Package 'rineq'

Calculates different type of concentration indexes

Description

This function calculates the relative concentration index (Kakwani et al.), the generalized concentration index (Clarke et al., 2002), the Wagstaff index for bounded variables (Owen et al. 2016), and the concentration index with Erreygers' correction (Erreygers et al., 2009). It returns an object of class hci for which confidence intervals, summaries and plots are defined.

Usage

ci(
  ineqvar,
  outcome,
  weights = NULL,
  type = c("CI", "CIg", "CIc", "CIw"),
  method = c("linreg_delta", "linreg_convenience", "cov_convenience", "direct"),
  df_correction = TRUE,
  robust_se = FALSE,
  rse_type = "HC3",
  rank_function = rineq::rank_wt
)

Arguments

ineqvar	Used for ranking, usually relates to the socioeconomic position, for example income.
outcome	The variable in which the inequality should be measures, for example health.
weights	Optional, used to weigh the observations. Defaults to equal weights for all observations.
type	Character, the type of concentration index to be calculated: relative concentration index (CI, default), generalized concentration index (CIg), concentration index with Erreygers Correction CIc, or Wagstaff concentration index suitable for bounded and binary outcomes CIw
method	Character, defines the calculation method. One of: linreg_delta: Based on linear regression without transforming the left hand side variable. Computes correct standard errors that take into account the sampling variability of the estimate of the mean (O’Donnell et al. 2008, Owen et al. 2016) linreg_convenience): Based on simpler regression with transformed left hand side variable. Standard errors do not take into account the sampling variability of the estimate of the mean(O’Donnell et al. 2008, Owen et al. 2016) cov_convenience: Based on covariance. Equivalent to linreg_convenience (O’Donnell et al. 2008, Owen et al. 2016) direct: Using direct formula, standard errors do no take weighting appropriately into account (O’Donnell et al. 2008, Kakwani et al. 1997)
df_correction	If TRUE (default), calculates the concentration index based on the population variance (derived from the sample variance).
robust_se	Uses robust standard errors if TRUE. Only available for the ⁠linreg_*⁠ type methods. Requires the sandwich package.
rse_type	Character, type argument for the vcovHC(). HC3' is suggested as default, set to HC1 for Stata compatibility. See ?sandwich::vcovHC() for options.
rank_function	Function to calculate the weighted rank of ineqvar. Takes two arguments: the variable that holds the rank order information, and the weights for the ranks. rineq currently provides two, rank_wt (default, corresponds to code provided in the World Bank report by O’Donnell et al.,2008) and rank_gwt (generalized handling of ties as also used by the Stata Conindex command, initially published by van Ourti, (2004)).

Value

An S3 object of class hci. Contains:

• concentration_index The concentration index

• type The type

• method The method used for calculation

• variance The variance,used for calculation of confidence intervals

• fractional_rank Computed fractional rank NA

• outcome Outcome after removing NA

• call Call signature

• n Number of observations after removing NA

• robust_se Were robust standard errors calculated?

• rse_type Type of robust standard errors.

• df_correction Do the degrees of freedom correspond to a sample?

References

Clarke, P. M., Gerdtham, U. G., Johannesson, M., Bingefors, K., & Smith, L. (2002). On the measurement of relative and absolute income-related health inequality. Social Science & Medicine, 55(11), 1923-1928

Erreygers, G. (2009). Correcting the concentration index. Journal of health economics, 28(2), 504-515

Kakwani, N., Wagstaff, A., & Van Doorslaer, E. (1997). Socioeconomic inequalities in health: measurement, computation, and statistical inference. Journal of econometrics, 77(1), 87-103.

O'Donnel, O., O'Neill S., Van Ourti T., & Walsh B. (2016). Conindex: Estimation of Concentration Indices. The Stata Journal, 16(1): 112-138.

O’Donnell, O., Van Doorslaer, E. , Wagstaff, A., Lindelow, M., 2008. Analyzing Health Equity Using Household Survey Data: A Guide to Techniques and Their Implementation, World Bank Publications. The World Bank.

van Ourti, T., 2004. Measuring horizontal inequity in Belgian health care using a Gaussian random effects two part count data model. Health Economics, 13: 705–724.

