Computes the Ferreira-Gignoux (2014) measure of inequality of educational opportunity (IOp): the share of achievement inequality attributable to pre-determined circumstances (family background, gender, migration status, and similar variables outside the student's control). It is the \(R^2\) of a regression of achievement on the circumstance vector, using the variance as the inequality index.
Arguments
- data
A data frame of student-level records.
- achievement
Character vector of achievement plausible-value columns.
- circumstances
Character vector of circumstance columns (the vector \(C\)). Character/factor columns are treated as categorical; numeric columns enter linearly. The canonical PISA application (Ferreira & Gignoux, 2014) uses gender, parental education and occupation, books and household possessions, language at home and migration status.
- weight
Name of the final student weight column. If
NULL, equal weights are used (with a message).- repweights
Optional character vector of replicate-weight columns.
- rep_method, fay
Replication design and Fay factor; see
rep_factor().- design
Optional
lsa_design()bundlingweight,repweights,rep_methodandfay; when supplied it overrides those arguments.- statistics
Which quantities to return:
"iop"(relative IOp = raw \(R^2\)),"iop_adj"(df-adjusted relative IOp),"explained_var"(absolute IOp) and"total_var"(total achievement variance).- level
Confidence level for the stored interval (default
0.95).
Value
An object of class "ieop" / "lsastrat_estimate" (see
lsastrat_estimate for methods).
Details
Following Ferreira & Gignoux, the variance is the inequality measure of choice because, unlike the Gini coefficient or the mean log deviation, it is ordinally invariant under the affine standardisation that assessment scores undergo. The relative IOp is $$IOp = \mathrm{Var}(\hat y) / \mathrm{Var}(y) = R^2,$$ the ratio of circumstance-predicted ("smoothed") achievement variance to total achievement variance. The absolute IOp is \(\mathrm{Var}(\hat y)\) in squared score points.
Overfitting and the adjusted measure
The raw \(R^2\) is an upper bound: it is mechanically inflated when the
circumstance vector expands into many dummy columns (Ferreira & Gignoux,
2014). ieop() therefore also reports iop_adj, a degrees-of-freedom
adjusted relative IOp,
$$1 - (1 - R^2)\,\frac{n_\mathrm{eff} - 1}{n_\mathrm{eff} - k},$$
where \(n_\mathrm{eff}\) is Kish's weighted effective sample size and
\(k\) the number of estimated parameters, and warns when \(k\) is large
relative to \(n_\mathrm{eff}\). Cross-fitted / sample-split IOp is planned
for a future release.
Achievement is supplied as plausible values and pooled with Rubin's rules; standard errors are design-based when replicate weights are given.
References
Ferreira, F. H. G. & Gignoux, J. (2014). The measurement of educational inequality: Achievement and opportunity. World Bank Economic Review, 28(2), 210-246.
Examples
data(pisa_mini)
ieop(pisa_mini,
achievement = paste0("PV", 1:10, "MATH"),
circumstances = c("IMMIG", "parental_edu", "books"),
weight = "W_FSTUWT",
repweights = paste0("W_FSTURWT", 1:64))
#> Inequality of educational opportunity (Ferreira-Gignoux)
#> 10 plausible value(s) | n = 2048 | 3 circumstance(s): IMMIG, parental_edu, books
#> Variance: BRR, 64 replicate weights (+ PV imputation); t reference
#>
#>
#> Estimate Std.Error t df p
#> iop 0.209 0.018 11.79 612.6 4.6e-29
#> iop_adj 0.206 0.018 11.58 610.7 3.7e-28
#> explained_var 1518.225 143.167 10.60 1204.1 3.5e-25
#> total_var 7253.731 228.176 31.79 517.7 8.7e-124
#>
#> iop = share of achievement variance explained by circumstances (raw R^2, upper bound); iop_adj = df-adjusted