A Simple Measure of Catastrophic Health Expenditures

ABSTRACT

In this paper, we propose a simple Watts‐type measure of catastrophic health expenditure (CHE) which is an adaptation of the classic Watts poverty measure. The appeal of the proposed measure stems from the fact that it is both additively decomposable (i.e., it provides information on the contributions of the various population subgroups of interest, e.g., as categorized by gender, race, region, etc., to the overall level of CHE), and multiplicatively decomposable (i.e., it enables identification of three key drivers of CHE, namely, CHE incidence, CHE intensity and CHE inequality). We also describe how the Watts‐type CHE measure can be estimated and additively decomposed using the widely available ordinary least squares regression packages. The empirical example provided shows the policy value of the Watts‐type CHE measure, which makes it a useful supplement to the Foster‐Greer‐Thorbecke type measures of CHE recently proposed by Ogwang and Mwabu. Temporal dynamics in the Watts‐type CHE measures are also introduced.

1. Introduction

Catastrophic health expenditure (CHE) is an issue that commonly arises in the discussions surrounding out‐of‐pocket healthcare financing. A particular healthcare spending unit of interest (e.g., an individual or a household) is deemed to incur CHE in some reference period when the spending unit's out‐of‐pocket healthcare expenditure as a proportion of its total monetary endowment surpasses a prescribed threshold. The total monetary endowment can be total household expenditure (e.g., Wagstaff and van Doorslaer ²⁰⁰³; Ogwang and Mwabu ²⁰²⁴); total discretionary expenditure (e.g., K. Xu et al. ^{2003, 2007}; Y. Xu et al. ²⁰¹⁵; Quintal ²⁰¹⁹; Ssewanyana and Kasirye ²⁰²⁰; Edeh ²⁰²²); or total income (e.g., Wagstaff ²⁰¹⁹; Nguyen et al. ²⁰²³).

CHE is of concern to health policy makers owing to its potentially negative impact on the welfare of the healthcare spending units involved. The negative welfare impact could arise when the healthcare spending units avoid excessive out‐of‐pocket healthcare expenditures by foregoing necessary healthcare altogether or by settling for cheaper, albeit inferior, healthcare options (see, e.g., Mwabu et al. ¹⁹⁹³; Bremer ²⁰¹⁴; Rahman et al. ²⁰²²; Kakietek et al. ²⁰²²). Negative welfare impact could also arise when the healthcare spending units are forced to borrow on very unfavorable terms to finance current healthcare expenditure shortfalls making them vulnerable to poverty. Moreover, a negative welfare impact can arise from successful but costly treatment for which positive financial returns cannot be realized due to meager wages or unemployment, especially in some of the developing countries where labor markets are beset by many imperfections ranging from job‐skill mismatch to lack of entrepreneurial capital to start a business. Even when healthcare expenditure does not lead to cure, it has the benefit of reducing sickness pain but this utility gain from reducing sickness pain can be offset by disutility associated with the catastrophic level of the expenditure.

Considering its negative welfare impact mentioned above, it is not surprising that CHE is highly relevant as far as the United Nations Sustainable Development Goal No. 3 (i.e., good health and promotion of well‐being for all), with the achievement of universal health coverage being one of its key targets.

On the empirical front, Azzani et al. (2019), Wagstaff et al. (2020), Eze et al. (2022), Nguyen et al. (2023), and Ataguba et al. (2024), among many others, provide overviews of the salient issues in the empirical analysis of CHE. Also, many empirical papers (e.g., Y. Xu et al. ²⁰¹⁵; Wagstaff et al. ²⁰¹⁸; Quintal ²⁰¹⁹; Ssewanyana and Kasirye ²⁰²⁰; Edeh ²⁰²²; Sanoussi et al. ²⁰²³) have examined the status of CHE in a wide diversity of countries around the world.

Formulating appropriate CHE reduction policies necessitates developing credible measures of CHE. In a recent paper Ogwang and Mwabu (2024) stressed the need to develop CHE measures that do not only gauge the overall level of CHE but also possess two important normative properties, namely (i) additive/subgroup decomposition (i.e., the possibility of the overall level of CHE being broken down into the contributions of the various mutually exclusive and exhaustive population subgroups where subgroup formation could be by ethnicity, gender, region, etc.) and (ii) multiplicative decomposition (i.e., the feasibility of isolating CHE incidence, CHE intensity and CHE inequality from the overall CHE measure). The three components of the multiplicative decompositions of the overall CHE measure (i.e., CHE incidence, CHE intensity and CHE inequality) are collectively known as the Three I's of CHE (TICHE), following Ogwang and Mwabu who also introduced the graphical representations of TICHE using TICHE curves. It is worth emphasizing that CHE incidence indicates the percentage of the healthcare spending units that incur CHE, whereas CHE intensity shows the extent to which the healthcare expenditure proportion of a healthcare spending unit experiencing CHE surpasses the prescribed threshold proportion for CHE characterization; and, finally, CHE inequality depicts the disparity in the healthcare expenditure proportions among the healthcare spending units that have incurred CHE.

On the policy front, the additive decomposition property of CHE measures enables policy makers to identify the subgroups of healthcare spending units experiencing CHE that contribute most to overall CHE. Hence, the results of the additive decompositions of CHE measures could inform the formulation of CHE reduction policies that specifically target the disadvantaged subgroups. The multiplicative decomposition property enables policy makers to determine whether the observed changes in the magnitude of the overall CHE measure are largely driven by changes in CHE incidence, CHE intensity or CHE inequality. Hence, the results of multiplicative decompositions could inform the targeting and prioritization of interventions for reducing the risk of exposure to catastrophic health expenditure, tackling the extent of burden upon exposure and narrowing the disparity of the burden.

One notable aspect of the similarity between poverty measures and their CHE counterparts is the focus on an appropriate threshold. For the poverty measures, the focus is on the income shortfall from the threshold poverty line whereas for the CHE measures the focus is on the overshoot of the relevant healthcare expenditure threshold proportion beyond the socially prescribed threshold. This similarity makes it possible to adapt poverty measures for the purpose of measuring and characterizing CHE, and for addressing extreme poverty and the potentially impoverishing healthcare expenditures.

One recent initiative to adapt poverty measures for purposes of measuring CHE is by Ogwang and Mwabu who developed a family of CHE measures that is an adaptation of the highly popular family of poverty measures proposed by Foster et al. (1984). Ogwang and Mwabu's adaptation entails the introduction of a single CHE aversion parameter in the Foster‐Greer‐Thorbecke‐type CHE measure whose value can be increased to put greater emphasis on the healthcare spending units whose healthcare expenditure proportions surpass the relevant threshold for CHE characterization by the greatest margins. The appeal of the resulting Foster‐Greer‐Thorbecke‐type CHE measures stems not only from their additive decomposability and multiplicative decomposability but also from the estimation convenience using the widely available ordinary least squares (OLS) regression software packages. Ogwang and Mwabu highlighted other attractive features of the Foster‐Greer‐Thorbecke‐type family of CHE measures. They also provided an illustrative example to demonstrate the efficacy of the Foster‐Greer‐Thorbecke‐type family of CHE measures.

