Confirmatory Factor Analysis CFA

1
Confirmatory Factor Analysis (CFA)

• CFA is used when strong theory and/or when a
  strong empirical base is available
• Specify relations a priori
• number of factors
• relations among factors (i.e., correlated vs.
  uncorrelated)
• variables specified as fixed or free on a
  respective factor(s)

2
Confirmatory Factor Analysis

• a priori measurement model is specified and
  tested
• factor loadings
• direct relations between observed and latent
  variables are modeled
• error terms are the leftovers
• estimate the variance of these things
• factor variances/covariances
• typically we are interested in the standardized
  covariances between factors
• factor correlations

3
Two-factor correlated CFA model
?
Relationship Self-Esteem
?
?
?
?
?
?
?
?
e1
e5
e7
e8
e6
4
The Process

• Specifying the pattern matrix
• parameter we will estimated (free)
• 0 parameter noted estimated (fixed)

Variables Academic Relationship S1 0 S2
0 S3 0 S4 0 S5 0 S6 0 S7 0
S8 0
5
The Process

• Model specification via the Bentler-Weeks model
• all variables in a model are categorized as IVs
  or DVs
• DV variable with unidirectional arrows aiming
  at it
• want to explain the variance in these variables
  with other variables (IVs)
• e.g., our eight observed variables
• IV no unidirectional arrows aiming at it
• but there can be an unanalyzed association
• mathematically we estimate the error variances
  from slide 3 as well

6
The Process

• you write an equation for each DV
• S1 ?(Academic Self-Esteem) e1
• S2 ?(Academic Self-Esteem) e2
• .....
• S5 ?(Relationship Self-Esteem) e5
• S6 ?(Relationship Self-Esteem) e6
• ....
• ? represents a regression coefficient (factor
  loading)
• e represents error (or residual), this path is
  not directly estimated (fixed)
• predetermined by the factor loading

7
The Process

• core parameters in CFA are these factor loadings
  and variances/covariances for IVS
• notice that the latter are not directly specified
  in these equations
• e.g., covariance between Academic and
  Relationship Self-Esteem
• they are a function of the equations that you see
• via some complicated matrix algebra
• however, you must specify them to solve the
  equations
• software does this for us

8
The Process

• Estimating the model
• using primarily maximum likelihood
• estimation produces a fit function
• Determination of model fit
• done at two levels
• overall model fit
• individual parameter fit
• parameters generally factor loadings in CFA
• but include factor covariances (correlations) if
  specified

9
The Process

• Overall model fit (Goodness of fit)
• tells us if the model should be accepted or
  rejected
• if model is accepted, interpret model parameters
• if model is rejected, do not interpret model
  parameters
• Determining goodness of fit
• test statistic
• ?2 provides a statistical test of fit
• ?2 ( fit function ) ( N 1)
• we want this to be nonsignificant

10
The Process

• types of descriptive indices
• absolute fit indices
• indexes the amount of variance/covariance
  accounted for by a model
• goodness of fit index (GFI) and adjusted GFI
• want values gt .90
• root mean square residual (RMSR)
• average size of residuals generated by a model
• want standardized values lt .05 if model is good

11
The Process

• comparative fit indices (CFI)
• compare target model to a baseline model
• baseline model null or independence model
• null model specifies no factors
• CFI values gt .90 are good, .93 better, .95 great
• parsimony adjusted fit indices
• adjusts fit by weighting values by the number of
  parameters estimated
• root mean square error of approximation (RMSEA)
  is best
• values less than .08 are good, .05 are better

12
The Process

• Fit of individual parameters
• we have statistical tests for each factor loading
  and each factor co(variance)
• evaluate the critical ratios (CR)
• these are distributed as z-values
• What if my model and/or individual parameters do
  not fit?
• report that and stop, or
• go to the model modification phase
• the LaGrange Multiplier test
• the Wald test

13
Practical Issues

• Identification
• also needed to mathematically solve the
  equations
• based largely on degrees of freedom (df) for the
  model
• df nonredundant elements in ? - parameters
  estimated
• elements in ? variances covariances
• this equals p (p1) / 2, where p observed
  variables
• parameters estimated
• count up factor loadings, factor covariances, and
  IV variances estimated

14
Identification of a one-factor model

• e.g., 4 MVs ? 4 (4 1) /2 10
  variances/covariances
• e.g., 4 factor loadings, 4 error variances
• e.g., df 10 8 2

Depression
?
?
?
?
Scale 1
Scale 2
Scale 3
Scale 4
e1 (d1)
e3
e2
e4
15
Practical Issues

• over-identified (the ideal)
• positive df more information than parameters to
  estimate
• can determine overall model fit
• under-identified
• too many parameters, not enough information
• model cannot be estimated
• just-identified (df 0)
• parameters to be estimated amount of
  information
• no overall model fit, but you can interpret
  parameter estimates

16
Practical Issues

• EQS will present unstandardized factor loadings
  and factor covariances
• remember, the analyses are based on ?
• however, we generally interpret standardized
  solutions
• this makes factor loadings range (roughly)
  between 1 and 1
• and makes factor covariances into factor
  correlations

17
The Structural Model

• Testing the directional relations among latent
  variables
• This is just path analysis with latent variables
• Latent variables are developed through
  confirmatory factor analysis (CFA)
• General modeling process is the same as with CFA

18
Structural Model Equations
Two equations (1)Depression ?31(Illness
Perceptions) ?32(Coping) d3 (2) Coping
?21(Illness Perceptions) d2
d3
Illness Perceptions (1)
?31
Depression (3)
?21
?32
d2
Coping (2)
19
Comparing nested models
d3
Illness Perceptions (1)
Depression (3)
?21
?32
d2
Coping (2)
Model 1
20
Comparing nested models
d3
Illness Perceptions (1)
?31
Depression (3)
?21
?32
d2
Coping (2)
Model 2
21
Comparing nested models

• we can statistically compare models 1 and 2
• model 1 does not have the direct effect
• model 2 does have the direct effect
• both models have the mediated or indirect effect
• model 1, then, is nested within model 2, and thus
  they can be statistically compared
• we do this using the ?2 difference test (? ?2 )

22
?2 Difference Test

• Statistically compares nested models
• nested lower-order models that contain a subset
  of the parameters from a target higher-order
  model
• e.g., model 1 is nested within model 2 (target)
• ??2 ? 2nested - ? 2target ?df dfnested
  dftarget
• notice that the nested model will always
• have worse overall model fit (higher ?2 )
• and more degrees of freedom
• because we are estimating fewer things
• ??2 and ?df will always be positive because of
  this

23
?2 Difference Test

• if ? ?2 is not significant...
• there is no difference between models
• the simpler or more parsimonious model fits
  "better"
• if ? ?2 is significant...
• target model fits better and...

24
Comparing nonnested models

• This is a direct comparison between models that
  have at least a subset of variables that differ
• We typically use other descriptive fit indices
  for these purposes
• Akaike Information Criterion (AIC)
• Bayesian Information Criterion (BIC)
• Expected Cross-Validation Index (ECVI)
• For all of the above indices, the model with the
  smaller index value is the better-fitting model

25
Setting the Scale (Metric) for an Endogenous
Latent Variable

• We need to do this to mathematically solve the
  equations of the model
• Two options
• fix variance of latent variable to 1
  (standardize)
• fix a factor loading for each LV to 1
• For an endogenous LV, you can only use the second
  option
• we want to predict the variance of the endogenous
  LV
• setting this value to 1 does not allow for this
  possibility