Analyzing longitudinal orthodontic data. Part 4: Latent growth curve models
American Journal of Orthodontics and Dentofacial Orthopedics, 20131201, Volume 144, Issue 6, Pages 927930, Copyright © 2013 American Association of Orthodontists
In previous statistical articles, we demonstrated how to use multilevel linear or nonlinear models to analyze longitudinal orthodontic growth data. Here, we will explain how to use latent growth curve models for data with repeated measurements. From a statistical perspective, latent growth curve models and multilevel models for longitudinal data analysis are actually equivalent. However, because these 2 methods have been implemented in software packages in a different manner, they have their own advantages and limitations in practical data analyses.
Multilevel models are most useful when the number of repeated measurements is large and these measurements have been undertaken at different times for different subjects. Current software packages for latent growth curve models do not provide the same level of flexibility in coping with different time intervals between measurements among subjects. Latent growth curve models and multilevel modeling are equally useful, when (1) the repeated measurements were undertaken at the same intervals for all subjects, and (2) the number of repeated measurements is limited. When the number of repeated measurements is limited but the outcome shows a nonlinear change pattern, latent growth curve models can be a better choice. This scenario usually occurs when interventions are given to the subjects at a baseline or over a period of time, such as in clinical trials.
Latent growth curve models have been popular in social and psychological sciences for analyzing experimental and observational data. For this article, we used a simple example to show how latent growth curve models can be used for analyzing orthodontic pain data. Since latent growth curve models can be viewed as a special application of structural equation models, all software packages for structural equation models can be used for latent growth curve models analysis. All of our analyses were undertaken with the software Mplus (version 7.11; Muthen & Muthen, Los Angeles, Calif).
The data consist of repeated pain measurements on a visual analog scale (VAS) for 53 patients after placement of 2 types of orthodontic appliances, A (28 subjects) and B (25 subjects), at 4 hours, 24 hours, 3 days, and 7 days. The aim of our analysis was to evaluate whether there were differences in the pain levels at baseline and the changes in pain between the 2 groups.
Statistical analysis
We first plotted the VAS scores for the 53 patients. Figure 1 shows individual VAS scores over 7 days for each patient in the 2 groups. For both groups, the pain levels seemed to increase with time until day 3 but slightly decreased at day 7. Table I summarizes the VAS scores on the 4 occasions for each group. Although group B showed greater levels of pain on the 4 occasions than did group A, the 2sample t test shows that their differences were not statistically significant at the baseline (8.74; 95% confidence interval [CI], −8.3425.82) and at 24 hours (8.67; 95% CI, −8.3825.72), but they were significant on day 3 (20.28; 95% CI, 4.1136.45) and day 7 (20.28; 95% CI, 4.1934.26). When the 4 measurements of pain were summed as a total score, the differences in the total scores between the 2 groups were also statistically significant (56.91; 95% CI, 6.69107.14). These results are statistically significant when the significance level is set at 5%. As we carried out multiple testing procedures for the same data, a more stringent significance level (such as the adjustment made by the Bonferroni correction) might be required to control the falsepositive rates.
Variable  Mean  SD  Minimum  Maximum 

