‘How can we model the form of change in an outcome as time passes by?’, ‘Which statistical technique helps us to describe individual growth trajectory’s over time?’, ‘Can individual differences in an initial state and in change over time be analyzed?’ These questions are of importance to researchers who examine developmental, longitudinal, or consecutive measurements across multiple occasions. What solves their problems is a statistical technique called Latent Growth Curve Modeling (LGCM). The good news is that the recent release of JASP offers the Latent Growth option to use LGCM, within the SEM module! This tutorial guides readers from the concept of LGCM to interpreting results with intuitive examples.
The idea of the Latent Growth Curve Model
The key idea of LGCM is to model change over time in either a linear, quadratic, or another form of trend (Geiser, 2012).
So how does LGCM capture a growth trend? The answer is to use latent variables (also referred to as factors) with fixed factor loadings, and with the repeated measures of interest as indicators. Each of the latent factors represents a component of the growth trajectory. A latent intercept factor with fixed factor loadings of 1 models the so-called initial state, i.e. the baseline level of the scores on the outcome. A linear growth trend can be added by including a latent factor with loadings fixed to 0, 1, 2, 3, and so on. This factor represents how much the outcome changed from the initial state. For the initial state itself, the slope is 0, since at the initial there has obviously no change occurred yet. For the measurement occasions to follow, the slope increases linearly, meaning that the change from the initial state will become evenly larger with each additional measurement occasion. If a quadratic growth trend is to be modeled, factor loadings can for instance be set to 0, 1, 4, and 9 (0², 1², 2², 3²) on a quadratic slope factor. These factors, such as the intercept factor, the linear slope factor, and the quadratic slope factor, are called growth factors because they represent change over time through the estimation of latent variables (Felt, Depaoli, & Tiemensma, 2017).
As an example, let’s consider the GPA dataset from Hox, Moerbeek, & Van de Schoot (2017). Click here to download the dataset. The spreadsheet should show the variable names as column names once you successfully load the dataset.
The GPA dataset is collected from 200 college students. The students’ GPA (1 = lowest, 4 = highest) has been measured across six consecutive semesters (gpa1, gpa2, gpa3, gpa4, gpa5, gpa6). The students’ job status indicates how many hours they worked during the corresponding semesters (job1, job2, job3, job4, job5, job6). Other variables are identification number (student), gender (sex; 0 = male, 1 = female), high school GPA (highgpa; 1 = lowest, 4 = highest), and whether the students have entered the university of their choice (admitted; 0 = no, 1 = yes). Note how the data are structured. Each row contains repeated measurements per student. This type of dataset is called a wide or multivariate format, which is typical for longitudinal data.
Modeling Linear Latent Growth Curve
Let’s say we are interested in modeling the linear growth trend of GPAs across six semesters. For a linear growth curve model, we model successive GPA measurements with an intercept factor and a linear slope factor.
The intercept factor I represents the expected score of individual GPAs at the initial state, where the slope factor has a loading set to 0. All indicators (gpa1 though gpa6) have factor loadings of 1 on the intercept factor. The linear slope factor L, on the other hand, represents individual differences of GPA in the rate of linear change. If we consider the first GPA measurement (gpa1) as the reference time point, the factor loading of gpa1 is fixed to 0. The second measurement (gpa2) has a loading of 1, so the loading of the final measurement (gpa6) becomes 5.
To begin LGCM to model the linear trend, move six GPA measurements (gpa1 through gpa6) into the Variables section.
By default, the first measurement is set as the reference time point. You can check which measurement is the reference by looking at the numbers under the Timings section. We see that gpa1 has the number 0 in the corresponding right blank, hence being the reference time point. Another measurement can be set as a reference depending on how we change the numbering. If you want to set the last measurement (gpa6) as the reference point, the numbers corresponding to gpa1, gpa2, …, gpa5, gpa6 are -5, -4, …, -1, 0. For now, we go with the default setting (i.e., gpa1 as the reference point), and check the estimated parameters.
What matters most in interpreting the results of LGCM is the mean and the variance of the factors. For the intercept factor, the mean is estimated to be 2.598, which represents the average GPA of students at the first measurement. Assuming an alpha level of 0.05 the mean of the intercept factor is significantly different from 0. The variance of the intercept factor, which is estimated to be 0.035 (again statistically significant assuming an alpha of 0.05), represents the individual differences in GPAs at the initial state. There is thus a small, but statistically significant variance between individuals in their initial GPAs.
