Background literature

Image schemas and preposition learning

CL posits that language is grounded in our cognitive experiences and conceptual systems. That is, our bodily experiences shape how we understand and use language to communicate meanings. As human beings, we continuously learn about objects not only as objective forms but also by conceptualizing how they can impact us (Holme, Reference Holme2009; Talmy, Reference Talmy and Hampe2005; Tyler, Reference Tyler2012). This body–mind awareness begins developing as early as infancy, preceding even language development (Mandler, Reference Mandler1992). Conceptual notions obtained through this body–mind awareness, such as animacy, inanimacy, agency, and containment, will be accumulated in the form of image schemas. These are structural contours that exist as recurring and analogue patterns beneath conscious awareness, prior to language acquisition (Hampe, Reference Hampe2005). Johnson (Reference Johnson2005) emphasizes the critical role of image-schematic structures in deciphering concepts ranging from spatial relations to abstract notions of reason, mind, knowledge, justice, rights, and values. Importantly, Langacker (Reference Langacker2008) argues that an imagistic approach is as capable as a propositional one in depicting complex structures. Consequently, the notion of image schema has been found to be particularly compatible with the teaching and learning of prepositions (Brugman, Reference Brugman1988; Lam, Reference Lam2009; Tyler, Reference Tyler2012; Tyler & Evans, Reference Tyler and Evans2003; Wong et al., Reference Wong, Zhao and MacWhinney2018), whose primary function is to convey spatial and temporal relationships.

Alongside compelling behavioral evidence, a recent event-related potential study confirmed the neurocognitive advantages of using diagram-based instruction for acquiring English prepositions (Zhao, Huang, Zhou, & Wang, Reference Zhao, Huang, Zhou and Wang2020). It is crucial to note that schematic diagrams and pictorial illustrations in CL-informed instruction serve a purpose beyond being visual aids. Their effectiveness in facilitating learning is maximized when they closely resemble the psychological gestalts stored in our mental lexicons. To demonstrate how schematization can enhance the learning of English prepositions, we will explore the CONTAINMENT schema (see Figure 1). The CONTAINMENT schema is one of the earliest acquired by infants, even before language development (Mandler, Reference Mandler1992). Figure 1 illustrates how the CONTAINMENT schema can convey both spatial and nonspatial senses of the preposition in. To grasp the schematic structure, one must identify two essential elements that realize an image schema: the Trajector (TR), spatially related to but more movable than the Landmark (LM). In the sentence “Mary is talking in her room,” Mary is the TR, and her room is the LM. The TR, Mary, is spatially bounded by the exterior of the LM, such as the walls and the door of her room. Similarly, in the sentence “Joe is trapped in his marriage,” Joe, the TR, is metaphorically bounded by the LM, his marriage, conceptualized metaphorically as a container limiting the movement of the TR. Hence, both spatial and nonspatial senses of the preposition in can be elucidated using the same schematization.

Figure 1. The CONTAINMENT schematization for the preposition in.

The difficulty in acquiring prepositions lies not in learners’ inability to understand these schematizations but often in their uncertainty about which preposition to choose when faced with various options. For instance, the preposition at is commonly viewed as a typical English preposition not found in many other languages, such as Chinese or Italian. The vague definition of at as indicating a particular location provides minimal guidance when it competes with in. In contrast, the CL definition clarifies that the LM is conceptualized as a point along the route (see Figure 2), enabling learners to recognize cross-linguistic differences and the semantic distinctions conveyed by competing forms. Emphasizing conceptual understanding in this way directs learners’ attention to how preposition forms convey various meanings. Thus, a meaning-oriented approach, supported by schematic diagrams in CL instruction, empowers learners to have greater control over their preposition knowledge and increased confidence in their preposition usage.

Figure 2. The POINT-ALONG-THE-ROUTE schematization for the preposition at.

