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st: Latent Growth model and mixed effects random slopes model - equivalent ?
From
W Robert Long <[email protected]>
To
"[email protected]" <[email protected]>
Subject
st: Latent Growth model and mixed effects random slopes model - equivalent ?
Date
Sun, 28 Apr 2013 23:19:40 +0100
Dear all
What am I missing here ? I'm looking at latent growth models in -sem-
and mixed effects / multilevel models in -xtmixed- which I thought
should be equivalent.
For a random intercepts model, everything agrees 100%, but with random
slopes, the variances of the random effects differ.
use http://www.stata-press.com/data/r12/pig.dta, clear
* first a random intercepts model:
xtmixed weight week ||id: , var
------------------------------------------------------------------------------
weight | Coef. Std. Err. z P>|z| [95% Conf.
Interval]
-------------+----------------------------------------------------------------
week | 6.209896 .0390124 159.18 0.000 6.133433 6.286359
_cons | 19.35561 .5974059 32.40 0.000 18.18472 20.52651
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf.
Interval]
-----------------------------+------------------------------------------------
id: Identity |
var(_cons) | 14.81751 3.124226 9.801716 22.40002
-----------------------------+------------------------------------------------
var(Residual) | 4.383264 .3163348 3.805112 5.04926
------------------------------------------------------------------------------
* now with -sem-
reshape wide weight, i(id) j(week)
sem (Intercept@1 Slope@0 -> weight1) ///
(Intercept@1 Slope@1 -> weight2) ///
(Intercept@1 Slope@2 -> weight3) ///
(Intercept@1 Slope@3 -> weight4) ///
(Intercept@1 Slope@4 -> weight5) ///
(Intercept@1 Slope@5 -> weight6) ///
(Intercept@1 Slope@6 -> weight7) ///
(Intercept@1 Slope@7 -> weight8) ///
(Intercept@1 Slope@8 -> weight9) ///
,nocons means(Intercept Slope) ///
cov(Slope@0 e.weight1@fix1 e.weight2@fix1 e.weight3@fix1
e.weight4@fix1 ///
e.weight5@fix1 e.weight6@fix1 e.weight7@fix1 e.weight8@fix1
e.weight9@fix1 ///
Intercept*Slope@0)
--------------+----------------------------------------------------------------
Mean |
Intercept | 25.56551 .585829 43.64 0.000 24.41731 26.71371
Slope | 6.209896 .0390124 159.18 0.000 6.133433 6.286359
--------------+----------------------------------------------------------------
Variance |
e.weight1 | 4.383264 .3163348 3.805112 5.049261
e.weight2 | 4.383264 .3163348 3.805112 5.049261
e.weight3 | 4.383264 .3163348 3.805112 5.049261
e.weight4 | 4.383264 .3163348 3.805112 5.049261
e.weight5 | 4.383264 .3163348 3.805112 5.049261
e.weight6 | 4.383264 .3163348 3.805112 5.049261
e.weight7 | 4.383264 .3163348 3.805112 5.049261
e.weight8 | 4.383264 .3163348 3.805112 5.049261
e.weight9 | 4.383264 .3163348 3.805112 5.049261
Intercept | 14.81749 3.12422 9.801705 22.39999
Slope | 0 (constrained)
-------------------------------------------------------------------------------
* so. -xtmixed- and -sem- agree 100% with a random intercepts model
*now with random slopes....
use http://www.stata-press.com/data/r12/pig.dta, clear
xtmixed weight week ||id:week , var
------------------------------------------------------------------------------
weight | Coef. Std. Err. z P>|z| [95% Conf.
Interval]
-------------+----------------------------------------------------------------
week | 6.209896 .0906819 68.48 0.000 6.032163 6.387629
_cons | 19.35561 .3979159 48.64 0.000 18.57571 20.13551
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf.