One potential drawback of the Foster‐Greer‐Thorbecke‐type family of CHE measures proposed by Ogwang and Mwabu is that it entails subjective choice of the CHE aversion parameter owing to the fact that the information provided by the observed data is not used to estimate its value. Considering the policy significance of CHE measures in the healthcare industry and especially the advocacy role of these measures in protecting households from impoverishment owing to exorbitant health maintenance costs, there is need to expand the CHE measurement options by considering other measures that eschew the choice of CHE aversion parameter. One natural and practical way to expand these options is to adapt other readily available poverty measures for purposes of measuring CHE. In this regard, we believe that a highly popular poverty measure proposed by Watts (1968) is an excellent candidate for adaptation.

The Watts poverty measure is renowned not only for its computational simplicity but also for its well‐behaved additive and multiplicative decomposability properties. The Watts poverty measure is also valued for its ability to address many issues of policy significance both within and outside the health sector, such as poverty duration or persistence (Morduch 1998; Israeli and Weber ²⁰¹¹; Villar ²⁰²⁴) and the low rate of pro‐poor growth (Ravallion and Chen 2003), among others.

We are not aware of any salient endeavors in the health economics literature to adapt the Watts poverty measure for purpose of measuring CHE, despite its potential efficacy and simplicity in application. The purpose of this paper therefore is to fill this gap in the literature by constructing a simple measure of CHE, which is an adaptation of the Watts poverty measurement procedure.

As will be seen below, the proposed Watts‐type CHE measure is appealing on the following six grounds. First, it is a simple measure by construction. Second, the proposed Watts‐type CHE measure, like the Foster‐Greer‐Thorbecke‐type CHE measures proposed by Ogwang and Mwabu (2024), has the advantage of being both additively and multiplicatively decomposable thereby enhancing its policy significance. Furthermore, as explained below, some empirically popular measures proposed by Wagstaff and van Doorslaer (2003) are linked to the multiplicative decompositions of the Watts‐type CHE measure. Third, unlike the Foster‐Greer‐Thorbecke‐type measure of CHE proposed by Ogwang and Mwabu (2024), the Watts type measure eschews complications surrounding the choice of the CHE aversion parameter. Fourth, with respect to the additive decompositions, it is apparent from the detailed derivations presented in Appendix B that the Watts index always provides information on all the three key drivers of CHE (i.e., CHE incidence, CHE intensity and CHE intensity). In contrast, the Foster‐Greer‐Thorbecke measure provides information on all the three key drivers only if the CHE aversion parameter is 2 or greater. Fifth, the Watts measure enables the introduction of temporal dynamics to the analysis of CHE in several ways, including the average time to exit CHE under certain assumptions regarding CHE reduction efforts. Sixth, the Watts‐type measure of CHE, like its Foster‐Greer‐Thorbecke counterpart, can be conveniently estimated using widely available OLS regression packages.

Considering the many attractive features mentioned above, the proposed Watts‐type CHE measure is a valuable supplement to the Foster‐Greer‐Thorbecke‐type CHE measures for purpose of conducting a comprehensive and timely empirical analysis of the status of CHE.

The Watts poverty measure also possesses several other normative properties that are frequently mentioned in the literature (see, e.g., Blackburn ¹⁹⁸⁹; Zheng ^{1993, 1997}; Muller ²⁰⁰¹; Saisana ²⁰¹⁴). These properties are also transferable to the associated Watts‐type CHE measure. However, for the sake of maintaining focus on the policy relevant additive and multiplicative decomposition properties, these additional normative properties are not emphasized in the adaptation entailed in this paper.

The rest of the paper is structured as follows. In Section 2, we introduce the overall Watts‐type measure of CHE as well as its additive and multiplicative decomposition properties and temporal dynamics. The regression approach to the estimation of the overall Watts‐type CHE measure and its additive decomposition are explained in Section 3. An illustrative empirical example demonstrating the efficacy and versatility of the Watts‐type CHE measure is presented in Section 4. Concluding remarks are in Section 5, and detailed derivations of the key results pertaining to Watts‐type CHE measures are presented in technical appendices.

2. The Overall Watts‐Type Measure of CHE, Additive and Multiplicative Decompositions and Temporal Dynamics

The construction of the classic Watts poverty measure where the focus is below the threshold line entails sums of the natural logarithm of the ratio of the poverty line to the incomes of only the income receiving units that are below the poverty threshold (see, e.g., Watts ¹⁹⁶⁸; Simler and Arndt ²⁰⁰⁷; Saisana ²⁰¹⁴; Ogwang ²⁰²²). Hence, the construction of the Watts‐type CHE measure where the focus is above the threshold proportion for CHE characterization entails summing up the natural logarithms of the ratio of the threshold proportion for CHE characterization to the healthcare expenditure proportions for only the healthcare spending units that are above the CHE threshold.

The finer details regarding the construction of the Watts‐type CHE measure presented below are relevant regardless of whether the healthcare expenditure proportions are computed using total expenditure, discretionary expenditure or income as mentioned above.

1. The overall Watts‐type measure of CHE ($WCHE$)

Let $N$ denote the totality of healthcare spending units (e.g., individuals or households) of interest in the estimation of the Watts‐type measure of CHE regardless of their CHE status (i.e., catastrophic if the prescribed threshold proportion is surpassed or not catastrophic if the prescribed threshold proportion is not surpassed). Hereafter, it is understood that the healthcare expenditure proportions of all the $N$ healthcare spending units and the relevant pre‐specified threshold proportion for CHE characterization, can be based on total expenditure, discretionary expenditure or income.

Let $T>0$ denote the threshold proportion of healthcare expenditures against which the CHE status of each healthcare spending unit is judged. Let $H$ denote the number of healthcare spending units that are deemed to have incurred CHE for which healthcare expenditure proportions are greater than the prescribed threshold proportion ($T>0$) as noted above. Also, let $E_i$ denote the healthcare expenditure proportion of the ith healthcare spending unit regardless of CHE status, $i=1,2,\mathit{\text{…}},N$. Finally, let $E_i^{\ast }$ denote the healthcare expenditure proportion of the ith healthcare spending unit whose healthcare expenditure is deemed catastrophic ($i=1,2,\mathit{\text{…}},H;H\mathit{\le }N$). Let ${\stackrel{\sim }{E}}_i^{\ast }=\frac{E_i^{\ast }}{T},i=1,2,\mathit{\text{…}},H$, denote the normalized healthcare expenditure proportion for the ith healthcare spending unit whose healthcare expenditure proportion is deemed catastrophic. This normalization ensures that expressing the healthcare expenditure proportions and the associated thresholds in either percentages or in decimals does not affect the value of the overall Watts‐type CHE measure or the corresponding values of its additive and multiplicative decompositions.

