Chapter 8 Advanced Multivariate Longitudinal SEM designs
Thus far our longitudinal SEM designs have mostly dealt with single growth curves. These models assume and underlying trajectory that is unitary across time. In other words, the trajectory may not be linear but it is thought to be resulting from a singular process. What is somewhat unsatisfactory about this is that change in constructs may happen at a different timescale. With growth curves the time scale of change is the entire period.
An alternative approach is to look at repeated processes of change, thereby getting at more complex theories that can look at a) what construct changes prior to another b) whether changes in a construct occur for the same reasons at different times and c) whether or not the dynamics of change among multiple constructs are similar across time. These more advanced questions necessarily involve breaking the change process down into smaller change quantities. Instead of one growth curve, there may be multiple change (difference) scores. Or there may be multiple regressions associating one construct to another.
How are these different than the cross-lagged panel model that Josh said was mostly worthless prior in the semester? Well they still aren’t great, and that is because it is difficult to address chicken and egg questions that the cross-lag panel model gets used for. However, they do attempt to overcome some of the limitations of those models. Moreover, some of the questions that they attempt to answer side step the difficulties of cross-lagged panel models.
8.0.1 Cross-lagged panel model
As an aside, what are the difficulties of cross-lagged panel models? To list a few:
- Arbitrary starting point can change association.
- Time between lags can influence results because changes may not be aligned with assessment. Think of a wheel rolling and measuring the location of a point at at every quarter rotation compared to 2 rotations.
- Different constructs may change at different rates, influencing results.
- Theoretically, the model suggest that one point in time influences change in some other construct. Why would Tuesday at 2 be so important, why not Thursday at 4?
- typically need to constrain cross loadings to be the same to ensure quality estimates. But doing so makes it impossible to see if the association differ across time.
8.1 Difference scores
The stepping stone to these more advanced models that move beyond cross lagged panel models are built on the humble difference score.
A long time ago I said that these are not great. Indeed, there is a great history about how difference scores are problematic. This changed the way that experimental psychology analyzed and conducted their experiments in the 60s and early 70s. Cronbach of the alpha fame had a wonderful missive about the lack of utility of difference scores.
He is right, every difference scores contain measurement error that cannot be separated from true change. In place, people have regressed t2 on t1 and used the residual for a residual change score. However, there are problems such as a loss of information where 2 variables become 1. Low reliability. And there might be a different factor structure at time 2 from that at time 1.
What to do? Don’t use difference scores, use something else. Maybe latently?
8.2 Analyzing change with two time points
This occurs a lot. Pre-post design. Only have enough money to follow up participants once. Getting an itchy publishing-finger while waiting for your longitudinal study to mature.
There are a few options (See McArdle Annual Review of Psych for more details).
The most basic (and most extendable) is really similar to a growth model. To understand where it is coming from:
(delta T) = T2 - T1 delta T is going to be a latent variable, created by factor loading of 1 from T1 -> T2, meaning that delta T is thought of as an error variance term; what is left over after saying T1 perfectly causes T2. Can now relate that to T1, something the difference score and residual change score models could not do simultaneously.
