The equity analysis was threefold. The study first assessed the progressivity of the NHIS premium and measured the degree of progressivity. Second, the study computed the redistributive effect of the premium, a measure that involves the computation of the horizontal inequity. Lastly the study computed the incidence and intensity of catastrophic expenditure of the premium.

To assess the progressivity of the premium, households in the data were categorized into ability to pay quintiles. Concentration curve and Lorenz curves were drawn on the same graph for the ability to pay quintiles and compared. The concentration curve plotted the cumulative percentage of the premium contribution against the cumulative percentage of the sample according to ability to pay in increasing order. If the premium contribution was the same regardless of ability to pay the concentration curve would be equal to the 45° line. If the premium contributions of the poor exceeded that of the rich then the curve would lie above the 45 degree line, otherwise it would lie below the line. The Lorenz curve is the representation of the cumulative distribution of ability to pay among the group. If ability to pay was evenly distributed the Lorenz curve would be equal to the 45 degree line otherwise it would be convexly sloped under the 45 degree line. If the concentration curve was everywhere below the Lorenz curve then the premium collection would be progressive. The premium collection would be regressive if the concentration curve was everywhere above the Lorenz curve [17].

The method introduced by [

18] was used to measure the ability to pay redistributive effect of the premium. The redistribution is measured as:

$\begin{array}{l}\mathit{RE}=V+H+R\\ \phantom{\rule{1.2em}{0ex}}=\left(\frac{g}{1-g}\right){K}_{p}-{\displaystyle \sum}{\alpha}_{x}{G}_{x-p}-\left[{G}_{x-p}-{C}_{x-p}\right]\end{array}$

where *g* is the average share of ability to pay taken up by the premium, *K*_{
p
} is the Kakwani index of premium progressivity, *α*_{
x
}, weight, is equal to the product of the square of population share of those with pre-premium ability to pay *x* and post premium ability to pay share of the pre-premium ability to pay of the group, *G*_{
x-p
} is the post premium Gini coefficients of those with prepayment ability to pay *x*, *C*_{
x-p
} is the post premium concentration index which is obtained by first ranking the households by their pre-premium ability to pay and then within each equal ability to pay group rank them by their post premium ability to pay. The first term on the right hand side (*V*) measures the vertical redistribution or the inequality reduction that would occur in the absence of horizontal inequity. The second term (*H*) measures horizontal inequity and is equal to the weighted sum of the post-premium ability to pay Gini coefficients of the ability to pay groups. The third term, (*R*) measures re-ranking that occurs from the move from an ability to pay group to another as a result of the premium contribution. If *R* is zero there is no re-ranking. According to Aronson et al., *H* increases while *R* falls as the range of income used to define ‘equals’ widens. Thus a distinction between *H* and *R* is not interesting. The focus of the analysis was on the component of *V* versus *H* + *R* in income redistribution. However reranking is typically caused by horizontal inequity and so *H + R* was considered as capturing horizontal inequity [4].

In general Kakwani index equals twice the area between the premium concentration curve and the Lorenz curve and is computed as

*K*_{
p
} =

*C - G* where

*C* is the pre-premium concentration index and

*G* is the pre-premium Gini coefficient. The -2 <

*K*_{
p
} < 1. If

*K*_{
p
} < 0, then the premium is regressive. The study applied a convenient regression below found in [

17] for the estimation of the Kakwani index:

$2{\sigma}_{R}^{2}\left[\frac{{h}_{i}}{\eta}-\frac{{y}_{i}}{\mu}\right]=\alpha +\beta {R}_{i}+{e}_{i}$

where

*R* is the fractional rank of the ability to pay variable,

*σ*_{
R
}^{2} is the variance of

*R*_{
i
},

*h*_{
i
} is the premium paid by household,

*η* is the mean of premiums paid,

*y*_{
i
} is the ability to pay variable and

*μ* is the mean of ability to pay. Kakwani index is the coefficient of

*R*. The formula for Gini coefficient is:

