# State space specification of Random Intercept CLPM

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Joined: 08/15/2021 - 05:55
State space specification of Random Intercept CLPM

Hi, I'm trying to specify a RI-CLPM (Hamaker et al., 2015; Mulder et al. 2021) as a state-space model, but I don't know how to specify the two random intercepts used to capture stable between-individual differences in change. I've tried different ways of including them but none of them worked. Could you please help me identify which matrices should I modify to include the effect of the random intercepts? I leave my code below. Thank you very much for your attention!

### Drift matrix
Amat <- mxMatrix(name="A", nrow=2, ncol=2, free = c(T,T,T,T),
values = c(.5, 0, 0, .5),
labels = c("a11","a21","a12","a22") )

### Input effects on the latent variables
Bmat <- mxMatrix(name = "B", "Zero", 2, 1)

### Measurement model
Cmat <- mxMatrix(name = "C", "Full", 2, 2, free = FALSE, values = c(1,0,0,1),
dimnames = list(c("Y1", "Y2"), c("y1", "y2")) )

### Input effects on the observed variables
Dmat <- mxMatrix("Zero", 2, 1, name = "D")

### Dynamic error
Qmat <- mxMatrix("Full", 2, 2, name = "Q",
free = c(T,T,T,T),
values = c(.6,0,0,.6),
labels = c("y1der","dercv","dercv","y2der"))

### Measurement error
Rmat <- mxMatrix("Zero", 2, 2, F, name = "R")

### Mean vector
x0mat <- mxMatrix("Zero", name='x0', nrow=2, ncol=1)

### Covariance matrix
P0mat<- mxMatrix(name='P0', nrow=2, ncol=2,
free = TRUE,
values=c(1,0,0,1),
labels=c("y1v","y12cv","y12cv","y2Av"))

### Covariates
Umat <- mxMatrix("Zero", 1, 1, name = "u")

### Specification of the time index
Tmat <- mxMatrix('Full', 1, 1, name='time', labels='data.time')

References:
Hamaker, E. L., Kuiper, R. M., & Grasman, R. P. (2015). A critique of the cross-lagged panel model. Psychological methods, 20(1), 102.
Mulder, J. D., & Hamaker, E. L. (2021). Three extensions of the random intercept cross-lagged panel model. Structural Equation Modeling: A Multidisciplinary Journal, 28(4), 638-648.
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Joined: 07/31/2009 - 15:26
Extend the latent state

The general idea is to add "dummy" latent variables to the latent state vector that do not change over time. In discrete time, these "dummy" variables take the form x_t = 1*x_{t-1} + 0. In continuous time, these "dummy" variables take the form dx/dt = 0*x + 0.

I've attached an R script I have laying around that does some random intercept models.

I've also attached a (slightly reduced form) presentation I gave at M3 around six years ago that shows a bunch of "longitudinal" model are easier to understand as state space models.

Hunter, M.D. (2016). That's a State Space Model too! Talk presented at the Modern Modeling Methods (M3) Conference; Storrs, CT; May 24-25, 2016.

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Joined: 08/15/2021 - 05:55
Solved.

Aha! I understand. I included the random intercepts in the drift matrix as latent states with 0 dynamics and that did the trick. Thanks a lot for the explanation and the examples.