Considering that in CHE measurement the focus is on the healthcare spending proportions that overshoot the prescribed threshold healthcare expenditure proportion, a Watts‐type measure of CHE is given by the formula

$$WCHE=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{H}}\ln \left(\frac{{\mathrm{E}}_{\mathrm{i}}^{\ast }}{\mathrm{T}}\right)=\frac{1}{\mathrm{N}}\sum _{\mathrm{i}=1}^{\mathrm{H}}\left(\ln \left({\mathrm{E}}_{\mathrm{i}}^{\ast }\right)-\text{In}\left(\mathrm{T}\right)\right)$$ (1)

A higher value of $WCHE$ in Equation (1) signifies a greater degree of CHE, that is, the extent to which healthcare is an excess burden to a healthcare spending unit.

• b.Additive decomposition of $WCHE$

As noted above, the additive decomposition property of a measure of CHE has policy significance since it enables the contribution of each population subgroup to the overall level of CHE to be computed, provided that the subgroups are mutually exclusive and exhaustive of the population.

To see how additive decomposition of $WCHE$ in Equation (1) is accomplished, let us consider a breakdown of the total population of $N$ healthcare spending units into $k$ mutually exclusive and exhaustive subgroups, where subgroup formation could be by race, gender, region, etc. Let $N_j$ denote the number of healthcare spending units in the jth population subgroup ($j=1,2,\mathit{\text{…}},k$). Also let, $E_{ij}$ denote the healthcare expenditure proportion of the ith healthcare spending unit in the jth population subgroup ($j=1,2,\mathit{\text{…}},k;i=1,2,\mathit{\text{…}},N_j$) regardless of the spending unit's CHE status. Also, let $H_j$ denote the number of healthcare spending units in the jth subgroup that incurred CHE, that is, $E_{ij}>T,j=1,2,\mathit{\text{…}},k;i=1,2,\mathit{\text{…}},N_j$, in which case the total number of healthcare spending units that incurred CHE in all the $k$ subgroups is given by $H=\sum _{j=1}^k{H}_j$. Finally, let $E_{ij}^{\ast }$ denote the healthcare expenditure proportion of the ith spending unit that incurred CHE in the jth population subgroup ($j=1,2,\mathit{\text{…}},k;i=1,2,\mathit{\text{…}},N_j$).

The detailed derivation provided in Appendix A gives rise to the following simple additive decomposition of $WCHE$ as set out in Equation (1):

$$WCHE=\frac{1}{N}\sum _{i=1}^H\mathit{ln}\left(\frac{E_i^{\ast }}{T}\right)=\frac{H}{N}\sum _{j=1}^k{s}_j{\widehat{\beta }}_j$$ (2)

where $s_j=H_j/H$ is the share of $H_j$ healthcare spending units in the jth population subgroup that incurred CHE in relation to the totality of $H=\sum _{j=1}^k{H}_j$ healthcare spending units in all the $k$ subgroups that incurred CHE; $H/N$ is the overall CHE headcount ratio, the proportion of the totality of $N$ healthcare spending units that incur CHE; and ${\widehat{\beta }}_j=\frac{1}{H_j}\sum _{i=1}^{H_j}\mathit{ln}\frac{E_{ij}^{\ast }}{T}$ is a numerical quantity which can be conveniently obtained by running an artificial regression as explained in Section 3 of this paper.

It follows from Equation (2) that the contribution of the jth population subgroup to $WCHE$ is $\left(H/N\right)s_j{\widehat{\beta }}_j,j=1,2,\mathit{\text{…}},k$.

• c.Multiplicative decomposition of $WCHE$

As noted above, the multiplicative decomposition property of a measure of CHE also has policy significance since it allows policy makers to determine whether changes in CHE are largely driven by changes in CHE incidence, CHE intensity or CHE inequality.

The exposition of the multiplicative decomposition of the overall Watts‐type CHE measure is facilitated by taking a closer look at the CHE intensity measure, $I$, defined as the average amount by which the normalized catastrophic healthcare expenditure proportions ${\stackrel{\sim }{E}}_i^{\ast }=E_i^{\ast }/T,\mathrm{i}=1,2,\text{…},\mathrm{H}$ exceed, $T/T=1$, the normalized threshold expenditure proportion, based solely on the healthcare expenditure proportions of the healthcare spending units that incurred CHE that is, $I=\frac{1}{H}\sum _{i=1}^H\left(\frac{E_i^{\ast }-T}{T}\right)=\frac{1}{H}\sum _{i=1}^H\left(\frac{E_i^{\ast }}{T}-1\right)=\frac{{\bar{E}}^{\ast }}{T}-1$, in which case $\frac{{\bar{E}}^{\ast }}{T}=\left(I+1\right)$.

Now let us consider the normalized healthcare expenditure proportions of the healthcare spending units that incurred CHE, obtained by dividing each healthcare expenditure proportion by the threshold healthcare expenditure proportion that is ${\stackrel{\sim }{E}}_i^{\ast }=E_i^{\ast }/T,i=1,2,\mathit{\text{…}},H$. It turns out that the mean of the normalized healthcare expenditure proportions is given by $\widehat{\mu }\left({\stackrel{\sim }{E}}^{\ast }\right)=\frac{1}{H}\sum _{i=1}^H\frac{E_i^{\ast }}{T}=\frac{{\bar{E}}^{\ast }}{T}=\left(I+1\right)$.

The detailed derivation provided in Appendix B in conjunction with the above information gives rise to the following simple multiplicative decomposition of $WCHE$ as set out in Equation (1):

$$WCHE=\frac{1}{N}\sum _{i=1}^H\mathit{ln}\left(\frac{E_i^{\ast }}{T}\right)=\frac{H}{N}\left[\left(-GE\left(0,E^{\ast }\right)\right)+\mathit{ln}\left(I+1\right)\right]$$ (3)

It is apparent from Equation (3) that $CHEW$ can be multiplicatively decomposed into CHE incidence measure ($H/N$), CHE intensity measure, $I=\frac{1}{H}\sum _{i=1}^H\left(\frac{E_i^{\ast }-T}{T}\right)$, and a Generalized Entropy CHE inequality measure $GE\left(0,E^{\ast }\right)=\frac{1}{H}\sum _{i=1}^H\mathit{ln}\left(\frac{{\bar{E}}^{\ast }}{E_i^{\ast }}\right)$. Hence, the multiplicative decomposition of $WCHE$ as set out in Equation (3) is fully consistent with TICHE and the associated TICHE curve can be constructed using the procedure proposed by Ogwang and Mwabu (2024; Appendix C).

