PART 1

Question 1

SEM steps

Answer 1

• start with a sample (observed) covariance matrix 𝑺 that we want to model
• specify a model based on theory and hypotheses we want to test
• ensure the model is over identified
• estimate values for the specified model parameters
• evaluate the fit of the specified model given the estimated parameter values
• extend the SEM framework with mean structure and multiple group analysis
• test for measurement invariance and conduct homogeneity testing

Question 2

model parameter

Answer 2

• a specific characteristic of a statistical model that quantifies some aspect of the underlying population
• a fixed value for the entire population, but unknown to the researcher
• we aim to estimate it using sufficient statistics
• typically using Maximum Likelihood estimation

Question 3

1. **We start with observed covariance matrix

Answer 3

p = (k * (k +1)) / 2
unique pieces of information

For with mean structure k + 3, or + k at the end

Question 4

estimate parameters

Answer 4

we usually estimate fewer parameters than p

Question 5

estimated values of model parameter

Answer 5

1. We estimate values for the model parameters
   - by means of numerical optimization
   - often using the Maximum Likelihood estimation method
   - the vector $\hat\theta$ holds the estimated value for each model parameter

$\Sigma(\hat\theta)$

Question 6

when Is a model identified

Answer 6

fewer estimated parameters (the ones you count) than the number of observations / unique elements in the observed covariance matrix

Question 7

Latent variables must have a scale. Three options.

Answer 7

• option 1: Fix one loading per latent variable to 1
  = marker variable approach
  sets the scale of the latent variable to be the same as the observed variable with the fixed loading
• option 2: fix the variance of the latent variable to 1
  = standardized latent variable approach
  useful when we are interested in estimating values for all factor loadings (measurement invariance testing)
  sets the scale of the latent variable in a 𝑧-score metric
  we standardize the latent variable —> 𝜇 = 0 and 𝜎! = 1
• option 3: constrain the set of loadings of the latent variable to average to 1
  = effects coding approach
  identical to having them sum to the number of unique items for the latent variable
  sets the scale of the latent variable as the average of its indicators

Question 8

three identifications based on the degrees of freedom…

Answer 8

1. over identified model ⇒ what we want!
   = there are more equations than unknows
   for 𝑑𝑓 > 0
2. under identified model
   = there are more unknows than equations
   for 𝑑𝑓 < 0
3. just identified model
   = there are as many equations as unknows = saturated model
   for 𝑑𝑓 = 0

Question 9

problem with just identified

Answer 9

• We don’t want that, it is just a sample and we can’t do GoF statistics.
• You cannot look at goodness of fit, byt you can test factor loadings. they are based on standard errors.
• there is only one set of values for the model parameters
  
  • they reproduce the observed covariance matrix perfectly
• cannot make meaningful model fit statistical inferences