Despite the fruitful discussion of CL on preposition acquisition and successful applications of CL approaches to teaching prepositions, the diverse learning outcomes (Boieblan, Reference Boieblan2023; Jacobsen, Reference Jacobsen2018; Lam, Reference Lam2009; Tyler, Reference Tyler2012; Wong, Reference Wong2023), as well as the limited number of experimental research studies, fail to offer robust empirical support for the pedagogical advantages associated with CL. Many CL-informed experimental studies conducted in the earlier phases, including some of the above-mentioned studies, primarily concentrated on the quality of pedagogical materials, such as CL explanations. However, there was often a lack of emphasis on robust research design and appropriate statistical procedures, as pointed out by Jacobsen (Reference Jacobsen2018), raising concerns about the generalizability of the corresponding findings. Wong et al. (Reference Wong, Zhao and MacWhinney2018) also raised their concern regarding the replicability of many CL-informed experimental studies due to (a) low transparency of adopted CL materials, (b) a lack of information regarding the selection of instructional paradigm, and (c) mostly researcher-fronted training.

The initial study

To address the multiple research gaps identified in applied CL research, Wong et al. (Reference Wong, Zhao and MacWhinney2018) developed an experimentalized CALL (eCALL) system called the English Preposition Tutor, to investigate the effectiveness of the CL approach in the CALL environment. While CL-informed studies focused on the pedagogical content of training tasks, studies of eCALL applications, motivated by the CM (MacWhinney, Reference MacWhinney, de Groot and Kroll1997), have examined other aspects of language teaching, including the role of cue focusing (i.e., presenting two competing cues only at a time), the types of feedback, and practice effects (Wong, Reference Wong2023; Wong et al., Reference Wong, Zhao and MacWhinney2018; Zhao & MacWhinney, Reference Zhao and MacWhinney2018). Combining the two usage-based frameworks (i.e., CL and CM), the authors investigated whether the CL-informed approach, delivered via the eCALL tutor, had improved learnability of both spatial and nonspatial preposition polysemes.

Sixty-three Chinese-speaking learners of English, ages 15 to 16, were assigned to three treatment groups (schematic diagram feedback group, rule and exemplar feedback group, and corrective feedback group) and one test-only control. The three treatment groups received training on 12 prepositional polysemes selected from the prepositions in, at, and over. For each training trial, a picture stimulus and two contrasting sentences were provided to participants (see supplementary material S1 and S2 for details). The cue-focusing design, motivated by CM, aimed to draw participants’ limited attentional resources to processing the two most competing cues, the target and the distractor prepositions. The major differences between the three treatment groups lay within the feedback that learners received: (a) the schematic diagram feedback was composed of a schematic diagram to reflect the spatial configuration of the target preposition, accompanied by a brief explanation on usage; (b) the rule and exemplar feedback consisted of a metalinguistic rule explanation paired with three example sentences to illustrate usage; and (c) in the corrective feedback group, learners would only receive feedback on whether their answer provided was correct. The test control group received online training on English articles instead.

A sentence-level cloze task and a translation task were adopted as measures. Pretest, training, and posttest were all delivered online (See Wong et al., Reference Wong, Zhao and MacWhinney2018 for a full description of the instruction and testing). Repeated measures analysis of variance (ANOVA) confirmed that cue focusing was an effective instructional method. In addition, the gains of the three treatment groups were significant at the posttest. However, there was no interaction between time and types of instruction on the cloze task. The superiority effect of the CL approach was observed only on the translation task where the Bonferroni post hoc tests indicated the main source of variation was between the schematic feedback group and the corrective feedback group only. In other words, rich feedback (schematic feedback; rule and exemplar feedback) contributed to an increase in productive knowledge, as measured by the translation test, whereas minimal feedback (corrective feedback) would already be sufficient to promote receptive knowledge, as measured by the cloze test. For the learning of spatial and nonspatial polysemes, learners across the three treatment groups demonstrated significantly better learning for spatial polysemes over nonspatial polysemes on the cloze test. Yet, the results from the translation test demonstrated significant yet comparable gains between the two types of polysemes.

Rationale for the replication study

Findings from the original study demonstrated a superiority effect of CL-informed instruction solely over one of the traditional groups (i.e., minimal feedback), specifically in the translation task. Further, no delayed posttest was administered. Therefore, the authors could not confirm whether the positive learning outcomes were sustained over time. Replicating the study would therefore enable us to elicit more evidence for interpretation.