Examples

# Direct
data(housing)
ci.bmi <- ci(ineqvar = housing$income, outcome = housing$bmi, method = "direct")
summary(ci.bmi)

# retrieve value
ci.bmi$concentration_index

# obtain confidence intervals
confint(ci.bmi, level = 0.95) 
plot(ci.bmi)

# Wagstaff type with binary outcome and robust standard errors 
# that should correspond to Stata (depends on 'sandwich'):
ci.bmi.b <- ci(housing$income, housing$high.bmi, type = "CIw", robust_se = TRUE, 
   rse_type = "HC1")

---

Confidence intervals for hci objects

Description

Confidence intervals for hci objects

Usage

## S3 method for class 'hci'
confint(object, parm = NULL, level = 0.95, ...)

Arguments

object	An object of class hci
parm	Unused
level	Confidence interval level defaults to 0.95
...	Unused

Value

A confidence interval in a numeric vector of length 2

Examples

data(housing)
ci.bmi <- ci(ineqvar = housing$income, outcome = housing$bmi, method = "direct")
confint(ci.bmi)

---

Function to decompose the Relative Concentration Index into its components

Description

Currently compatible with lm, glm logit and probit, svyglm, coxph and mfx marginal effects probit.

Usage

contribution(object, ranker, correction = TRUE, type = "CI", intercept = "exclude")

Arguments

object	The model result object. class coxph, glm, lm or svyglm, probitmfx, logitmfx; the outcome should be the health variable and the predictors the components.
ranker	Ranking variable with the same length as the outcome.
correction	A logical indicating whether the global and partial confidence should be corrected for negative values using imputation.
type	Character, concentration index type that the decomposition should be applied to. Defaults to CI. Use CIw for binary outcomes.
intercept	Character, one of exclude or include, defaults to exclude. If exclude, the intercept coefficient will not included in the decomposition analysis, if set to include, it will be included.

Details

These functions decompose the Relative Concentration Index into its components using a (generalized) linear model, optionally using a survey design, or a Cox Proportional Hazards model. Print, summary and plot methods have been defined for the results.

If correction is TRUE negative values of components or outcome are corrected using correct_sign() with option shift = FALSE.

For non-linear models the decomposition needs to rely on a linear approximations of the effects. There are different approaches. One is to work on the scale of the glm coefficients and calculate the concentration index based on the predicted outcome. (Konings et al., 2010, Speybroeck et al., 2010). Another approach is to use marginal effects as beta coefficients and the original outcome (O'Donnel et al. 2008).

This function supports both. For glm, coxph, and svyglm models, the first approach is used. The second approach is implemented for model objects of type probitmfx and logitmfx from the 'mfx' package. See examples.

Per default, the intercept in models is excluded, but this can be changed by setting the the intercept argument to include, but this may conceptually make less sense and is more appropriate if the model does not contain an intercept.

Use decomposition() function directly to manually specify coefficients, outcomes, and model matrices for arbitrary models.

NOTE: Be careful with automatically omitted rows in models. Only models with data with ordinary indexes are supported (starting from 1, sequentially increasing by increments of 1). For the case were rows with NA are automatically omitted by the model function, the used indices are guessed based on the row names of the model matrix and then used for accessing the ranker variable. However, this may lead to issues if the row names do not correspond to ordinary integer indexes. For example, if a model such as lm uses the default na.omit action and removes rows, the data in the model might not be consistent with the ranker vector anymore.

Value

An object of class decomposition containing the following components:

• betas A numeric vector containing regression coefficients

• partial_cis A numeric vector containing partial confidence intervals

• confints A numeric vector containing 95\

• averages Weighted averages of every variable in the model

• ci_contribution Confidence intervals for contributions

• overall_ci Confidence intervals for the concentration index

• corrected_coefficients Corrected coefficients using correct_sign() if, requested FALSE otherwise

• outcome_corrected Corrected outcome correct_sign() if requested, FALSE otherwise

• rows Rownames of used rows in the model

Warning

ranker should be chosen with care. Ideally, it is a variable from the same dataframe as the other variables. If not, redefine the row names in the model.

