Comparing Intersectional Inequality Across Groups
Introduction
MAIHDA is often used to estimate how much outcome variation lies between intersectional strata in a single population. The group comparison workflow asks a related contextual question: is intersectional inequality larger in some higher-level groups than in others?
Examples include comparing gender-by-class inequality across
countries, race-by-gender inequality across regions, or intersectional
inequality across survey waves. In the MAIHDA package, this is handled
by maihda() with a group argument or, when you
want only the group comparison table, by
compare_maihda_groups().
Example data: countries, gender, and socioeconomic status
The package includes maihda_country_data, a teaching
dataset based on a balanced subset of OECD PISA 2018 data. Each row is a
student, the outcome is a mathematics score, and the intersectional
strata are formed by gender and ses.
The higher-level grouping variable is country, which
lets us ask whether the VPC/ICC for gender-by-SES inequality differs
across countries.
library(MAIHDA)
data("maihda_country_data")
country_counts <- as.data.frame(table(maihda_country_data$country))
names(country_counts) <- c("country", "n")
country_counts
#> country n
#> 1 Finland 600
#> 2 Germany 600
#> 3 United Kingdom 600
#> 4 Italy 600
#> 5 Japan 600
#> 6 Mexico 600
table(maihda_country_data$gender, maihda_country_data$ses)
#>
#> Low Medium High
#> female 619 571 588
#> male 581 629 612One-call workflow
Use maihda() when you want the usual model summary and
the group comparison in one object. The formula below defines six
intersectional strata (gender:ses) and asks for a separate
VPC/ICC estimate within each country.
analysis <- maihda(
math ~ 1 + (1 | gender:ses),
data = maihda_country_data,
group = "country"
)
analysis
#> MAIHDA Analysis
#> ===============
#>
#> Formula: math ~ (1 | stratum)
#> Engine: lme4 | Family: gaussian
#> VPC/ICC: 0.1493
#> Strata: 6
#>
#> Group comparison by 'country':
#> MAIHDA Group Comparison
#> =======================
#>
#> Group variable: country
#> Engine: lme4 | Family: gaussian | Strata: shared/global
#>
#> group n n_strata vpc var_between var_other var_residual status
#> Finland 600 6 0.10994 785.8 0 6361 ok
#> Germany 600 6 0.14448 1271.6 0 7529 ok
#> Italy 600 6 0.11890 1065.3 0 7895 ok
#> Japan 600 6 0.13344 1032.3 0 6704 ok
#> Mexico 600 6 0.13649 771.5 0 4881 ok
#> United Kingdom 600 6 0.06011 470.5 0 7357 ok
#>
#> Use summary() for variance components and plot(type = ...) for figures.The printed report includes the overall VPC/ICC first, then the
country-level comparison. The group comparison is also stored in
analysis$groups, so it can be inspected or saved as a
regular data frame.
group_results <- as.data.frame(analysis$groups)
group_results[order(group_results$vpc, decreasing = TRUE),
c("group", "n", "n_strata", "vpc", "var_between",
"var_residual", "status")]
#> group n n_strata vpc var_between var_residual status
#> 2 Germany 600 6 0.14448319 1271.5810 7529.311 ok
#> 5 Mexico 600 6 0.13649162 771.5419 4881.127 ok
#> 4 Japan 600 6 0.13344144 1032.3346 6703.902 ok
#> 3 Italy 600 6 0.11889899 1065.3186 7894.544 ok
#> 1 Finland 600 6 0.10994297 785.7610 6361.226 ok
#> 6 United Kingdom 600 6 0.06011138 470.5471 7357.372 okIn this example all countries have 600 students and all six gender-by-SES strata are populated. Higher VPC/ICC values indicate that a larger share of mathematics score variation lies between the intersectional strata within that country.
Visualizing the comparison
The group VPC plot orders countries by the estimated intersectional
VPC/ICC. If the comparison was run with bootstrap = TRUE,
the same plot will include per-country confidence intervals.
The variance-components view shows the same comparison as shares of the total model variance. The orange segment is the between-stratum component, which is the numerator of the VPC/ICC.
Direct group comparison
If you only need the group comparison table and plots, call
compare_maihda_groups() directly. This fits the same
stratified per-country models used by
maihda(group = "country").
group_cmp <- compare_maihda_groups(
math ~ 1 + (1 | gender:ses),
data = maihda_country_data,
group = "country"
)
group_cmp
plot(group_cmp, type = "vpc")
plot(group_cmp, type = "components")By default, shared_strata = TRUE. That means the
gender:ses strata are defined once on the pooled data, then
reused in every country. A “female:Low” stratum therefore refers to the
same type of stratum in each country, which makes the stratum
definitions comparable across countries.
Shared definitions are necessary but not by themselves sufficient for
the VPCs to be directly comparable. A given country may not
contain every stratum, so two countries’ VPCs can be estimated over
different sets of populated strata. When that happens,
compare_maihda_groups() issues a warning, and the VPCs
should be read with that caveat in mind (alongside the
n_strata column) rather than as strictly like-for-like.
Set shared_strata = FALSE only when you intentionally
want each group to rebuild its own strata. In that case, stratum
identities are no longer shared across groups, so interpretation should
focus on within-group structure rather than direct stratum-to-stratum
comparison.
Adding bootstrap intervals
For publication-oriented summaries, you can request per-group
parametric bootstrap confidence intervals with the lme4
engine. This can take noticeably longer because each group requires
repeated model refits, so the example is not run when the vignette is
built.