Re: st: decrease ll nested models sem

[jpitblado@stata.com (Jeff Pitblado, StataCorp LP)]

Volker Lang <v.lang@uni-tuebingen.de> is using -sem- and noticed that the
log-likelihood decreased when adding an exogenous variable.

> I'm fitting a log-linear regression model using "sem".
> (It shall later be extended to a path model with robust s.e.'s,
> therefore "sem".)
> 
> Question:
> Why does the log likelihood in comparison of the two nested models
> estimated with "sem" decrease?
> (It does not happen if I use a different estimation command, e.g.,
> "sureg .. , isure".
> Also, in this case coefficients and s.e.'s estimated by "sem" and
> "sureg .. , isure" are identical,
> so why does the log likelihood differ between these estimation
> commands at all?)
> 
> Here is the output:
> 
> . sem (grade_l <- infov_l)
> ..
> Iteration 1:   log likelihood =  -175.8425
> 
> Structural equation model                       Number of obs = 1281
> Estimation method  = ml
> Log likelihood     =  -175.8425
> ..
> 
> . sem (grade_l <- infov_l ued_l)
> ..
> Iteration 1:   log likelihood =  -513.0155
> 
> Structural equation model                       Number of obs = 1281
> Estimation method  = ml
> Log likelihood     =  -513.0155
> ..
> 
> . sureg (grade_l infov_l), isure
> Iteration 1:   tolerance =  7.234e-17
> ..
> . di e(ll)
> -677.17747
> 
> . sureg (grade_l infov_l ued_l), isure
> Iteration 1:   tolerance =  1.447e-16
> ..
> . di e(ll)
> -656.64981

Notice that the reported log-likelhoods between the "equivalent" -sem- and
-sureg, isure- models do not agree.

	Model 1:	grade_l <- infov_l
	sem:		-175.8425
	sureg, isure:	-677.17747

	Model 1:	grade_l <- infov_l ued_l
	sem:		-513.0155
	sureg, isure:	-656.64981

This hints that -sem- is computing a difference log-likelihood from -sureg-.
In fact, -sem-'s log-likelihood is not conditional on the observed exogenous
variables while -sureg, isure-'s is.

If Volker intends to make direct comparisons between the reported
log-likelihoods from -sem- output, such as the -lrtest- command, then all
observed exogenous variables must be present in the fitted models; otherwise,
the likelihoods are not comparable.

Here is a silly example using the auto data.  We will use -lrtest- to test the
significance of the -trunk- coefficient.

	. sysuse auto

	# constrained model
	. sem (mpg <- turn trunk@0)
	. est store sem0

	# full model
	. sem (mpg <- turn trunk)
	. est store sem1

	. lrtest sem0 sem1

We can easily verify/validate the results from -lrtest- by using -regress-:

	. regress mpg turn
	. est store reg0
	. regress mpg turn trunk
	. est store reg1
	. lrtest reg0 reg1

--Jeff
jpitblado@stata.com
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