Latent growth mirrors linear mixed and multilevel models

[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 3^rd, 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:

     y_it = β_0i. + β_1i.∙Time._t + β_2i.∙X_it + 1∙ε_it

Level-2 Model

     β_0i. = γ₀₀ + γ₀₁*A_i. + u_0i.

     β_1i. = γ₁₀ + γ₁₁*A_i. + u_1i.  

β_2i. = γ₂₀

τ₀₀ = σ²(u_0i.)_i  ; τ₁₁ = σ²(u_1i.)_i  ; τ₀₁ = σ(u_0i., u_1i.)_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:

y_it = 1∙η_YIntercept,i. + η_YSlope,i ∙ λ_Slope,._t + β_YX,..X_it   +  1∙ε_Yit            [β_YX.. 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:

η_{YIntercept, i}. = α_IntY,.. + β_AInt,..A_i.  + uY_Ii.                                    ~ B0 = G00 + G01*(A) + U0

η_{YLSlope, i}.    = α_SloY,.. + β_ASlo,..A_i.  + uY_Si.                                   ~ B1 = G10 + G11*(A) + U1

β_YX,.. = β_YX,..                                                                          ~ B2 = G20

The major insights gained are (this comes through in the table of meanings above too):

(1). G00 or γ₀₀ is LGM’s  α_IntY,..

(2). G10 or γ₁₀ is LGM’s α_SloY,..

(3). G01 or γ₀₁ is LGM’s β_AInt,..

(4). G11 or γ₁₁ is LGM’s β_ASlo,..

(5) G20 or γ₂₀ is LGM’s β_YX,..  

(6) Tau00 or τ₀₀ is LGM’s σ²(η_{YIntercept, i.})_i 

(7) Tau11 or τ₁₁ is LGM’s σ²(η_{YLSlope, i.})_i  

(8) Tau01 or τ₀₁ is LGM’s σ(η_{YIntercept, i.} ,η_{YLSlope, 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 X_it 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: β_YX, two more pointers, from what to what: e.g. β_YTime,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 β_YTime’ 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 4. Is it really ‘this simple’? Well… the setup for LGM I was lucky to get right before figuring out the equations, there are several settings that one can miss or mess up easily: I got the ‘all Y intercepts set @0’ insight from the online codes accompanying Kevin Grimm, Nilam and Ryne’s  growth modeling ‘bible’ (Grimm et al., 2017) .

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

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            

MLM Mplus

                    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                   

LGM <plus

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;

Covid-19 infections in CT, April 10

I updated the numbers of infections, by town, in CT, from the daily reporting site of DPH. Gender and race/ethnic differences are detailed at https://bit.ly/covid_hds_ct 
This time, one can see a 'dynamic view', or a 'movie', of both Covid-19 infection rates (per 10,000 residents), starting with towns with lowest values (green...) and going up; however, below it one can also see 'where those cases are' in both the infection histogram, AND in the histogram of a second town-level variable, percent non-White residents. At the same time, one can see the values in the data behind these, in a table.

 Note that the infection rate was nonsignificant initially, now it became statistically significantly different from 0 on April 10.
A static' map and 'scatter plot matrix':

More data on Covid-19 infections and deaths in CT, and differences by gender and race/ethnicity are posted at: https://public.tableau.com/profile/coman#!/vizhome/Covid-19infectiondisparitiesinCTuptoApr_10/Dashboard1

Epidemic modeling

I just came across an amazing modeling tool built by Gabriel Goh from UC Davis, a  mathematician; https://gabgoh.github.io/COVID/index.html Pretty flexible and amazing!

Less complicated that that, here is a snapshot of the local Covid-2019 (known) infection cases in CT is shown in a GIF below, with data from the CT Department of Public Health "COVID-19 Update March 31, 2020" done with free software GeoDa and then for GIF/screen capture with free software ShareX. This shows number of cases as of March 31, 2020 by town, per 10,000 residents. 

Mixed emotions

A post from the ISPR Presence Community FB page (by Matt Lombard) brings up an interesting question: what is an emotion (human)? How many are there? Is Longing one of them?

It does not appear in the list by Smith:

But it does, in another from Clore, which shows how emotions may be ‘organizing themselves’…:

Clore, G. L., & Schnall, S. (2005). The influence of affect on attitude. In D. Albarracin, B. T. Johnson, & M. P. Zanna (Eds.), The handbook of attitudes (pp. 437-489). Mahwah, New Jersey: Lawrence Erlbaum Associates.

Shaver, P. R., Wu, S., & Schwartz, J. C. (1992). Cross-cultural similarities and differences in emotion and its representation. In M. S. Clark (Ed.), Emotion. Review of personality and social psvchology (pp. 175-212). Thousand Oaks, CA: Sage.

Smith, H., & Schneider, A. (2009). Critiquing Models of Emotions. Sociological Methods & Research, 37(4), 560-589.

