model estimation & fit

Question 1

what is estimation

Answer 1

we need to find values for the model parameters such that 𝚺 and 𝑺 are as similar as possible → using Maximum Likelihood estimation

Question 2

the discrepancy between 𝚺(𝜽) and 𝑺 is operationalized by

Answer 2

he fit function

Question 3

fit function his expression is derived based on the assumption of

Answer 3

multivariate normality y ∼ N (0 ,Σ)

Question 4

When does the fit function yield the lowest value

Answer 4

when the model-implied and sample covariance matrices are identical

Question 5

A value of 0 for the fit function means

Answer 5

that the model implied covariance matrix reproduces the observed covariance matrix perfectly
→ this means our model is just identified!
→ which means no meaningful way to asses model fit!

Question 6

the Maximum Likelihood (ML) approach is

Answer 6

• robust to violations
  
  • parameter estimates will be correctly obtained
  • but standard errors and model fit may be affected

Question 7

we have to alternatives estimation approaches related to ML

Answer 7

Satorra-Bentler → MLM

Question 8

why is ML not enough

Answer 8

the optimization can only provide the best set of values it can find for our model parameters

• we still need to assess the quality of the model specification and estimated parameter values
  
  • to determine how well they explain observed covariance matrix
  • this is where model fit and fit indices come into play

Question 9

model fit =

Answer 9

• evaluate the fit of the specified model given the estimated parameter values
• checking whether there is evidence of model misspecification

Question 10

The $F_{ml}$ is proportional to the likelihood ratio…

Answer 10

• the likelihood of the specified (hypothesized) model divided by the likelihood of the saturated model
  
  • i.e., difference in the log case

Question 11

The $F_{ml}$ tells us

Answer 11

• how well the specified model fits compared to the best possible fit → i.e., the saturated model
  
  • we translate it into a summary test statistic that is central to model fit

Question 12

a value of $F_{ml}$ = 0 is unlikely since we

Answer 12

1. do not know the population covariance matrix → we only have a sample 𝑺
   even if our model is correctly specified → $F_{ml}$ ≠ 0 due to sampling error
2. are not interested in the saturated model → we usually want positive degrees of freedom

Question 13

T test statistic formula

Answer 13

T = n*Fml

Question 14

T test follows chi square distribution if

Answer 14

• • if the model-implied covariance matrix 𝚺 = the population (true) covariance matrix
  • then, 𝑇 has a central $𝜒^2$distribution with as many degrees of freedom as the
    specified (hypothesized) model

Question 15

Types of model fit

Answer 15

Exact, close

Question 16

For exact fit, we want Chi square value to be…

Answer 16

LOW, we do not want to reject H0

Question 17

caveats of exact fit

Answer 17

• for small sample sizes we have a poor approximation of the $𝜒^2$distribution
• large sample size test is overpowered

Question 18

Two models we compare our model with

Answer 18

baseline, only item variances
Saturated = DF = 0, T = 0

Question 19

types of fit indices

Answer 19

Incremental (compare to baseline) - How much better does the specified model fits compared to the baseline model? - CFI

absolute (compare to saturated) - how close the specified model $𝑇_m$ is to the saturated model - RMSEA

Question 20

incremental fit indices rules

Answer 20

CFI, NFI, -> the higher the better!
between 0 and 1
.95 = good
.90 = acceptable

Question 21

absolute fit indices

Answer 21

RMSEA = measures the amount of misfit per degree of freedom. allows us to quantify if the specified model is close to the saturated model

• smaller values are better
  proposed benchmarks → i.e., rules of thumb
  
  • < 0.5 à very good fit or close fit
  • . 05 − .08 à good fit or fair fit
  • . 08 − .1 à mediocre fit or good
  • . 05 − .08 à good fit or fair fit
  • > .10 à poor or unacceptable

Question 22

Two RMSEA tests

Answer 22

Exact fit - RMSEA = 0 ( same as T test statistic)

close fit - RMSEA = good 0.05
sig -> not good

poor fit - RMSEA = 0.08
sig -> good!

Question 23

SRMR

Answer 23

• compares 𝑺 and 𝚺 based on the residuals
• it is the average of the squared values in the residual correlation matrix

a cutoff value < 0.08 is considered a good fit

Question 24

Goodness of fit

Answer 24

how much variance in 𝑺 is explained by 𝚺…

the proportion of variance in the sample covariance matrix 𝑺 accounted for by the model-implied covariance matrix 𝚺

• takes values in the range 0 to 1
  
  • higher values are better
  • > .90 is considered acceptable fit

Question 25

Select competing models based on…

Answer 25

Nested -> Likelihood ratio test
NOT nested -> information criteria

Question 26

Types of models to compare

Answer 26

Simple model (more restrictions)
vs.
complicated model (more general, less restrictions)

Question 27

complicated model has

Answer 27

additional parameter, so 1 DF less.

Question 28

Which model always fits better? simple or complicated (general) model?

Answer 28

general model!

Question 29

what if simple and complicated model have similar fit

Answer 29

If they indicate similar model fit → we prefer the simpler model

Question 30

what are information criteria

Answer 30

Information criteria balance model fit and complexity

by weighing the log-likelihood function with model complexity

AIC, BIC ->

Question 31

what do we want AIC and BIC to be?

Answer 31

we prefer models that have lower information criteria values

Question 32

model modification
- mi =
- epc =

Answer 32

→ modification index → the expected decrease in the 𝑇 test statistic

expected parameter change → the approximate value for the added model parameter

Question 33

• we can think of 𝜽 as

Answer 33

• a vectorized version of the full model specification of 𝚺

Question 34

• 𝚺(𝜽) contains

Answer 34

• only the free model parameters for which we want to estimate values