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Advanced Psychometrics and Latent Variable Modeling

Manual: General · Subject: Psychology

Explore the mathematical foundations of Item Response Theory and measurement invariance in psychological testing.

Item Response Theory (IRT)

The 2-Parameter Logistic Model

Unlike Classical Test Theory (CTT), IRT models the non-linear relationship between a latent trait θ\theta and the probability of endorsing a specific response. The 2-Parameter Logistic (2PL) model is defined mathematically as: P(Xij=1∣θj)=11+e−ai(θj−bi)P(X_{ij} = 1 | \theta_j) = \frac{1}{1 + e^{-a_i(\theta_j - b_i)}} where aia_i represents item discrimination and bib_i represents item difficulty.

IRT Parameters

a-parameter
Discrimination: How well the item differentiates between individuals with different levels of the latent trait.
b-parameter
Difficulty: The point on the latent trait scale where the probability of endorsement is 0.5.
c-parameter
Pseudo-guessing: The baseline probability of a correct response for individuals with very low trait levels (used in 3PL).
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Measurement Invariance

A test must measure the same construct in the same way across different subpopulations. If item parameters vary across groups (e.g., gender or culture) when controlling for the latent trait, the test exhibits Differential Item Functioning (DIF).

In the 2PL Item Response Theory model, what does the aia_i parameter represent?

What acronym is used to describe when an item functions differently for different demographic groups who are matched on the underlying latent trait?

Steps for Testing Measurement Invariance (MGCFA)

  1. 1

    Configural Invariance: Test if the basic factor structure is the same across groups.

  2. 2

    Metric (Weak) Invariance: Constrain factor loadings to be equal across groups.

  3. 3

    Scalar (Strong) Invariance: Constrain item intercepts to be equal across groups.

  4. 4

    Strict Invariance: Constrain residual variances to be equal across groups.