$G=1-\frac{\frac{2}{T}\left({{\displaystyle \sum}}_{i=1}^{n-1}{x}_{i}\right)+1}{n}$ where

*x*_{
i
} is the cummulated values of the premiums for individuals,

*T* is the last value of the cummulative column, and

*n* is the sample size. Gini coefficient is between zero and one with zero meaning perfect equity and one implying all income is owned by one person and nothing for others (perfect inequity). A convenient regression below, also from [

17] was used to estimate the Concentration Index:

$2{\sigma}_{R}^{2}\left[\frac{{h}_{i}}{\eta}\right]=\vartheta +\gamma {R}_{i}+{\epsilon}_{i}$

where the *γ* is the concentration index and all other variables are as defined above.

Following [17], catastrophic expenditure was computed as the ratio of the premium and the ability to pay variable: *P*_{
i
}*/y*_{
i
} where *P*_{
i
} is the premium for household i and *y*_{
i
} is the household’s ability to pay variable. A household is said to incur catastrophic expenditure if the ratio exceeds a threshold, *z*. Following the literature, a range of values were used as threshold: *5, 10, 15*, and *20*%. For example, a threshold of 10% means that if the premium payment forms more than 10% of the household expenditure on food and others the premium payment causes household to sacrifice essential needs, sell assets, or become impoverished. Where there is catastrophic expenditure, information on incidence and the intensity of the catastrophic expenditure would be important. The incidence simply means the number of households who incurred catastrophic expenditure for a given threshold. It is computed as the fraction of the households who incurred catastrophic expenditure: $H=\frac{{{\displaystyle \sum}}_{i=1}^{N}{E}_{i}}{N}$*N* is the sample size, *E*_{
i
} equals one when *P*_{
i
}*/y*_{
i
} > *z* and zero otherwise. This measurement of incidence does not take into account the distribution of catastrophic expenditure and so gives the same weight to households who incurred catastrophic expenditure regardless of ability to pay. A weighted incidence, *H*^{
W
} was thus used: *H*^{
W
} = *H*(1 − *C*_{
E
}) where; *C*_{
E
} is the concentration index for *E*. *H*^{
W
} is the rank weighted incidence or head count and it takes into account the distribution of the catastrophic expenditure. The rank weighted head count puts a greater weight on the poor household that incur catastrophic payment than the rich. Thus, the *H* < *H*^{
W
}. A negative *C*_{
E
} means that the poor are more likely to exceed the threshold than the rich.

The intensity of any catastrophic expenditure (i.e., the amount by which catastrophic expenditure exceeds the threshold) that might exist was measured as: $O=\frac{{{\displaystyle \sum}}_{i=1}^{N}{O}_{i}}{N}$ where ${O}_{i}={E}_{i}\left[\frac{{P}_{i}}{{y}_{i}}-z\right]$. Again, to adjust for the distribution of catastrophic expenditure the rank weighted overshoot was used: *O*^{
W
} = *O*(1 − *C*_{
O
}), where *C*_{
O
} is the concentration index for *O.* The mean positive overshoot (*MPO*) was also computed to provide information on the average overshoot among those who exceeded the threshold. $\mathit{\text{MPO}}=\frac{O}{H}$, thus *z* + *MPO* represents the average expenditure on premium as a share of ability to pay by those whose premium exceeded the threshold.

To find the characteristics of those who are likely to incur catastrophic expenditure multivariate logistic regressions were run: *E*_{
i
} = *α*_{1} + *α*_{2}*X*_{2} + *α*_{3}*X*_{3} + *α*_{4}*X*_{4} + *α*_{5}*X*_{5} + *e*_{
i
}

where *X*_{
2
} is a vector of demographic characteristics: age and gender (female); *X*_{
3
} is a vector of variables for marital status, *X*_{
4
} is the ability to pay variable, and finally, *X*_{
5
} is the location dummy for Kumasi. Three regressions were run; one for each of the thresholds: *5*, *10*, and *15*.