Moreover, in the original analysis, repeated measures ANOVAs were used, and the dependent variable was the aggregated mean scores. These traditional approaches to variance analysis continue to be widely adopted in applied CL research (Boieblan, Reference Boieblan2023; Colasacco, Reference Colasacco2019; Jacobsen, Reference Jacobsen2018; Qin et al., Reference Qin, Wu and Zhong2023). However, repeated measures ANOVA are only suitable for simple experimental designs, as several assumptions have to be met before conducting the analysis, including normally distributed variance, sphericity (e.g., constant variance across time points), no outliers in any of the repeated measurements, and balanced numbers across comparison groups (Jaeger, Reference Jaeger2008). These assumptions are often unrealistic and thus pose great challenges on L2 interventional studies. Given the recent advances in computational software, we believe a computational turn is necessary and inevitable not only for CL (Divjak & Milin, Reference Divjak, Milin, Díaz-Campos and Balasch2023) but also for applied CL. Our focus of the reanalysis was to compare the outcomes derived from Bayesian MEM with those obtained through traditional ANOVA in the original study. To utilize the Bayesian mixed effects logistic models for the present study, the reanalysis was also a necessary step to facilitate prior distribution setting (Bürkner, Scholz, & Radev, Reference Bürkner, Scholz and Radev2023).

Unlike repeated measures ANOVA, MEM can address concerns, such as nonspherical error variance and heteroscestadicity, which are commonly observed in natural datasets. More importantly, MEM accounts for subjective variation by calculating several intercepts (means), one for each subject (patient), thereby teasing apart random effects from the fixed effects of parameters (e.g., predictors of an outcome variable), increasing the accuracy of interpretation on fixed effects. Because MEM can model variation explicitly (e.g., adjust estimates for imbalanced sampling or to study variation), preaveraging or averaging data to construct variable would not be necessary (McElreath, Reference McElreath2015). Also, the sigmoid function in logistic regression tapers the outliers and thus the presence of outlier data points does not exert great impact on the model performance compared to traditional ANOVA.

There is an increasing number of instructed L2 studies adopting MEM (Bovolenta & Williams, Reference Bovolenta and Williams2023; Hwang, Reference Hwang2023; Puimege, Perez, & Peters, Reference Puimege, Perez and Peters2023); however, most of these adopted a frequentist perspective, for example, NHT. Data analysis and interpretation using the NHT approach have begun to draw concerns in applied linguistics research (Norouzian, de Miranda, & Plonsky, Reference Norouzian, de Miranda and Plonsky2018; Norris, Reference Norris2015; Plonsky, Reference Plonsky and Plonsky2015). A frequentist approach draws conclusions based on facts obtained from the observed data at hand only (i.e., excluding any prior information) and identifies probability with frequency, premising on imaginary data resampling (McElreath, Reference McElreath2015). This frequentist perspective continues to motivate researchers and reviewers to aim for large sample sizes. For example, if the coin toss experiment was performed 3,000 times or tossing continued until 300 tails were observed, then the total probability, often denoted as the p value, could fall below a critical threshold (typically set at 0.05). In this context, p < .05 suggests that the likelihood of obtaining such results by random change is low, allowing a rejection to the null hypothesis. Hence, in the framework of NHT, a power analysis must be conducted to determine the optimal sample size of a study before data collection begins. In contrast, the Bayesian analysis yields a posterior distribution that updates its belief with every observation, making it less sensitive to sample size. Therefore, taking a Bayesian approach would be more appropriate and realistic for L2 intervention studies (Garcia, Reference Garcia2021, Reference Garcia2020; Wirtz & Pfenninger, Reference Wirtz and Pfenninger2023), considering the logistic and cost constraints inherent in the real-world scenarios.

From a frequentist perspective, the parameter used (e.g., θ = 0.5 the two-sided coin is believed fair) to estimate the population is assumed to be fixed where there is only a single true parameter that is estimated and is not modeled as a probability distribution. When new data becomes available, it is used to perform statistical tests and predictions. On the other hand, Bayesian inference takes into the account initial estimates, the prior (i.e., prior information from the original study), which are both expressed in terms of probability distribution. In replication studies, incorporating prior information from the original research is crucial and is best achieved through Bayesian analysis. Neglecting such valuable information would be counterproductive, provided that the researcher acknowledges and appreciates the significance of the original study.