Author(s)

Peter Konings

References

Konings, P., Harper, S., Lynch, J., Hosseinpoor, A.R., Berkvens, D., Lorant, V., Geckova, A., Speybroeck, N., 2010. Analysis of socioeconomic health inequalities using the concentration index. Int J Public Health 55, 71–74. https://doi.org/10.1007/s00038-009-0078-y

Speybroeck, N., Konings, P., Lynch, J., Harper, S., Berkvens, D., Lorant, V., Geckova, A., Hosseinpoor, A.R., 2010. Decomposing socioeconomic health inequalities. Int J Public Health 55, 347–351. https://doi.org/10.1007/s00038-009-0105-z

O’Donnell, O., Doorslaer, E. van, Wagstaff, A., Lindelow, M., 2008. Analyzing Health Equity Using Household Survey Data: A Guide to Techniques and Their Implementation, World Bank Publications. The World Bank.

Examples

data(housing)

## Linear regression direct decomposition
fit.lm <- lm(bmi ~ sex + tenure + place + age,data = housing)
       
# decompose relative concentration index
contrib.lm <- contribution(fit.lm, housing$income) 
summary(contrib.lm)
plot(contrib.lm, decreasing = FALSE, horiz = TRUE)

    
# GLM: Decomposition based on predicted outcome
fit.logit <-glm(high.bmi ~ sex + tenure + place + age, data = housing)

contrib.logit <- contribution(fit.logit, housing$income) 
summary(contrib.logit)
plot(contrib.logit, decreasing = FALSE,horiz = TRUE)


# GLM probit: Decomposition based on predicted outcome
fit.probit <-glm(high.bmi ~ sex + tenure + place + age, data = housing, 
                family = binomial(link = probit))

# binary, set type to 'CIw'
contrib.probit <- contribution(fit.probit, housing$income, type = "CIw") 
summary(contrib.probit)
plot(contrib.probit, decreasing = FALSE,horiz = TRUE)


# Marginal effects probit using package 'mfx': Decomposition based on predicted outcome
fit.mfx <-mfx::probitmfx(high.bmi ~ sex + tenure + place + age, data = housing)

contrib.mfx <- contribution(fit.mfx, housing$income, type = "CIw") 
summary(contrib.mfx, type="CIw")
plot(contrib.mfx, decreasing = FALSE, horiz = TRUE)


# package survey svy 
des = survey::svydesign(~1, data= housing, weights = rep(1, NROW(housing)))
fit.svy = survey::svyglm(bmi ~ tenure+height+weight, design = des)
contrib.svy = contribution(fit.svy, housing$income)


# adapted from the `coxph` example in survival package 
testcph <- data.frame(time = c(4,3,1,1,2,2,3), 
              status = c(1,1,1,0,1,1,0), 
              x      = c(0,2,1,1,1,0,0), 
              sex    = c(0,0,0,0,1,1,1),
              income = c(100,50, 20, 20, 50, 60,100))
              
# Fit a stratified model 
fit.coxph = survival::coxph(Surv(time, status) ~ x + strata(sex), testcph) 
contrib.coxph = contribution(fit.coxph, testcph$income)

---

Corrects negative values in the health variable

Description

The Relative Concentration Index is not bonded between $[-1,1]$ if the health variable contains both negative and positive values. This function corrects for this either by imputing a value of 0 for all negative values or by subtracting the minimum value.

Usage

correct_sign(x, shift = TRUE)

corrected_value(x)

is_corrected(x)

Arguments

x	A numeric vector, typically representing health.
shift	If FALSE (the default), 0 is imputed for all negative values in x. If TRUE the minimum value of x is subtracted from it.