Interval]
-----------------------------+------------------------------------------------
id: Independent |
var(week) | .3680668 .0801181 .2402389 .5639103
var(_cons) | 6.756364 1.543503 4.317721 10.57235
-----------------------------+------------------------------------------------
var(Residual) | 1.598811 .1233988 1.374358 1.85992
------------------------------------------------------------------------------
* now with -sem-
sem (Intercept@1 Slope@0 -> weight1) ///
(Intercept@1 Slope@1 -> weight2) ///
(Intercept@1 Slope@2 -> weight3) ///
(Intercept@1 Slope@3 -> weight4) ///
(Intercept@1 Slope@4 -> weight5) ///
(Intercept@1 Slope@5 -> weight6) ///
(Intercept@1 Slope@6 -> weight7) ///
(Intercept@1 Slope@7 -> weight8) ///
(Intercept@1 Slope@8 -> weight9) ///
,nocons means(Intercept Slope) ///
cov(e.weight1@fix1 e.weight2@fix1 e.weight3@fix1 e.weight4@fix1 ///
e.weight5@fix1 e.weight6@fix1 e.weight7@fix1 e.weight8@fix1
e.weight9@fix1 ///
Intercept*Slope@0)
--------------+----------------------------------------------------------------
Mean |
Intercept | 25.56551 .4016519 63.65 0.000 24.77829 26.35273
Slope | 6.209896 .0919231 67.56 0.000 6.02973 6.390062
--------------+----------------------------------------------------------------
Variance |
e.weight1 | 1.592648 .1225009 1.369772 1.851787
e.weight2 | 1.592648 .1225009 1.369772 1.851787
e.weight3 | 1.592648 .1225009 1.369772 1.851787
e.weight4 | 1.592648 .1225009 1.369772 1.851787
e.weight5 | 1.592648 .1225009 1.369772 1.851787
e.weight6 | 1.592648 .1225009 1.369772 1.851787
e.weight7 | 1.592648 .1225009 1.369772 1.851787
e.weight8 | 1.592648 .1225009 1.369772 1.851787
e.weight9 | 1.592648 .1225009 1.369772 1.851787
Intercept | 7.141896 1.575664 4.634674 11.00545
Slope | .3790488 .0825236 .2473877 .5807808
--------------+----------------------------------------------------------------
Covariance |
Intercept |
Slope | 0 (constrained)
-------------------------------------------------------------------------------
* The random effects are different.
* now with correlated random effects
------------------------------------------------------------------------------
weight | Coef. Std. Err. z P>|z| [95% Conf.
Interval]
-------------+----------------------------------------------------------------
week | 6.209896 .0910745 68.18 0.000 6.031393 6.388399
_cons | 19.35561 .3996387 48.43 0.000 18.57234 20.13889
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf.
Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
var(week) | .3715251 .0812958 .2419532 .570486
var(_cons) | 6.823363 1.566194 4.351297 10.69986
cov(week,_cons) | -.0984378 .2545767 -.5973991 .4005234
-----------------------------+------------------------------------------------
var(Residual) | 1.596829 .123198 1.372735 1.857505
------------------------------------------------------------------------------
* and with -sem-
reshape wide weight, i(id) j(week)
sem (Intercept@1 Slope@0 -> weight1) ///
(Intercept@1 Slope@1 -> weight2) ///
(Intercept@1 Slope@2 -> weight3) ///
(Intercept@1 Slope@3 -> weight4) ///
(Intercept@1 Slope@4 -> weight5) ///
(Intercept@1 Slope@5 -> weight6) ///
(Intercept@1 Slope@6 -> weight7) ///
(Intercept@1 Slope@7 -> weight8) ///
(Intercept@1 Slope@8 -> weight9) ///
,nocons means(Intercept Slope) ///
cov(e.weight1@fix1 e.weight2@fix1 e.weight3@fix1 e.weight4@fix1 ///
e.weight5@fix1 e.weight6@fix1 e.weight7@fix1 e.weight8@fix1
e.weight9@fix1)
--------------+----------------------------------------------------------------
Mean |
Intercept | 25.56551 .3979436 64.24 0.000 24.78555 26.34546
Slope | 6.209896 .0910744 68.18 0.000 6.031393 6.388398
--------------+----------------------------------------------------------------
Variance |
e.weight1 | 1.59683 .1231981 1.372736 1.857506
e.weight2 | 1.59683 .1231981 1.372736 1.857506
e.weight3 | 1.59683 .1231981 1.372736 1.857506
e.weight4 | 1.59683 .1231981 1.372736 1.857506
e.weight5 | 1.59683 .1231981 1.372736 1.857506
e.weight6 | 1.59683 .1231981 1.372736 1.857506
e.weight7 | 1.59683 .1231981 1.372736 1.857506
e.weight8 | 1.59683 .1231981 1.372736 1.857506
e.weight9 | 1.59683 .1231981 1.372736 1.857506
Intercept | 6.997989 1.55229 4.530646 10.80902
Slope | .3715242 .0812954 .2419528 .5704842
--------------+----------------------------------------------------------------
Covariance |
Intercept |
Slope | .2730897 .2523778 1.08 0.279 -.2215616
.767741
-------------------------------------------------------------------------------
* here the difference is even more marked - different signs for the
covariance of the intercept and slope. While not statistically
significant in this example, in the data I am working with there is
significant positive covariance: 0.38 (0.26 - 0.51) from -sem- but a
negative significant covariance from -xtmixed- : -0.18 (-0.32 -0.04)
What am I missing here ?
Many thanks
Robert Long
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