## T1X1 T1X2 T1X3 T2X1 T2X2 T2X3
## 4.820205 5.470553 4.191088 5.483167 6.165787 4.832366change <-
'C_T1=~1*T1X1 + L2*T1X2 + L3*T1X3
C_T2=~1*T2X1 + L2*T2X2 + L3*T2X3
C_T2 ~ 1*C_T1 # Fixed regression of C_T2 on C_T1
dC =~ 1*C_T2 # Fixed regression of dC on C_T2
C_T2 ~ 0*1 # This line constrains the intercept of C_T2 to 0
C_T2 ~~ 0*C_T2 # This fixes the variance of the C_T2 to 0
dC ~ 1 # This estimates the intercept of the change score
C_T1 ~ 1 # This estimates the intercept of C_T1
dC ~~ dC # This estimates the variance of the change scores
C_T1 ~~ C_T1 # This estimates the variance of the COG_T1
dC ~ C_T1 # This estimates the self-feedback parameter, which for now is just the covaration with change and T1
T1X1~~T2X1 # This allows residual covariance on indicator X1 across T1 and T2
T1X2~~T2X2 # This allows residual covariance on indicator X2 across T1 and T2
T1X3~~T2X3 # This allows residual covariance on indicator X3 across T1 and T2
T1X1~~T1X1 # This allows residual variance on indicator X1
T1X2~~T1X2 # This allows residual variance on indicator X2
T1X3~~T1X3 # This allows residual variance on indicator X3
T2X1~~T2X1 # This allows residual variance on indicator X1 at T2
T2X2~~T2X2 # This allows residual variance on indicator X2 at T2
T2X3~~T2X3 # This allows residual variance on indicator X3 at T2
T1X1~0*1 # This constrains the intercept of X1 to 0 at T1
T1X2~M2*1 # This estimates the intercept of X2 at T1
T1X3~M3*1 # This estimates the intercept of X3 at T1
T2X1~0*1 # This constrains the intercept of X1 to 0 at T2
T2X2~M2*1 # This estimates the intercept of X2 at T2
T2X3~M3*1 # This estimates the intercept of X3 at T2
'
change.fit <- sem(change, data=simdat, missing="ML")
summary(change.fit, fit.measures=TRUE, standardized=TRUE, rsquare=TRUE)## lavaan (0.5-23.1097) converged normally after 56 iterations
##
## Number of observations 500
##
## Number of missing patterns 1
##
## Estimator ML
## Minimum Function Test Statistic 5.327
## Degrees of freedom 9
## P-value (Chi-square) 0.805
##
## Model test baseline model:
##
## Minimum Function Test Statistic 1003.549
## Degrees of freedom 15
## P-value 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 1.000
## Tucker-Lewis Index (TLI) 1.006
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -4826.130
## Loglikelihood unrestricted model (H1) -4823.467
##
## Number of free parameters 18
## Akaike (AIC) 9688.260
## Bayesian (BIC) 9764.123
## Sample-size adjusted Bayesian (BIC) 9706.990
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000
## 90 Percent Confidence Interval 0.000 0.032
## P-value RMSEA <= 0.05 0.994
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.014
##
## Parameter Estimates:
##
## Information Observed
## Standard Errors Standard
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## C_T1 =~
## T1X1 1.000 1.029 0.729
## T1X2 (L2) 1.022 0.048 21.300 0.000 1.051 0.721
## T1X3 (L3) 0.819 0.041 19.775 0.000 0.842 0.638
## C_T2 =~
## T2X1 1.000 1.137 0.756
## T2X2 (L2) 1.022 0.048 21.300 0.000 1.161 0.780
## T2X3 (L3) 0.819 0.041 19.775 0.000 0.931 0.665
## dC =~
## C_T2 1.000 0.521 0.521
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## C_T2 ~
## C_T1 1.000 0.905 0.905
## dC ~
## C_T1 -0.055 0.059 -0.939 0.348 -0.096 -0.096
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .T1X1 ~~
## .T2X1 -0.123 0.060 -2.035 0.042 -0.123 -0.129
## .T1X2 ~~
## .T2X2 -0.098 0.061 -1.602 0.109 -0.098 -0.104
## .T1X3 ~~
## .T2X3 -0.065 0.056 -1.157 0.247 -0.065 -0.061
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .C_T2 0.000 0.000 0.000
## .dC 0.963 0.285 3.376 0.001 1.625 1.625
## C_T1 4.803 0.058 82.284 0.000 4.669 4.669
## .T1X1 0.000 0.000 0.000
## .T1X2 (M2) 0.555 0.251 2.210 0.027 0.555 0.381
## .T1X3 (M3) 0.292 0.217 1.350 0.177 0.292 0.222
## .T2X1 0.000 0.000 0.000
## .T2X2 (M2) 0.555 0.251 2.210 0.027 0.555 0.373
## .T2X3 (M3) 0.292 0.217 1.350 0.177 0.292 0.209
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .C_T2 0.000 0.000 0.000
## .dC 0.348 0.079 4.375 0.000 0.991 0.991
## C_T1 1.058 0.105 10.068 0.000 1.000 1.000
## .T1X1 0.934 0.081 11.535 0.000 0.934 0.469
## .T1X2 1.017 0.088 11.537 0.000 1.017 0.480
## .T1X3 1.033 0.077 13.340 0.000 1.033 0.593
## .T2X1 0.968 0.087 11.170 0.000 0.968 0.428
## .T2X2 0.865 0.082 10.539 0.000 0.865 0.391
## .T2X3 1.095 0.082 13.402 0.000 1.095 0.558
##
## R-Square:
## Estimate
## C_T2 1.000
## dC 0.009
## T1X1 0.531
## T1X2 0.520
## T1X3 0.407
## T2X1 0.572
## T2X2 0.609
## T2X3 0.442This gives you everything you need for two time points. How would you introduce a predictor variable? Multiple group model? Bi-variate latent change model?
