iDIFr

Intersectional Differential Item Functioning Analysis

iDIFr is an R package for detecting Differential Item Functioning (DIF) using Logistic Regression, IRT Likelihood Ratio Tests, and model-based recursive partitioning (MOB) — with first-class support for intersectional group designs and built-in Intersectional Contrast Analysis (ICA).

Why iDIFr?

Most DIF packages focus on two-group comparisons along a single demographic dimension. iDIFr is built around the idea that test-takers belong to multiple groups simultaneously, and that DIF sometimes only appears at the intersection of those identities.

Key features:

• Intersectional group support — define groups using ~ gender * nationality * age_band
• Effect sizes as first-class outputs — results lead with Nagelkerke ΔR² and standardised chi, not just p-values
• Three methods in one interface — LR, LRT, and MOB with consistent output
• Built-in ICA — ica = TRUE classifies each item as amplified, pure intersection, obscured, or none by comparing single-variable and intersectional analyses
• Transparent cell-size guidance — check_groups() and merge_groups() help you manage sparse intersectional cells
• Tidy output — tidy() returns a flat data frame for use with dplyr and ggplot2

Installation

# From CRAN
install.packages("iDIFr")

# Development version from GitHub
# install.packages("remotes")
remotes::install_github("thmsrgrs/iDIFr")

Quick start

library(iDIFr)

# 1. Check your group structure first
check_groups(my_data, group = ~ gender * nationality * age_band)

# 2. Run DIF analysis — method selection is required
result <- idifr(
  data   = my_data,
  items  = 1:20,
  group  = ~ gender * nationality * age_band,
  method = c("LR", "LRT")
)

# 3. Explore results
print(result)                       # Flagged items with effect sizes
summary(result)                     # Full breakdown by method + concordance
plot(result)                        # Effect size heatmap
plot(result, type = "concordance")  # Method agreement
tidy(result)                        # Flat data frame
tidy(result, table = "direction")   # Group-level direction table

Methods

Argument	Method	Effect size	Best for
"LR"	Logistic Regression	Nagelkerke ΔR²	General use, no IRT assumptions
"LRT"	IRT Likelihood Ratio Test	Standardised chi (df-scaled)	IRT-based programmes
"MOB"	Model-based recursive partitioning	Standardised score difference	Intersectional designs, exploratory

Intersectional Contrast Analysis (ICA)

Pass ica = TRUE to idifr() to run ICA automatically. After the main analysis, iDIFr runs one additional idifr() per demographic variable and classifies each item by comparing where it was flagged:

Classification	Meaning
amplified	Flagged in single-variable and intersectional runs
pure_intersection	Flagged only in the intersectional run
obscured	Flagged in a single-variable run but not intersectionally
none	Not flagged anywhere

result <- idifr(
  data   = my_data,
  items  = 1:20,
  group  = ~ gender * nationality * age_band,
  method = "LR",
  ica    = TRUE
)

print(result)                  # ICA section printed automatically
tidy(result, table = "ica")    # Flat ICA classification table

Note: ICA runs N + 1 analyses without cross-analysis p-value correction. Interpret pure_intersection and obscured findings with caution in small samples.

Effect size thresholds

iDIFr requires both statistical significance (after p-value adjustment) and a meaningful effect size before flagging an item. This reduces false positives in large samples.

Method	Metric	Negligible	Moderate	Large
LR (uniform)	Nagelkerke ΔR²	< .035	.035–.070	≥ .070
LR (non-uniform)	MAPPD	< .05	.05–.10	≥ .10
LRT (uniform)	Std. chi (df-scaled)	< 0.10×√(df/2)	0.10–0.20×√(df/2)	≥ 0.20×√(df/2)
LRT (non-uniform)	MAPPD	< .05	.05–.10	≥ .10
MOB	Std. score difference	< .35	.35–.70	≥ .70

LRT thresholds are df-adjusted following Oshima et al. (1997) to maintain equivalent sensitivity across designs with different numbers of groups. The MOB threshold of 0.35 is intentionally conservative to avoid over-detection in multigroup designs.

Group management

# Inspect cell sizes before analysis
check_groups(my_data, group = ~ gender * nationality * age_band)

# Merge sparse cells
grp <- check_groups(my_data, group = ~ gender * nationality * age_band)
merged_data <- merge_groups(
  grp,
  nationality = list("Other" = c("DE", "FR", "ES"))
)

# Merge multiple variables in one call
merged_data <- merge_groups(
  grp,
  nationality = list("Other" = c("DE", "FR")),
  age_band    = list("18-30" = c("18-24", "25-30"))
)

# Exclude groups below a minimum size at run time
result <- idifr(
  my_data, 1:20,
  group            = ~ gender * nationality * age_band,
  method           = "LR",
  exclude_below_min = TRUE,
  min_cell_size    = 50
)

Simulating DIF data

simulate_dif() generates synthetic dichotomous item response data with known DIF structure, including intersection-only DIF for validating iDIFr on controlled data:

# Standard DIF
dat <- simulate_dif(n_persons = 1000, n_items = 20, dif_items = c(3, 7))

# DIF confined to a single intersectional cell
dat_ix <- simulate_dif(
  n_persons     = 2000,
  n_items       = 20,
  dif_items     = c(5, 12),
  dif_effect    = 1.5,
  dif_structure = "intersection",
  dif_group     = list(group = "G1", nationality = "UK", age_band = "Young"),
  demo_vars     = list(nationality = c("UK", "DE", "FR"),
                       age_band    = c("Young", "Old")),
  seed          = 42
)

# Mixed DIF — some items standard, some intersectional
dat_mixed <- simulate_dif(
  n_persons     = 2000,
  n_items       = 20,
  dif_items     = list(standard = c(3, 7), intersection = c(12, 15)),
  dif_effect    = 1.0,
  dif_structure = "mixed",
  dif_group     = list(group = "G1", nationality = "UK", age_band = "Young"),
  demo_vars     = list(nationality = c("UK", "DE", "FR"),
                       age_band    = c("Young", "Old")),
  seed          = 42
)

Citation

If you use iDIFr in published work, please cite:

Rogers, T. (2026). iDIFr: Intersectional Differential Item Functioning Analysis. R package version 1.0.1. 
https://CRAN.R-project.org/package=iDIFr

Contributing

Bug reports and feature requests are welcome via GitHub Issues.