# Methods for measuring horizontal equity in health resource allocation: a comparative study

- Yi Tao
^{1}, - Kizito Henry
^{2}, - Qinpei Zou
^{1}and - Xiaoni Zhong
^{1}Email author

**4**:10

https://doi.org/10.1186/s13561-014-0010-x

© Tao et al.; licensee Springer. 2014

**Received: **10 October 2013

**Accepted: **13 June 2014

**Published: **10 August 2014

## Abstract

### ᅟ

There are many existing methodologies on measuring health equity, while seldom has method aiming at health resource allocation. We collected 6 method of measuring equity in health resource allocation. This paper presents key contents of methods on measuring horizontal equity in health service allocation, yet each method has its advantages and disadvantages as well as range of application, which may help researchers or government to make wise decision when choosing appropriate method for measuring equity. Through comparative analysis, we concluded that socioeconomic factors were considered in concentration index; although the Lorenz curve and Gini-coefficient are widely used, which exist uncertainty and incompleteness; overall inequality can be decomposed by Theil index, which is of significance for the planning of urban and rural areas; preferences on a certain class can be set artificially by Atkinson index; it is easy for Chi-square to analyze aided with statistical software; specific regional differences can be calculated by index of dissimilarity.

### Classification codes

I1

## Keywords

## Introduction

Health resource allocation refers to the health resource which were distributed and flowed among health care industry (or departments), and also influenced by the factors as convenience level for medical service; hierarchy of needs and quantity; the quantity, quality, and scope of supply which could be actually provided by medical and health organization; and effective utilization degree etc. [1],[2].

From the view of [3],[4], the equity in the realm of health resource means the distribution of health resources should be based on the needs as the orientation, rather than depending on the social privilege or income difference; should share the results of social progress, rather than sharing inevitable misfortune and loss of health right [5]. Obviously, equivalent health service can hardly meet the need of every individual, which will lead too little health services utilization coexists with too much. Managing health resources and health care effectively and efficiently is an important part of promoting the development of public health. Experience has shown that, without strategic policies and focused spending mechanisms, the poor and other ordinary people are likely to get left out [6].

The issue of health resource allocation has become more and more concerned to scholars. Almost all scholars agree that equity in health resource is divided into vertical and horizontal dimensions. Horizontal equity refers to the social members who have equal need for health resource receive the same [7]; vertical equity emphasizes individuals with different levels of need can receive appropriately different amounts of health resources ([8]).

There are many methods for measuring health equity or equity first applied in the realm of economy, however, seldom has method aiming at health resource allocation. Under the consideration of simpleness, commonness and easy-comprehension, this paper summarized six methods for horizontal health allocation equity estimation, each method has its advantages and disadvantages, as well as applicable conditions, and been analyzed through definition, calculation method, application, data requirement and other aspect. Meanwhile, different research methods are also likely to produce different results.

## Review

### Methodology of measuring horizontal equity

In health resource delivery, inequity is means that discrimination for non-need factors, since we determined that only allocation according to need is equitable [9]. An easy method to test for the existence of inequity in health resource allocation is to test whether two (or more) groups (for instance the rich and the poor or different regions) receive the same amount of resource [10]. When we compare whether inequity is present, we need to take into account whether these two groups have the same amount of need (are equal) and therefore whether are completely comparable. This can be amended by correcting for the difference in need between the groups, either via direct or via indirect standardization method. They can be compared, when both groups have equal needs, or when standardized to equal need.

### Method of concentration curve and concentration index

#### Concentration curve

The concentration curve [11],[12], and related concentration index (CI), have now days attained the status of “workhorse” in most health economic studies” [13]. For example, it could be used to assess whether subsidies to the health sector are well targeted towards the poor among countries [14], or whether inequalities in health resource allocation are more pronounced in some countries than in others [15]. And other applications are also possible. When examining the equity of health care resource allocation, it uses the concept of horizontal equity, i.e. treating people with equal need the same and irrespective of their income [16].

