Modification Indices in umx
In umx: Structural Equation Modeling and Twin Modeling in R

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 6,
  fig.height = 5
)
library(umx)
library(OpenMx)
umx_set_auto_plot(FALSE)

Introduction

Modification indices (MIs) flag single parameters whose addition or removal would most improve model fit (if freed and the model re-estimated). They are useful for exploring where a misspecified model struggles, but they are not a substitute for theory-driven model comparison.

umxMI() wraps [OpenMx::mxMI()] and prints the largest indices (by default those above $\chi^2_1(0.01) = 6.63$). By default full = FALSE (only the tested parameter moves — fast). Set full = TRUE to allow all free parameters to re-adjust (slower).

This vignette uses the classic Holzinger & Swineford (1939) data (HS.ability.data in OpenMx): fit a deliberately poor one-factor model (the worst of the usual teaching sequence), run one umxMI() call, then show a better three-factor alternative with umxCompare() (no second MI pass).

Data

data("HS.ability.data", package = "OpenMx")

manifests = c("visual", "cubes", "flags", "paragrap", "sentence", "wordm",
  "addition", "counting", "straight")

# Scale for stable optimization (recommended for this dataset in umx)
df = umx_scale(HS.ability.data)

One-factor model (worst fit → one MI call)

m1 = umxRAM("Holzinger_1f", data = df,
  umxPath("g", to = manifests),
  umxPath(v.m. = manifests),
  umxPath(v1m0 = "g")
)

umxSummary(m1, refModels = FALSE)

Fit is poor (as expected for a single factor on these data). What does umxMI() suggest?

umxMI(m1)

Large indices often point to within-domain residual covariances (e.g. flags_with_visual) or cross-domain loadings that a single factor cannot represent. That pattern hints at multiple abilities rather than one g.

Three-factor alternative (no second MI)

Holzinger and Swineford's data are usually modeled with three correlated abilities:

model3f = "
  spatial =~ visual + cubes + flags
  verbal  =~ paragrap + sentence + wordm
  speed   =~ addition + counting + straight
"
m3 = umxRAM(model3f, data = df, name = "Holzinger_3f")

umxSummary(m3, refModels = FALSE)

umxCompare(m3, m1)

The three-factor model fits substantially better. Remaining residual associations are typically smaller; run umxMI(m3) locally if you want that table.

Bifactor / multi-group (optional, local)

# Bifactor + umxMI, or multi-group one-factor by sex
modelBifactor = "
  g =~ visual + cubes + flags + paragrap + sentence + wordm + addition + counting + straight
  spatial =~ visual + cubes + flags
  verbal  =~ paragrap + sentence + wordm
  speed   =~ addition + counting + straight
  spatial ~~ 0*verbal + 0*speed + 0*g
  verbal  ~~ 0*speed + 0*g
  speed   ~~ 0*g
"
mBio = umxRAM(modelBifactor, data = df, name = "Holzinger_bifactor")
umxMI(mBio)

mMG = umxRAM("Holzinger_1f_MG", data = df, group = "Gender",
  umxPath("g", to = manifests),
  umxPath(v.m. = manifests),
  umxPath(v1m0 = "g")
)
umxMI(mMG)

Summary

| Model | Role | |:-----------------------|:--------------------------------------------------| | One-factor | Baseline misspecification; one umxMI() call | | Three-factor | Theory-driven improvement via umxCompare() | | Bifactor / multi-group | Optional local demos |

Take-home: Use umxMI() to diagnose where a model misfits; prefer umxCompare() on theory-driven competing models for inference.

References

Holzinger, K. J., & Swineford, F. (1939). A study in factor analysis: The stability of a bi-factor solution. Supplementary Educational Monographs, 48. University of Chicago Press.