Degrees of freedom in example 6.14
Message/Author
 finnigan posted on Thursday, February 26, 2009 - 8:53 am
Linda/Bengt

I was going through the above example and would like to ask how the degrees of freedom 34 and 36 were calculated.

For example in a CFA model
using rules of path diagrams one counts factor loadings, residual variances, any factor covariances ,but not the factor intercepts and factor means.
The number of these parametres is then subtracted from p(p+1)/2)
to get the Degres of freedom. Iassume that in invariance testing under CFA that fctor intercepts

Is there a particular way that the degrees of freedom are calculated in the multiple indicator growth model that uses a similar approach to that in SEM.

Thanks
 Linda K. Muthen posted on Thursday, February 26, 2009 - 4:04 pm
The degrees of freedom for Example are computed in the regular way. There are 45 variances and covariances and 9 means for a total of 54 sample statistics. 54 minus 21 free parameters in the H0 model results in 33 degrees of freedom. The Baseline model has free parameters of nine means and nine variances. 54 minus 18 free parameters is 36 degrees of freedom.
 finnigan posted on Friday, February 27, 2009 - 6:53 am
Thanks

The trouble I am having is identifying and separating the variances, covariances and means included in 54(45+9) and free parametres which constitute the 21 and 18 under the Ho and baseline model
Ho model
45 variances and covariances
I counted
9 residual variancese
3 factor residual variances
1 slope residual variance
1 intercept residual variance
1 slope variance
1 intercept variance
0 correlated residuals by default
1 covariance between I and S

Total 17: clearly I am missing 28 items!

Means
Do the 9 means refer to indicator intercepts; what do they include?

Ho model
21 free parametres

Free parametres 21
9 indicator intercepts
1 intercept variance
1 slope variance
3 factor residuals
1 slope residual
1 intercept residual
1 slope mean
1 covariance

Total 24

Base line model 18 parametres(9vars and 9 means)
In the base line model of 18 free parametres, what 3 extra parametres were estimated
Do the 9 means refer to factor intercepts or include slope mean

9 variances - do these refer to the residual variance of y11- y33. If not what is included in the 9 varances and what about the covariance between I and S
 Linda K. Muthen posted on Friday, February 27, 2009 - 8:54 am
See TECH1 to see what the 21 free parameters are for the H0 model. The eighteen parameters for the Baseline model are nine means and nine variances, one each for each observed variable.
 finnigan posted on Saturday, February 28, 2009 - 1:17 am
Thanks

I have ran tech 1 and under parametre specification, I found the 21 parametres across theta delta, Psi Alpa ,lamda. Does Nu contain the indicator intercepts in 9 means

I looked at lamda and expected to see factor loadings,but instead saw 4 for y33 on F3.

How are these whole numbers interpreted?

Similarly in Nu y11 has a value of 1 which I assume arose becase y11 was set to 1 under Mplus default. What is the interpretation of the 2 and 3.
 Linda K. Muthen posted on Saturday, February 28, 2009 - 5:46 am
The intercepts of the first-order factor indicators are in nu. They are held equal across time as part of the growth model parameterization. See page 546 of the Mplus User's Guide for the BY language for this model specification. See also the Topic 4 course handout.
 Linda K. Muthen posted on Saturday, February 28, 2009 - 8:43 am
One further comment about TECH1. The numbers are simply numbers assigned to each parameter. These numbers are used in error messages. Starting values are given in the second part of TECH1. See pages 586-588 of the Mplus User's Guide.
 finnigan posted on Tuesday, March 03, 2009 - 8:23 am
Linda,
Many thanks for this.

I took a look at the topic four course handout about the estimation of the multiple indicator growth model.

I assume that the EFA at each time point is done to see what indicators load on what first order factors. Why is an efa using all time points not done?
Why is the growth model part done before the CFA ?

I 'd like to ask if there is a set of references available to explain the logic of each step and if there is a longitudinal CFA example available.

Thanks
 Linda K. Muthen posted on Tuesday, March 03, 2009 - 1:40 pm
EFA at each time point is done to determine that at a minimum the same number of factors is represented. If not, it would not make sense to go on to the CFA. The growth model is the last step after CFA.

I know of no set of references that explicitly discusses the strategy we recommend. Slides 77-96 of short course Topic 4 show the steps and the inputs involved. Example 6.14 is also multiple indicator growth.