1. CFA with Categorical Outcomes Psych 818 - DeShon

2. Big Picture The basic approach is to assume that the latent variable is continuous The basic approach is to assume that the latent variable is continuous However, the indicator of the latent variable is “course” resulting in an ordered set of categorical responses (e.g., 0, 1, 2). However, the indicator of the latent variable is “course” resulting in an ordered set of categorical responses (e.g., 0, 1, 2). Mathematically... Mathematically...

3. Math... For a single indicator, single factor model: For a single indicator, single factor model: Where y* is the continuous score for the i th person, ν is the intercept, λ is the loading or regression wieght, η is the score for the i th person on the latent variable, and ε is the error for the i th person Where y* is the continuous score for the i th person, ν is the intercept, λ is the loading or regression wieght, η is the score for the i th person on the latent variable, and ε is the error for the i th person

4. More Math... The mean and variance of y* are: The mean and variance of y* are: Where α is the mean of η, ψ is the variance of η, and θ is the variance of the errors Where α is the mean of η, ψ is the variance of η, and θ is the variance of the errors This is fine but we don't observe y* This is fine but we don't observe y*

5. Even More Math... Instead, we observe y as categorical indicator of y* Instead, we observe y as categorical indicator of y* y is related to the ordered polytomous variable y via a threshold function: y is related to the ordered polytomous variable y via a threshold function: Where the τ are the unknown thresholds yielding the categories c=1, 2, 3...C-1 Where the τ are the unknown thresholds yielding the categories c=1, 2, 3...C-1

6. Graphically... For dichotomous outcomes... For dichotomous outcomes...

7. Identification Y* is a latent variable caused by another latent variable. This means that the metric of Y* is not determined and must be set in some way. Y* is a latent variable caused by another latent variable. This means that the metric of Y* is not determined and must be set in some way. Common to set ν =0 and to σ *=1 Common to set ν =0 and to σ *=1 In this case the errors are determined and not estimated In this case the errors are determined and not estimated Alternatively, ν =0 and θ =1 Alternatively, ν =0 and θ =1 Different programs use different metrics Different programs use different metrics

8. Estimation Issues The proportion of individuals who fall into or endorse a particular categorical response provides information about the latent distribution The proportion of individuals who fall into or endorse a particular categorical response provides information about the latent distribution

9. Estimation Issues Now imagine that a person provides a response to two dichotomous questions or indicators Now imagine that a person provides a response to two dichotomous questions or indicators

10. Estimation Issues The proportion of responses in each cell of the contingency table is a function of both the thresholds and the underlying correlation between the latent variables The proportion of responses in each cell of the contingency table is a function of both the thresholds and the underlying correlation between the latent variables This is a very tough estimation problem This is a very tough estimation problem Vast array of approaches Vast array of approaches Documented in Wirth & Edwards (2007)‏ Documented in Wirth & Edwards (2007)‏

11. Mplus Approach Weighted Least Squares Weighted Least Squares

12. Good News! After all of this, the basic model may be constructed just like the standard CFA model. After all of this, the basic model may be constructed just like the standard CFA model. There is a stronger norm to set a path to zero to scale the latent variable. There is a stronger norm to set a path to zero to scale the latent variable. The parameter estimates are harder to interpret...just as in Probit or Logit regression The parameter estimates are harder to interpret...just as in Probit or Logit regression