Value

correct_sign() returns a list with 2 components:

• corrected: corrected version of x

• modified: logical, TRUE when any of the elements of x have been changed

corrected_value(): returns the corrected value if passed the result of 'correct_sign().

is_corrected(): returns TRUE if a modifications was made and FALSE otherwise. Takes as argument the result of correct_sign(),

Functions

• corrected_value(): Return the corrected value

• is_corrected(): Check if the sign was corrected

Author(s)

Peter Konings

Examples

data("housing")

# standardize & normalize bmi, will introduce negative values
housing$bmi.std <- (housing$bmi - mean(housing$bmi))/ sd(housing$bmi)

housing$bmi.std.shifted <- corrected_value(correct_sign(housing$bmi.std, shift = TRUE))
housing$bmi.std.imputed <- corrected_value(correct_sign(housing$bmi.std, shift = FALSE))

## compare the effect of both methods
plot(density(housing$bmi.std, na.rm = TRUE))
points(density(housing$bmi.std.shifted, na.rm = TRUE), col = 'blue')
points(density(housing$bmi.std.imputed, na.rm = TRUE), col = 'green')

---

Decomposition analysis

Description

Used by the wrapper contribution() but can be used manually. Calculates the decomposition for a given regression model.

Usage

decomposition(outcome, betas, mm, ranker, wt, correction, citype = "CI")

Arguments

outcome	Outcome variable
betas	Beta coefficients from regression.
mm	Model matrix from regression
ranker	Ranking variable
wt	Weights
correction	Apply sign correction?
citype	Character, CI type to be calculated, defaults to CI. Use CIw for binary outcomes.

Details

NOTE: Only models with data with ordinary indexes are supported (starting from 1, sequentially increasing by increments of 1). For the case were rows with NA are automatically omitted by the model function, the used indices are guessed based on the row names of the model matrix and then used for accessing the ranker variable. However, this may lead to issues if the row names do not correspond to ordinary integer indexes.

Value

S3 object of class decomposition

Examples

fit.lm = lm(mtcars$mpg ~ mtcars$cyl)
decomp = decomposition(mtcars$mpg, coefficients(fit.lm), fit.lm$model, 
                       mtcars$hp, wt = rep(1, nrow(mtcars)), correction = FALSE) 
summary(decomp)

---

Artificial example data on housing conditions

Description

Microdata with a permissive license that includes continuous data on health and income is hard to come by. In stead of real data, the package thus includes an imaginary dataset.

Usage

data(housing)

Format

data.frame object.

Variable list:

id

unique identifier per person

sex

female or male

age

integer, from 20 to 94

tenure

categorical. One of homeless, irregular, own_apartment, own_house, or rent

height

height in cm

weight

weight in kg

bmi

weight/(height/100)^2

income

continuous, imaginary currency without unit

Source

Artificially generated by the package authors

---

Plots a barplot of the contribution percentages in a decomposition object. Sets custom plot margins and uses the graphical parameters xlim, horiz, las and xlab which therefore cannot be customized

Description

Plots a barplot of the contribution percentages in a decomposition object. Sets custom plot margins and uses the graphical parameters xlim, horiz, las and xlab which therefore cannot be customized

Usage

## S3 method for class 'decomposition'
plot(x, decreasing = TRUE, horiz = FALSE, ...)

Arguments

x	Object returned from decomposition function
decreasing	Whether to sort contributions decreasing or not
horiz	If the barplots should be printed horizontally or vertically
...	Graphical parameter passed on to base::barplot()

Value

Invisibly returns x as the function is called for side effects (plotting).

Examples

data(housing)
# Linear regression & decompose 
fit.lm <- lm(bmi ~ sex + tenure + place + age,data = housing)
contrib.lm <- contribution(fit.lm, housing$income)

# plot horizontally, in increasing order
plot(contrib.lm, decreasing = FALSE, horiz = TRUE)

---

Plots the concentration curve for an hci object.

Description

Plots the concentration curve for an hci object.

Usage

## S3 method for class 'hci'
plot(x, ...)

Arguments

x	Object with of hci
...	Further arguments passed to base::plot()

Value

Invisibly returns x as the function is called for side effects (plotting).

Examples

data(housing)
ci.bmi <- ci(ineqvar = housing$income, outcome = housing$bmi, method = "direct")
plot(ci.bmi)

---

Print function for decomposition objects.

Description

Print function for decomposition objects.

