DescriptiveRepresentationCalculator: An R package for quantifying observed and expected descriptive representation

What is the DescriptiveRepresentationCalculator? | Key Definitions | Main Package Functions | Installation | Example Use | Tutorial | Other Package Uses | Data | CRAN | References | Documentation

What is the DescriptiveRepresentationCalculator?

How much representation should we expect in a political body, given the composition of the population and the size of that body? How much variability should we expect to see around mean descriptive representation? How can we quantify observed representation in a political body and compare it to the expected value? This package helps provide answers to these questions based on Gerring, Jerzak, and Öncel (2023).

Key Definitions

Descriptive Representation

Definition: The degree to which the identity characteristics present in an organization or political body align with those of the population they serve.

Details: Descriptive representation is often contrasted against substantive representation, which concerns the degree to which the preferences of those in the body align with those of the population. Descriptive representation is thought to be connected with substantive representation in that individuals with a given set of identity characteristics may be better positioned to understand the needs of that population compared to others. Descriptive representation would be low, for example, if a political body contained few woman (since the population contains around 50% women). Representation would be high if the political body contained 50% women.

Rose Index of Proportionality

Definition: A quantitative measure of descriptive representation.

Details: The quantity is calculated by taking the body shares of each social group, finding the absolute difference compared with the population shares, summing, and re-scaling so that the measure is between 0 and 1, with 1 indicating perfect descriptive representation. Perfect descriptive representation implies that the population and body shares are perfectly balanced.

Main Package Functions

Characterizing Expected Representation: Among other things, this package provides a function, ExpectedRepresentation, that calculates the degree of representation under random sampling, where, by default, representation is calculated using the Rose Index of Proportionality. This quantifies the value of descriptive representation we should expect in a political body, given population and body characteristics.

Characterizing Representation Variability: The package contains another function, SDRepresentation, that calculates the degree of representation we would expect not to be explained by the average discrepancy value; this quantity is the standard deviation of the Rose Index of Proportionality under the random sampling model. It captures how much uncertainty we would expect over the expected representation value under the random sampling model.

Quantifying Relative Representation: RelativeRepresentation compares the observed representation in a body to its expected value under random sampling. It optionally standardizes this difference by the variability captured through SDRepresentation.

Characterizing Observed Representation: The package contains third function, ObservedRepresentation, that computes the Rose Index of Proportionality using observed data. Observed representation index scores can be compared against the expected value of the index under random sampling and also against the variability of observed represenation under that model.

Installation

You may download via the devtools package. In particular, use

```

install stable package version from CRAN

install.packages("DescriptiveRepresentationCalculator")

install development version

devtools::install_github(repo = "cjerzak/DescriptiveRepresentationCalculator-software/DescriptiveRepresentationCalculator")

```

Then, to load the software in, use library( DescriptiveRepresentationCalculator )

Example Use

```

Compute the expected amount of representation

(i.e. the expected Rose Index of Proportionality:)

- for a population of group proportions (1/3, 2/3, 1/3)

- in a political body of size 50

ExpectedRep <- ExpectedRepresentation( PopShares = c(1/3, 2/3, 1/3), BodyN = 50) print( ExpectedRep )

Compute the amount of representation left unexplained under

the random sampling model for the same body and population

ResidualRep <- SDRepresentation( PopShares = c(1/3, 2/3, 1/3), BodyN = 50) print( ResidualRep ) ```

Tutorial

Now, let's do a more complex application. Let's compare the observed representation in a hypothetical supreme court with the expected representation index we would see under random sampling. ```

Generate synthetic body with group types A and B

MemberIdentitiesInSupremeCourt <- c("A","B","A","A", "A","A","A","A","A") bodyN <- length( MemberIdentitiesInSupremeCourt )

Population proportions of A and B (e.g., from census data)

PopulationProportions <- c("A"=0.8, "B"=0.2)

Compute the observed representation index

ObservedIndex <- ObservedRepresentation( BodyMemberCharacteristics = MemberIdentitiesInSupremeCourt, PopShares = PopulationProportions) print(ObservedIndex) # 0.91

Compute expected representation index

ExpectedRep <- ExpectedRepresentation( PopShares = PopulationProportions, BodyN = bodyN) print( ExpectedRep ) # 0.89

The hypothetical body is actually somewhat more representative

than would be expected under random sampling

variability analysis

SDRep <- SDRepresentation(PopShares = PopulationProportions, BodyN = bodyN) ExpectedRepCI <- c(ExpectedRep-1.96SDRep, ExpectedRep + 1.96SDRep) print(ExpectedRepCI) # -> 0.736855 1.048397

Conclusion:

Observed representation is well-within

confidence intervals under the random sampling model

(i.e., the representation index so observed could have

plausibly been generated under random sampling)

```

Other Package Uses

There are several other ways in which this package could be helpful in practice: - Quantifying expected and observed descriptive representation: If you want to explain what determines descriptive representation outcomes, you can use the package to compute relative representation---i.e., the level of representation relative to what we would expect based on random sampling alone. The RelativeRepresentation function computes this quantity directly. - Control variable creation: In other situations, researchers may want to use the expected degree of representation under the random sampling model in order to control for compositional factors such as body size and population composition. This expected representation measure is a direct quantification of how those factors should, under the random sampling model, affect representation and therefore is a useful control variable.

Feature Requests & Development Plan

Don't hesitate to reach out to Connor Jerzak at connor.jerzak@gmail.com for package support or feature requests.

In future releases, we will allow users to compute expected and residual representation under different institutional arrangements. We will also release country-level descriptive representation data.

References

John Gerring, Connor T. Jerzak, Erzen Öncel. "The Composition of Descriptive Representation." American Political Science Review, 118(2): 784-801, 2024. [PDF] [Dataverse] @article{GJO-CompDR, title={The Composition of Descriptive Representation}, author={Gerring, John and Connor T. Jerzak and Erzen Öncel}, journal={American Political Science Review}, year={2024}, volume={118}, number={2}, pages={784-801} }

Related Work

John Gerring, Allen Hicken, Connor T. Jerzak, Robert G. Moser, and Erzen Öncel. "Where Minorities are the Majority: Electoral Rules and Ethnic Representation." OSF Preprint, 2024. [PDF]