For the sake of comprehensiveness in the exposition, it makes sense to compare the three measures entailed in the multiplicative decomposition of $WCHE$ with other measures suggested in the literature. In this regard, it is interesting to note that two of the three measures entailed in the multiplicative decomposition of $WCHE$ as per Equation (3) are identical to the empirically highly popular CHE measures proposed by Wagstaff and van Doorslaer (2003). More specifically, the CHE incidence measure (H/N) is identical to Wagstaff and van Doorslaer's catastrophic payment headcount measure (hcat) whereas the CHE intensity measure $\left(I\right)$ is identical to their mean positive gap measure (mpgcat) when normalized healthcare expenditure proportions (i.e., healthcare expenditure proportions divided by the threshold proportion for CHE characterization) and the normalized threshold proportion, 1, are used in the computation of mpgcat, in which case mpgcat becomes unit‐free. Furthermore, multiplying the CHE incidence value (hcat) by its CHE intensity counterpart (mpgcat) gives rise to Wagstaff and van Doorslaer's average catastrophic payment gap measure (gcat), which is consistent with the prominent relationship $gcat=hcat\times mpgcat$ that characterizes Wagstaff and van Doorslaer's (2003) paper. The Generalized Entropy CHE inequality measure associated with the multiplicative decomposition of $WCHE$ is not linked to Wagstaff and van Doorslaer's hcat, gcat or mpgcat. Clearly, the insights into CHE inequality that $CHEW$ provides extend beyond what Wagstaff and van Doorslaer's hcat, gcat and mpgcat measures can provide.

Another notable aspect of the multiplicative decompositions of $WCHE$ is that the incidence and intensity components are identical to their counterparts in the multiplicative decompositions of the Foster‐Greer‐Thorbecke‐type CHE measures proposed by Ogwang and Mwabu when the CHE aversion parameter is 2. Hence, as will be apparent from the illustrative example provided below, we would expect the values of the CHE incidence and CHE intensity measures entailed in $WCHE$ to be identical to their counterparts in the Foster‐Greer‐Thorbecke‐type CHE measures with a CHE aversion parameter of 2. In contrast, we would expect the value of the CHE inequality measure entailed in $WCHE$ to be different from its counterpart in the Foster‐Greer‐Thorbecke‐type CHE measure with CHE aversion parameter of 2. This difference arises because the CHE inequality measure entailed in $WCHE$, as set out in Equation (3), is of the Generalized Entropy type whereas its counterpart entailed in the Foster‐Greer‐Thorbecke‐type CHE measure, as set out in Ogwang and Mwabu (2024; Appendix A) is either the variance or the squared coefficient of variation.

• d.Temporal dynamics in $WCHE$

As noted above, one of the strengths of the Watts‐type index of CHE is the ability to introduce temporal dynamics by examining changes in CHE over time. In this regard, there are several ways in which temporal dynamics can be introduced in relation to the Watts‐type index of CHE, including CHE duration (i.e., the average time to exit CHE) and CHE persistence (i.e., the analysis of CHE spells), among others. In this section, we focus solely on the average time to exit CHE, the empirical implementation of which does not require panel data, which builds on earlier endeavors to measure CHE exit times (e.g., Mussa ²⁰¹⁶; Frimpong et al. ²⁰²¹). The treatment of temporal dynamics in the presence of panel data is briefly explained in Section 5 below.

It is apparent from the detailed derivations presented in Appendix C that the average time it would take the healthcare spending units to exit CHE if it were possible to ensure that all CHE proportions are reduced at the constant positive rate $\delta$ over time, taking into account the fact that the CHE exit times for the ($N-H$) healthcare spending units that do not incur CHE is zero, is given by

$$ATE1=\frac{1}{N}\sum _{i=1}^H\left(\frac{{lnE}_i^{\ast }-lnT}{\delta }\right)=\left(\frac{1}{\delta }\right)\left[\frac{1}{N}\sum _{i=1}^H\left({lnE}_i^{\ast }-lnT\right)\right]=\frac{WCHE}{\delta }$$ (4)

It is also apparent from the detailed derivations presented in Appendix C that the average time to exit CHE for only the $H$ healthcare spending units that incur CHE is given by

$$ATE2=\frac{1}{H}\sum _{i=1}^H\left(\frac{{lnE}_i^{\ast }-lnT}{\delta }\right)=\left(\frac{1}{\delta }\right)\left[\frac{1}{N}\sum _{i=1}^H\left({lnE}_i^{\ast }-lnT\right)\right]\left(\frac{N}{H}\right)=\frac{WCHE/\delta }{H/N}$$ (5)

where $H/N$ is the overall CHE headcount ratio as indicated above.

The expression for the average time to exit CHE as set out in Equation (4) resembles that for the average time to exit income poverty, reported by Morduch (1998), even though the two exit times focus on different sides of the relevant thresholds. Also, the fact that the average time to exit CHE as set out in Equation (5) utilizes only information pertaining to the victims of CHE makes it coherent with the CHE incidence, intensity and inequality measures associated with the multiplicative decompositions of $WCHE$.

3. The Regression Approach to the Estimation and Additive Decomposition of the Watts‐Type CHE Measure

As pointed out in Section 1, one of the advantages of the Watts‐type CHE measure is that it can be conveniently estimated using widely available OLS regression software packages. We first consider the regression approach to the estimation of the overall Watts‐type CHE measure followed by the regression approach to its additive decomposition.

1. Regression approach to the estimation of $WCHE$

A careful inspection of Equation (1) reveals that the estimation of $WCHE$ can be conveniently accomplished by setting up the following intercept‐suppressed artificial regression equation that applies to the totality of N healthcare spending units regardless of their CHE status:

$$\mathit{ln}\frac{E_i}{T}=\beta D_i+u_i,i=1,2,\mathit{\text{…}},N$$ (6)

where $D_i$ is a dummy variable which takes the value 1 if the ith healthcare spending unit incurred CHE (i.e., $E_i>T$, i = 1,2,…,N) during the reference period or the value 0 otherwise; $\beta$ is the slope parameter; and $u_i$ is the error term which is assumed to be homoscedastic (i.e., $E\left(u_i^2\right)={\sigma }^2,i=1,2,\mathit{\text{…}},N$) for the sole purpose of justifying the application of the artificial estimation procedure on WCHE.

