# Introduction to Latent Growth Curve Models

Latent growth curve models allow us to see the “growth” or change over numerous time points for a respondent.This type of analysis works well for longitudinal data collection, espe- cially with test-retest situations. If a respondent was measured at only two time points, we could use a two group analysis to determine differences of the two time points.When you have more than two time points, latent growth curve modeling can be quite handy. This type of modeling allows us to see the change over all the time points. Latent growth curve modeling works best for linear relationships, but it can determine the quadratic nature of the data if that is a concern.With a latent growth curve model, you are going to determine the intercept and slope of a value. If you are looking to forecast values into the future, it is advisable to have equal intervals between your time periods. Otherwise, the growth curve will have a “general- ized” prediction in trying to account for the diversity of time frames. Saying that, if you are more concerned with the actual growth over the time period than future prediction, then you can use unequal time intervals.The real benefit of latent growth curves is to determine if there is a significant difference between groups in the “growth curve—slope/intercept”. Obviously, understanding if change or “growth” over a period of time is important, but without a refer- ence group you are not sure if this change really matters. That is why latent growth curves are well suited for experimental design research that is longitudinal or is a test-retest format. Let’s look at an example about a snack shop at a golf course. The snack shop wants to know if using environmentally sustainable packaging for their snacks would increase pur- chase behaviors of golfers. To test this out, they ran an experiment with golfers who make a reservation to play golf every Saturday and who typically purchase snacks after their round is over. A random group of those golfers after completing a round of golf were presented snack options to purchase that were packaged in environmentally sustainable materials.The cost was initially the same; just the packaging changed. Another group was selected as a control group, and they were presented snack options with the original packaging that did not mention being environmentally sustainable. The golf shop was going to see if spending patterns changed for the group that had an environmentally sustainable package. The snack shop tracked the pur- chases over four Saturdays. Dollar amount spent on snacks was captured and gender was also recorded, with 1 denoting males and 0 denoting females. Lastly, the snack shop coded which experimental group the golfers were in by a column called “sustain”. Customers who got environmentally sustainable packaging for their snacks were coded as a “0”, and those that received the original package with no mention of sustainability were coded as a “1”.

Once our data is coded in SPSS, we are ready to draw a growth curve model in AMOS. To perform a latent growth curve, you need to go to the “plugins” tab at the top and then select “Growth Curve Model”.You will then see a pop-up window asking how many time points you have. Select four and hit OK.

Figure 9.1 Data Collected at Four Different Time Points

Figure 9.2 AMOS  Latent Growth  Curve Popup Window

After doing this, AMOS will populate a generic growth curve model with four time points. The growth model will have two unobservable variables listed as “ICEPT”, which stands for intercept, and the second variable is called “SLOPE”, which represents how steep or flat the “growth” or change is taking place. From the intercept construct, you will see parameters drawn to four different observa- bles. You will notice that the parameters from the intercept construct to each observable is con- strained to a value of “1”. This is done so that regard- less of the data point, the intercept (or starting point) should be the same for each time period. The slope has default values spaced out over four time periods (0, .33, .67, 1). I prefer to change these to meaningful time periods. The line to the first time period is listed as a “0”. We are going to keep this value as a “0” because we want the first time period to reflect the intercept. Subsequently, the parameters for the next time periods will be labeled 1, 2, and 3. AMOS will also label the intercept mean and variance along with the slope mean and variance (imean, ivariance, smean, svariance).You need to remove these labels. If you had only one group, the labels would be fine; but since we have two groups, we need to remove those labels. Our two groups are (1) the custom- ers who got an environmentally sustainable package; and (2) customers who did not get an environmentally sustainable package. The covariance between the slope and intercept constructs is also labeled, and we will remove that label as well. You can just double click into the labels which will bring up the Object Properties window.You will then just erase the labels for the mean and variance. As for the covariance label, do the same thing and remove that label.

Figure 9.3 Latent Growth Curve With Intercept and Slope Modeled

Figure 9.4 Object Properties Window With Mean and Variance Labeled

You will see four observables, X1–X4; these are the time points where we captured sales figures at the four different points of time. The sales figures are simply labeled Time 1–4 in the data file. In AMOS, you now need to view the variables of your data set and drag in the time points to the observables in the model. One last thing to note is that the error variances for the time periods are all equal. You will see that AMOS puts a label in the error vari- ance of “var”. This is to denote that the error variances for all time periods should be equal. I will remove this label as well so that the variance for each time period can be individually captured.

