In this chapter we look at a way of fitting a longitudinal growth model that allows for a nonlinear curve that you do not parameterize. This is a useful tool for longitudinal data that shows up a lot in the final projects. That said, this approach only works if you are working with your data in waves.
We will illustrate with the National Youth Survey (NYS) data as described in Raudenbush and Bryk, page 190. This data comes from a survey in which the same students were asked yearly about their acceptance of 9 “deviant” behaviors (such as smoking marijuana, stealing, etc.). We analyze the first 5 years of data, and have ATTIT (attitude towards deviance) and EXPO (“exposure”, based on asking the children how many friends they had who had engaged in each of the “deviant” behaviors). See Chapter 5 for more information on the data.
Our modeling approach has two key ideas. The first is to let each year have its own mean. The second is to then “tilt” our curves to fit each student as best we can.
For the first idea, we make each age a factor, and then fit our model:
M0 = lmer( ATTIT ~ 0 + age_fac + (1|ID), data=nys1 )
arm::display( M0 )lmer(formula = ATTIT ~ 0 + age_fac + (1 | ID), data = nys1)
coef.est coef.se
age_fac11 0.21 0.02
age_fac12 0.23 0.02
age_fac13 0.33 0.02
age_fac14 0.41 0.02
age_fac15 0.45 0.02
Error terms:
Groups Name Std.Dev.
ID (Intercept) 0.19
Residual 0.18
---
number of obs: 1079, groups: ID, 239
AIC = -192.2, DIC = -272
deviance = -239.2 Note how each wave (here age) has its own mean across our coefficients.
We can then plot our population average trajectory:
newdata <- nys1 %>%
dplyr::select( age_fac ) %>%
unique()
newdata$ID = -1
newdata$ATTIT <- predict(M0, newdata=newdata, re.form=NA)
ggplot(newdata, aes(age_fac, ATTIT, group=ID)) +
geom_line() +
geom_point() +
labs(title = "Population average trajectory of attitude towards deviance over time",
x = "Age",
y ="Attitude towards deviance") 
Our model does not allow for individual trajectories for each student, however. We are only allowing for an intercept shift. This is where the trick comes in: we are going to let each student have their own random slope for growth rate, which will “tilt” our curve for each student. We do this by having a random slope on the continuous age variable, even though our fixed effects are on the factor age variable. We center age around the beginning of the study, so our random intercepts correspond to ATTIT at age 11.
nys1$age_c = nys1$age - 11
M1 = lmer( ATTIT ~ 0 + age_fac + (1+age_c|ID), data=nys1,
control = lmerControl(optimizer = 'bobyqa') )
arm::display( M1 )lmer(formula = ATTIT ~ 0 + age_fac + (1 + age_c | ID), data = nys1,
control = lmerControl(optimizer = "bobyqa"))
coef.est coef.se
age_fac11 0.21 0.01
age_fac12 0.24 0.01
age_fac13 0.33 0.02
age_fac14 0.41 0.02
age_fac15 0.45 0.02
Error terms:
Groups Name Std.Dev. Corr
ID (Intercept) 0.12
age_c 0.05 0.47
Residual 0.16
---
number of obs: 1079, groups: ID, 239
AIC = -298.5, DIC = -384
deviance = -350.1 We can then plot individual trajectories to see how this model is working. We first make a set of 20 students to plot:
set.seed( 40440 )
smp <- sample( unique( nys1$ID ), 20 )
smp_dat <- nys1 %>% filter( ID %in% smp ) %>%
complete( ID, age ) %>%
mutate( age_fac = as.factor( age ),
age_c = age - 11 )Each student is five rows of data, sometimes with missing values due to the complete() method from above:
filter( smp_dat, ID == 52 ) %>%
dplyr::select( ID, age, age_fac, age_c, ATTIT )# A tibble: 5 × 5
ID age age_fac age_c ATTIT
<dbl> <dbl> <fct> <dbl> <dbl>
1 52 11 11 0 NA
2 52 12 12 1 0.29
3 52 13 13 2 0.2
4 52 14 14 3 0.44
5 52 15 15 4 0.44We next predict the values for each student and plot those predicted values:
smp_dat$pred <- predict(M1, newdata=smp_dat)
# Make a population reference curve
newdata$age = 11:15
newdata$age_c = 0:4
newdata$pred <- predict(M1, newdata=newdata, re.form=NA)
ggplot(smp_dat, aes(age, pred)) +
geom_line( aes( group=ID ), alpha=0.5) +
labs(title = "Individual trajectories",
x = "Age",
y ="Attitude towards deviance") +
geom_line( data = newdata, col="red", linewidth=1 )
Note how all our students have the same shape of our overall trajectory, but some are slightly steeper and some more shallow. This allows us to have a shared shape that we can shift (random intercept) and tilt (random slope) to fit the actual data.
Compare our trajectories to the measured values:
newdata$ID = NULL
ggplot(smp_dat, aes(age, ATTIT)) +
facet_wrap( ~ ID ) +
geom_point() +
geom_line( aes( y = pred ), col="blue" ) +
labs(title = "Comparing latent curves to observed values",
x = "Age",
y ="Attitude towards deviance") +
geom_line( data = newdata, col="red", lty=2 )
We can see how the latent curves are trying to get close to the observed data for each student. Our fit seems reasonable, but not perfect.
Hopefully this tool is useful for examining how different waves of data may be different from one another. For example, if COVID happened in the middle of your study, you might expect a big shift in the data. This model allows you to model that shift while still allowing for different individual growth trajectories over time.