The label “artificial regression” is used in this paper, as in other papers in the statistics and econometrics literature (e.g., MacKinnon ¹⁹⁹²; Davidson and MacKinnon ²⁰⁰³; Hoque and Clarke ²⁰¹⁵), to reflect the fact that Equation (6) must be specified as is for purpose of computing $WCHE$, with no freedom to deviate from this specification. For example, if an intercept is incorporated into Equation (6) the OLS estimates of the coefficient of the dummy variable representing CHE status becomes inappropriate for the purpose of computing $WCHE$. Likewise, if we relax the homoscedasticity assumption in Equation (6) and use weighted least squares (WLS) to account for this relaxation, the resulting WLS estimate of the slope parameter β is no longer appropriate for computing $WCHE$.

The OLS estimator of the slope parameter $\beta$ in Equation (6) (involving mandatory suppression of the intercept) is given by

$$\widehat{\beta }=\frac{1}{H}\sum _{i=1}^H\mathit{ln}\frac{E_i^{\ast }}{T}$$ (7)

Although Equation (6) pertains to all the $N$ healthcare spending units regardless of CHE status, the OLS estimator of $\beta$ presented in Equation (7) captures only the information pertaining to the $H<N$ healthcare spending units that incurred CHE, which is needed to compute $WCHE$.

It follows from Equations (1) and (5) that Watts‐type index of CHE is given by

$$WCHE=\frac{H}{N}\widehat{\beta }$$ (8)

It is apparent from Equation (8) that $WCHE$ is a product of the CHE headcount ratio ($H/N$), which is a CHE incidence measure, and the OLS estimate of the slope coefficient of the artificial regression specified in Equation (6).

• b.Regression approach to the additive decomposition of $WCHE$

A careful inspection of Equation (2) reveals that isolating the contribution of the jth population subgroup in the additive decomposition of $WCHE$ entails specifying the following intercept‐suppressed artificial regression involving all the healthcare spending units in the jth population subgroup regardless of their CHE status:

$$\mathit{ln}\frac{E_{ij}}{T}={\beta }_j{D}_{ij}+u_{ij},j=1,\mathit{\text{…}},k;i=1,2,\mathit{\text{…}},N_j$$ (9)

where $D_{ij}$ is a dummy variable which takes the value 1 if the ith healthcare spending unit in the jth population subgroup incurred CHE (i.e., $E_{ij}>T,j=1,\mathit{\text{…}},k;i=1,2,\mathit{\text{…}},N_j$) during the reference period or the value 0 otherwise; ${\beta }_j$ is the slope parameter; and $u_{ij}$ is the error term which is assumed to be homoscedastic (i.e., $E\left(u_{ij}^2\right)={\sigma }^2,i=1,2,\mathit{\text{…}},N_j$) for the sole purpose of isolating the contribution of the jth population subgroup to $WCHE$.

It turns out that the OLS estimator of ${\beta }_j$ in Equation (9) is given by the formula

$${\widehat{\beta }}_j=\frac{1}{H_j}\sum _{i=1}^{H_j}\mathit{ln}\frac{E_{ij}^{\ast }}{T}$$ (10)

Although the artificial regression in Equation (9) pertains to the total population of healthcare spending units in the jth population subgroup regardless of their CHE status, the OLS estimator of ${\beta }_j$ presented in Equation (10) incorporates only the information pertaining to the healthcare spending units that incurred CHE in that subgroup which are needed for purposes of carrying out the additive decompositions the Watts‐type index of CHE as defined in Equation (2).

A careful inspection of Equations (2) and (10) reveals the following regression‐based formula for the additive decomposition of $WCHE$:

$$WCHE=\left({\sum _{j=1}^k{s}_j\widehat{\beta }}_j\right)\frac{H}{N}$$ (11)

where the quantity (H/N) is the overall CHE headcount ratio for all the k population subgroups; $s_j=H_j/H$ is the share of healthcare spending units that incurred CHE in the jth subgroup in relation to all the healthcare spending units that incurred CHE in all the $k$ subgroups as noted above.

It is apparent from Equation (11) that the contribution of the jth population subgroup to the overall level of CHE is, in fact, $s_j{\widehat{\beta }}_j\frac{H}{N}$, where $\frac{H}{N}$ is the overall CHE headcount ratio as defined above in relation to Equation (8).

4. Illustrative Example

It should be emphasized that the Watts methodology proposed in this paper can be applied to any country or region with suitable data. However, considering our interest in comparing the Watts‐type CHE measure with the competing Foster‐Greer‐Thorbecke‐type measure proposed by Ogwang and Mwabu (2024) on an equal footing, we have opted to employ the same Canadian dataset previously used by Ogwang and Mwabu (2024) to illustrate the efficacy of their Foster‐Greer‐Thorbecke‐type CHE measures. For example, one of our goals is to confirm that the CHE incidence and CHE intensity estimates based on the Watts‐type measure are identical the corresponding estimates based on the Foster‐Greer‐Thorbecke‐type measure with a CHE aversion parameter of 2 when applied to the same data.

The dataset employed in this illustrative example is described in detail by Ogwang and Mwabu (2024). Hence, in the present application of the above dataset, we shall only mention some of its key features for the sake of providing context. First, the dataset is derived from the 2019 Canadian Survey of Household Spending conducted by Statistics Canada between January and December 2019. Second, the 2019 survey provides detailed data on household demographics, household health expenditures, and other household expenditures, among others, for approximately 11,000 households across Canada. Hence, the survey provides suitable information for the analysis of the status of household CHE in Canada. Hajizadeh et al. (2023) employed consolidated data from the pre‐2019 Canadian Surveys of Household Spending to undertake a comprehensive analysis of CHE in Canada.

We focus on 223 one‐person households living in the Canadian Province of Ontario, which is adequate for purposes of fulfilling our illustrative objective, bearing in mind that the sample size would have to be increased significantly if the objective is to conduct a comprehensive study that provides resolute policy prescriptions. Our choice of one‐person households was influenced by the fact that in Canada a large proportion of older people with typically more complex health needs, that increase their risk of exposure to CHE situations, tend to live in such households. Hence, it is not surprising that 109 of the 223 one‐person households (i.e., 48.4%) comprising this dataset are headed by individuals aged 65 and over.

We base our CHE analysis on the total healthcare expenditures as a percentage of the combined discretionary and non‐discretionary expenditures. Like in Ogwang and Mwabu (2024), we use 20% CHE threshold for the estimation and decompositions of $CHEW$.