Usage

## S3 method for class 'decomposition'
print(x, ...)

Arguments

x	Object of type decomposition
...	Currently unused

Value

Invisibly returns x as the function is called for side effects.

Examples

data(housing)
# Linear regression &  decompose 
fit.lm <- lm(bmi ~ sex + tenure + place + age,data = housing)
contrib.lm <- contribution(fit.lm, housing$income)

# print
print(contrib.lm)

---

Prints an hci object.

Description

Prints an hci object.

Usage

## S3 method for class 'hci'
print(x, ...)

Arguments

x	Object of type hci
...	Currently unused

Value

Invisibly returns x as the function is called for side effects.

Examples

data(housing)
ci.bmi <- ci(ineqvar = housing$income, outcome = housing$bmi, method = "direct")
print(ci.bmi)

---

Generalized weighted ranking function

Description

In the case of ties, the ordinary rank_wt() function uses the order in the original data. This is the same approach as in the Stata code provided by O’Donnell et al. (2008) in the original World Bank publication, but depends on the arbitrary initial order in the data The Stat conindex code however uses uses the generalized weighted rank implementation published by van Ourti (2004). For Stata compatibility use rank_gwt()

Usage

rank_gwt(x, wt)

Arguments

x	numeric vector
wt	weights

Details

The formula notation in van Ourti (2004) seems to rely on absolute an absolute deduction of 1 unit of monetary income value. This only works in the integer case. Instead, this this implementation uses the next lowest x value, respectively the next lowest rank, to calculate the proportion of the inequality variable up to the respective value

Value

A numeric vector containing weighted fractional ranks of the elements of x.

References

van Ourti, T., 2004. Measuring horizontal inequity in Belgian health care using a Gaussian random effects two part count data model. Health Economics, 13: 705–724.

Examples

x <- sample(1:10, size = 10, replace = TRUE)
x.weight <- seq(0, 1, length.out = 10)
rank_gwt(x, wt = x.weight)

---

Calculates the weighted rank

Description

Calculates the weighted rank

Usage

rank_wt(x, wt)

Arguments

x	numeric vector
wt	weights

Value

A numeric vector containing weighted fractional ranks of the elements of x.

Author(s)

Peter Konings

References

Kakwani et al., 1997.

Examples

x <- sample(1:10, size = 10, replace = TRUE)
x.weight <- seq(0, 1, length.out = 10)
rank_wt(x, wt = x.weight)

---

Prints and returns a summary for a decomposition object.

Description

Prints and returns a summary for a decomposition object.

Usage

## S3 method for class 'decomposition'
summary(object, digits = getOption("digits"), addcoefs = FALSE, ...)

Arguments

object	Result of a decomposition analysis, of class decomposition
digits	Number of digits, defaults to R digits option
addcoefs	Whether or not to add coefficients (defaults to FALSE)
...	Additional parameters, currently unused

Value

A data frame frame with columns for the absolute and relative contribution, elasticity, concentration index including confidence intervals, and whether correction was applied. If specified using addcoefs, the coefficients are included as the first column.

Examples

data(housing)
# Linear regression & decompose 
fit.lm <- lm(bmi ~ sex + tenure + place + age,data = housing)
contrib.lm <- contribution(fit.lm, housing$income)

# print
print(contrib.lm)

---

Prints the a summary of the concentration index object hci

Description

Prints the a summary of the concentration index object hci

Usage

## S3 method for class 'hci'
summary(object, ...)

Arguments

object	Object of type hci
...	Currently unused

Value

No returns value. Directly prints to the standard output connection.

Examples

data(housing)
ci.bmi <- ci(ineqvar = housing$income, outcome = housing$bmi, method = "direct")
summary(ci.bmi)

---

Calculates the weighted variance

Description

Calculates the weighted variance

Usage

var_wt(x, wt, na.rm = FALSE)

Arguments

x	numeric vector
wt	weights
na.rm	If TRUE, indices where x is NA will be removed

Value

A numeric vector containing weighted variance of the elements of x

Examples

x <- sample(1:10, size = 10, replace = TRUE)
x.weight <- seq(0, 1, length.out = 10)
var_wt(x, wt = x.weight)

---