Slope index of inequality (SII) and relative index of inequality (RII)

Introduction

In order to calculate the SII and RII for a given indicator and geographical area, data are required for subgroups of the area which can be ranked by level of deprivation. For each subgroup, the indicator value, its confidence interval or standard error, and population denominator must be known.

Commonly, the SII and RII are calculated from data broken down into 10 subgroups based on the Index of Multiple Deprivation (IMD). These subgroups are often referred to as deprivation deciles, and further guidance on these can be found in the assigning deprivation categories section. However, the SII and RII for an area can also be calculated from other subgroups, as long as they can be ranked by deprivation. For example, wards within a local authority, or deprivation quintiles rather than deciles, could be used to calculate the SII and RII for the area. Care should be taken when choosing appropriate subgroups, as too few, or too many, could reduce the robustness of the SII measure. Too few groups risks hiding the full extent of inequality by grouping data for areas that are quite different, whereas too many groups can lead to very wide confidence intervals for the very small contributing values, which will lead to an SII with very wide confidence intervals.

Linearity and the SII/RII

The standard approach for calculating the SII, described by Low and Low (2004), includes the assumption that there is a linear relationship between the indicator of interest and deprivation. This relationship doesn’t need to be perfect, but if there is a systematic non-linear element to the relationship then the SII will not fully reflect the extent of inequality.

One method of establishing whether the relationship is linear is to plot the data and assess linearity by eye. There are also a number of statistical techniques to assess whether a relationship between two variables is linear. However, there are no set criteria to determine whether the assumption of linearity is met in order to be able to calculate the SII using the standard approach.

There are two different aspects to consider: the first is non-linearity due to the nature of the statistics being analysed (eg rates, proportions, etc) and the second is non-linearity that is simply a fact of the relationship between the specific outcome and the measure of deprivation: there is no reason to assume that the difference in ‘deprivation’ between the most deprived and second most deprived deciles in an area is the same in a fundamental way as the difference between the fourth and fifth most deprived, for example. There is a tendency with health outcomes for the least deprived and, particularly, the most deprived, to show steeper gradients, with the relationship being relatively flat across the middle of the range. This pattern is observed frequently, but not universally.

The first, statistical, aspect is fairly easy to deal with: OHID’s approach is that rates are log-transformed and proportions logit-transformed prior to calculation of the SII/RII to account for the assumption of non-linearity with these statistics. Life expectancy and healthy life expectancy are not transformed, as the range of values for these statistics is relatively limited (they do not get close to zero) and linearity has been found generally not to be improved by using a log or logit transformation.

The second aspect, ie the true underlying distribution of deprivation and its relationship with health outcomes, is not so easily accounted for: it is important always to look at the data when the SII is fitted, and be aware of the limitations of the statistic when a systematic non-linear relationship is observed. If the regression line really doesn’t appear to reflect the distribution of the points, it is probably inappropriate to use the SII.

The sections below describe the calculation of the SII and RII both for indicators where a linear relationship is assumed, using the standard approach set out by Low and Low (2004) and for other indicators, using the transformation approach developed and used by OHID.

Calculation of the SII where there is a linear relationship

Figure 1 provides an example of the calculation of the slope index of inequality where it is assumed that there is a linear relationship between the indicator and deprivation. In this example, the SII for life expectancy in England is being calculated.

The chart shows life expectancy for 10 deprivation deciles within England. These have been plotted to take account of their population size. The horizontal x-axis along the bottom of the chart represents the whole population of England. If Decile 1 includes exactly 10% of the population, the first point is positioned at 5%, the mid-point of the range of population covered by that decile. If the second decile includes 11% of the population, this would cover the range from 10% to 21%, so the midpoint is 15.5%, and that is where the point would be located on the x-axis.

The orange line on the chart is a linear regression line of best fit for the data, calculated by the ordinary least squares method. A transformed regression equation is used to account for potential heteroskedasticity (different variances arising from different sized groups) of the error terms, as described by Low and Low (2004). The regression formula is:

\[ Y \sqrt a = 0 + \sqrt a + b\sqrt a \] where:

• \(Y\) is the the indicator value
• \(a\) is the proportion of population in the decile (or other subgroup)
• \(b\) is the relative rank (cumulative proportion of the population in the decile (or other subgroup))

The SII is the \(b\) coeffient of the regression equation, which is the gradient of the regression line. Since the x axis is plotted on a scale from 0 to 1, the SII is also equal to the difference between the top of the line (at x=1 on the horizontal axis) and the bottom (at x=0 on the horizontal axis).

The SII is an absolute measure of inequality (see the inequality measurement overview for a description of absolute and relative inequality). In this example, the SII is the absolute difference between 84.1 years and 74.4 years, giving an SII of 9.7 years.

There is a clear non-linear pattern in this example: the differences between the deciles are greater at the extremes of the deprivation distribution and smaller in the middle, as is often observed. In this example this is not due to the statistical nature of the outcome (life expectancy) and there is no simple transformation that could be applied, and the SII is not considered to be misleading: it is a conservative estimate of the range of inequality, but is a fair reflection of the gradient across all groups.

