[slides, data and more SAS details at https://bit.ly/LGM_Mplus_Stata] This
post deals with analyzing many-time-points data, intensive longitudinal (ILD,
some use ecological momentary assessment, EMA (Shiffman, Stone,
& Hufford, 2008), or experience sampling method (Larson &
Csikszentmihalyi, 1983) an example to the right, 28 waves; note: when
analyzing 1 case only, these are labeled time series).
I extend the the excellent crosswalk between 2
less intuitive statistical models of changes for many time points of data posted
by the UCLA folks: the Linear mixed Model (LMM) and the MultiLevel Model
(MLM). I add here a 3rd,
which I didn’t see until trying to replicate the 2 they linked: the Latent
Growth Model (LGM, in structural equation modeling SEM = latent variables
tradition). What I add here then are: SAS+Stata+Mplus code to run the LGM that
replicates the LMM and the MLM analyses and then: set them side-by-side for
direct matching that can allow for more informed interpretation. I also add
some help with interpretation of results, heavily drawn from ch. 4 from the
intensive longitudinal analysis (ILA) ‘bible’ by (Bolger & Laurenceau, 2013).
Several points for ease of
navigation: (1). Follow the subscripts! I am using
generously subscripts, and insist on keeping a dot in there when that
variability dimension has been muted,
either because we averaged across it, or because that variable or parameter
does not vary across it; (2). In
addition to using the original Greek letters in setting up the mathematical
models, I then revert to their simpler labels, as the UCLA folks did too: G00
for e.g. instead of γ00; (3).
I use the same data, publicly
available online (in *.dat format, ready for Mplus, Stata too: this means
is does not have any variable labels or codes in it, it’s raw numbers data); (4). I add the SAS LMM MIXED and LGM CALIS syntaxes,
because SAS is often omitted in such head-to-head comparisons; e.g. in a ‘growth
bible’ book (Grimm, Ram, & Estabrook, 2017) (with
lots of online codes) SAS (nlmixed)
and R codes are used for LMM models, then Mplus and R (but not SAS anymore) for
LGM ones.
Let’s dive in:
(A). The biggest barrier in easily analyzing data with many
time points (>5, e.g.) with these models is the naming of things, and then the Greek letters, and then (non)use of
subscripts.
(B). To interpret consistently across models, and obtain the
same results, one needs to use a time
setting that is common in LGM, but not otherwise: first time point is 0,
last one 1, and each wave is set to the % of time it passed from 0, for 4 waves
this comes out as 0, .333, .666, 1 (see ch. 4 in (Bolger & Laurenceau, 2013)).
The biggest challenge
in walking across the 3 models is the language/labeling/meanings. My insights
are captured in the table below; also, Mplus/LGM/SEM talks about the ‘random
coefficient for the time slope’ in terms of a ‘latent slope varying across
cases’ instead, although equation-wise we end up in the same spot.
Symbol
|
General meaning
|
LGM parameter meaning
|
MLM parameter meaning
|
LMM parameter meaning
|
|
LEVEL 1
|
|
|
|
S^2
|
Y residual variance
|
σi2(Yit) Y0-YT S^2 Residual variances (set @=)
|
Residual Y variance
|
Residual Y variance
|
|
LEVEL 1 &
2
|
|
Group by time
|
Group by time
|
G01
|
Ai.
effect
on Y intercept
|
Ai.
-> ‘LGM Y Intercept’ beta
|
L2 Ai.
->Y effect
|
Ai.
-> Y
effect
|
G11
|
Ai.
effect
on time slope
|
Ai.
->
‘LGM Y Slope’ beta
|
L2 Ai.
->Slopes effect
|
Ai.