8.3 multi wave latent change model
The real benefit of this type of framework is building on this type of change model to incorporate multiple change models simultanously. This allows one to look at multiple change processes in a single model, opening the door to analyzing more dynamic processes. One can fit two latent change scores (LCSs) with 3 repeated assessments, 3LCSs with 4 repeated measures, etcetera.
This is similar to the Steyer Model’s that Leah will present.
One top of this type of model you can fit a growth curve. Why? Can capture both general trends and more fine-grained temporal dynamics. This can be especially useful when trying to separate a known, more stable change occurring during development (e.g.natural agining process) from more specific fluctuations due to external factors (e.g., life events).
## COG_T1 COG_T2 COG_T3 COG_T4 NEU_T1 NEU_T2 NEU_T3
## 2.122489 4.942527 9.222710 14.925048 1.894173 4.573540 8.417871
## NEU_T4
## 13.275938BDCS<-'
COGlv_T1=~1*COG_T1 # Defining the COG latent variables
COGlv_T2=~1*COG_T2 # Defining the COG latent variables
COGlv_T3=~1*COG_T3 # Defining the COG latent variables
COGlv_T4=~1*COG_T4 # Defining the COG latent variables
NEUlv_T1=~1*NEU_T1 # Defining the NEU latent variables
NEUlv_T2=~1*NEU_T2 # Defining the NEU latent variables
NEUlv_T3=~1*NEU_T3 # Defining the NEU latent variables
NEUlv_T4=~1*NEU_T4 # Defining the NEU latent variables
COGlv_T2 ~ 1*COGlv_T1 # This parameter regresses COG_T2 perfectly on COG_T1
COGlv_T3 ~ 1*COGlv_T2 # This parameter regresses COG_T3 perfectly on COG_T2
COGlv_T4 ~ 1*COGlv_T3 # This parameter regresses COG_T4 perfectly on COG_T3
NEUlv_T2 ~ 1*NEUlv_T1 # This parameter regresses NEU_T2 perfectly on NEU_T1
NEUlv_T3 ~ 1*NEUlv_T2 # This parameter regresses NEU_T3 perfectly on NEU_T2
NEUlv_T4 ~ 1*NEUlv_T3 # This parameter regresses NEU_T4 perfectly on NEU_T3
dCOG1 =~ 1*COGlv_T2 # This defines the change score as measured perfectly by scores on COG_T2
dCOG2 =~ 1*COGlv_T3 # This defines the change score as measured perfectly by scores on COG_T3
dCOG3 =~ 1*COGlv_T4 # This defines the change score as measured perfectly by scores on COG_T4
dNEU1 =~ 1*NEUlv_T2 # This defines the change score as measured perfectly by scores on NEU_T2
dNEU2 =~ 1*NEUlv_T3 # This defines the change score as measured perfectly by scores on NEU_T3
dNEU3 =~ 1*NEUlv_T4 # This defines the change score as measured perfectly by scores on NEU_T4
COG_T1~~COG_T1 # This estimates the COG residual variances
COG_T2~~COG_T2 # This estimates the COG residual variances
COG_T3~~COG_T3 # This estimates the COG residual variances
COG_T4~~COG_T4 # This estimates the COG residual variances
