Step 1.

In the first step, LCA was applied to each wave separately to identify distinct latent classes that capture salient features in the indicator variables. In short, LCA corresponds to fitting a mixture model, assuming that the data follow a distribution consisting of a mixture of several normal distributions, interpreted as the underlying distributions that generate the latent classes. Estimation aims to find the model parameters that assign the highest likelihood to the data (the global maximum of the likelihood function is being searched for), and it is commonly carried out for a fixed number of latent classes, which needs to be specified in advance. However, multiple models for different choices of the number of latent classes may easily be fitted and subsequently compared. In our analysis, LCA models were fitted assuming 2 to 5 latent classes [15]. To avoid ending up with sub-optimal model fits due to local maxima found during the estimation, which is not uncommon to LCA, one approach is to estimate the LCA models multiple times with different starting values for the parameter estimates [15]. Trying out a range of starting values should be done routinely when fitting LCA models.

The optimal number of latent classes is often decided on using an information criterion, such as Akaike information criteria (AIC) and Bayesian information criteria (BIC), or adjusted versions of AIC and BIC [16]. Following the approach of the original study, AIC and the sample-size adjusted BIC were reported [17]. A difference in an information criterion less than 10 means that the difference between the two model fits is negligible [18]. However, other considerations, such as the interpretability of results, may also play an important role in deciding on the optimal number of latent classes [2, 9]. LCA produced prevalences of identified latent classes and item-response probabilities (conditional on a latent class). Item-response probabilities denote the likelihood of responding yes/no to a specific indicator variable, given the latent class status and they were used to distinguish between latent classes and, consequently, to assign a meaningful label to each latent class. In addition, item-response probabilities at wave 1 and 2 were compared visually to assess measurement invariance across time, i.e., similar probabilities would support the measurement invariance assumption that the same latent class structure was observed both at wave 1 and 2 [6]. As measurement invariance is a modelling assumption, a visual assessment seems sensible and in line with common practice for other statistical models such a linear regression; statistical tests for model assumptions should be avoided and an element of subjective judgment should be accepted [19, 20]. In case the assumption of measurement invariance is not tenable then the proposed approach is still applicable. The interpretation of results may, however, become more challenging (but in some cases most likely also more realistic) as it would mean that individuals transition between different classes over time. For instance, it might be difficult to entertain measurement invariance for large gaps between waves.

In Steps 2 and 3, as detailed below, multiple logistic regression models were fitted as a flexible alternative to multinomial regression models. It has been shown that there is almost no loss in efficiency when using logistic regression instead of multinomial regression [21].

Step 2.

Logistic regression models were used to estimate transition probabilities. Specifically, a binary outcome variable was defined for each class at wave 2, indicating whether or not the participant belongs to that particular class. Separate logistic regression models were fitted for each of these binary outcomes, with class membership at wave 1 as the only independent (categorical) variable. The number of logistic regression models that were fitted corresponded to the number of classes at wave 2. As results from logistic regression, by tradition, are reported on a logarithmic scale, a subsequent back-transformation step was needed. Specifically, back-transformation from the log-odds scale was used to obtained estimated transition probabilities for changing from one latent class at wave 1 to any of the three latent classes at wave 2; corresponding 95% confidence intervals were also obtained through back-transformation.

Step 3.

Logistic regression models were fitted to estimate ORs that quantify associations between transitions from wave 1 to wave 2 and relevant predictors. Specifically, for each transition of interest, a binary outcome variable was defined indicating whether or not the participant underwent the transition or remained in the same class at wave 2 as at wave 1. Separate logistic regression models were fitted for each of these derived binary outcomes and for each of the following five predictors: age, gender, migration, school, and socio-economic status. Each of these logistic regression models included an interaction term between the class membership at wave 1 and the predictor in question. The number of logistic regression models fitted for each predictor corresponded to the number of transitions of interest. In this tutorial, three transitions were considered: from low to middle wellbeing, low to high wellbeing, and middle to high wellbeing [14]. However, it should be noted that there would be three more transitions to consider (middle to low, high to low, and high to middle), but they seem less relevant in the present context as they corresponded to transitions towards reduced instead of improved well-being.

Handling missing values.