* Time
interaction effect
|
|
LEVEL 2
|
|
|
|
G00
|
Intercept (Y level at time 0)
|
αI.. Intercept/mean of Y
|
L2 Y Intercept
|
Y Intercept (when all X’s are 0)
|
G10
|
Time ‘effect’
|
αS.. Intercept/mean of LGM slope
|
L2 Intercept of Y slopes
|
Time effect = time slope
|
Tau11
|
Variance of time slope
|
σ2(uIi.) Residual variance of LGM Y Slope
|
L2 Slopes residual variance
|
Time slopes variance
|
Tau00
|
Variance of Y intercept
|
σ2(uSi.) Residual variance of LGM Y Intercept
|
L2 Slopes residual variance
|
Y Intercepts variance
|
Tau01
|
Covariance time slope* Y intercept
|
σ(uIi.,uSi.) Covariance LGM Y Intercept*LGM Y Slope
|
L2 slopes*intercepts covariance
|
Time slopes with Y intercepts covariance
|
G20
|
Effect of X on Y
|
Contemporaneous effects set equal Xi= -> Yi=
|
Within level Xit -> Yit effect
|
Xit
-> Yit effect
|
Formally:
1. LMM and MLM setup is:
Level-1 Model
Y =
B0 + B1*(TIME) + B2*(X) + E
Level-2 Model
B0
= G00 + G01*(A) + U0
B1
= G10 + G11*(A) + U1
B2 = G20
Which can be ‘combined’ into 1 equation by
bringing B0, B1, and B2 from level-1 into the level-1 equation (which is what
makes the LMM harder to intuitively decipher!)”:
Y = G00 + G20*(X) + G01*A + G10*TIME + G11*A*TIME
+ [U0 + U1*TIME + e]
+ with some additional parameters that appear now at level-2:
Tau00 = sigma^2(U0,U0); Tau11 =
sigma^2(U1,U1); Tau01 = sigma^2(U0,U1)
or with Greek and subscripts:
yit
= β0i. + β1i.∙Time.t + β2i.∙Xit
+ 1∙εit
Level-2 Model
β0i.
= γ00 + γ01*Ai. + u0i.
β1i.
= γ10 + γ11*Ai. + u1i.
β2i. = γ20
τ00 = σ2(u0i.)i ; τ11 = σ2(u1i.)i ; τ01 = σ(u0i.,
u1i.)i
(I add a ‘redundant’ last subscript outside () to
indicate variability dimension that gives rise to the co/variance)
2. The LGM setup is:
Level-1 Model
Y = 1*B0 + B1*(TIME.) + E (time
here are fixed ‘loading’ numbers; can also be freed)
or with Greek and subscripts:
yit = 1∙ηYIntercept,i. + ηYSlope,i ∙ λSlope,.t + βYX,..Xit + 1∙εYit [βYX..
forced equal across time and persons!]
Level-2 Model
B0 = G00 + G01*(A) + U0
B1
= G10 + G11*(A) + U1
B2 = G20
or with Greek and subscripts:
ηYIntercept, i. = αIntY,.. + βAInt,..Ai.
+ uYIi. ~
B0 = G00 + G01*(A) + U0
ηYLSlope, i.
= αSloY,.. + βASlo,..Ai.
+ uYSi. ~
B1 = G10 + G11*(A) + U1
βYX,.. = βYX,.. ~
B2 = G20
The major insights gained are (this comes through in the
table of meanings above too):
(1). G00 or γ00 is LGM’s αIntY,..
(2). G10 or γ10 is LGM’s αSloY,..
(3). G01 or γ01 is LGM’s βAInt,..
(4). G11 or γ11 is LGM’s βASlo,..
(5) G20 or γ20 is LGM’s βYX,..
(6) Tau00 or τ00 is LGM’s σ2(ηYIntercept,
i.)i
(7) Tau11 or τ11 is LGM’s σ2(ηYLSlope,
i.)i
(8) Tau01 or τ01 is LGM’s σ(ηYIntercept,
i. ,ηYLSlope, i.)i
- I use A as a time-fixed (non-varying) predictor, A.t is sometimes used for
‘treatment’ (T would not work here, because with t time, T is maximum waves…)
and Xit for a
time-varying predictor (I switched the UCLA notation, sorry). Note that the
gamma, beta, lambda, eta Greek labels don’t make our job of understanding
easier, I would want to have a simpler labeling, common across all traditions;
for now I think the UCLA G00 etc. that comes out of LMM (and MLM) is good
enough to show commonalities between models: this means I will drop the LGM specific
labels here, to make it clear what LGM estimates that matches the LMM/MLM
approaches.
- Note that the ‘intercept/slope language becomes quickly a
problem, unless we complete the phrase’: whose intercept/slope… much like a
regression coefficient β for that matter: βYX, two more pointers, from
what to what: e.g. βYTime,1i.is the time
slope of Y, the outcome (Y and ‘Time’ can be dropped often), and because it
becomes itself a variable at level-2, it will have its own intercept estimated G10
or γ10,
and its own slope (if a predictor of it is modeled)… G11 or γ11,
hence G10/γ10
is the ‘level-2 intercept or conditional mean, of the Y time slope’, while G11/γ11
is the ‘level-2 slope of βYTime’ or ‘the effect
of the time-invariant predictor A on the Y time slope’: yes, 2 mouthfulls!