Calculation of the RII where there is a linear relationship

The RII is a summary measure of inequality related to the SII. While the SII measures the absolute difference between the most and least deprived, the RII measures the relative difference.

In the example above, the RII is the realitive difference between 84.1 years and 74.4 years, giving an RII of 84.1/74.4 = 1.13.

Calculation of the SII and RII where there is a non-linear relationship

Figure 2 provides an example of the calculation of the slope index of inequality where there is assumed to be a non-linear relationship between the indicator and deprivation. In this example, the SII for cardiovascular disease mortality in England is being calculated.

Figure 2 (a) shows the directly standardised cardiovascular disease mortality rate for each deprivation decile in England, and the SII regression line, using the standard linear approach described in the Calculation of the SII where there is a linear relationship section above. Here, the line slopes downwards, since mortality rates decrease with decreasing level of deprivation, meaning that the SII value is negative. In this example, the SII value is –110.2 per 100,000 population and the RII value is 0.17.

The decile values are clearly not linear though, and the SII line does not fit well to the data. Since this is to be expected for rates over a wide range of values, OHID’s recommended approach is to log transform the decile values (where they are rates), or logit transform the decile values (where they are proportions) prior to calculation of the SII and RII to account for this systematic non-linearity in the data.

Figure 2 (b) shows the mortality rates for each decile after log transformation. The SII has been calculated using the same regression equation as above, but using the log transformed rates as the \(Y\) values. As before, the SII can be calculated from the \(Y\) value at x=0 and x=1 on the horizontal axis. In this case, the \(Y\) value at x=0 is 5 and the \(Y\) value at x=1 is 3.6, giving an SII value of –1.4 and an RII value of 0.72.

In this case the SII line fits the data better. However, it is difficult to interpret the SII and RII values when they are presented on a log scale. Therefore, OHID recommends converting these values back into the original units of the indicator.

Figure 2 (c) shows the mortality rates for each decile, and represents the SII line following conversion back to the original units of the indicator. The top and bottom values of the SII line are calculated by taking the antilog of the \(Y\) value at x=0 of chart b) (\(e^{5} = 148.4\)) and the antilog of the \(Y\) value at x=1 of chart b) (\(e^{3.6} = 36.6\)). This gives an SII value of –111.8 and an RII value of 0.25.

In this case the SII line better fits the data, and the SII and RII values can be interpreted in the same way as the standard linear approach.

Where there is a linear relationship between the indicator and deprivation

Confidence intervals for the SII and RII are calculated using a simulation approach (introduced in the Overview of confidence intervals). Simulation is a method used to estimate the degree of uncertainty for measures where the statistical distributions underpinning the measure are too complex to analyse mathematically.

For each decile (or other subgroup), the indicator values are required, along with their standard errors. Where confidence intervals are provided, rather than standard errors, the standard error can be estimated from the confidence intervals, as described in the Overview of confidence intervals.

The standard errors give information about the degree of uncertainty around each of the indicator values: essentially they summarise the statistical distribution for each decile. Using a random number generating algorithm, a random value is taken from each decile indicator value distribution and the SII recalculated. This is repeated many times (1,000,000 repetitions are often necessary to get adequate precision for the SII), to build up a distribution of SII values based on random sampling from the decile indicator value distributions.

The 2.5% and 97.5% values from this distribution of SII values are then reported as the 95% confidence interval for the SII.

Where there is not a linear relationship between the indicator and deprivation

When the indicator has been log or logit transformed prior to calculation of the SII, confidence intervals can be calculated using the simulation approach described above with the transformed values. These can then be converted back to the original units of the indicator for presentation, as for the transformed SII itself illustrated in Figure 2 (c). Converting the confidence intervals for the transformed SII back to the original units requires a few steps. Conceptually we treat the simulations as providing a multitude of gradients, all pivoted around the central point of the slope: the overall level of the line doesn’t change, just the slope.

1. Calculate the central point of the original SII regression line, ie the value when x=0.5

2. Calculate the intercept corresponding to the lower confidence limit slope by subtracting half the LCL for the transformed SII from the central point calculated in step 1 and the intercept corresponding to the upper confidence limit slope by subtracting half the UCL for the transformed SII from the central point calculated in step 1

3. Calculate the value when x=1 corresponding to the lower confidence limit slope by adding half the LCL for the transformed SII from the central point calculated in step 1 and the value when x=1 corresponding to the upper confidence limit slope by adding half the UCL for the transformed SII from the central point calculated in step 1

4. Calculate the antilogs (or antilogits, depending on the original transformation used) of the 4 values from steps 2 and 3 to give the intercepts and values when x=1 corresponding to the lower and upper confidence intervals for the slope - these are now on the original scale of the indicator

5. Subtract the intercept values from the corresponding values when x=1 to give the LCL and UCL for the SII on the original scale of the indicator