phe_sii: Calculate Slope Index of Inequality using phe_sii

Description

phe_sii()returns the slope index of inequality (SII) statistic for each subgroup of the inputted dataframe, with lower and upper confidence limits based on the specified confidence.

Usage

phe_sii(
  data,
  quantile,
  population,
  x = NULL,
  value = NULL,
  value_type = 0,
  transform = FALSE,
  lower_cl = NULL,
  upper_cl = NULL,
  se = NULL,
  multiplier = 1,
  repetitions = 1e+05,
  confidence = 0.95,
  rii = FALSE,
  intercept = FALSE,
  reliability_stat = FALSE,
  type = "full"
)

Value

The SII with lower and upper confidence limits for each subgroup of the inputted data.frame.

Arguments

data: data.frame containing the required input fields, pre-grouped if an SII is required for each subgroup; unquoted string; no default
quantile: field name within data that contains the quantile label (e.g. decile). The number of quantiles should be between 5 and 100; unquoted string; no default
population: field name within data that contains the quantile populations (ie, denominator). Non-zero populations are required for all quantiles to calculate SII for an area; unquoted string; no default
x: (for indicators that are proportions) field name within data that contains the members of the population with the attribute of interest (ie, numerator). This will be divided by population to calculate a proportion as the indicator value (if value field is not provided); unquoted string; no default
value: field name within data that contains the indicator value (this does not need to be supplied for proportions if count and population are given); unquoted string; no default
value_type: indicates the indicator type (1 = rate, 2 = proportion, 0 = other). The value_type argument is used to determine whether data should be transformed prior to calculation of the standard error and/or SII. See the Tansformations section for full details; integer; default 0
transform: option to transform input rates or proportions prior to calculation of the SII. See the Transformations section for full details; logical; default FALSE
lower_cl: field name within data that contains 95 percent lower confidence limit of indicator value (to calculate standard error of indicator value). This field is needed if the se field is not supplied; unquoted string; no default
upper_cl: field name within data that contains 95 percent upper confidence limit of indicator value (to calculate standard error of indicator value). This field is needed if the se field is not supplied; unquoted string; no default
se: field name within data that contains the standard error of the indicator value. If not supplied, this will be calculated from the 95 percent lower and upper confidence limits (i.e. one or the other of these fields must be supplied); unquoted string; no default
multiplier: factor to multiply the SII and SII confidence limits by (e.g. set to 100 to return prevalences on a percentage scale between 0 and 100). If the multiplier is negative, the inverse of the RII is taken to account for the change in polarity; numeric; default 1;
repetitions: number of random samples to perform to return confidence interval of SII (and RII). Minimum is 1000, no maximum (though the more repetitions, the longer the run time); numeric; default 100,000
confidence: confidence level used to calculate the lower and upper confidence limits of SII, expressed as a number between 0.9 and 1, or 90 and 100. It can be a vector of 0.95 and 0.998, for example, to output both 95 percent and 99.8 percent CIs; numeric; default 0.95
rii: option to return the Relative Index of Inequality (RII) with associated confidence limits as well as the SII; logical; default FALSE
intercept: option to return the intercept value of the regression line (y value where x=0); logical; default FALSE
reliability_stat: option to carry out the SII confidence interval simulation 10 times instead of once and return the Mean Average Difference between the first and subsequent samples (as a measure of the amount of variation). Warning: this will significantly increase run time of the function and should first be tested on a small number of repetitions; logical; default FALSE
type: "full" output includes columns in the output dataset specifying the parameters the user has input to the function (value_type, multiplier, CI_confidence, CI_method); character string either "full" or "standard"; default "full"

Calculation

The SII is calculated using linear regression (1). To allow for differences in population size between quantiles (e.g. deprivation deciles), each is given a rank score (or relative rank) based on the midpoint of its range in the cumulative distribution of the total area population. The quantiles are first ordered (e.g from 1 most deprived to 10 least deprived for deprivation deciles). If quantile 1 then contains 12 percent of the total population, its relative rank is 0.12/2=0.6. If quantile 2 includes 10 percent of the population, its relative rank is 0.12+(0.10/2)=0.17. A square root transformation is applied to the regression to account for heteroskedasticity (the tendancy for the variances of the quantile values to be related to the size of the values, ie larger values will tend to have larger variances). A regression model is fitted to the transformed data: \(Y * \sqrt a = \sqrt a + b * \sqrt a\), where Y is the value of the indicator for the quantile, a is the proportion of the total population in the quantile and b is the relative rank. The SII is the gradient of the resulting fitted line, and could be positive or negative according to the indicator polarity. Since the relative ranks, by definition, range from 0 to 1, the SII is the difference between the fitted value at x=1 and x=0. The RII is the ratio of the fitted value at x=1,Y1 and the fitted value at x=0,Y0. which can be calculated as: RII = (Y0 + SII)/Y0

Transformations

The indicator type can be specified as 1 (rate), 2 (proportion) or 0 (other), using the value_type parameter. This setting determines the data transformations that will be applied in the following two parts of the method.