(C). The SAS Calis option is pretty flexible, see e.g. URL
,and it has several flavors, of which we showcase 2: the lineqs option, which
is an equation-like coding (like y1 = 1*yintLGM + Lambda * ysloLGM + e1), and the path option (like y <--- yintLGM
+ Lambda * ysloLGM)
(D). LGMs differ in Mplus from the Stata and the SAS ones;
among the reasons may be … “Mplus does not consider a growth model to be a
two-level model as in multilevel modeling but a single-level model. With
longitudinal data, the number of levels in Mplus is one less than the number of
levels in conventional multilevel modeling.” (Mplus Guide, p. 113; needs
further investigation). Not sure; even a simple unconditional LGM differs (see
the detailed outputs at https://bit.ly/LMM_MLM_LGM
)
(E). SAS has a ‘proc’ that is underutilized: CALIS. It of
course can run LGMs (as well as LCSs). E.g.: URL1;
URL2;
URL3.
Mplus has a unique feature that
other software needs a different procedure for: it can rename the variables you
have and use in an analysis, on the spot, without changing the data itself.
Meaning, one can recycle the same analysis over and over, which comes in handy
for long code (like LGM, or Latent Change
Score LCS models).
Q&A
You might have some questions, I suspect (so I label them
‘predicted Q…):
pQ&A 1. What if we add auto-regressive (AR) paths in
LGM? Go ahead, go wild! You will then run a Autoregressive latent trajectory
(ALT) (Bollen & Curran, 2004) or LV-ALT (Bianconcini & Bollen, 2018).
pQ&A 2. Can we add nonlinear slopes in this LGM? Yes, of
course, or even let the trajectory be ‘free’, i.e. follow the data (go up and
down as in the data). Just free the Lambda loadings of the latent slope (except
the first @0 and the last @1)
pQ&A 3. Is this ‘new’? Certainly not, many before
pointed to this possibility, but I have not seen a ‘how to’ and side-by-side
crosswalk like this anywhere. Excellent
detailed appendix (with SAS, Mplus and Mx codes) accompanies (Mehta & Neale, 2005); also see the
appendix for (McNeish & Matta, 2018)
pQ&A 5. Can this be made even simpler? Likely yes, using
the graphical instead of equational display/tool: e.g. for the LGM:
pQ&A 6. Is there more to this? Certainly, Tihomir
Asparouhov told me first this is not so such a surprising find (but his brain
resonates at other wavelengths than normal ones), it's just a lond to wide change... and in fact the Mplus team
has recently merged the classical ‘very many time points’ (econometric time series)
tradition with the latent variable (SEM) one recently, under the comfortably large and colorful
umbrella of DSEM. Much more possible here to explore.
pQ&A 7. (I am
asking this, sorry) Isn’t this a good candidate for a ‘teacher’s corner’ paper
for a journal like SEM ?It sure looks like to me (thanks for asking!), however, a bad experience that
got me caught in some ideological disputes that the editors seem to be part of
(regarding the ‘existence’ of potential outcomes in SEM) made me promise that I
won’t knock on that door ever again: so I prefer to lay out things in the open,
both for critiques, and for free open use and enjoyment.
Some
limitations/invitations for more:
(i). I tried for many hours to replicate the simple LMM
models from Stata in SAS: there are many little quirks that made this pretty
hard (and got the dreaded “WARNING: Stopped because of infinite likelihood” and
another like it in response too: if someone knows the easy way to do this,
email me; UCLA has
a page, and URL2,
and URL3,
etc.); eventually Dan
McNeish’s code did it for me!
(ii). The SAS MLM setup is also a next ‘must’: so that SAS
users can see the link in their own preferred output format (the logic and
interpretation should not change): if someone knows a quick way to do this, email
me.
(iii). Can LCS replicate these, and then go wilder beyond?
Most likely, that’s for another post.
References
Bianconcini, S., & Bollen, K. A.
(2018). The Latent Variable-Autoregressive Latent Trajectory Model: A General
Framework for Longitudinal Data Analysis. Structural
Equation Modeling: A Multidisciplinary Journal, 25(5), 791-808.
Bolger, N., &
Laurenceau, J. (2013). Intensive longitudinal methods: an introduction to diary
and experience sampling research. New
York: Guilford Press.