#### Concentration index

The concentration index is an index to investigate the unfair degree of a certain variable associated with social and economic status, which dynamically reflects the effect of the variable influenced by income.

_{exp}(Figure 1), and the line of equality (the 45°line running from the bottom-left corner to the top-right). So, in the case where there is no income-related inequality, the concentration index is zero [17]. When computing, firstly, rank by social class with corresponding rank (X); then calculate a certain variable level (H) of one social class level, and according to this variable mean (M), can finally work out CI.

Where, COV(XH) is the covariance of (X) and (H), E (XH) is the mathematical expectation of the product for (X) and (H), E(X) is the mathematical expectation of (X), E(H) is the mathematical expectation of (H).

C lies in range (−1—1). CI < 0, indicates health resource variable is disproportionately concentrated on poor; CI > 0, indicates health resource variable is disproportionately concentrated on rich; C = 0, when distribution in proportionate; the further the CI deviate from 0, the higher level of unfair will be. C = 1, if richest person has the entire health resource variable; C = −1, if poorest person has the entire health resource variable.

### Method of Lorenz curve and Gini-coefficient

#### Lorenz curve

#### Gini-coefficient

As a foundation of welfare economics to measure inequity in health resource, the primary measure of income inequality, Gini-coefficient (GINI), has been widely used to test the relationship between inequality and health [18],[19]. A region with no inequality will have a value of 0 while a region with complete inequality will be denoted by a Gini-coefficient of 1. Given a Lorenz-curve plot, we can measure the degree of inequality of the distribution of health resource by a one-dimensional number, the so-called Gini-coefficient ([8]).

[1] GINI-coefficient = Area A/(Area A + Area B)

The higher the GINI-coefficient is, the more unequal is the resource being distributed across the population in question.

Because 100% is equal to 100/100 = 1, and the two axes in the Lorenz curve goes from 0% to 100%, the area of the entire box must be 1. It follows that Area A + Area B must equal ½.

The Gini-coefficient therefore also can be written as,

[2] Gini-coefficient = (Area A)/(1/2) = 2 × (Area A)

It is the metric you see when Gini-coefficients are shown.

The Gini-coefficient ranges from 0 to 1, with 0 signifying perfect equality (the Lorenz curve coincides with the diagonal line in Figure 3) and with 1 signifying perfect inequality. The standard of Gini-coefficient in health resource allocation refers to income distribution fairness in economics. Gini-coefficient < 0.3, indicates in perfect equity condition; 0.3-0.4, indicates in normal condition; >0.4, indicates in alert condition, >0.6, indicates in highly inequity perilous condition [20].

#### Theil index

*P*

_{ i }represents the proportion of some place’s population accounts for total population;

*Y*

_{ i }represents the proportion of health resources owned by some place accounts for the total number of health resources. A weighted sum of inter-unit inequality within each group, called the “within group” component, and a “between-group” component that measures inequality due solely to variations in health resource density across groups. The decomposition formula is;

In the above formulas, ‘T_{intra-class}’ in this article means the differences of health resource allocation in the area; ‘T_{inter-class}’ means the differences of health resource allocation between areas; *P*_{
g
} represents the proportion of some place’s population accounts for total population; *Y*_{
g
} represents the proportion of health resources owned by someplace accounts for the total number of health resources. The contribution rate of the difference between each part on total theil index can be calculated after decomposing the theil index. For health resource allocation, if the TI = 0, means equity in allocation, the smaller the value is, the more equity in allocation will be, and vice versa.

#### Atkinson index

*ε* is a parameter related to the external clearer display of regional imbalance, called inequality aversion. The parameter reflects social equal degree for inequality aversion (or preferences). 0 < *ϵ* < + ∞, the higher the *ε* is, the display of imbalance will be more obvious, when *ε* = 2, Atkinson index can reveal moderate imbalance. For assessing equity of health resource, *Y*_{
i
} is the health resource gotten by individuals in the ith health resource range (N ranges altogether), *f*_{
i
} is the proportion of the population with health resource in the ith range, $\overline{\mathit{Y}}$ is the mean health resource in group.