Latent change score model in Mplus

A similar post was done for R, and here's an Mplus code; requests for such simple code come along periodically.
It is for 2 waves only, with 1 indicator latents (and with measurement error estimated for both 'cause' and 'effect' sets of variables: yes, it runs!), and for observed variables only. It was posted on SEMNET, and there is a lot more on this 'change modeling' (another posting on Semnet sends to lots of materials, full texts too).

Try it on your data (no changes needed: just add your own variable names in DEFINE), and let me know what you see in your data, in one of mine I saw the SIGN of the effect flipping when using latents!!!

1. With LVs

  DEFINE:
  X1 = YourXatTime1;
      X2 = YourXatTime2;
  Y1 = YourYatTime1;
      Y2 = YourYatTime2;

  MODEL:
  !X part
  ! first create LV behind each observed indicator:
  ! time 1
  LX1 by X1; ! define new 1 indicator LV
  [X1@0] ; ! set the intercept of the X1 to 0 to identify/define the meal of its latent
  X1 (measerrX);! name the residual error to be set equal across time and estimated
  ! time 2
  LX2 by X2; ! define new 1 indicator LV
  [X2@0] ; ! set the intercept of the X1 to 0 to identify/define the meal of its latent
  X2 (measerrX);! name the residual error to be set equal across time and estimated

  LCS21X by LX2@1; !defined by the 2nd variable
  LX2 @0; !residual variance of 2nd measure @0: note this is true residual, with measurement
  LCS21X ON LX1;  !PROPORTIONAL-GROWTH estimated: can omit it too, by adding ! or adding @0:
  [LX2 @0] ; !intercepts of 2nd measure @0
  LX2 on LX1@1; !autoregressive path set @1

  !Y part
  ! first create LV behind each observed indicator:
  ! time 1
  LY1 by Y1; ! define new 1 indicator LV
  [Y1@0] ; ! set the intercept of the X1 to 0 to identify/define the meal of its latent
  Y1 (measerrY);! name the residual error to be set equal across time and estimated
  ! time 2
  LY2 by Y2; ! define new 1 indicator LV
  [Y2@0] ; ! set the intercept of the X1 to 0 to identify/define the meal of its latent
  Y2 (measerrY);! name the residual error to be set equal across time and estimated

  LCS21Y by LY2@1; !defined by the 2nd variable
  LY2 @0; !residual variance of 2nd measure @0: note this is true residual, with measurement
  LCS21Y ON LY1;  !PROPORTIONAL-GROWTH estimated: can omit it too, by adding ! or adding @0:
  [LY2 @0] ; !intercepts of 2nd measure @0
  LY2 on LY1@1; !autoregressive path set @1

  ! bivariate section

  LCS21Y ON LCS21X; ! CHANGE IN X LEADING TO CHANGE IN Y
  ! COVARIANCES BETWEEN EXOGENOUS VARIABLES ARE ADDED BY DEFAULT: YOU MAY WANT TO DEFLECT TH
  ! LX1 WITH LY1@0 ;
  ! BE PREPARED TO MAKE THESE @0 IT MAY NOT RUN OTHERWISE
  LCS21X @0;
  LCS21Y @0;
   OUTPUT: TECH1 TECH4 stand ;

2. With observed variables:

DEFINE:
X1 = YourXatTime1;
    X2 = YourXatTime2;
Y1 = YourYatTime1;
    Y2 = YourYatTime2;

MODEL:
!X part

LCS21X by X2@1; !defined by the 2nd variable
X2 @0; !residual variance of 2nd measure @0: note this is true residual, with measurement error of X1 partialled out
LCS21X ON X1;  !PROPORTIONAL-GROWTH estimated: can omit it too, by adding ! or adding @0: different models!!!
[X2 @0] ; !intercepts of 2nd measure @0
X2 on X1@1; !autoregressive path set @1

!Y part

LCS21Y by Y2@1; !defined by the 2nd variable
Y2 @0; !residual variance of 2nd measure @0: note this is true residual, with measurement error of X1 partialled out
LCS21Y ON Y1;  !PROPORTIONAL-GROWTH estimated: can omit it too, by adding ! or adding @0: different models!!!
[Y2 @0] ; !intercepts of 2nd measure @0
Y2 on Y1@1; !autoregressive path set @1

! bivariate section

LCS21Y ON LCS21X; ! CHANGE IN X LEADING TO CHANGE IN Y
! COVARIANCES BETWEEN EXOGENOUS VARIABLES ARE ADDED BY DEFAULT: YOU MAY WANT TO DEFLECT THEM INITIALLY
! X1 WITH Y1@0 ;
! BE PREPARED TO MAKE THESE @0 IT MAY NOT RUN OTHERWISE
!LCS21X @0; 
!LCS21Y @0; 
 OUTPUT: TECH1 TECH4 stand ;

The science of cause and effect

An 'introduction' to causality for high schoolers, developed at the invitation of the Connecticut Academy of Science and Engineering, to contribute to their NIE (News in Education) Hartford Courant activities offerings (available on the Hartford Courant site, see http://bit.ly/causalityscience ). As with many recent 'ideas of mine' this flows directly from the wisdom of Judea Pearl (see some recent insights of his on his blog).
Comments and suggestions are welcome!