Table 1 shows the estimates of the overall Watts‐type CHE measure, the three components of its multiplicative decomposition and CHE exit time estimates using the relevant formulas mentioned in this paper. Several aspects of the results reported in the table are particularly noteworthy. First, the results are highly plausible. Second, the reported CHE incidence and CHE intensity estimates associated with $WCHE$ are identical to their counterparts associated with the Foster‐Greer‐Thorbecke‐type CHE measure with CHE aversion parameter 2 as reported by Ogwang and Mwabu (2024, Table 1). As noted above, the CHE incidence and CHE intensity estimates for the Foster‐Greer‐Thorbecke‐type CHE measure with CHE aversion parameter 2 should be identical to the corresponding estimates for $WCHE$ if applied to the same dataset. This is because the CHE incidence and CHE intensity formulas associated with both measures are identical.

TABLE 1.

Estimates of $WCHE$, the components of its multiplicative decomposition and CHE exit times ($T=0.2$).

Measure	Estimate
Overall CHE index ($WCHE$)	0.146075
CHE incidence ($H/N$)	0.251121
CHE intensity ($I$)	0.993050
CHE inequality $GE\left(0,E^{\ast }\right)$	0.107980
Average CHE exit time ($ATE1$; Equation (4) using $\delta =0.02$)	7.3 years
Average CHE exit time ($ATE2$; Equation (5) using $\delta =0.02$)	29.1 years

Note: For this dataset, $H=56$, $N=223$ and $\widehat{\beta }=\frac{1}{H}\sum _{i=1}^H\mathit{ln}\frac{E_i^{\ast }}{T}=0.581690$. Hence, the CHE incidence value is $\frac{H}{N}=0.251121$; the overall CHE measure is given by $WCHE=\widehat{\beta }\frac{H}{N}=0.581690\times 0.251121=0.146075$; the CHE intensity value is given by $I=\frac{1}{H}\sum _{i=1}^H\left(\frac{E_i^{\ast }-T}{T}\right)=\frac{{\bar{E}}^{\ast }}{T}-1=\frac{0.39861}{0.2}-1=0.99305$; and $GE\left(0,E^{\ast }\right)=\frac{1}{H}\sum _{i=1}^H\mathit{ln}\left(\frac{{\bar{E}}^{\ast }}{E_i^{\ast }}\right)=0.107980$.

Although the actual numerical values of the CHE inequality measure associated with $CHEW$ cannot be directly compared with the corresponding value associated with the Foster‐Greer‐Thorbecke‐type CHE measure with a CHE aversion parameter of 2 owing to the fact that the multiplicative decompositions give rise to different inequality measures as explained above, it turns out that the CHE inequality value reported in Table 1 like the two corresponding values reported in Ogwang and Mwabu's paper are greater than zero which indicates the existence of inequality in the normalized healthcare expenditure proportions among the households that incur CHE. This inequality is captured by the downward concavity of the non‐flat portion of a TICHE curve. Even though we do not present the TICHE curve associated with $WCHE$ in this paper, the CHE incidence, CHE intensity and CHE inequality values reported in Table 1 for $WCHE$ are also consistent with the TICHE curve for the Foster‐Greer‐Thorbecke‐type measures of CHE reported by Ogwang and Mwabu (2024, fig. 1) based on the same dataset. Clearly, these observations confirm the status of $CHEW$ as a valuable supplement to the Foster‐Greer‐Thorbecke‐type CHE measures.

It is also interesting to note from Table 1 that the average time to exit CHE would be approximately 7 years if the annual 2% rate of decrease in the CHE proportions (i.e., $\delta =0.02$) was sustained and uniform among all the healthcare spending units that incurred CHE, with the exit time for those who did not incur CHE staying at zero. However, the average exit time increases significantly to approximately 29 years if only the healthcare spending units that incurred CHE in the initial period are considered.

Table 2 shows the results of the additive decomposition of $WCHE$ by population subgroups, where subgroup formation is by gender of the household head. As explained above, subgroup formation by gender enables estimation of the contributions of male‐headed households and their female‐headed counterparts to $WCHE$. In this regard, it is interesting to note from the entries in Table 2 that female‐headed households contribute more to $WCHE$ than their male‐headed counterparts. Furthermore, the greater contribution of females to CHE is consistent with the general literature on gender differences in healthcare expenditures and healthcare utilization in Canada (e.g., Kazanjian et al. ²⁰⁰⁴; Forget et al. ²⁰⁰⁵; Hajizadeh et al. ²⁰²³) and other countries (e.g., Owens ²⁰⁰⁸).

TABLE 2.

Contributions of male‐headed and female‐headed households to $WCHE$ ($T=0.2$).

Subgroup (j)	$N_j$	$H_j$	${\widehat{\beta }}_j$	$\left(H/N\right)$	$s_j=\left(H_j/H\right)$	Contribution: $s_j{\widehat{\beta }}_j\left(H/N\right)$
1 (Male headed)	86	13	0.826029	0.251121	0.232143	0.048154
2 (Female headed)	137	43	0.507820	0.251121	0.767857	0.097921
Total	N = 223	H = 56			$\sum _{j=1}^2{s}_j=1$	$CHEW=\left(\sum _{j=1}^2{s}_j{\widehat{\beta }}_j\right)\left(H/N\right)=0.146075$

Note: In line with the notations used in the main text, $N_j$ is the number of healthcare spending units (i.e., households in the case of this dataset) in the j‐th population subgroup (j = 1,2); $H_j$ is the number of households in the j‐th population subgroup (j = 1,2) that incurred CHE; $N_j$ is the number of healthcare spending units in the j‐th population subgroup (j = 1,2); ${\widehat{\beta }}_j$ is the estimated slope coefficient of β_j as defined in Equation (10); $H/N$ is the overall CHE headcount ratio; $s_j=\left(H_j/H\right)$ is the share of healthcare spending units in the j‐th population subgroup (j = 1,2) that incurred CHE in relation to all the H healthcare spending units in all the population subgroups that incurred CHE; and $s_j{\widehat{\beta }}_j\left(H/N\right)$ is the contribution of the j‐th population subgroup (j = 1,2) to $WCHE$.

Using the same dataset, Ogwang and Mwabu (2024) also found that female‐headed households contribute more to the Foster‐Greer‐Thorbecke‐type CHE measures than their male‐headed counterparts, when the CHE aversion parameter is 0 or 1. However, they found that the situation is reversed in the case of the Foster‐Greer‐Thorbecke‐type measure with CHE aversion parameter 2. These conflicting results about the relative contributions to the overall CHE measures under consideration of male‐headed versus female‐headed households heighten the need for empirical researchers to employ more than one measure to analyze their CHE data from various perspectives.

Following a referee's suggestion, we also checked for sensitivity of the results with respect to the choice of alternative CHE thresholds. Table 3 shows the estimates of the overall $WCHE$, the three components of its multiplicative decomposition and CHE exit time estimates using 40% threshold for CHE characterization. The corresponding additive decomposition results are reported in Table 4. A comparison of the entries in Table 1 for $T=0.2$ with the corresponding entries in Table 3 for T = 0.4 reveals that increasing the threshold from 20% to 40% results in reductions in the overall $WCHE$, all components of its multiplicative decompositions and the CHE exit times, as would be expected. The additive decomposition results reported in Table 4 also show greater contribution of females to $WCHE$ than their male counterparts.