Bollen, K. A.,
& Curran, P. J. (2004). Autoregressive latent trajectory (ALT) models a
synthesis of two traditions. Sociological
Methods & Research, 32(3), 336-383.
Grimm, K. J.,
Ram, N., & Estabrook, R. (2017). Growth
modeling: Structural equation and multilevel modeling approaches: Guilford Publications.
Larson, R., &
Csikszentmihalyi, M. (1983). The experience sampling method. New Directions for Methodology of Social
& Behavioral Science.
McNeish, D.,
& Matta, T. (2018). Differentiating between mixed-effects and latent-curve
approaches to growth modeling. Behavior
Research Methods, 50(4), 1398-1414.
Mehta, P. D.,
& Neale, M. C. (2005). People are variables too: Multilevel structural
equations modeling. Psychological
Methods, 10, 259-284.
Shiffman,
S., Stone, A. A., & Hufford, M. R. (2008). Ecological momentary assessment.
Annu. Rev. Clin. Psychol., 4, 1-32.
Summary of outputs for
illustration of identical results and how to read/what:
LMM (Stata) results:
. xtmixed y x1 a time
timeBYx1 || id: time, cov(un) var mle
y | Coef.
Std. Err. z P>|z|
[95% Conf. Interval]
-------------+----------------------------------------------------------------
G01 x1 |
.611028 .062607 9.76
0.000 .4883205 .7337355
G20 a |
.2978672 .0215842 13.80
0.000 .255563 .3401715
G10 time |
3.217683 .0972144 33.10
0.000 3.027147 3.40822
G11 timeBYx1 | .8891406
.0974282 9.13 0.000
.6981847 1.080096
G00 _cons |
.6647655 .0624683 10.64
0.000 .5423298 .7872011
------------------------------------------------------------------------------
Random-effects Parameters |
Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Unstructured |
Tau11 var(time) | 3.724995
.3006731 3.179938 4.363477
Tau00 var(_cons) | 1.561595
.1239062 1.336685 1.824349
Tau01 cov(time,_cons) | 1.111405
.1396993 .8375996 1.385211
-----------------------------+------------------------------------------------
S^2 var(Residual) | .5415599
.0242323 .4960884 .5911993
------------------------------------------------------------------------------
LR test vs. linear model:
chi2(3) = 2209.97 Prob >
chi2 = 0.0000
Estimate S.E.
Est./S.E. P-Value
Within
Level
Y
ON
G20 A 0.298 0.022
13.352 0.000
Residual Variances
S^2 Y 0.542
0.024 22.149 0.000
Between
Level
S
ON
G11 X1 0.889
0.101 8.836 0.000
Y
ON
G01 X1 0.611 0.063
9.739 0.000
Y
WITH
Tau01 S
1.112 0.145
7.685 0.000
Intercepts
G00 Y 0.665 0.062
10.683 0.000
G10 S 3.218 0.097
33.025 0.000
Residual Variances
Tau00 Y
1.562
0.118 13.202 0.000
Tau11 S 3.725 0.284
13.121 0.000
YINTLGM ON
G01 X1 0.611 0.063
9.760 0.000
YSLOLGM
ON
G11 X1 0.889 0.097
9.126 0.000
YY0-3
ON
G20 AX0-3
0.298 0.022
13.772 0.000
YSLOLGM
WITH
Tau01 YINTLGM 1.111 0.140
7.956 0.000
Intercepts
YY0-3
0.000
0.000 999.000 999.000
G00 YINTLGM
0.665 0.062
10.642 0.000
G11 YSLOLGM
3.218 0.097
33.099 0.000
Residual Variances
S^2 YY0-3
0.542 0.024
22.349 0.000
Tau00
YINTLGM 1.562
0.124 12.603 0.000
Tau11
YSLOLGM 3.725
0.301 12.389 0.000
Summary input/syntax/software
codes for replications (but ZIP with full list of: data, codes, outputs are
posted here https://bit.ly/LMM_MLM_LGM
)
1. SAS
1.1. SAS LMM
* 1. Read data first LONG/Vertical
proc
import out=
ex6long datafile = "P:\ex6.10L.dta";
run;
proc
contents data=ex6long;
run;
/* NOT completely sure about this MIXED model */
proc
mixed data=
ex6long covtest method=reml;
*Method=REML uses restricted ML;
model
y = time | x1 / s ddfm=kr;*ddfm
= KR uses Kenward-Roger correction;
random
int time / subject=id type=vc;
*Uncorrelated random effects for the intercept and slope;
solution adds MANY estimates: what are they?