For assessing the equity of health resource allocation, an intuitive interpretation of this index is possible: Atkinson values can be used to calculate the proportion of total health resource that would be required to achieve an equal level of allocation as at present if health resources were perfectly distributed. For example, an Atkinson index value of 0.20 suggests that we achieve the level of allocation with only 1–0.20=80% of health resource. The theoretical range of Atkinson values is 0 to 1, with 0 being a state of equal distribution [31]. The smaller the index is, the more equal the allocation will be, and vise verse.

#### Chi-square value method

*f*i ( i = 1, 2, ,, k) means there are k actual frequencies, which can be gotten through the investigation and experiment; $\overline{\mathit{f}}$ i ( i = 1, 2, ,, k ) means there are k expectation frequencies, which should be calculated according to the statistical hypothesis or refer to relevant data.

- (2)
X2 test steps

- 1)
Establish the null hypothesis H

_{0}and alternative hypothesis H_{1}; - 2)
Calculate expectation frequency according to the theoretical distribution (or empirical distribution);

- 3)
Calculate sample chi-square value according to the actual frequency and expectation frequency (formula 4);

- 4)
Find corresponding chi-square critical value According to the degree of freedom and significance level a in the chi-square distribution list. If the calculated value is less than chi-square critical value, accept the null hypothesis; otherwise, accept alternative hypothesis.

- (3)
Evaluation of X2 value on equity of health service allocation

Analyze form the formula, the general term ${\frac{\left({\mathit{f}}_{\mathit{i}}-\overline{{\mathit{f}}_{\mathit{i}}}\right)}{\overline{{\mathit{f}}_{\mathit{i}}}}}^{2}$ (*i =* 1,2…,k) is a ratio of the square of *f*_{
i
} deviated from $\overline{{\mathit{f}}_{\mathit{i}}}$ with $\overline{{\mathit{f}}_{\mathit{i}}}$, the sum of which reflects the difference between distribution of *f*_{
i
} and distribution of $\overline{{\mathit{f}}_{\mathit{i}}}$. The bigger the chi square value, the more significant difference between distributions will be, and vice versa. Hence, the variation of chi-square value can reflect variable trend of the difference degree between variables’ actual distribution and theoretical distribution. For health resource allocation, if the chi-square value less than critical value, indicates that in a certain significance level α, this kind of health resource is allocated fairly, and vice verse; from the perspective of the variable trend, by comparing a health resource variate in different years, which can also indicates the trend of the equity of health resource allocation.

#### Index of dissimilarity

*S*_{
jh
} means the proportion of a certain variable which can reflect the equity of health resource allocation of jth region (or jth region of a certain socioeconomic level); *S*_{
jp
} means the population proportion of jth region (or in a certain socioeconomic level). The greater the differences between *S*_{
jh
} and *S*_{
jp
}, the higher the health resource inequality degree is. The ID value is between 0 and 1, if the ID = 0, means equity in allocation, the smaller the value is, the more equity in allocation will be, and vice versa. The index of dissimilarity is large, when large parts of the population are in low and high socioeconomic groups and there are few people in intermediate groups [32].

## Conclusions

**Summary of health resource allocation equity measures**

Measure | Definition | Complexity of calculation | Application | Required data | Benefits | Caveats |
---|---|---|---|---|---|---|

Concentration curve, concentration index | Calculate and compare cumulative percentage of population (ranked by socioeconomic factors) and health resource. CI $\mathrm{CI}=\phantom{\rule{0.5em}{0ex}}\frac{2\times \mathrm{COV}\phantom{\rule{0.25em}{0ex}}\left(\mathrm{X},\mathrm{H}\right)}{\mathit{M}}$ | Complex but aided by statistical software | Systematic assessment, and can be a rough estimation on equity of differences between different regions. | Income of individual, health resource of individual | -not only represent overall inequity, also reflect accurately which social classes allocated with more resource and which less via positive or negative CI value | -incapable of considering the other variables, especially the resource delivery itself. |