Missing values for the latent class membership were handled using multiple imputation through chained equations (MICE) based on the six input variables at wave 1 and 2, predictors to be investigated (age, gender, socio-economic status, migration status, and school level), and the latent class membership variables at wave 1 and wave 2 (in total 19 variables). Based on this dataset, MICE was used to generate 10 complete, imputed datasets. Often a small number of imputed datasets, such as 5 or 10, are used as it suffices for capturing the uncertainty in the imputation step, but also avoids that the computational burden becomes too large [22]. Note that a pre-specified random seed was given to ensure reproducibility of results, as was also the case for LCA in the above Step 1. For each imputed dataset, separate model logistic regression models fits were obtained. Subsequently, individual parameter estimates were combined using Rubin’s rule, which makes due allowance for the uncertainty introduced through use of multiple imputation [23]. The MICE approach is advantageous as it can handle arbitrary types of variables (e.g., continuous and categorical variables), ensuring that imputed values will be of the same type as the observed values [23]. In this tutorial MICE was applied after LCA but before any fitting logistic regression models as multiple imputation is useful for statistical inference, obtaining estimates and standard errors that reflect the uncertainty in the imputation step.

Software used.

Statistical analyses were carried out in R using the extension packages "mice" (version 3.15) and "poLCA" (version 1.6) [12, 15, 23]. Supplementary material containing an annotated R script, including precise definitions of binary outcome variables used and detailed explanations of model specifications, is available online at zenodo.org (https://doi.org/10.5281/zenodo.10794077). Key steps using R functionality are also explained and shown in boxes in the Results section. A significance level of 0.05 was assumed.

Approach in the original study.

LTA was carried out using the commercial software Mplus. Specifically, latent classes were identified and the corresponding transition probabilities were estimated. Missing values in wave 1 were imputed although the imputation method was not described. Subsequently, for each wave, a multivariate multinomial regression model was fitted with latent class membership as a multinomial outcome and gender, migration background (yes or no), school level, and socio-economic status as predictors. However, these analyses, which were carried out using SPSS (version 25), did not investigate the effects of predictors on transitions but only evaluated cross-sectional effects [14].

LTA step 1—estimating latent classes.

Box 1 shows the R code for carrying out LCA, exemplified for wave 1; it can briefly be explained as follows: the LCA requires that input variables are initially combined into a model formula, which will be the first argument for the function poLCA() from the extension package of the same name. The second argument of poLCA() needs to be the data set where the variables in the model formula can be found. Finally, the third argument specifies the number of latent classes to be assumed. The random number generator is initiated by means of the function set.seed() to ensure reproducibility.

Based on the fitted LCA models, relevant fit statistics were extracted as summarized in Table 1. The model fit statistics indicated that either 2 or 3 latent classes were optimal. This result was found both for the proposed alternative approach and for the original study despite the discrepancy in AIC and aBIC values for wave 1 where missing values were imputed in the original study. Specifically, for both waves, AIC values did not appreciably differ for 3, 4, and 5 classes (as differences were not larger than 10) but AIC was appreciably larger for 2 classes. A similar picture was observed for aBIC. Consequently, a parsimonious choice was to select the smallest number of latent classes among the options with similar AIC and aBIC values, resulting in a three latent class solution; this choice also ensured that sufficient participants were included in each class [16]. The above analysis, as described in Box 1, included the argument nrep = 10 in the function poLCA(), implying that LCA was repeated 10 times with different starting values to ensure that the final model fit corresponded the global maximum and not some local maximum [15].

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Table 1. Model fit statistics for the proposed approach and as reported in the original study by Kassis et al. (n = 377).

https://doi.org/10.1371/journal.pone.0317617.t001

The three latent classes were well-separated as item response probabilities group in a low, middle, and high trend (Fig 1). Moreover, the same latent classes were found for both waves, supporting the measurement invariance model assumption [10]. For wave 1, predicted class membership percentages were 21%, 48%, and 31% for the low, middle, and high well-being classes, respectively, compared to 24%, 46%, and 30% in the original study; the small discrepancy was due the missing values being imputed in the original study. For wave 2, predicted class membership percentages were 31%, 31%, and 38% for the low, middle, and high well-being classes, respectively, compared to 31%, 30.5%, and 38.5% in the original study.

