality. Our analysis reveals the contribution of each health determinant to income-related health mobility, which in turn are shown to depend on the progressivity and scale of the health changes attributable to that determinant. Moreover, our dynamic modelling framework also serves to identify how each health determinant contributes to long-run or structural IRHI.

We applied our procedures to the investigation of the causes of changes in IRHI in Great Britain over the five year period 1999 to 2004, using Quality Adjusted Life Years (QALYs) as our health measure. Health changes due to expected mortality and expected morbidity changes both had a disequalising effect over this period, with the overall effect dominated by health losses due to expected deaths. This finding points to the importance of understanding the determinants of mortality in the evaluation of policies designed to tackle health inequalities. Consistent with Balia and Jones (2008) we find that the major driver of the disequalising effects of mortality was the positive association between (old) age and poverty given that the old were at greater risk of death, with other significant contributors including initial health status, advanced levels of educational attainment, gender and smoking. Health service interventions that improve the survival rates of the old and chronically sick are likely to reduce excess deaths among the poor yet, paradoxically, may also result in higher future levels of cross-sectional health inequalities if they do not also improve their average levels of morbidity. Programmes that act on the distribution of health determinants in the population, for example targeted smoking cessation programmes, may reduce both excess deaths among the poor and long term levels of IRHI.

The disequalising effect of morbidity related health changes was driven by the lagged process of adjustment to changes in health determinants and health shocks prior to 1999. However, this effect was moderated by the poor enjoying a disproportionately large share of the contemporaneous gains in relative health due to pro-poor real income growth between 1999 and 2004. This period largely coincided with New Labour's second term in office, which was characterised by income growth rates that were highest at the very bottom of the income distribution (Joyce et al., 2010: 25; see also Jenkins and Van Kerm, 2011) as a result of a range of factors including low unemployment, the introduction of the minimum wage and an assortment of new and enhanced social security benefits.

The findings further indicate that just under half of the equilibrium level of IRHI in 1999 was attributable to income inequality, providing some support for the conclusions of the Independent Inquiry into Inequalities in Health Report (Acheson, 1998) ‘that without a shift of resources to the less well off, both in and out of work, little will be accomplished in terms of a reduction of health inequalities by interventions addressing particular “downstream” ‘influences’. However, like Adams et al. (2003), we are unable to show a significant direct influence of income on the risk of mortality after controlling inter alia for the protective effect of initial health.³⁵

These empirical findings illustrate the value of the proposed approach for understanding how changes in individuals’ determinants of health impact on health inequalities in order to shape the design of an effective set of public policies to tackle the issue. Nevertheless, the proposed approach offers only a partial analysis of changes in IRHI in that it establishes the causes of income-related health mobility but not of health-related income mobility, with the latter also playing a significant role in the determination of the overall rise in IRHI in Great Britain between 1999 and 2004. Extension of the approach to incorporate a decomposition of health-related income mobility by the determinants of changes in income (ranks) would require a joint model of the determination of health and income changes that accounts for dual causality, but this lies beyond the scope of the current paper.

Funding

Chief Scientist Office (CSO) Grant 2012CB416603CZG/2/451: ‘Development of tools to measure and explain changes in health inequalities in Scotland and benchmark performance”.

Appendix A. Linearisation of the Two-Part Model

The linearisation of the TPM is based on the Taylor series expansion of the standard normal cumulative distribution function $\Phi \left(z\right)$ about zero (see e.g. Marsaglia, 2004):

$$\Phi \left(z\right)=\frac{1}{2}+\frac{1}{\sqrt{2\pi }}\left(\sum _{j=0}^{\infty }\frac{{\left(-1\right)}^j{z}^{2j}}{(2j+1)2^{j}j!}\right)z\equiv \frac{1}{2}+\frac{1}{\sqrt{2\pi }}\left(\sum _{j=0}^{\infty }{\theta }_j\left(z\right)\right)z$$

Hence $E\left(\Delta h_{if}^{MB}\right)$ and $E\left(\Delta h_{if}^{MB}\right)$ in (13) may be written as³⁶

$$\begin{array}{l}E\left(\Delta h_{if}^{MB}\right)=\left\{\sum _{k=1}^K{\rho }_{kis}\Delta x_{kif}+{\rho }_{EqE,is}\left(h_{is}^{*}-h_{is}\right)\right\}+\left\{{\pi }_{0is}+\sum _{k=1}^K{\pi }_{kis}{x}_{kis}+{\pi }_{his}{h}_{is}\right\}+{\varpi }_{if};\\ E\left(\Delta h_{if}^{MT}\right)={\tau }_{0is}+\sum _{k=1}^K{\tau }_{kis}{x}_{kis}+{\tau }_{his}{h}_{is}+{\omega }_{if};\end{array}$$ (A1)

where

$${\rho }_{kis}=\left\{\frac{1}{2}+\frac{1}{\sqrt{2\pi }}\left(\sum _{j=0}^J\left(\frac{1}{2j+2}\right){\theta }_j\left(z_{is}\right)\right)z_{is}\right\}{\delta }_k;k=\mathrm{1,...},K$$