repeated
/ type=vc subject=id;
*homogeneous diagonal error structure;
run;
1.2. SAS LGM
* 1. Read data first WIDE/Horizontal
proc
import out=
ex6wide datafile = "P:\ex6.10W.dta";
run;
proc
contents data=ex6wide;
run;
title
"Linear Latent Growth Model LGM Model replicating the
Unconditional Linear Mixed Model LMM and the MultiLevel Model MLM";
proc
calis
data= ex6wide method=ml
pall;
lineqs
y1 = 0.
* Intercept + 1*F_yintLGM + 0
* F_ysloLGM + e1,
y2 = 0.
* Intercept + 1*F_yintLGM + 0.3333333
* F_ysloLGM + e2,
y3 = 0.
* Intercept + 1*F_yintLGM + 0.6666666
* F_ysloLGM + e3,
y4 = 0.
* Intercept + 1*F_yintLGM + 1
* F_ysloLGM + e4;
variance
F_yintLGM
F_ysloLGM,
e1-e4;
mean
F_yintLGM
F_ysloLGM;
cov
F_yintLGM
F_ysloLGM;
run;
* OR run like:
title
" LGM replicating the LMM and the MLM";
proc
calis
data= ex6wide method=ml
PSHORT ;
path
y1 <---
F_yintLGM = 1, /*
constrained = 1 */
y1 <--- F_
ysloLGM = 0, /*
constrained = 0 */
y2 <---
F_yintLGM = 1, /*
constrained = 1 */
y2 <--- F_
ysloLGM = 0.3333333, /*
constrained = 0.33333334*/
y3 <---
F_yintLGM = 1, /*
constrained = 1 */
y3 <--- F_
ysloLGM = 0.6666666, /*
constrained = 0.66666669 */
y4 <---
F_yintLGM = 1, /*
constrained = 1 */
y4 <--- F_
ysloLGM = 1; /*
constrained = 1 */
pvar
y1-y4 =
reserr, /* variancee
estimated but equal*/
F_yintLGM =
varint,
F_ysloLGM =
varSlo;
pcov
F_yintLGM F_
ysloLGM ;
mean
y1 = 0,
/* intercept of the 2nd constrained = 0 */
y2 = 0,
/* intercept of the 2nd constrained = 0 */
y3 = 0,
/* intercept of the 2nd constrained = 0 */
y4 = 0;
/* intercept of the 2nd constrained = 0 */
run;
title
"Conditional x1 LGM replicating the LMM and the MLM with
time varying predictor too a1-4";
proc
calis
data= ex6wide method=ml
PSHORT ;
path
y1 <---
F_yintLGM = 1, /*
constrained = 1 */
y1 <---
F_ysloLGM = 0, /*
constrained = 0 */
y1 <--- a1 =
bx,
y2 <---
F_yintLGM = 1, /*
constrained = 1 */
y2 <---
F_ysloLGM = 0.3333333, /*
constrained = 0.33333334*/
y2 <--- a2 =
bx,
y3 <---
F_yintLGM = 1, /*
constrained = 1 */
y3 <--- F_ysloLGM
= 0.6666666, /* constrained =
0.66666669 */
y3 <--- a3 =
bx,
y4 <---
F_yintLGM = 1, /*
constrained = 1 */
y4 <---
F_ysloLGM = 1, /*
constrained = 1 */
y4 <--- a4 =
bx,
F_yintLGM
<--- x1,
F_ysloLGM
<--- x1;
pvar
y1= reserr,
/* variance estimated but equal*/
y2= reserr,
/* variance estimated but equal*/
y3= reserr,
/* variance estimated but equal*/
y4= reserr,
/* variance estimated but equal*/
F_yintLGM =
varint,
F_ysloLGM =
varSlo;
pcov
F_yintLGM F_
ysloLGM ;
mean
y1 = 0,
/* intercept of the 2nd constrained = 0 */
y2 = 0,
/* intercept of the 2nd constrained = 0 */
y3 = 0,
/* intercept of the 2nd constrained = 0 */
y4 = 0,
/* intercept of the 2nd constrained = 0 */
F_yintLGM = mnInt, /* declare them to be
estimated*/
F_ysloLGM =
mnSlo; /* declare them to be estimated*/
run;
SELECTION of output to show identical results:
|
Absolute
Index
|
Fit
Function
|
0.0724
|
|
|
Chi-Square
|
36.1498
|
|
|
Chi-Square
DF
|
25
|
|
|
Pr
> Chi-Square
|
0.0694
|
|
Path
|
Param.
