means & groups

Question 1

What do we mean when we say we are
setting the scale of a latent variable?

-

Answer 1

We set its variance → the width of its
distribution.

Question 2

What do we mean when we say we are
setting the location of a latent variable?

Answer 2

• We set its mean → the balance point of its
  distribution.

Question 3

without vs. with mean structure difference

Answer 3

y (mu, Sigma)
Zeta (alpha, psi)
mu = tao + Lambda*alpha

Question 4

How to calculate number of observations for mean structure

Answer 4

k = p * (p+ 1) /2 + p
p *( p+3)

Question 5

what changes in model parameters with mean structure

Answer 5

the number of free model parameters
𝑞 now contains additional

• 𝑝 intercept terms in the vector 𝝉
• and the latent means in vector 𝜶
• i.e., one mean for each latent variable

Question 6

what do we need to do for model identification

Answer 6

We need to set the location (set the mean) of the latent variable for model identification… for identification purposes -> Alpha = 0

Question 7

equal mean structure across groups

Answer 7

alpha

Question 8

equal covariance structure across groups

Answer 8

psi

Question 9

benefits of multiple group like this

Answer 9

Benefits

• this is a powerful technique that allows us to take measurement error into account

Question 10

challenges of measuring multiple group like this

Answer 10

however, there are some challenges
- the latent means are not identified
- the latent variances are arbitrary due to scaling
- e.g., fixed to one or to the scale of the marker indicator
- the test may not measure the same latent construct across groups
- i.e., the test may be measurement non-invariant

• we can solve these problems by
  
  1. imposing equality constraints
  2. and testing for increasing levels of measurement invariance

Question 11

o compare groups on latent means and variances

Answer 11

the test must measure the same latent construct across groups and must not be biased
group membership may influence the latent trait, but should not influence individual items
-> measurement invariant!

Question 12

• f measurement invariance does not hold

Answer 12

• ## test scores (e.g., sum-scores) must not be used to compare groups → the test is biased

Question 13

if partial measurement invariance holds

Answer 13

• most parameters are equal across groups → we can test for homogeneity in means and variances

Question 14

4 levels of equality constrains

Answer 14

configural = same zero’s (factor loadings are significantly different from 0 in both groups)
weak = same loadings (test psi)
strong = same intercepts (test alpha)
strict = same residual (co)variance (test full homogeneity)

Question 15

What does full homogeneity imply?

Answer 15

That the correlation structure is exactly the same across groups

Question 16

• we want our measurement model to be invariant across groups. Why?

Answer 16

it is necessary if we want to make group comparisons on the latent variables

otherwise, we can’t trust differences in latent variable means and variances

are there actual differences in the latent constructs? or simply an artifact of the test being unfair and biased?

therefore, before performing any comparisons
- we first need to establish measurement invariance (e.g. differential item functioning

Question 17

What does full homogeneity imply?

Question 18

For categorical observed predictors assumes…

Answer 18

Strict invariance

Question 19

we can get around violations of …

we cannot get around…

Answer 19

… strong invariance (i.e., equal intercepts) by allowing direct effect on the observed indicators
- i.e., a significant direct effect is indicative of Differential Item Functioning (DIF)

…violations of measurement invariance in the factor loadings and residuals