Table 1 Summary of CI-based index properties and their health variable requirements
Properties of the health variable (hi) | Properties of the index | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
Intervalb | Ratio | Unbounded | Bounded | Absolute | Relative | Mixed | Quasi-absolute | Mirror | Transfer | Weighting scheme | Index equationa | |
Standard CI | ✓ | ✓ | ✓ | ✓ | Fixed. Inequality aversion parameter (v) = 2 | \(\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left\{ {\frac{{h_{i} }}{{\overline{h}}}\left( {2R_{i} - 1} \right)} \right\}\) | ||||||
Modified CI | ✓ | ✓ | ✓ | ✓ | Inequality aversion parameter (v) = 2 | \(\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left\{ {\frac{{h_{i} }}{{\overline{h} - a_{h} }}\left( {2R_{i} - 1} \right)} \right\}\) | ||||||
Generalized CI | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Inequality aversion parameter (v) = 2 | \(\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left\{ {h_{i} \left( {2R_{i} - 1} \right)} \right\}\) | ||||
Extended CI | ✓ | ✓ | ✓ | ✓ | Asymmetric Inequality aversion parameter (v) can be varied | \(\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left[ {\frac{{h_{i} }}{{\overline{h}}} \left\{ {1 - v\left( {1 - R_{i} } \right)^{v - 1} } \right\}} \right]\) | ||||||
Generalized extended CI | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Inequality aversion parameter (v) can be varied | \(\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left[ {\left( {\frac{{v^{{\frac{v}{v - 1}}} }}{v - 1}} \right)\left( {\frac{{h_{i} - a^{h} }}{{b_{h} - a_{h} }}} \right)\left\{ {1 - v\left( {1 - R_{i} } \right)^{v - 1} } \right\}} \right]\) | |||
Wagstaff index | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Fixed | \(\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left[ {h_{i} \left\{ {\frac{{h_{i} \left( {b_{h} - a_{h} } \right)}}{{\left( {b_{{\text{h}}} - \overline{h}} \right)\left( {\overline{h} - a_{{\text{h}}} } \right)}}} \right\}\left( {2R_{i} - 1} \right)} \right]\) | ||||
Erreygers index | ✓ | ✓ | ✓b | ✓ | ✓ | ✓ | Fixed | \(\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left\{ {4\frac{{h_{i} }}{{\left( {b_{h} - a_{{\text{h}}} } \right)}}\left( {2R_{i} - 1} \right)} \right\}\) | ||||
Symmetric index | ✓ | ✓ | ✓ | ✓ | Symmetric Inequality aversion parameter (β) can be varied β can be varied | \(\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} \left( {\frac{{h_{i} }}{{\overline{h}}}} \right)\left[ {\beta 2^{\beta - 2} \left\{ {\left( {R_{i} - \frac{1}{2}} \right)^{2} } \right\}^{{\frac{\beta - 2}{2}}} \left( {R_{i} - \frac{1}{2}} \right)} \right]\) | ||||||
Generalized symmetric index | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Symmetric Inequality aversion parameter (β) can be varied β can be varied | \(\frac{1}{n}\mathop \sum \limits_{i = 1}^{n} 4\left( {\frac{{h_{i} - a^{{\text{h}}} }}{{b_{{\text{h}}} - a_{{\text{h}}} }}} \right)\left[ {\beta 2^{\beta - 2} \left\{ {\left( {R_{i} - \frac{1}{2}} \right)^{2} } \right\}^{{\frac{\beta - 2}{2}}} \left( {R_{i} - \frac{1}{2}} \right)} \right]\) | |||