STRUCTURAL EQUATION MODELLING
improving model fit
let residuals correlate
improves model fit
SEM model components
path models get upgraded to latent variables
error variances become parameters within the model
we estimate the error variance
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model estimation
maximum likelihood
assumption that data is multivariate normally distributed
Confirmatory Factor Analysis
estimate covariances between latent variables
add regressions
SEM model
model the variance-covariance matrix of latent variables
allows for testing of causal hypotheses
sufficient statistics
covariance matrix can be shared
research replication
setting the scale
marker variable approach
one loading per latent variable fixed to 1
latent variable scale = scale of observed variable
standardised latent variable approach
variance of the latent variable fixed to 1
when we are interested in estimating values for all factor loadings
z-score metric
mean = 0 sd = 1
effects coding approach
constrain the set of loadings of the latent variable to average to 1
scale of latent variable is the average of its indicators
model identification
over-identified model
more equations than unknowns
optimisation
minimising fit criterion
underidentified model
more unknowns than equations
cannot make meaningful goodness-of-fit inferences
fit criterion = perfect reproduction of the observed covariance matrix
just-identified model
as many equations as unknowns
values reproduce the observed covariance matrix perfectly
cannot make meaningful goodness-of-fit inferences
useful if interested in significance of only specific model parameters
sample
observed covariance matrix S
specification of over-identified model
estimate values for specified model parameters
numerical optimisation methods (ML)
estimate values for theta model parameters
model-implied covariance matrix
mean structures
multiple group analysis
measurment invariance
homogeneity testing
disturbance terms
unexplained variance of endogenous variables
loading of disturbance term is 1
disturbance term perfectly predicts the unexplained variance
theta matrix
variance of error terms are mostly found on the diagonal
if not
violation of local independence
cross-loadings
latent variable doesn't fully account for the variance in observed variable