Next, we interpret the mean and the variance of the linear slope factor. The mean implies the average linear growth rate over time. The estimate of the mean, 0.106, is positive with the p-value lower than 0.001. Therefore, there exists a significant linear increase by 0.106, on average, in GPAs across semesters. The variance, which is estimated to be 0.003 (statistically significant assuming an alpha of 0.05), represents the individual differences in the linear growth trend. That is, the variance shows how individual students differ in their trajectories of GPA across six semesters. This implies that statistically significant individual differences exist in the linear growth of GPA. Given the small estimate, and the scale of the outcome, these differences do, however, not appear to be very large. Taken together, the students differ in terms of both their initial GPAs and the linear growth rates over time. Some students exhibit a stronger increase or a decrease in their GPAs compared to other students.
Another important value is the covariance between the intercept factor and the growth factor. The covariance tells us whether there is any meaningful relationship between GPAs at the reference point and how they change over time. Since the estimate is positive, students who have higher GPAs at the initial point are more likely to have a steeper linear growth trend as the semester passes by than students with lower GPAs at the beginning. However, the covariance does not differ significantly from 0, so it should be interpreted with care.
If we check R-Squared in the Additional Output tab, it is possible to examine how much the growth factors explain the variance of indicators.
Between 30.4% and 90.0% of the observed variance in the repeated measures is explained by the growth factors.
Next, we can assess whether our model fits the data well in general. JASP, by default, provides the model fit from the chi-square test in the output panel.
In the chi-square test of model fit, the fitted model is rejected in favor of the saturated, perfectly fitting model if the p-value is lower than 0.05. This happened with our model! According to the Chi-square Test table, the p-value of the Growth curve model is lower than 0.001. We thus say the model does not fit the data well. To assess the model fit with more fit indices, we request JASP to give us various fit measures. In the control panel, find the Additional Output tab and check Additional Fit Measures.
There are a lot of fit measures that we can refer to. Here, we use the Comparative Fit Index (CFI) and Tucker-Lewis Index (TLI) under the table of Fit indices and Root mean square error of approximation (RMSEA) and Standardized root mean square residual (SRMR) under the table of Other fit measures. The values of CFI and TLI are 0.965 and 0.967, respectively. These values are greater than 0.90, which indicates good model fit. For RMSEA and SRMR, the respective values are 0.093 and 0.098. Because the value of RMSEA is greater than 0.05 and that of SRMR is greater than 0.08, our model has a poor fit to the data. Hence, various fit measures lead to divergent conclusions regarding model fit in the sense that some show good fit while others do not. In this case, researchers should contemplate from a theoretical viewpoint. If their model represents the phenomenon under interest based on a firm theory, they are fine to go with the model.
So far, we have worked with numbers. Some visual aid would enhance our understanding. JASP offers two options for visual investigation: one is a curve plot and the other is a model plot. The curve plot is the graphical analysis of growth trajectories. It presents the shape of the estimated growth curve. The model plot is a path diagram that contains the indicators and factors. It shows a picture of how we made the latent growth curve model.
Let’s draw a curve plot. For this, check Curve Plot in the Plots Tab, and the curve plot will appear in the output panel.
We can see that there are individual differences in the initial state, which are being represented by the different points at time 0. The positive average linear growth in GPA over time is represented by the fact that the cluster of points lies a little higher on the y-axis with every time point. Next, the variance in the linear slope factor L, is represented by the fact that the lines, representing the linear growth over time, are not parallel.
Next, let’s draw a model plot. Check Model plot in the Plots tab. Check also Show parameters. Again, the model plot will appear in the output panel.
Note that the values are unstandardized. These values indicate variance and covariance of latent variables (I for the intercept factor and L for the linear slope factor), factor loadings, and residual variances of indicators. Factor loadings are overlapping, but you can trust us in that all factor loadings on the intercept factor are fixed to 1 and those on the slope factor are set to 0, 1, 2, 3, 4, and 5 from gpa1 to gpa6. We can at least check that the factor loading of gpa1 on the intercept factor is 1 and that of gpa6 on the linear slope factor is 5!
