1 Overview

In this tutorial we walk through the very basics of testing measurement invariance in the context if a longitudinal factor model - and how a 2nd order growth model can then be used to describe change in an invariant factor.

This tutorial follows the example in Chapter 14 of Growth Modeling: Structural Equation and Multilevel Modeling Approaches (Grimm, Ram & Estabrook, 2017). Using 3-occasion data from the ECLS-K, we test for factorial invariance, and then use a second-order growth model to describe change in the factor scores across time.

2 Introduction to the Common Factor Model

The basic factor analysis model can be written as a matrix equation …

\[ \boldsymbol{Y_{i}} = \boldsymbol{\tau} + \boldsymbol{\Lambda}\boldsymbol{F_{i}} + \boldsymbol{U_{i}} \]

where \(\boldsymbol{Y_{i}}\) is a \(p\) x 1 vector of observed variable scores, \(\boldsymbol{\Lambda}\) is a p x q matrix of factor loadings, \(\boldsymbol{F_{i}}\) is a \(q\) x 1 vector of common factor scores, and \(\boldsymbol{U_{i}}\) is a p x 1 vector of unique factor scores.

We can rewrite the model in terms of variance-covariance and mean expectations. For example, the expected covariance matrix, \(\boldsymbol{\Sigma} = \boldsymbol{Y}'\boldsymbol{Y}\), becomes

\[ \boldsymbol{\Sigma} = \boldsymbol{\Lambda}\boldsymbol{\Psi}\boldsymbol{\Lambda}' + \boldsymbol{\Theta} \] where \(\boldsymbol{\Sigma}\) is a p x p covariance (or correlation) matrix of the observed variables, \(\boldsymbol{\Lambda}\) is a p x q matrix of factor loadings, \(\boldsymbol{\Psi}\) is a q x q covariance matrix of the latent factor variables, and \(\boldsymbol{\Theta}\) is a diagonal matrix of unique factor variances.

and the expected p x 1 mean vector, $ $ becomes \[ \boldsymbol{\mu} = \boldsymbol{\tau} + \boldsymbol{\Lambda}\boldsymbol{\alpha} \]

where \(\boldsymbol{\tau}\) is a p x 1 vector of manifest variable means, \(\boldsymbol{\Lambda}\) is a p x q matrix of factor loadings, and \(\boldsymbol{\alpha}\) is a q x 1 vector of latent variable means.

We can then extend the model to multiple-occasion settings with occasion-specific subscript, so that

\[ \boldsymbol{\Sigma_t} = \boldsymbol{\Lambda_t}\boldsymbol{\Psi_t}\boldsymbol{\Lambda_t}' + \boldsymbol{\Theta_t} \] and \[ \boldsymbol{\mu_t} = \boldsymbol{\tau_t} + \boldsymbol{\Lambda_t}\boldsymbol{\alpha_t} \]

Different levels of measurement invariance are established by testing (or requiring) that various matrices are equal.

Specifically, …
For Configural Invaraince we establish that the structure of \(\boldsymbol{\Lambda_t}\) is equivalent across occasions.

For Weak Invariance we additionally establish that the factor loadings are equivalent across occasions, \(\boldsymbol{\Lambda_t} = \boldsymbol{\Lambda}\).

For Strong Invariance we additionally establish that the manifest means are equivalent across occasions, \(\boldsymbol{\tau_t} = \boldsymbol{\tau}\). and

For Strict Invariance we additionally establish that the residual/unique variances are also equivalent across occasions \(\boldsymbol{\Theta_t} = \boldsymbol{\Theta}\).

Measurement Invariance testing is usually conducted within a Structural Equation Modeling (SEM) framework. Here we illustrate how this may be done using the lavaan package.

Lots of good information and instruction (including about invariance testing - in a multiple-group setting) can be found on the package website … http://lavaan.ugent.be.

library(psych)
library(ggplot2)
library(corrplot) #plotting correlation matrices
library(lavaan)  #for fitting structural equation models
library(semPlot)  #for automatically making diagrams 

#set filepath for data file
filepath <- "https://raw.githubusercontent.com/LRI-2/Data/main/GrowthModeling/ECLS_Science.dat"
#read in the text data file using the url() function
dat <- read.table(file=url(filepath),na.strings = ".") 

names(dat) <- c("id", "s_g3", "r_g3", "m_g3", "s_g5", "r_g5", "m_g5", "s_g8", 
                "r_g8", "m_g8", "st_g3", "rt_g3", "mt_g3", "st_g5", "rt_g5", 
                "mt_g5", "st_g8", "rt_g8", "mt_g8")

#selecting only the variables of interest
dat <- dat[ ,c("id", "s_g3", "r_g3", "m_g3", "s_g5", "r_g5", "m_g5", "s_g8", 
                "r_g8", "m_g8")]

#data structure
head(dat,10)

#descriptives (means, sds)
psych::describe(dat[,-1]) #-1 to remove the id column

#correlation matrix
round(cor(dat[,-1], use = "pairwise.complete"),2)

##      s_g3 r_g3 m_g3 s_g5 r_g5 m_g5 s_g8 r_g8 m_g8
## s_g3 1.00 0.76 0.71 0.85 0.73 0.68 0.75 0.68 0.66
## r_g3 0.76 1.00 0.75 0.73 0.85 0.70 0.70 0.76 0.68
## m_g3 0.71 0.75 1.00 0.70 0.72 0.88 0.71 0.66 0.81
## s_g5 0.85 0.73 0.70 1.00 0.77 0.74 0.81 0.73 0.70
## r_g5 0.73 0.85 0.72 0.77 1.00 0.75 0.74 0.80 0.71
## m_g5 0.68 0.70 0.88 0.74 0.75 1.00 0.74 0.68 0.85
## s_g8 0.75 0.70 0.71 0.81 0.74 0.74 1.00 0.78 0.78
## r_g8 0.68 0.76 0.66 0.73 0.80 0.68 0.78 1.00 0.75
## m_g8 0.66 0.68 0.81 0.70 0.71 0.85 0.78 0.75 1.00

#visusal correlation matrix
corrplot(cor(dat[,-1], use = "pairwise.complete"), order = "original", tl.col='black', tl.cex=.75)