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Fig 1. Item-response probabilities for the low, middle, and high well-being latent classes.Solid and dashed lines connect item-response probabilities between the 6 input variables (indicators) for wave 1 and wave 2, respectively.

https://doi.org/10.1371/journal.pone.0317617.g001

LTA step 2—estimating transition probabilities.

In R multiple imputation is carried out using the package "mice" as shown in Box 2. The first argument of the function named mice() was the dataset containing the variables that should be used for the imputation step. In this case all variables needed for the subsequent logistic regression analyses were provided. The second argument specified that 10 complete, imputed datasets should be generated. Note that the seed for the random number generator is again specified using the function set.seed(). Once these datasets were constructed, logistic regression models were fitted to each of these 10 datasets using the function with() with arguments "with" and "exp", specifying the list of imputed datasets and the logistic regression model to be fitted. The logistic regression model is fitted using the model fitting function glm() with an outcome that is defined as being in high well-being class at wave 2 (yes or no), a single predictor that is the variable denoting the class membership at wave 1, and, finally, the argument needed to inform glm() that a logistic regression model is specified: link = binomial, ie., logistic regression is a generalized linear model (glm) with a link function that accommodates a binomial distribution [24]. Subsequently, the use of summary(pool()) implied that results from the 10 separate model fits were combined into a single result, in this case three parameter estimates corresponding to the three latent classes at wave 1. These estimates were then back-transformed to probabilties.

Both for the low and high well-being classes, the majority of students were most likely to remain in the same class in both waves (Table 2). For the high well-being class, there was a 67% probability of remaining in that class, whereas it was somewhat smaller, only 56% for the low well-being class. For the middle well-being class, transition to either low or high well-being classes at wave 2 was almost equally likely, with probabilities of 27% and 33%, respectively, corresponding to a 40% probability of remaining in middle well-being class.

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Table 2. Transition probabilities between wave 1 and 2 expressed as percentages (n = 377) for the proposedapproach and as reported in the original study by Kassis et al.

https://doi.org/10.1371/journal.pone.0317617.t002

LTA step 3—investigating predictors of transition.

Box 3 shows the R code for fitting a logistic regression model where the effect of a specific predictor on a specific transition was investigated. In this case it was the effect of age on the transition from low to middle (or remain in low); for the other predictors the R code may be found in the online supplementary material. As in Step 2 the logistic regression model was fitted to each of the imputed datasets. The logistic regression model was fitted to the subset corresponding to low or middle well-being at wave 2, and, consequently, the outcome, which was class membership at wave 2, became binary and suitable for logistic regression. Moreover, the model included an interaction between class membership at wave 1 and the predictor age. The model specification involved both the main effect of the class membership variable and the interaction term specified using ":" and not "*". This parameterization ensured that the output contained results that could be directly back-transformed into interpretable odds ratios reported in Table 3 below.

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Table 3. Associations between selected transition and predictors (n = 377).

https://doi.org/10.1371/journal.pone.0317617.t003

There was a significant association between age and the low to middle transition, where the odds of transition were increased by 179% (95% CI: [27, 511] %; p = 0.01) per year, whereas odds for transitions from low to high and middle to high well-being class were reduced by 20% and 1% per year, respectively (Table 3).

No other associations between predictors and transitions were significant. Other notable findings included: Male students had higher odds for the transition from low to middle and low to high well-being classes compared to female students (64% and 29%, respectively) but 28% reduced odds for the transition from middle to high. Migration only affected transitions slightly (-14% to 2% change in odds). Students at higher secondary schools had 79% and 87% higher odds for transitioning from low to middle well-being and from low to high well-being, respectively, but 52% reduced odds for transitioning from middle to high well-being.

The high SES group had 20% and 10% higher odds for the transition from low to middle and from low to high well-being, respectively, compared to the middle SES group, but only 1% higher for the transition from middle to high well-being. In contrast, the low SES group had 78% and 55% reduced odds for the transitions from low to middle and from low to high, respectively, but 465% increased odds for the transition from middle to high.

No similar results were reported in the original study where only cross-sectional associations of predictors of class membership at wave 1 and wave 2, respectively, were investigated [14]. It is also worth noting that the effect of age was not investigated at all in the original study. However, in the original study, a gender effect was found at wave 2 but not at wave 1.