NEU_T1~~NEU_T1 # This estimates the NEU residual variances
NEU_T2~~NEU_T2 # This estimates the NEU residual variances
NEU_T3~~NEU_T3 # This estimates the NEU residual variances
NEU_T4~~NEU_T4 # This estimates the NEU residual variances
#Dynamics
dNEU1~B1*NEUlv_T1 # This estimates the NEU self-feedback parameter
dNEU2~B1*NEUlv_T2 # This estimates the NEU self-feedback parameter
dNEU3~B1*NEUlv_T3 # This estimates the NEU self-feedback parameter
dCOG1~B2*COGlv_T1 # This estimates the COG self-feedback parameter
dCOG2~B2*COGlv_T2 # This estimates the COG self-feedback parameter
dCOG3~B2*COGlv_T3 # This estimates the COG self-feedback parameter
dNEU1~C1*COGlv_T1 # This estimates the COG to NEU coupling parameter
dNEU2~C1*COGlv_T2 # This estimates the COG to NEU coupling parameter
dNEU3~C1*COGlv_T3 # This estimates the COG to NEU coupling parameter
dCOG1~C2*NEUlv_T1 # This estimates the NEU to COG coupling parameter
dCOG2~C2*NEUlv_T2 # This estimates the NEU to COG coupling parameter
dCOG3~C2*NEUlv_T3 # This estimates the NEU to COG coupling parameter
iCOG=~1*COGlv_T1 # This defines the COG intercept measurement model
sCOG=~1*dCOG1+1*dCOG2+1*dCOG3 # This defines the COG slope measurement model
iCOG~1 # This estimates the COG intercept intercept (mean)
iCOG~~iCOG # This estimates the COG intercept variance
sCOG~1 # This estimates the COG slope intercept
sCOG~~sCOG # This estimates the COG slope variance
iNEU=~1*NEUlv_T1 # This defines the NEU slope measurement model
sNEU=~1*dNEU1+1*dNEU2+1*dNEU3 # This defines the NEU slope measurement model
iNEU~1 # This estimates the NEU intercept intercept (mean)
iNEU~~iNEU # This estimates the NEU intercept variance
sNEU~1 # This estimates the NEU slope intercept
sNEU~~sNEU # This estimates the NEU slope variance
iNEU~~sNEU # This estimates the iNEU sNEU covariance
iNEU~~sCOG # This estimates the iNEU sCOG covariance
iNEU~~iCOG # This estimates the iNEU iCOG covariance
iCOG~~sCOG # This estimates the iCOG sCOG covariance
iCOG~~sNEU # This estimates the iCOG sNEU covariance
sCOG~~sNEU # This estimates the sCOG sNEU covariance
'
fitBDCS <- lavaan(BDCS, data=simdatBD, missing="ML")## Warning in lav_object_post_check(object): lavaan WARNING: covariance matrix of latent variables
## is not positive definite;
## use inspect(fit,"cov.lv") to investigate.summary(fitBDCS, fit.measures=TRUE, standardized=TRUE) ## lavaan (0.5-23.1097) converged normally after 956 iterations
##
## Number of observations 500
##
## Number of missing patterns 1
##
## Estimator ML
## Minimum Function Test Statistic 203.204
## Degrees of freedom 18
## P-value (Chi-square) 0.000
##
## Model test baseline model:
##
## Minimum Function Test Statistic 3145.600
## Degrees of freedom 28
## P-value 0.000
##
## User model versus baseline model:
##
## Comparative Fit Index (CFI) 0.941
## Tucker-Lewis Index (TLI) 0.908
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) -10755.946