Use in conjunction with the transform parameter in calculation of the SII: It is recommended that rates and proportions are transformed prior to calculation of the SII by setting the transform parameter to TRUE for these indicator types. This will perform a log transformation for rates, or logit for proportions, and return outputs transformed back to the original units of the indicator. These transformations are recommended to improve the linearity between the indicator values and the quantile, which is an assumption of the method. A user-provided standard error will not be accepted when the transform parameter is set to TRUE as the confidence limits are required for this transformation.

Use in calculation of the standard error: Rates and proportions, and their confidence limits, are transformed prior to calculation of the standard error for each quantile. This is because it is assumed that the confidence interval around the indicator value is non-symmetric for these indicator types. Note that this transformation is not controlled by the transform parameter and is applied based on the value of the value_type parameter only. A user-provided standard error will not be accepted when the transform parameter is set to TRUE as the confidence limits are required for this transformation.

Warning

The SII calculation assumes a linear relationship between indicator value and quantile. Where this is not the case the transform option should be considered. Small populations within quantiles can make the SII unstable. This function does not include checks for linearity or stability; it is the user's responsibility to ensure the input data is suitable for the SII calculation.

Details

The Relative Index of Inequality (RII) can also be returned via an optional argument.

The SII and RII are two measures of health inequality. They show the relation between the level of health or frequency of a health problem in different population groups and the ranking of these groups on the social scale.

The input dataframe should be grouped before passing to the function if an SII/RII for each subgroup is required, and quantiles ordered from least to most advantaged.

References

(1) Low A & Low A. Measuring the gap: quantifying and comparing local health inequalities. Journal of Public Health; 2004;26:388-395.

Examples

Run this code

library(dplyr)

data <- data.frame(area = c(rep("Area1", 10), rep("Area2", 10)),
                   decile = c(1:10, 1:10),
                   population = c(7291, 7997, 6105, 7666, 5790, 6934, 5918, 5974, 7147, 7534, 21675,
                                  20065, 19750, 24713, 20112, 19618, 22408, 19752, 18939, 19312),
                   value = c(75.9, 78.3, 83.8, 83.6, 80.5, 81.1, 81.7, 84.2, 80.6, 86.3, 70.5,
                              71.6, 72.5, 73.5, 73.1, 76.2, 78.7, 80.6, 80.9, 80),
                   lowerCL = c(72.7,75.3,80.9,80.2,77.1,78,79,81.4,75.8,83.2,
                               70.1,71.1,72,73.1, 72.7, 75.7, 78.2,80.1,80.4,79.5),
                   upperCL = c(79.1,81.4,86.8,87.1,83.8,84.2,84.4,86.9,85.4,
                                89.4,71,72.1,73.2,73.7,75.8,78.8,79.8,81.2,81.3,80.9),
                   StandardError = c(1.64,1.58,1.51,1.78,1.7,1.56,1.37,1.4,2.43,
                                     1.57,0.23,0.26,0.3,0.16,0.79,0.78,0.4,0.28,0.23,0.35)
                   )


# Run SII function on the two areas in the data
phe_sii(group_by(data, area),
        decile,
        population,
        value_type = 0, # default normal distribution
        value = value,
        lower_cl = lowerCL,
        upper_cl = upperCL,
        confidence = 0.95,
        rii = TRUE,
        type = "standard")

# Supplying the standard error instead of the indicator 95 percent confidence limits
# gives the same result
phe_sii(group_by(data, area),
        decile,
        population,
        value_type = 0,
        value = value,
        se = StandardError,
        confidence = 0.95,
        rii = TRUE,
        type = "standard")

# multiple confidence intervals, log transforming the data if they are rates
phe_sii(group_by(data, area),
        decile,
        population,
        value_type = 1,
        transform = TRUE,
        value = value,
        lower_cl = lowerCL,
        upper_cl = upperCL,
        confidence = c(0.95, 0.998),
        repetitions = 10000,
        rii = TRUE,
        type = "standard")

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