1 Steps of conducting a multilevel LCA

  • Firstly, we ignored the multilevel structure of the data and estimated a series of traditional LC models to determine the number of classes at the observational-level;
  • Next, a series of MLCA models were fitted to account for the multi-level structrure of the data. In these models, the number of observational-level classes was based on the best fitting LCA model from the first step, and the LCA model at the individual-level was estimated to identify the number of individual-level latent classes;
  • Thirdly, when number of individual-level latent classes is defined based on the previous stage, observational-level classes was modified (one class lower and one class higher than in the second step), to see the effect of changing level 1 classes and confirm the best fitting model.

1.1 How to decide the number of classes in level 1 and/or level 2 data

The number of classes in level 1 were determined by

  1. the evaluation of model fit indices, including the Akaike information criterion (AIC), Bayesian information criterion (BIC), adjusted Bayesian information criterion (aBIC) where smaller values indicate better, and entropy which is a statistic that summarizes latent class probabilities where values near 1 indicate better latent class separation;

  2. the Lo-Mendell-Rubin Likelihood Ratio Test (LMR-LRT) which compares \(q\) vs. \(q-1\) class models, where \(q\) is the number of latent classes;

  3. pattern interpretability.

  4. In the step of performing multilevel LCA, where LMR-LRT was not available, same rules of model fit indices and pattern interpretability were used to determine the optimal combination of latent classes in level 1 and level 2.

2 Level 1 classes selection

  • In this analysis, within each hour of the day carbohydrates intake was defined as:
    • not eating;
    • eating and carbohydrates contributed less than 50% of energy;
    • eating and carbohydrates contributed higher or equal to 50% of energy.
Level 1 class selection. (All data, n = 6155, 24483 data points)
N of classes N of free parameters log-likelihood AIC BIC aBIC Entropy Lo-Mendel-Rubin LRT
1 48 -372017.3 744130.6 744519.7 744367.1 – –
2 97 -368913.7 738021.4 738807.7 738499.4 0.777 < 0.0001
3 146 -366665.0 733621.9 734805.4 734341.4 0.666 < 0.0001
4 195 -365528.6 731447.1 733027.7 732408.0 0.658 0.8478
5 244 -364901.2 730290.3 732268.1 731492.7 0.648 0.7602
6 293 -363641.8 727869.5 730244.5 729313.4 0.701 0.7632
7 342 -362789.9 726263.7 729035.9 727949.0 0.729 0.7702
8 391 -362047.9 724877.9 728047.2 726804.6 0.737 0.8261
Note:
Abbreviation: N, number; AIC, Akaike information criterion; BIC, Bayesian information criterion; aBIC, adjusted BIC; Entropy, a pseudo-r-squared index; Lo-Mendel-Rubin LRT, likelihood ratio test comparing q classes models with q-1 classes models.

From what is shown in the table that, although AIC, BIC, aBIC or entropy are improving with increasing n of classes, however, the Lo-Mendell-Rubun likelihood ratio test suggested that n of classes higher than 4 did not fit the data necessarily better. The null hypothesis of the LRT is that the fit of a n-class model is equal to that of the (n-1)-class model. So that p-value less than 0.05 indicates that the more complex model (the one with more latent classes) provides better fit to the data. We show the results from 2 classes (CW = 2) below.

2.1 Fit information for each model (combination of level 1 and level 2 classes)

Models Specification
Number of level 1 classes
Model 1 class 2 classes 3 classes 4 classes 5 classes 6 classes
Fixed effects model
No. of free parameters 48 97 146 195 244 293
Log-likelihood -372017.293 -368913.708 -366664.971 -365528.553 -364901.166 -363641.759
BIC 744519.661 738807.672 734805.379 733027.723 732268.13 730244.498
Lo-Mendell-Rubun LRT — < 0.0001 < 0.0001 0.8478 0.7602 0.7632
Entropy 1 0.777 0.666 0.658 0.648 0.701
Random effects model
2 between classes
No. of free parameters 195 293 391
Log-likelihood -363460.153 -361571.164 -360297.394
BIC 728890.925 726103.308 724546.13
Entropy 0.834 0.798 0.784
3 between classes
No. of free parameters 293 440 587
Log-likelihood -360910.13 -358902.821 -357404.521
BIC 724781.241 722252.166 720741.109
Entropy 0.824 0.793 0.824
4 between classes
No. of free parameters 391 587 783
Log-likelihood -358684.859 -356668.354 -355235.955
BIC 720078.473 719268.774 718384.699
Entropy 0.816 0.817 0.806
Note:
Abbreviation: No, number; BIC, Bayesian information criterion; Entropy, a pseudo-r-squared index; Lo-Mendel-Rubin LRT, likelihood ratio test comparing q classes models with q-1 classes models.

3 (CW = 2) Level 1 latent classes

3.1 Visualisation of level 1 latent classes

3.2 Visualisation of level 2 latent classes (CB = 2)

3.3 Visualisation of level 2 latent classes (CB = 3)

4 (CW = 3) Level 1 latent classes

4.1 Visualisation of level 1 latent classes

4.2 Visualisation of level 2 latent classes (CB = 2)

4.3 Visualisation of level 2 latent classes (CB = 3)

5 (CW = 4) Level 1 latent classes

5.1 Visualisation of level 1 latent classes

5.2 Visualisation of level 2 latent classes (CB = 2)