$${\rho }_{EqE,is}=\left\{\frac{1}{2}+\frac{1}{\sqrt{2\pi }}\left(\sum _{j=0}^J\left(\frac{1}{2j+2}\right){\theta }_j\left(z_{is}\right)\right)z_{is}\right\}\lambda ;$$

$${\pi }_{mis}=\left\{\frac{1}{\sqrt{2\pi }}\left(\sum _{j=0}^J\left(\frac{2j+1}{2j+2}\right){\theta }_j\left(z_{is}\right)\right)\left(\sum _{k=1}^K{\delta }_k\Delta x_{kif}+\lambda \left(h_{is}^{*}-h_{is}\right)\right)\right\}{\gamma }_m;m=\mathrm{0,1,...},K,h$$

$${\tau }_{kis}=\left\{\frac{1}{\sqrt{2\pi }}\left(\sum _{j=0}^J\left(\frac{2j+1}{2j+2}\right){\theta }_j\left(z_{is}\right)\right)h_{is}\right\}{\gamma }_k;k=\mathrm{0,1,...},K$$

$${\tau }_{his}=\left\{\frac{1}{\sqrt{2\pi }}\left(\sum _{j=0}^J\left(\frac{2j+1}{2j+2}\right){\theta }_j\left(z_{is}\right)\right)h_{is}\right\}{\gamma }_h+\left\{-\frac{1}{2}+\frac{1}{\sqrt{2\pi }}\left(\sum _{j=0}^J\left(\frac{1}{2j+2}\right){\theta }_j\left(z_{is}\right)\right)z_{is}\right\};$$

and the size of the approximation errors ϖ_if and ω_if can be controlled by the choice of J, which determines the order of the expansions. Table A1

Table A1.

Average linearization parameter estimates.

Dependent variable		$E\left(\Delta h_{if}^{MB}\right)$		$E\left(\Delta h_{if}^{MT}\right)$
Explanatory variables		Av. coeff		Av. coeff
		Std error		Std error
ΔLNINCOME	${\overline{\hat{\rho }}}_{\Delta LNINCOME}$	0.0065034***		–
		0.0019225
ΔSMOKING	${\overline{\hat{\rho }}}_{\Delta SMOKING}$	−0.0001762		–
		0.0002636
Equilibrium error	${\overline{\hat{\rho }}}_{EqE}$	0.4050026***		–
		0.0108085
HEALTH99	${\overline{\hat{\pi }}}_{HEALTH99}$	−0.0015027***	${\overline{\hat{\tau }}}_{HEALTH99}$	−0.0729326***
		0.0004362		0.0154532
LNINCOME99	${\overline{\hat{\pi }}}_{LNINCOME99}$	−0.0000543	${\overline{\hat{\tau }}}_{LNINCOME99}$	0.0048212
		0.0000559		0.0044940
SMOKING99	${\overline{\hat{\pi }}}_{SMOKING99}$	0.0000110**	${\overline{\hat{\tau }}}_{SMOKING99}$	−0.0009746***
		0.0000048		0.0002588
AGE99	${\overline{\hat{\pi }}}_{AGE99}$	−0.0000219*	${\overline{\hat{\tau }}}_{AGE99}$	0.0019456**
		0.0000121		0.0009053
AGESQ99	${\overline{\hat{\pi }}}_{AGESQ99}$	0.0000005***	${\overline{\hat{\tau }}}_{AGESQ99}$	−0.0000420***
		0.0000002		0.0000077
MALE	${\overline{\hat{\pi }}}_{MALE}$	0.0002459***	${\overline{\hat{\tau }}}_{MALE}$	−0.0218441***
		0.0000902		0.0039315
NONWHITE	${\overline{\hat{\pi }}}_{NONWHITE}$	−0.0000876	${\widetilde{\overline{\hat{\tau }}}}_{NONWHITE}$	0.0077807
		0.0001211		0.0098915
ADVEDUC	${\overline{\hat{\pi }}}_{ADVEDUC}$	−0.0001577*	${\overline{\hat{\tau }}}_{ADVEDUC}$	0.0140080**
		0.0000915		0.0062602
STDEDONLY	${\overline{\hat{\pi }}}_{STDEDONLY}$	−0.0001055	${\overline{\hat{\tau }}}_{STDEDONLY}$	0.0093661
		0.0000741		0.0057983
Constant	${\overline{\hat{\pi }}}_0$	−0.0004662	${\overline{\hat{\tau }}}_0$	0.0414092
		0.0004105		0.0277510

Source: Authors’ calculations based on Eqs. (14a) and (14b) as further defined in Appendix A. The reported coefficient values are sample weighted averages of the individual-specific coefficient values obtained from the Taylor series expansion with J = 20. Bootstrapped standard errors based on 2000 replications. Statistical significance at 1%, 5% and 10% levels are denoted by ***, ** and *, respectively.

reports estimates of the linearisation parameters with J = 20, which serves to reduce the magnitude of the contributions of the approximation errors to income-related health mobility reported in Table 4 to the order of 10⁻⁸. Note that ${\overline{\hat{\tau }}}_{HEALTH99}$ is approximately equal to minus the average probability of death given that the better an individual's initial health the more they lose from dying and that the protective effect of better initial health on the risk of mortality is small.