Verbal learning and memory tests are highly varied among older-aged adults due to various influences. Early cognitive intervention in older adults is a critical program to reduce the future risk of dementia (Thomas et al., 2019). The efficacy of the Chinese form Hopkins Verbal Learning Test (HVLT) for screening dementia and mild cognitive impairment in a Chinese population showed that HVLT scores were affected by age, education, and sex (Shi et al., 2012). The dataset of Advanced Cognitive Training for Independent and Vital Elderly (ACTIVE) study consists of two hierarchies in which four different repeated measures are nested within each participant (Luo and Wang, 2014). The outcome measures of the cognitive training interventions were the total HVLT from three learning trials and the baseline measure (Gross, 2011).

Bayesian logistic and hierarchical probit models of accuracy data that allow two levels of mixed-effects in repeated-measures designs have been implemented. The Bayes factor through the Bayesian information criterion estimate and the Widely applicable information criterion (WAIC) model selection techniques were used (Song et al., 2017). Duff (2016) used stepwise regression model to scrutinize the effect of age, education, and gender on HVLT scores in 290 cognitively intact older adults. The study revealed that age was negatively correlated with the HVLT score, while education status was positively correlated. Moreover, there were fewer gender differences among four repeatedly measured verbal learning tests (Lekeu et al., 2009).

Another study showed that besides capabilities through training, personal characteristics like age, unmarried status, and lower occupational cognitive requirements increased the likelihood of cognitive risk (Silva et al., 2012). Higher educational levels and active engagement in exercise may contribute to cognitive reserve and have a protective effect on cognitive decline in late life (Shen et al., 2021).

Gender effects on neuropsychological performance were negligible when the age and educational status of elderly people were controlled (Welsh-Bohmer et al., 2009). Recently, the Markov chain Monte Carlo (MCMC) methods have been widely used to generate samples from complicated and high-dimensional distributions (Hadfield, 2017). Among all Bayesian computational methods, the Hamiltonian Monte Carlo (HMC) (Almond, 2014) approach is the most efficient for approximating complex data structure models and converges faster than the traditional Metropolis-Hastings and Gibbs methods (Kruschke and Vanpaemel, 2015). The common MCMC approaches show poor performance and tremendously slow convergence in complex parameter structures (Yao and Stephan, 2021).

The HVLT is the ultimate in situations calling for multiple neuropsychological assessments (Benedict et al., 1998). Classical statistical inferences and single-level models have limitations for predicting naturally nest data. Bayesian hierarchical models (Congdon, 2020) were able to predict verbal learning test and memory scores from baseline personal characteristics, such as age, gender, cognitive status [mini-mental state exam (MMSE) score], years of education, and participants’ booster training and reasoning ability measured by training progress (Kuslansky et al., 2004).

In Bayesian inference, the WAIC, the leave one out information criterion (LOO-IC), and K-fold cross-validation (K-fold-CV) are recently developed measures of complexity penalized fitting models (Almond, 2014; Sivula et al., 2020). In this article, model comparisons and model selections were performed using these three methods under the Bayesian Regression Models using ‘Stan’ (BRMS) package of R (Bürkner, 2018). In most cases, WAIC and LOO-IC showed a slight preference for the random slope model over other models (Bürkner, 2018). However, the general model selection principle shows to choose the null model when diffuse priors are used in the parameters to be included or rejected by the algorithms (Liu, 2000). Therefore, in this article, we used the HMC approach to fit the three different Bayesian hierarchical models and select the best predictive model.

In practice, the three basic Bayesian hierarchical models have been fitted in BRMS default settings, and population-level (fixed) effects and subject-level (random) effects were obtained (Luo et al., 2021). All three models (Models 1, 2, and 3) had both fixed and random (mixed) parts but with different estimated parameter types. In the result, the estimate shows the posterior mean and Est. Error is the SD for each parameter. Model convergence was achieved well enough both the bulk effective sample size (Bulk_ESS) and the tail effective sample size (Tail_ESS) for the 95% CIs were adequate (Vehtari et al., 2017). In general, every parameter is summarized using the posterior distribution’s mean (“Estimate”) and SD (“Est. Error”), as well as two-sided 95% credible intervals as lower and upper bounds based on quintiles.

Table 1 of the fixed effects shows that the posterior mean verbal testing score was estimated to be 26.33 with an SD of 0.73. The 95% credible interval shows that the posterior distribution mean (intercept) was significant. On the other hand, the random effect showed significant verbal score test variation between groups (participant subjects) and within-subjects (between different measurements of different time points). Thus, according to the null model, the HVLT score showed more between-group/subject variation than within-group (between repeated measurements) variation.

Table 2 showed that the coefficient of booster training was positive with a zero overlapping 95% CI. This indicates that, on average, there is little evidence that taking booster training increases elderly adults’ verbal learning and memory test scores by 0.1865, but the evidence-based on the data and random intercept model. On the other hand, adults’ years of education (edu) estimate was negative with a zero overlapping 95% CI. This negative estimate indicates that, on average, in the random intercept model, there is little evidence that increasing the years of education decreases elderly adults’ verbal learning and memory test scores by 0.0034 units.