TABLE 3.

Estimates of $WCHE$, the components of its multiplicative decomposition and CHE exit times ($T=0.4$).

Measure	Estimate
Overall CHE index ($WCHE$)	0.033153
CHE incidence ($H/N$)	0.080717
CHE intensity ($I$)	0.590075
CHE inequality $GE\left(0,E^{\ast }\right)$	0.053407
Average CHE exit time ($ATE1$; Equation (4) using $\delta =0.02$)	1.7 years
Average CHE exit time ($ATE2$; Equation (5) using $\delta =0.02$)	20.5 years

Note: For this dataset, $H=18$, $N=223$ and $\widehat{\beta }=\frac{1}{H}\sum _{i=1}^H\mathit{ln}\frac{E_i^{\ast }}{T}=0.410734$. Hence, the CHE incidence value is $\frac{H}{N}=0.080717$; the overall CHE measure is given by $WCHE=\widehat{\beta }\frac{H}{N}=0.410734\times 0.080717=0.033153$; the CHE intensity value is given by $I=\frac{1}{H}\sum _{i=1}^H\left(\frac{E_i^{\ast }-T}{T}\right)=\frac{{\bar{E}}^{\ast }}{T}-1=\frac{0.63603}{0.4}-1=0.590075$; and $GE\left(0,E^{\ast }\right)=\frac{1}{H}\sum _{i=1}^H\mathit{ln}\left(\frac{{\bar{E}}^{\ast }}{E_i^{\ast }}\right)=0.053407$.

TABLE 4.

Contributions of male‐headed and female‐headed households to $WCHE$ ($T=0.4$).

Subgroup (j)	$N_j$	$H_j$	${\widehat{\beta }}_j$	$\left(H/N\right)$	$s_j=\left(H_j/H\right)$	Contribution: $s_j{\widehat{\beta }}_j\left(H/N\right)$
1 (Male headed)	86	7	0.508423	0.080718	0.388889	0.015959
2 (female headed)	137	11	0.348568	0.080717	0.611111	0.017194
Total	N = 223	H = 18			$\sum _{j=1}^2{s}_j=1$	$CHEW=\left(\sum _{j=1}^2{s}_j{\widehat{\beta }}_j\right)\left(H/N\right)=0.033153$

Note: Refer to the notes in Table 2.

5. Conclusions

In this paper we introduced the Watts‐type measure of CHE, which is an adaptation of the empirically popular Watts poverty measure. The illustrative example provided clearly shows the role of the Watts‐type CHE measure as a useful supplement to the Foster‐Greer‐Thorbecke‐type measures of CHE proposed by Ogwang and Mwabu (2024).

On the practical front, there is no doubt that the regression approach developed in this paper tremendously facilitates the process of estimating the overall Watts‐type CHE measure and decomposing it by population subgroups using widely available OLS regression packages. For the benefit of readers, we have provided supplementary materials with an additional illustrative example showing a step‐by‐step implementation of the regression approach for estimating the overall Watts‐type CHE index and decomposing it by population subgroups.

In principle, credible CHE measures can be formed by adapting other poverty measures. Hence, for the sake of expanding the choices among alternative CHE measures, it makes sense to adapt several other additively and multiplicatively decomposable poverty measures for purposes of measuring CHE.

As noted in this paper, the numerical estimates of the incidence and intensity components of the multiplicative decompositions of the Foster‐Greer‐Thorbecke‐type and Watts‐type CHE measures are the same. However, the numerical estimates of the inequality component are different owing to differences in the inequality measures associated with the multiplicative decompositions of the two CHE measures. It is also possible to compare the trends in these CHE inequality measures over time. Regardless of the CHE measure employed, the associated TICHE curve should reveal the same observations regarding the status of CHE incidence, CHE intensity and CHE inequality since CHE inequality is captured by the curvature of a TICHE curve.

The fact that the discussions surrounding CHE extend beyond health expenditures makes it necessary to develop multidimensional indices of CHE. In this regard, the multidimensional Watts index considered by Chakravarty et al. (2008) and Ogwang (2022) could, in principle, be adapted for purposes of measuring multidimensional CHE. Hence, our future goal is to adapt the multidimensional Watts poverty index for purposes of constructing a multidimensional Watts‐type CHE measure that involves two or more healthcare expenditure types.

Finally, considering the vast literature on poverty persistence (e.g., Alkire et al. ²⁰¹⁷; Fusco and Van Kerm ²⁰²³; Wang et al. ²⁰²³; Villar ²⁰²⁴) and the limited literature on CHE persistence (e.g., Garcia‐Diaz et al. ²⁰²⁴) it would be nice to extend the applications of the Watts methodology to the analysis of CHE persistence. For example, replacing the CHE dimensions in a multidimensional Watts index of CHE with the time dimension and incorporating the minimum number of time periods for CHE persistence characterization provides a decomposable Watts‐type index of CHE persistence which can serve as a credible numerical yardstick for informing policy. This extension would require the availability of full‐fledged panel datasets for which the same healthcare spending units are followed over time.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Supporting Information S1

Appendix A. Additive Decomposition of $WCHE$

We know from Equation (1) in the main document that

$$WCHE=\frac{1}{N}\sum _{i=1}^H\mathit{ln}\left(\frac{E_i^{\ast }}{T}\right)$$

$$=\frac{1}{N}\sum _{j=1}^k\sum _{i=1}^{H_j}\mathit{ln}\frac{E_{ij}^{\ast }}{T}$$

$$=\frac{H}{N}\sum _{j=1}^k\frac{H_j}{H}\frac{1}{H_j}\sum _{i=1}^{H_j}\mathit{ln}\frac{E_{ij}^{\ast }}{T}$$

$$=\frac{H}{N}\sum _{j=1}^k{s}_j\frac{1}{H_j}\sum _{i=1}^{H_j}\mathit{ln}\frac{E_{ij}^{\ast }}{T}$$

$$=\frac{H}{N}\sum _{j=1}^k{s}_j{\widehat{\beta }}_j$$ (A1)

where $s_j=H_j/H$ is the share of $H_j$ healthcare spending units in the jth population subgroup that incurred CHE in relation to the totality of $H=\sum _{j=1}^k{H}_j$ healthcare spending units in all the k subgroups that incurred CHE; $\frac{H}{N}$ is the overall CHE headcount ratio, the proportion of the totality of $N$ healthcare spending units that incur CHE; and ${\widehat{\beta }}_j=\frac{1}{H_j}\sum _{i=1}^{H_j}\mathit{ln}\frac{E_{ij}^{\ast }}{T}$ is a numerical quantity which can be conveniently obtained by running OLS regression as explained in Section 3 of the main document.