-socioeconomic factors are taken into consideration when measure the inequity. And which is very sensitive to different social classes | ||||||

-simple to calculate | ||||||

-simple to interpret when combine with corresponding curve | ||||||

-the concentration index must be interpreted with the curve | ||||||

-does not allow for within or between income group comparisons | ||||||

Lorenz curve, Gini index | Calculate and compare cumulative percentage of population (ranked by how much resources obtained) and health resource. $\mathit{G}={\displaystyle \sum _{\mathit{i}=1}^{\mathit{k}}{\mathit{P}}_{\mathit{i}}}{\mathit{S}}_{\mathit{i}+1}-{\displaystyle \sum _{\mathit{i}=1}^{\mathit{k}}{\mathit{P}}_{\mathit{i}+1}}{\mathit{S}}_{\mathit{i}}$ | Complex but can aided by statistical software | Systematic assessment, and can be a rough estimation on equity of differences between different regions. | Health resource of individual, total health resource, population of area | -a graphical representation of allocation inequality that can be compared over time and between geographic areas | -incapable of showing different kinds of inequality represented by various shapes of Lorenz curves [34] |

-simple to calculate | ||||||

-data readily available | ||||||

-can be calculated for individual and household level data | ||||||

-easily interpreted when combine with Gini coefficients | ||||||

-does not emphasize inequalities in the top or bottom of the spectrum (polarization) | ||||||

-shows the direction of allocation redistribution but does not indicate where the redistributions are occurring | ||||||

-does not allow for within or between income group comparisons | ||||||

-overlook socioeconomic factors | ||||||

Theil index | Calculate the equity of health resource by population (area) in each region. $\mathit{T}={\displaystyle \sum _{\mathit{i}=1}^{\mathit{n}}{\mathit{P}}_{\mathit{i}}}log\frac{{\mathit{P}}_{\mathit{i}}}{{\mathit{Y}}_{\mathit{i}}}$ | complex | Measure equity of the allocation of health resources between different regions or the units. | Population of units or regions, total population , health resources in units or regions, total resource | -shows the contributions to inequality by within group and between group components | -complex to calculate and interpret. |

-varies greatly when the distribution varies regardless of the change in distribution occurs at the top, middle or bottom | ||||||

-high sensitivity to the efficiency of health resource allocation | ||||||

-resource redistributions will impact the calculation irrespective of whether the redistribution takes place between top and bottom or top and middle | ||||||

-cannot directly compare populations with different sizes as calculation is dependent on number of individuals in the population or group | ||||||

Atkinson index | Calculate the health resources of ith region and the proportion of population in which people get the resources. ${\mathit{I}}_{\mathit{R}}=1-{\left[{\displaystyle \sum _{\mathit{i}=1}^{\mathit{n}}{\left(\frac{{\mathit{Y}}_{\mathit{i}}}{\overline{\mathit{Y}}}\right)}^{1-\mathit{\epsilon}}{\mathit{f}}_{\mathit{i}}}\right]}^{\frac{1}{1-\mathit{\epsilon}}}$,if | complex | Assess the inequity of allocation, address needs of inequity assessment in health benefits analysis | Health resource of ith region, the proportion of population in ith region who get the resource, inequality aversion | -incorporates a sensitivity parameter directly into the equation. | -sensitivity parameter means that a subjective judgment has been made about inequality |

-not intuitive | ||||||

Chi-square Value Method | Calculate the actual and theoretical frequency of health resources. ${\mathit{X}}^{2}={\displaystyle \sum _{\mathit{i}=1}^{\mathit{k}}\frac{{\left({\mathit{f}}_{\mathit{i}}-\overline{{\mathit{f}}_{\mathit{i}}}\right)}^{2}}{\overline{{\mathit{f}}_{\mathit{i}}}}}$ | Easy when analyze aided with statistical software | Assess the difference between actual allocation of health resource with the expected allocation | Actual resources in ith region, the total resource, expected frequency of health resource allocation | -sensitive to reflect the inequity of allocation | -always need to standardize the data, otherwise may influence the results |