According to the predictive effects of each explanatory variable shown in Figure 2 and Table 3, taking booster training, age, and gender were the most influential factors affecting participants’ cognitive verbal test and memory ability. Table 3 reveals that there is also an adverse association between the intercepts and coefficients for reasoning ability, which implies reasoning ability has a large average score value showing additional variability by poor reasoning ability than by good reasoning ability. Nevertheless, it can be seen that the slope estimate of such a model is even further unreliable than that of the preceding models, as it can be clearly understood from the associated standard error and the size of the 95% CIs. Table 3 also showed that booster training had a significant positive predictive effect on elderly adults’ verbal learning and memory test scores. In contracts, adults’ years of education had a significant negative impact on elderly adults’ verbal learning and memory test scores.

We also noticed in Figure 2 and Figure 3 below that adding any term to the early model showed predictive performance improvements on the fitted models are ordered from Models 1 to 3 (full model). However, such a result may not be interpreted as a universal rule, subsequent adding extra terms to a unique model may also result in overfitting, which corresponds to a condition in which the fitted model is over-specified about the data, making the model good at clarifying the sample dataset but poor at predicting no observed data. The model convergence diagnosis plots are hairy caterpillars which showed the model converged. On the other hand, the models have well converged based on the estimated statistical values. This means that the R-hat (R^) statistics were close to 1 and the (bulk and tail) ESSs values were sufficiently high when ESS > 100 was chosen as the cutoff (Vehtari et al., 2021). The majority of parameters still showed sufficiently high ESS values when more conservative cutoffs were chosen (i.e., ESS > 400 or even 1,000, see Zitzmann and Hecht, 2019).

Based on the fitted varying slope model, which accounted for six predictors from the data, fixed effects showed that age, gender, reasoning ability, and booster training were significant predictors of verbal learning and memory test scores, whereas random-effect showed that much of the variation in test scores occurred within-subjects (between measurement time points) than between subjects.

After we have built the three different models, it is necessary to identify relatively the best model that can be used to predict the outcome variable and make inferences. However, choosing the model that has the best predictive and a better fit on the actual data is complicated with diverse information criteria since all selected models on the actual data might not essentially achieve as fit on a different dataset. In its place, it is necessary to decide on a model that fits best in terms of predicting new data which had not been practiced.

In case of the non-existence of extra information, cross-validation methods such as WAIC and LOO-CV can be used. According to Table 4, the varying slope model has the lowest WAIC, LOO-IC, and 10-fold estimates. However, the difference is relatively small when we compare the difference in estimates of criteria for each model and the corresponding standard errors (in the column SE).

Among the fitted models above, it looks like the final model (Model 3) in the HMC algorithm is the best model. Therefore, as a function of the six explanatory variables and the random coefficient for age and reasoning ability, Model 3 has the best predictive performance for the cognitive HVLT.

According to Figure 4, the varying slope and intercept model fit well and produced nearly identical posterior observed density and posterior predictive distribution plots of the outcome variable of THVLTS from the ACTIVE study.

Furthermore, the marginal effect of each predictor variable revealed (Figure 5) that age and reasoning skills are the most significant explanatory variables that predict the THVLTS of the ACTIVE study.

Based on the selected sample participants in the ACTIVE study dataset (Willis et al., 2015), the Bayesian hierarchical linear models of three types were fitted by considering only six explanatory variables as predictors of the cognitive verbal learning test. The null model without any predictor effect but with only the intercept term was fitted, and it shows a mass of cognitive verbal learning ability variability across subjects. The varying intercept model with the addition of all predictor variables was fitted; and getting booster training, age, and reasoning ability were significant predictor of verbal test scores (Duff, 2016). The varying coefficient/slope model (i.e., Model 3) is the best-fitted model than the other fitted models since it had the lowest WAIC, LOO-IC, and 10-fold estimates (Bafumi and Gelman, 2011). A bulk of participants’ cognitive verbal test scores variations were observed between subjects (Ryoo, 2011). The full hierarchical model with varying intercepts and slopes has the best performance for predicting verbal learning tests (from ACTIVE study dataset) using the hybrid Hamiltonian Markov Chain Monte Carlo approach.

Socio-demographic and training-related characteristics influence elderly verbal learning tests that can be measured in multiple occupations (Welsh-Bohmer et al., 2009).

Total Hopkins Verbal Learning Test Score from the ACTIVE study can be used as a measure of elderly adults’ cognitive verbal learning ability. Four demographic characteristics of adults, such as age, gender, educational status, and cognitive status (MMSE score), were measured at the baseline, and characteristics measured after cognitive training such as reasoning ability and booster training were considered. THVLTS from the ACTIVE study can be used as a measure of elderly adults’ cognitive verbal learning ability. According to the findings, the varying intercept and slope model fit best, and age, gender, booster, and reasoning ability are the main significant predictors for THVLTS, which measures cognitive verbal learning. Taking booster training had a positive significant predictive effect, while years of education (edu) had a negative significant predictive effect on THVLTS.

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found in the article/supplementary material.

Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. Written informed consent for participation was not required for this study in accordance with the national legislation and the institutional requirements. Written informed consent was not obtained from the individual(s) for the publication of any potentially identifiable images or data included in this article. This is because this quantitative analysis and modeling paper used open-access secondary data on repeated measurements.

EE participated in all aspects of the study: designing the study, performing data management, conducting the data analysis, writing the first draft of the manuscript, and discussing with MC to improve the manuscript, as it is a part of the first author’s Ph.D. dissertation. MC participated in revising the manuscript, commenting, and proofreading. Both authors listed have made a substantial, direct, and intellectual contribution to the manuscript and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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