|
Est.
|
SE
|
t Value
|
p
|
|
y1-4
|
<===
|
a4
|
G20
|
0.298
|
0.022
|
13.787
|
<.0001
|
|
F_yintLGM
|
<===
|
x1
|
G01
|
0.611
|
0.063
|
9.750
|
<.0001
|
|
F_ysloLGM
|
<===
|
x1
|
G11
|
0.889
|
0.098
|
9.117
|
<.0001
|
|
Variance Param.s
|
|
Variance
|
Variable
|
Param.
|
Est.
|
SE
|
t Value
|
p
|
|
Error
|
y1-4
|
S^2
|
0.543
|
0.024
|
22.338
|
<.0001
|
|
|
F_yintLGM
|
Tau00
|
1.565
|
0.124
|
12.591
|
<.0001
|
|
|
F_ysloLGM
|
Tau11
|
3.733
|
0.301
|
12.387
|
<.0001
|
|
Means and Intercepts
|
|
Type
|
Variable
|
Param.
|
Est.
|
SE
|
t Value
|
p
|
|
|
F_yintLGM
|
G00
|
0.665
|
0.063
|
10.620
|
<.0001
|
|
|
F_ysloLGM
|
G10
|
3.218
|
0.097
|
33.033
|
<.0001
|
|
Covariances Among Errors
|
|
Error of
|
Param.
|
Est.
|
SE
|
t Value
|
p
|
|
F_ysloLGM
|
_Tau01
|
1.114
|
0.140
|
7.9518
|
<.0001
|
2. Stata
2.1. Stata LMM
xtmixed y x1 a time timeBYx1 || id: time, cov(un) var mle
(yes, that’s all)
2.2. Stata LGM
* one would need to switch
the X and A to mirror the equations above! The original data had X as time-fixed
and A as time invariant: bad choices
* Unconditional Mplus eg
ex6.10W
sem (y1 <- IntY@1
SloY@0 _cons@0)
///
(y2 <- IntY@1
SloY@.33333334 _cons@0) ///
(y3 <- IntY@1
SloY@.66666669 _cons@0) ///
(y4 <- IntY@1 SloY@1 _cons@0)
, latent(IntY SloY) ///
var(e.y1@var e.y2@var
e.y3@var e.y4@var ) /// /*set = means (IntY SloY ) */
cov(IntY*SloY) means (IntY
SloY ) method(mlmv)
* Time-fixed &
time-varying predictor Mplus eg ex6.10W
* Beware: if not declaring _cons in Intercept & Slope line, df's will be larger by 2, and estimates off
sem (y1 <- a1@beta IntY@1 SloY@0 _cons@0)
///
(y2 <- a2@beta IntY@1
SloY@.3333333 _cons@0) ///
(y3 <- a3@beta IntY@1
SloY@.6666666 _cons@0) ///
(y4 <- a4@beta IntY@1
SloY@1 _cons@0)
///
(IntY <- x1 _cons) (SloY <-
x1 _cons), latent(IntY SloY) ///
var(e.y1@var e.y2@var
e.y3@var e.y4@var ) /// /*set = means (IntY SloY ) */
cov(e.IntY*e.SloY)
method(mlmv)
estat gof
3. Mplus
3.1. Mplus MLM
3.2. Mplus LGM
! one would need to switch the X and A to mirror the
equations above! The original data had X as time-fixed and A as time invariant:
bad choices
! time-varying outcome part
yintLGM BY
y0-y3@1;
yintLGM;
[yintLGM];
ysloLGM BY
y0@0
y1@.3333333
y2@.6666666
y3@1;
yintLGM on
x1 ;
ysloLGM on
x1 ;
[ysloLGM];
intLGM WITH
sloLGM; ! because sloLGM ERROR variance was>0 and is so small
yy0-yy3
(v_u);! set equal across time
[yy0-yy3@0]; ! set all intercepts to 0 as in Grimm
! make it like
LMM&MLM:
y0 on a0 (bx);
y1 on a1 (bx);
y2 on a2 (bx);
y3 on a3 (bx);
OUTPUT:
SAMPSTAT standardized tech4 tech1;