## Loglikelihood unrestricted model (H1) -10654.344
##
## Number of free parameters 26
## Akaike (AIC) 21563.891
## Bayesian (BIC) 21673.471
## Sample-size adjusted Bayesian (BIC) 21590.946
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.143
## 90 Percent Confidence Interval 0.126 0.162
## P-value RMSEA <= 0.05 0.000
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.068## Warning in sqrt(ETA2): NaNs produced## Warning in sqrt(ETA2): NaNs produced
## Warning in sqrt(ETA2): NaNs produced##
## Parameter Estimates:
##
## Information Observed
## Standard Errors Standard
##
## Latent Variables:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## COGlv_T1 =~
## COG_T1 1.000 1.900 0.606
## COGlv_T2 =~
## COG_T2 1.000 3.246 0.805
## COGlv_T3 =~
## COG_T3 1.000 5.499 0.847
## COGlv_T4 =~
## COG_T4 1.000 7.362 0.726
## NEUlv_T1 =~
## NEU_T1 1.000 1.503 0.503
## NEUlv_T2 =~
## NEU_T2 1.000 3.089 0.772
## NEUlv_T3 =~
## NEU_T3 1.000 5.431 0.908
## NEUlv_T4 =~
## NEU_T4 1.000 8.640 0.953
## dCOG1 =~
## COGlv_T2 1.000 0.608 0.608
## dCOG2 =~
## COGlv_T3 1.000 0.424 0.424
## dCOG3 =~
## COGlv_T4 1.000 NaN NaN
## dNEU1 =~
## NEUlv_T2 1.000 0.582 0.582
## dNEU2 =~
## NEUlv_T3 1.000 0.444 0.444
## dNEU3 =~
## NEUlv_T4 1.000 0.375 0.375
## iCOG =~
## COGlv_T1 1.000 1.000 1.000
## sCOG =~
## dCOG1 1.000 9.109 9.109
## dCOG2 1.000 7.699 7.699
## dCOG3 1.000 NaN NaN
## iNEU =~
## NEUlv_T1 1.000 1.000 1.000
## sNEU =~
## dNEU1 1.000 0.894 0.894
## dNEU2 1.000 0.666 0.666
## dNEU3 1.000 0.495 0.495
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## COGlv_T2 ~
## COGlv_T1 1.000 0.585 0.585
## COGlv_T3 ~
## COGlv_T2 1.000 0.590 0.590
## COGlv_T4 ~
## COGlv_T3 1.000 0.747 0.747
## NEUlv_T2 ~
## NEUlv_T1 1.000 0.487 0.487
## NEUlv_T3 ~
## NEUlv_T2 1.000 0.569 0.569
## NEUlv_T4 ~
## NEUlv_T3 1.000 0.629 0.629
## dNEU1 ~
## NEUlv_T1 (B1) 0.668 0.074 9.071 0.000 0.559 0.559
## dNEU2 ~
## NEUlv_T2 (B1) 0.668 0.074 9.071 0.000 0.856 0.856
## dNEU3 ~
## NEUlv_T3 (B1) 0.668 0.074 9.071 0.000 1.118 1.118
## dCOG1 ~
## COGlv_T1 (B2) -7.279 0.094 -77.707 0.000 -7.009 -7.009
## dCOG2 ~
## COGlv_T2 (B2) -7.279 0.094 -77.707 0.000 -10.124 -10.124
## dCOG3 ~
## COGlv_T3 (B2) -7.279 0.094 -77.707 0.000 NaN NaN
## dNEU1 ~
## COGlv_T1 (C1) -0.295 0.052 -5.678 0.000 -0.312 -0.312
## dNEU2 ~
## COGlv_T2 (C1) -0.295 0.052 -5.678 0.000 -0.397 -0.397
## dNEU3 ~
## COGlv_T3 (C1) -0.295 0.052 -5.678 0.000 -0.500 -0.500
## dCOG1 ~
## NEUlv_T1 (C2) 8.380 0.273 30.656 0.000 6.387 6.387
## dCOG2 ~
## NEUlv_T2 (C2) 8.380 0.273 30.656 0.000 11.091 11.091
## dCOG3 ~
## NEUlv_T3 (C2) 8.380 0.273 30.656 0.000 NaN NaN
##
## Covariances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## iNEU ~~
## sNEU 0.556 0.211 2.629 0.009 0.230 0.230
## sCOG ~~
## iNEU -15.778 2.365 -6.670 0.000 -0.584 -0.584
## iCOG ~~
## iNEU 0.134 0.014 9.806 0.000 0.047 0.047
## sCOG 26.658 2.449 10.884 0.000 0.781 0.781
## sNEU 2.318 0.247 9.382 0.000 0.760 0.760
## sCOG ~~
## sNEU 14.761 0.100 147.330 0.000 0.511 0.511
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## iCOG 2.042 0.127 16.073 0.000 1.075 1.075
## sCOG 2.200 1.629 1.351 0.177 0.122 0.122
## iNEU 1.874 0.123 15.187 0.000 1.246 1.246
## sNEU 2.108 0.224 9.416 0.000 1.312 1.312
## .COG_T1 0.000 0.000 0.000
## .COG_T2 0.000 0.000 0.000
## .COG_T3 0.000 0.000 0.000
## .COG_T4 0.000 0.000 0.000
## .NEU_T1 0.000 0.000 0.000
## .NEU_T2 0.000 0.000 0.000
## .NEU_T3 0.000 0.000 0.000
## .NEU_T4 0.000 0.000 0.000
## COGlv_T1 0.000 0.000 0.000
## .COGlv_T2 0.000 0.000 0.000
## .COGlv_T3 0.000 0.000 0.000
## .COGlv_T4 0.000 0.000 0.000
## NEUlv_T1 0.000 0.000 0.000
## .NEUlv_T2 0.000 0.000 0.000
## .NEUlv_T3 0.000 0.000 0.000
## .NEUlv_T4 0.000 0.000 0.000
## .dCOG1 0.000 0.000 0.000
## .dCOG2 0.000 0.000 0.000
## .dCOG3 0.000 NaN NaN
## .dNEU1 0.000 0.000 0.000
## .dNEU2 0.000 0.000 0.000
## .dNEU3 0.000 0.000 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .COG_T1 6.216 0.512 12.142 0.000 6.216 0.633
## .COG_T2 5.727 0.503 11.375 0.000 5.727 0.352
## .COG_T3 11.932 0.719 16.602 0.000 11.932 0.283
## .COG_T4 48.543 0.511 94.934 0.000 48.543 0.472
## .NEU_T1 6.672 0.502 13.293 0.000 6.672 0.747
## .NEU_T2 6.466 0.496 13.031 0.000 6.466 0.404
## .NEU_T3 6.307 0.524 12.029 0.000 6.307 0.176
## .NEU_T4 7.462 0.967 7.716 0.000 7.462 0.091
## iCOG 3.608 0.329 10.966 0.000 1.000 1.000
## sCOG 322.867 0.041 7856.080 0.000 1.000 1.000
## iNEU 2.260 0.334 6.765 0.000 1.000 1.000
## sNEU 2.581 0.350 7.377 0.000 1.000 1.000
## COGlv_T1 0.000 0.000 0.000
## .COGlv_T2 0.000 0.000 0.000
## .COGlv_T3 0.000 0.000 0.000
## .COGlv_T4 0.000 0.000 0.000
## NEUlv_T1 0.000 0.000 0.000
## .NEUlv_T2 0.000 0.000 0.000
## .NEUlv_T3 0.000 0.000 0.000
## .NEUlv_T4 0.000 0.000 0.000
## .dCOG1 0.000 0.000 0.000
## .dCOG2 0.000 0.000 0.000
## .dCOG3 0.000 NaN NaN
## .dNEU1 0.000 0.000 0.000
## .dNEU2 0.000 0.000 0.000
## .dNEU3 0.000 0.000 0.0008.4 ALT and ALT-SR
ANother way to build both a growth model and look at more specific time associations.