-reveal the trend of equity over time | ||||||

-the judgment is subjective when based on a certain significance level α | ||||||

index of dissimilarity | Calculate the health resources and population in each socioeconomic level (region). $\mathit{ID}=\frac{1}{2}{\displaystyle \sum _{\mathit{j}=1}^{\mathit{k}}\left|{\mathit{S}}_{\mathit{jh}}-{\mathit{S}}_{\mathit{jp}}\right|}$ | easy | assess the differences of resource allocation in different economic level(region), and calculate the degree of variance | Resource in jth region(or in a certain socioeconomic level), the population in jth region | -can know the differences between the situation of health resource allocation in each region (level) and the proportion of the population in relative region (level) | -can’t reflect the socioeconomic status influence on health resource allocation. |

-not intuitive |

Each indicator has its merits and demerits and each serves different purposes. The most commonly used measures are concentration curve combined with concentration index and Lorenz curve combined with Gini index, which are easy to calculate; and intuitive reflection could be made with corresponding curve; concentration index can be used to reflect the unequal distribution caused by socioeconomic factors, however, this measure only calculates income-related inequity without considering the other casual variable and not inequity in health service delivery per se [10]. Gini-coefficient allows direct comparison between units with different size of populations, nevertheless, which overlook socioeconomic status [31]. Calculation of Theil index is complex, however, it can avoid the demerits of uncertainty, imperfection, and incomparability when describe Lorenz curve and calculate Gini index; Theil index can also divide the overall fairness, which can better reflect the differences of distribution within and between groups. Atkinson index has an inequality aversion *ε*, preferences on certain people could be made artificially, and this enables to define how sensitivity the Atkinson index should react to inequalities [35]. Nonetheless Chi-square Value Method is not widely used, it’s convenient to analyze aided by statistical program; which not only can compare the equity condition in different regions, but also can reflect the trend of equity over time. Index of dissimilarity can used to calculate the accurate degree of difference.

### Example: measuring the equity of health resource allocation in Chongqing (China)

We illustrate the measures with health resource data in Chongqing (China) from 1998–2012, here we used Gini-index and Thiel index as examples.

**Gini-index of health resource allocation from 1998-2012**

Year | Allocation by population | Allocation by area | ||||
---|---|---|---|---|---|---|

Beds | Doctor | Nurse | Beds | Doctor | Nurse | |

1998 | 0.3300 | 0.2373 | 0.4407 | 0.5709 | 0.4934 | 0.5906 |

2002 | 0.3456 | 0.2728 | 0.4099 | 0.5832 | 0.5273 | 0.6389 |

2007 | 0.3071 | 0.2725 | 0.4141 | 0.5640 | 0.5326 | 0.6352 |

2012 | 0.2389 | 0.2843 | 0.3715 | 0.5019 | 0.5494 | 0.6049 |

**Theil-index of health resource allocation from 1998-2012**

Year | Allocation by population | Allocation by area | ||||
---|---|---|---|---|---|---|

Beds | Doctor | Nurse | Beds | Doctor | Nurse | |

1998 | 0.0778 | 0.0438 | 0.1149 | 0.2559 | 0.1935 | 0.3195 |

2002 | 0.0847 | 0.0537 | 0.0980 | 0.2675 | 0.2160 | 0.3362 |

2007 | 0.0670 | 0.0543 | 0.1108 | 0.2531 | 0.2209 | 0.3315 |

2012 | 0.0415 | 0.0579 | 0.0859 | 0.1982 | 0.2408 | 0.3013 |

According to the methods of evaluation on horizontal equity of health resource allocation, as well as the availability of the data, we select most commonly used and appropriate methods to study. Sometimes a single index cannot reflect all the allocation disparity problem, you can construct a comprehensive index, or use one of them as key index, and supported by a number of secondary indices, to more comprehensive, in-depth evaluate the equity of health resource allocation.

## Declarations

## Authors’ Affiliations

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