Next, you need to set up the two groups in AMOS. For a detailed discussion on how to set up groups in AMOS, see page 149. I am going to call one group “SustainPackage” for the group of individuals that were presented snacks in environmentally sustainable packages.The second group, labeled “NoSustain”, was the group that was presented snacks in the original package that did not have any reference to environmental sustainability.

In the data, the individuals who had sustainable packaging were listed as a “0” in the sustain column and the individuals who had no environmental labeling on their snacks were listed as a “1”. After setting up the groups, you need to read in the data for each group.The next step is to let AMOS uniquely label the mean, variance, and covariance for each group.To do this, select the “multi-group analysis” button .This will give you a prompt that it is going to label all the parameters and suggest potential models with constrained parameters to test. Hit OK and let AMOS label all the parameters.

Figure 9.5 Denoting Data File Location for Each Group

Figure 9.6 Labeled Latent Growth Model

This is what your model should look like after the parameter labeling of both groups. AMOS has labeled the mean for the intercept as m1_1 for the “SustainPackage” group. If you selected the other group (NoSustain), that mean value would be labeled as m1_2.You will see that the mean, variance, and covariance parameters all have a unique label name. Once you let AMOS label the parameters, it will suggest model comparisons across the groups. One of the potential models that AMOS suggests is called “Structural Means”.This model comparison will constrain the means for the intercept and slope to be equal across the groups. This is the model comparison test that we are primarily concerned with going forward.We want to know if the intercept and slope is significantly different across the groups. See Figure 9.7 to see what the structural means test looks like.

Figure 9.7 Structural Means Comparison Test

After setting up the groups and labeling the parameters, we now need to ask for AMOS to estimate the means and intercepts in the output. When we go to the Analysis Properties window under the Estimation tab, there is a checkbox called “Estimate means and intercepts” that we need to select. After doing this, we can cancel out of the window and run the analysis. Let’s initially go to the model fit statistics in the output. In the model fit output (Figure 9.8), we are only concerned with the unconstrained model. Based on these results, the model fit is acceptable, so we can move on to the next analysis, which is the model comparison test. In the model comparison test option, we are going to examine the structural means test, which constrained the means for the intercept and slope to be equal across the groups.

If we see a significant chi-square test, then we can say with confidence that the groups are significantly different. The results of the structural means comparison (Figure 9.9) shows a significant difference between the groups. With 2 degrees of freedom constrained, the chi- square difference was 222.85, which is significant at the p < .001 level. Now that we know that the groups are different, we need to see the intercept and slope for each group to deter- mine where the differences lie. In the “Estimates” link in the output, we are going to examine the means for the intercept and slope for each group.

Figure 9.8 Model Fit Statistics Across the Model Comparisons

Figure 9.9 Chi-Square Difference Test from the Model Comparison Results

Figure 9.10 Estimates Output for the Sustainable Packaging Group

Figure 9.11 Estimates Output for the No Sustainable Packaging Group

The results of our experiment show that the group that had their snacks in environmentally sustainable packaging purchased snacks at a significantly higher rate than those customers whose snacks did not list the packaging as being sustainable. Over the four-week time period, there was a significant change in spending behavior.

The reason I prefer to do a comparison with latent growth curves is that it gives you greater context to your research question. We could have run a latent growth curve with just the environmentally sustainable packaging group and the slope would be significant, so we see a change in behavior; but I do not know if that change is significant compared to a control group. You could make the argument that the first time period could act like a control group and then see if there is a significant change over time. I don’t prefer this option because it assumes that the intercept would be the exact same for each group. Especially with experimental manipu- lations, the first time point or intercept might be dramatically different. Ultimately, we are looking to see if the growth/change over time is significant in comparison to a group in an initial status and one that has experienced a manipulation. For instance, you see many latent growth curve examples that use student testing. Let’s say we instituted a new curriculum for half of a high school’s seniors in a school and the other half received the existing curriculum. If we just run a latent growth curve on the seniors who got the new curriculum, we could see if growth in test scores took place, but we don’t know if the new curriculum is better than the old one without a comparison. If we compare the groups and see no significant difference in slopes across either curriculum, this will let us know that growth/change may be taking place, but it is not significantly better than the curriculum they started with before the test. Saying all this, latent growth curves are beneficial, but comparison of latent growth curves is where you can get real insight.

Source: Thakkar, J.J. (2020). “Procedural Steps in Structural Equation Modelling”. In: Structural Equation Modelling. Studies in Systems, Decision and Control, vol 285. Springer, Singapore.