Appendix B. Multiplicative Decomposition of $WCHE$

We know from Equation (1) in the main document that

$$WCHE=\frac{1}{N}\sum _{i=1}^H\mathit{ln}\left(\frac{E_i^{\ast }}{T}\right)$$

$$=\frac{H}{N}\left(\frac{1}{H}\sum _{i=1}^H\mathit{ln}\frac{E_i^{\ast }}{T}\right)$$

$$=\frac{H}{N}\left[\frac{1}{H}\sum _{i=1}^H\mathit{ln}\left(\frac{E_i}{{\bar{E}}^{\ast }}\frac{{\bar{E}}^{\ast }}{T}\right)\right]$$

$$=\frac{H}{N}\left[\left(\frac{1}{H}\sum _{i=1}^H\mathit{ln}\left(\frac{E_i^{\ast }}{{\bar{E}}^{\ast }}\right)\right)+\mathit{ln}\left(\frac{{\bar{E}}^{\ast }}{T}\right)\right]$$

$$=\frac{H}{N}\left[\left(-\frac{1}{H}\sum _{i=1}^H\mathit{ln}\left(\frac{{\bar{E}}^{\ast }}{E_i^{\ast }}\right)\right)+\mathit{ln}\left(\frac{{\bar{E}}^{\ast }}{T}\right)\right]$$

$$=\frac{H}{N}\left[\left(-GE\left(0,E^{\ast }\right)\right)+\mathit{ln}\left(I+1\right)\right]$$ (A2)

where $GE\left(0,E^{\ast }\right)=\frac{1}{H}\sum _{i=1}^H\mathit{ln}\left(\frac{{\bar{E}}^{\ast }}{E_i^{\ast }}\right)$ is a member of the Generalized Entropy family of inequality members known as the Mean Logarithmic Deviation (MLD), applied to the normalized health expenditure proportions for the health spending units that incurred CHE.

It is apparent from Equation (A2) that WCHE can be multiplicatively decomposed into CHE incidence measure ($H/N$), CHE intensity measure, $I=\frac{1}{H}\sum _{i=1}^H\left(\frac{E_i^{\ast }-T}{T}\right)$, and CHE inequality measure which happens to be a Generalized Entropy inequality measure $GE\left(0,E^{\ast }\right)=\frac{1}{H}\sum _{i=1}^H\mathit{ln}\left(\frac{{\bar{E}}^{\ast }}{E_i^{\ast }}\right)$.

Appendix C. $WCHE$ and the Average Time to Exit CHE

All the notations in this appendix are exactly as stated in Section 2 of the main document, that is N is the total number of healthcare spending units regardless of CHE status; $T>0$ is the threshold proportion for CHE characterization; $H$ is the number of healthcare spending units that incurred CHE; $E_i$ is the healthcare expenditure proportion of the ith healthcare spending unit regardless of CHE status, $i=1,2,\mathit{\text{…}},N$; and $E_i^{\ast }$ is the healthcare expenditure proportion of the ith healthcare spending unit that incurred CHE ($i=1,2,\mathit{\text{…}},H;H\mathit{\le }N$).

The derivation of the average time to exit CHE takes into account the fact that that the time to exit CHE for each of the ($N-H$) healthcare spending units that do not incur CHE is 0.

For the H healthcare spending units that incur CHE, let $\delta >0$ designate an appropriate proportional decrease in the CHE proportion each period (e.g., year), if it could be achieved by manipulating the expenditures or incomes that are used to compute the relevant healthcare expenditure proportions. We also assume that the same value of $\delta$ applies to each of the $H$ healthcare spending units that incur CHE in the starting period and stays constant over time. For example if $\delta =0.01$, a healthcare spending unit that incurs CHE with 20% healthcare expenditure proportion in the initial period will have its healthcare expenditure proportion reduced to $20\left(1-0.01\right)=19.8$ percent in the next period, $20{\left(1-0.01\right)}^2=19.602$ percent in the following period, with the reduction process continuing until the threshold proportion for CHE characterization is reached, and the healthcare spending unit exits CHE. Under this CHE reduction scheme, the time it takes for the ith healthcare spending unit to exit CHE can be obtained using the following formula $T=E_i^{\ast }{\left(1-\delta \right)}^{{te}_i}$ or, equivalently,

$$lnT={lnE}_i^{\ast }+{te}_i\mathit{ln}\left(1-\delta \right)$$ (A3)

where ${te}_i$ is the time it takes for the healthcare spending proportion of ith healthcare spending unit to fall to $T$, the threshold proportion for CHE categorization.

Invoking the well‐known mathematical relationship $\mathit{ln}\left(1-\delta \right)\mathit{\approx }-\delta$ for very small values of $\delta$ (i.e., very close to 0), it follows from Equation (A3) that the time it takes for the ith healthcare spending unit to exit CHE is approximately given by

$${te}_i=\frac{lnT-{lnE}_i^{\ast }}{-\delta }=\frac{{lnE}_i^{\ast }-lnT}{\delta }$$ (A4)

It then follows from Equation (A4) that the average time to exit CHE for all the N healthcare spending units taking into account the fact that the CHE exit time for each of the $\left(N-H\right)$ healthcare spending units that do not incur CHE is zero is given by

$$ATE1=\frac{1}{N}\sum _{i=1}^H\left(\frac{{lnE}_i^{\ast }-lnT}{\delta }\right)=\left(\frac{1}{\delta }\right)\left[\frac{1}{N}\sum _{i=1}^H\left({lnE}_i^{\ast }-lnT\right)\right]=\frac{WCHE}{\delta }$$ (A5)

It follows from Equation (A5) that the average time to exit CHE is obtained by dividing WCHE by $\delta$, the proportional decrease.

It also follows from Equation (A5) that the average time to exit CHE for only the $H$ healthcare spending units that incur CHE is given by

$$ATE2=\frac{1}{H}\sum _{i=1}^H\left(\frac{{lnE}_i^{\ast }-lnT}{\delta }\right)=\left(\frac{1}{\delta }\right)\left[\frac{1}{N}\sum _{i=1}^H\left({lnE}_i^{\ast }-lnT\right)\right]\left(\frac{N}{H}\right)=\frac{WCHE/\delta }{H/N}$$ (A6)

where $H/N$ is the overall CHE headcount ratio in Equation (2) of the main document.

The expression for average time to exit CHE as set out in Equation (A5) resembles that for the average time to exit poverty, reported by Morduch (1998). The average time to exit CHE indicated in Equation (A6) is new.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.