Group A (n = 28)  
4 hours  40.54  33.03  0  98.2 
24 hours  56.45  34.13  0  100 
Day 3  40.17  31.29  0  99.1 
Day 7  22.07  23.57  0  84.68 
Group B (n = 25)  
4 hours  49.28  28.35  5.86  94.59 
24 hours  65.12  26.72  4.5  99.55 
Day 3  60.45  26.82  1.35  99.55 
Day 7  41.30  30.82  0.45  94.59 
For latent growth curve models, it is helpful to use a path diagram ( Fig 2 ), commonly used to portray a structural equation model, to explain the statistical concepts behind latent growth curve models. In our analysis, 4 VAS scores were made at 4 hours, 24 hours, day 3, and day 7. These 4 variables are represented by X1, X2, X3, and X4, respectively, in Figure 2 . Latent growth curve models use the 4 variables to estimate the baseline VAS and the change in VAS over the observation period by estimating first the individual baseline VAS and the change in VAS. This is exactly the same approach as multilevel models that we demonstrated previously. In multilevel models, the variations in baseline VAS are denoted as random intercepts (each patient’s baseline VAS score), whereas the variations in slopes for the changes in VAS are denoted as random slopes (each patient’s VAS score changed pattern with time); they were treated as 2 random variables in the models. In latent growth curve models, these 2 random variables are explicitly depicted as 2 latent (unobserved and to be estimated) variables, “Intercept” and “Slope.” Following the convention in structural equation models, latent variables are in circles, whereas observed variables such as VAS scores are in squares.
For the 2 latent variables Intercept and Slope in Figure 2 to represent the estimated intercepts and slopes, we needed to carefully set up their relationships to the 4 VAS scores; this was done by the arrows from the 2 latent variables to each X. The direction of the arrows in Figure 2 from Intercept and Slope to X and their associated factor loadings (the number next to an arrow) define their functions in the models. For instance, the relationship between VAS scores and intercept/slopes can be written as the following regression equations:
For the latent growth curve model, however, Figure 1 shows that the change in pain level does not appear to be linear, and this was confirmed by the results from the model fit statistics: the model’s chisquare value was 52.34 with 10 degrees of freedom (df); this was highly significant. In latent growth curve models and structural equation models in general, the null hypothesis is that our model represents the same causal relationship among variables in the model, and the chisquare test evaluates the difference between estimated and observed relationships among variables in the model. Therefore, if our model is correct, the chisquare test should be nonsignificant, and a significant chisquare test suggests that something is wrong with our proposed model.
A quadratic curve model including another latent variable to capture the nonlinear change in pain did not resolve the problem because the chisquare test was still highly significant (χ ^{ 2} = 39.3; df = 8). Since there were only 4 measurements of VAS scores, it made little sense to estimate a cubic or even higherorder growth curve model. An alternative approach, which is not feasible in multilevel models, is to allow the 2factor loadings associated with the arrows from Slope to X2 and X3 to be freely estimated. If the change in VAS scores was indeed linear, the estimated factor loadings would be close to the time intervals when these measurements were made; otherwise, these estimated factor loadings would try to reproduce the real changes in the VAS scores.
The results from the final model, where a few other changes to the original model were made, including a correlated error covariance between E3 and E4, are shown in Table II . The factor loadings for the arrows from Slope to X2 and X3 were −8.30 and −2.54, respectively, and because the estimated Slope is −2.04 per day, indicating an overall decrease in VAS scores on day 7 compared with the baseline, the negative factor loadings mean that the pain level actually increased on days 1 and 3. The difference in baseline VAS scores between the 2 appliance groups was 13.81 (SE = 6.10; P = 0.024), and the difference in change in pain level was 0.2 per day (SE = 0.48; P = 0.63). This suggests that although patients in group B experienced significantly higher levels of pain than did those in group A, the overall change in pain remained similar. The chisquare test was 6.13 with 6 df, and the associated P value was 0.41, which was not significant.
Estimate  SE  P value  

Intercept ≤ Group  13.81  6.10  0.024 
Intercept for group A  38.12  5.23  <0.001 
Slope ≤ Group  0.23  0.48  0.633 
Slope for group A  −2.04  0.73  0.005 
X1 ≤ Slope  0  
X2 ≤ Slope  −8.30  4.53  0.067 
X3 ≤ Slope  −2.54  3.06  0.405 
X4 ≤ Slope  7  
Variance  SE  
E1  642.09  147.83  
E2  435.24  84.90  
E3  435.24  84.90  
E4  435.24  84.90  
Intercept  327.78  103.28  
Slope  0.45  0.87  
Covariance  SE  
X3 with X4  273.10  77.21  <0.001 
Intercept with Slope  −9.25  6.08  0.128 
Discussion
We demonstrated how to use latent growth curve models to analyze data with repeated measurements. Simple comparisons show that the pain levels of the 2 groups are different at 4 and 24 hours, and those differences become larger on days 3 and 7. The latent growth curve model analysis tests a slightly different hypothesis: the differences in the estimated baseline pain levels and the changes in pain. When there are only a few repeated measurements at the same time intervals across all subjects, latent growth curve models can be flexible in modeling nonlinear curves, as shown in our example. Another potential advantage is that it is quite straightforward for a latent growth curve model to test the relationship between growth parameters and distant outcomes. For instance, suppose we also measure patients’ overall experiences with the orthodontic appliances at the end of 7 days, and it would be straightforward to test their overall experiences, their baseline pain levels, and the changes in pain latent growth curve models; such an analysis could be quite cumbersome with multilevel models. Although the application of latent growth curve models might have a smaller scope than multilevel models, latent growth curve models can be a useful tool for analysis of data from clinical trials when a limited number of repeated observations of the same outcomes are undertaken after the intervention.
Acknowledgments
We thank Padhraig Fleming for kindly providing the example data.
References

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