8.1 IRT and Factor Analysis

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8.1 IRT and Factor Analysis#

Multidimensional IRT

EFA and CFA are designed for metric input variables (unless we use specific correlation coefficients suited for categorical data, like tetrachoric correlation). The main outcomes are factor loadings and factor scores. IRT models are designed for categorical data with item-category parameters (location, discrimination) and person parameters as main outcomes. However, it has been found that there is a strong parametric relationship between FA and IRT (Takane and De Leeuw, 1986).

Let us consider a simple unidimensional 2-PL, rewritten in logit form:

(1) \(logit(P(X_{vi})) = \alpha_i(\theta_v - \beta_i))\)

We can re-parameterize this model as

(2) \(logit(P(X_{vi})) = \alpha_i\theta_v - d_i\)

with \(d_i = -\beta_i\alpha_i\)

The second equation reflects a factor analytic intercept-slope representation of the 2-PL. The discrimination parameter \(\alpha_i\) can be interpreted as loading (i.e., slope as reflected in the ICC plot). The parameter \(d_i\) is the intercept. From the first equation, it follows that \(d_i\) = \(-\beta_i\alpha_i\) . Note that both equations fit the same model, they are just parameterized differently. Consequently, the parameters \(\theta_i\) are the same, regardless which expression we use. Within an IRT context (first equation), we call them “person parameters,” whereas within a FA context (second equation), we call them “factor scores”.

Fit the models#

To illustrate, we conduct a 2-PL analysis which involved dichotomous items on knowledge characteristics from the WDQ dataset. We use the ltm package for R and request the two different parameterizations.

First, import all the necessary packages for R and Python.

# General imports
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns

# Rpy2 imports
from rpy2 import robjects as ro
from rpy2.robjects import pandas2ri, numpy2ri
from rpy2.robjects.packages import importr

# Automatic conversion of arrays and dataframes
pandas2ri.activate()
numpy2ri.activate()

# Set random seed for reproducibility
ro.r('set.seed(123)')

# Ipython extension for magic plotting
%load_ext rpy2.ipython

# R imports
importr('base')
importr('mirt')
importr('Gifi')
importr('MPsychoR')
importr('ltm')

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In[1], line 8
      5 import seaborn as sns
      7 # Rpy2 imports
----> 8 from rpy2 import robjects as ro
      9 from rpy2.robjects import pandas2ri, numpy2ri
     10 from rpy2.robjects.packages import importr

ModuleNotFoundError: No module named 'rpy2'

Load and inspect the dataset RWDQ and remove the first item (first column). Name your subset RWDQ1.

ro.r("data(RWDQ)")

# Convert to Python
RWDQ = pandas2ri.rpy2py(ro.globalenv['RWDQ'])

# Eliminate first item (misfit)
RWDQ1 = RWDQ.drop(RWDQ.columns[0], axis=1)

# Put data into R
ro.globalenv['RWDQ1'] = RWDQ1

Now we fit the two models mentioned above.

# Fit the IRT model using lavaan's ltm
ro.r('irtpar <- ltm(RWDQ1 ~ z1)')
ro.r('fapar <- ltm(RWDQ1 ~ z1, IRT.param = FALSE)')

# Combine and compare parameter estimates
compare_params = ro.r('head(cbind(coef(irtpar), coef(fapar)))')

# Print comparison with 3 decimal precision
ro.r('print(round(head(cbind(coef(irtpar), coef(fapar))), 3))')

The first two columns are based on the parameterization in Eq. (1), involving \(\alpha_i\) and \(\beta_i\) . The last two columns correspond to \(d_i\) and \(\alpha_i\) from Eq. (2).
Let us compute the person parameters and confirm that they are the same for both models.

In theory, both set of person parameters should be the same. To investigate if this is true, use the identical function.

ro.r("irtppar <- factor.scores(irtpar)$score.dat$z1")
ro.r("fappar <- factor.scores(fapar)$score.dat$z1")

print(ro.r("identical(irtppar, fappar)"))

[1] TRUE

As indicated by the identical function, the two scores are in fact the same.

Besides IRT and Factor Analysis there is also Item Factor Analysis (IFA). IFA marries IRT and FA so that we get the best of both worlds: On the one hand, we have all the possibilities exploratory/confirmatory model specification, good opportunities for goodness-of-fit assessment, and options for rotation from the FA world. On the other hand, we can make use of the wide range of developed IRT models and get a deep probabilistic insight into the behavior of items and persons. Note that the terms “MIRT” and “IFA” are often used synonymously. Here, from now on, we use “MIRT.”

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