Extending LGCM with Predictors
We can extend the current latent growth curve model by adding predictors of the intercept factor and the linear slope factor. For illustrative purposes, let’s use highgpa and sex as predictors. In doing so, we can investigate whether highgpa and sex explain why students differ in the initial GPAs and their rate of linear growth. Because highgpa is a continuous variable and sex is a dichotomous variable, move highgpa to the Regressions section and sex to the Factor section.
The way to interpret results about latent means and variances, latent covariance, and model fit are the same as we have done previously. One new thing that we did not deal with before is the Regressions table.
We can interpret the estimates in this table as regression coefficients that we are familiar with. Note that the coefficient for sex is indicated as sexfemale. This happened because male students are coded as 0, which is a reference category. The estimate of sexfemale consequently indicates to what extent female students differ from male students.
Consider the regression coefficients modeling the effect of highgpa and sexfemale on the intercept factor. Both regression coefficients (i.e., Estimate in the table) are positive and significant with p-values lower than 0.05. This means that students with higher high school GPAs have higher initial GPAs and that female students have higher GPAs at the beginning than the male students.
How about the estimates from highgpa and sexfemale to the linear slope factor? Since the p-value of the regression coefficient from highgpa to the linear slope factor is 0.764, high school GPAs are not a significant predictor in explaining the linear growth trend of GPA. However, the p-value of the estimate from sexfemale to the linear slope factor is positive and significant with the p-value lower than 0.05. This implies that female students exhibit a steeper linear growth trend than male students as the semester passes by. What follows is a model plot of this extended latent growth curve model, which shows that we are using these variables to predict the latent trajectory for each person. Note again that these estimates are unstandardized.
Modeling Quadratic Latent Growth Curve
If a research question involves modeling the quadratic growth trend over repeated GPA measurements, we can easily add a new growth factor: the quadratic slope factor. The factor loadings on the quadratic slope factor are to be fixed as 0, 1, 4, 9, 16, and 25 (0², 1², 2², 3², 4², 5²) from gpa1 through gpa6. Of course, JASP does this job automatically. We proceed with modeling without highgpa and sex as predictors for simplicity, but you can always add them if you want! Keep the six GPA measurements (gpa1 through gpa6) in the Variables section and empty the Regressions and the Factor section. In the Model Options tab, check Quadratic. You now know how to interpret all the results since we have discussed them above! We only present the model plot to grasp how our model looks like. Note that Q refers to the quadratic slope factor.
When looking at the curve plot, we can see that this time the curves are bent, because we included a non-linear, quadratic trend. As in the previous model, there are differences in growth between individuals. This time, they not only differ in the steepness of the growth-curves, but also in how they are bent. Individuals starting out with a relatively low GPA, have a growth-curve that is bent upwards. Hence, these individuals first decreased in their GPA, but then increased again with subsequent measurements. In contrast, individuals who started off with a relatively high GPA have a growth-curve that is bent downwards. These individuals first increased in their GPA, but the rate at which their GPA increased got lower over time. This pattern highlights the importance of considering the curve plots on top of the numerical results, as curve plots allow for an intuitive interpretation of the complex patterns in which growth trajectories can unfold.
What Is Unavailable Currently
“I have a question! I am wondering what to do if I want to use the six job status variables (job1 through job6) as predictors?” We admit this is a great and sharp question! These six variables are time-varying predictors such that values that job status can take on differ across time. On the contrary, two variables that we used to predict growth factors, highgpa and sex, are time-invariant predictors since the values are constant over time. Whereas time-invariant predictors are modeled to predict the growth factors, time-varying predictors should be modeled to directly influence repeated GPA measurements (i.e., gpa1 through gpa6). The current release of JASP cannot handle the time-varying predictors, unfortunately. We intend to add time-varying predictors to the LGCM analysis in JASP soon.
This tutorial explained the idea of Latent Growth Curve Modeling and how to implement it and interpret the results in JASP. If your research question needs to be answered via LGCM, why not use JASP? [Download](https://jasp-stats.org/download/) JASP now to do LGCM!
Felt, J. M., Depaoli, S., & Tiemensma, J. (2017). Latent growth curve models for biomarkers of the stress response. Frontiers in neuroscience, 11, 315.
Geiser, C. (2012). Data analysis with Mplus. Guilford Press.
Hox, J. J., Moerbeek, M., & Van de Schoot, R. (2017). Multilevel analysis: Techniques and applications (3rd ed.). Routledge.