Another important type of latent variable models are latent growth curve
models. Growth modeling is often used to analyze longitudinal or developmental
data. In this type of data, an outcome measure is measured on several
occasions, and we want to study the change over time. In many cases, the
trajectory over time can be modeled as a simple linear or quadratic curve.
Random effects are used to capture individual differences. The random effects
are conveniently represented by (continuous) latent variables, often called
*growth factors*. In the example below, we use an artifical dataset called
`Demo.growth`

where a score (say, a standardized score on a reading ability
scale) is measured on 4 time points. To fit a linear growth model for these
four time points, we need to specify a model with two latent variables: a
random intercept, and a random slope:

```
# linear growth model with 4 timepoints
# intercept and slope with fixed coefficients
i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4
s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4
```

In this model, we have fixed all the coefficients of the growth
functions. If `i`

and `s`

are the only â€˜latent variablesâ€™ in the
model, we can use the `growth()`

function to fit this model:

```
model <- ' i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4
s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4 '
fit <- growth(model, data=Demo.growth)
summary(fit)
```

```
lavaan 0.6-11 ended normally after 29 iterations
Estimator ML
Optimization method NLMINB
Number of model parameters 9
Number of observations 400
Model Test User Model:
Test statistic 8.069
Degrees of freedom 5
P-value (Chi-square) 0.152
Parameter Estimates:
Standard errors Standard
Information Expected
Information saturated (h1) model Structured
Latent Variables:
Estimate Std.Err z-value P(>|z|)
i =~
t1 1.000
t2 1.000
t3 1.000
t4 1.000
s =~
t1 0.000
t2 1.000
t3 2.000
t4 3.000
Covariances:
Estimate Std.Err z-value P(>|z|)
i ~~
s 0.618 0.071 8.686 0.000
Intercepts:
Estimate Std.Err z-value P(>|z|)
.t1 0.000
.t2 0.000
.t3 0.000
.t4 0.000
i 0.615 0.077 8.007 0.000
s 1.006 0.042 24.076 0.000
Variances:
Estimate Std.Err z-value P(>|z|)
.t1 0.595 0.086 6.944 0.000
.t2 0.676 0.061 11.061 0.000
.t3 0.635 0.072 8.761 0.000
.t4 0.508 0.124 4.090 0.000
i 1.932 0.173 11.194 0.000
s 0.587 0.052 11.336 0.000
```

Technically, the `growth()`

function is almost identical to the `sem()`

function. But a mean structure is automatically assumed, and the observed
intercepts are fixed to zero by default, while the latent variable
intercepts/means are freely estimated. A slightly more complex model adds two
regressors (`x1`

and `x2`

) that influence the latent growth factors. In
addition, a time-varying covariate `c`

that influences the outcome measure at
the four time points has been added to the model. A graphical representation
of this model is presented below.

The complete R code needed to specify and fit this linear growth model with a time-varying covariate is given below:

```
# a linear growth model with a time-varying covariate
model <- '
# intercept and slope with fixed coefficients
i =~ 1*t1 + 1*t2 + 1*t3 + 1*t4
s =~ 0*t1 + 1*t2 + 2*t3 + 3*t4
# regressions
i ~ x1 + x2
s ~ x1 + x2
# time-varying covariates
t1 ~ c1
t2 ~ c2
t3 ~ c3
t4 ~ c4
'
fit <- growth(model, data = Demo.growth)
summary(fit)
```