While it is our goal to replicate Wong et al. (Reference Wong, Zhao and MacWhinney2018) as closely as possible in most major aspects of their research methodology in order to either confirm or expand on their findings, we make two deliberate modifications to the setup as follows:

• • Statistical data analysis: Rather than performing NHT for separate effects as in Wong et al. (Reference Wong, Zhao and MacWhinney2018), we use Bayesian MEM (Prange & Wong, Reference Prange and Wong2023). One of the biggest advantages of the Bayesian approach is that we can incorporate the prior information from the original study into the model through prior distributions, leading to more robust and realistic inferences. While one can primarily obtain point estimates and confidence intervals (CIs) using frequentist MEM, the p values and CIs could be misleading and do not provide useful information for decision making in real life. Gelman and Hill (Reference Gelman and Hill2007) pointed out that “all multilevel models are Bayesian in the sense of assigning probability distributions to the varying regression coefficients” (p. 276). Therefore, the Bayesian perspective fits squarely with multilevel modeling and its interpretation. Moreover, the Bayesian method allows us to understand not only the central tendency (e.g., mean) but also the uncertainty and shape of the distribution of each parameter. More importantly, one can directly state the probability that a parameter is greater than zero or that one condition is better than another. As a result, this is more intuitive and informative for decision making, which is particularly useful for L2 interventional research.

• • Inclusion of a delayed posttest to examine whether the learning effect (if any) is sustainable over time.

Because the experimental manipulation of this replication is identical to that of the original study (as shown in Table 1), we addressed the research questions (RQs) as proposed in the original study but with greater precision and clarity:

Table 1. Comparison of methodologies in the original and replication studies

* See supplementary material S1 for more details regarding each pair of spatial and nonspatial polysemes of the three target prepositions.

^† The evaluation was provided by the English Panel Heads of the schools based on participants’ internal English test performances.

Response ~ Condition.F * Test.F * Idiomaticity.F *School.F + (1 + Condition.F | Participant) + (1 | ItemNumber)

Because we recruited participants from two different EMI schools (see Participants section for more details), we included School as a fixed factor in our model. Using leave-one-out (LOO) cross-validation as the predictive model checking, our model demonstrated the best predictive performance for the cloze test data, with the highest expected log predictive density (elpd_loo = –5949.322) and the lowest leave-one-out information criterion (LOOIC = 11898.64). Similarly, for the translation data, our model was the best fit, with the highest elpd_loo (–7948.36) and the lowest LOOIC (15896.71). In other words, our model strikes the best balance between model fit and complexity. Markov chain Monte Carlo sampling was conducted with four chains, each running for 4,000 iterations for cloze test data and 5,000 iterations for translation test data. The adapt_delta parameter was set to 0.95 to enhance convergence due to the complexity of the full models. Computational resources were allocated with six cores. We report mean posterior point estimates and associated probabilities for each parameter, along with the 95% highest density interval, which is a type of credible interval (CrI). A CrI is a range of values within which an unknown parameter θ lies with a certain probability, given the observed data and prior distribution. A CI, originating from the frequentist perspective, is a range of values that, if we were to repeat the experiment many times, would contain the true parameter θ value in a certain proportion of those experiments (e.g., 95% of the time). In other words, the CrIs from the Bayesian approach provide a probability statement about the parameter itself, while the CIs only offer a long-run frequency statement about the procedure. For instance, a 95% CrI directly informs us of the probability that the mean lies within that interval, given the observed data and prior information. However, upon calculating the sample mean and standard error (SE) from the observed data, a 95% CI only tells that if we repeated this sampling process many times, 95% of the calculated intervals would contain the true mean. Therefore, the Bayesian approach gives us a more intuitive understanding of the observed data (McElreath, Reference McElreath2015).

Obtaining highly informative priors from Wong et al. (Reference Wong, Zhao and MacWhinney2018)

One of the most important motivations of taking the Bayesian approach for this replication study is that it allows us to incorporate established knowledge from previous research into our Bayesian MEM, thereby improving the robustness and precision of the estimates (Kruschke, Reference Kruschke2018; McElreath, Reference McElreath2015). To the best of our knowledge, this study is among the first to employ highly informative priors in a replication study within L2 intervention research, particularly in applied CL. In this replication study, we have selected a normal distribution for all our priors with means and standard deviations specified for each parameter, as indicated in Table 2. Additional normal priors were specified for interaction terms and variance components. The data and code necessary to reproduce the analyses reported in this article are available at Open Science Framework (OSF) https://osf.io/fxb89/.

Table 2. Normal priors obtained for the cloze and translation test

In addition to the LOO analysis, posterior predictive checks were conducted to assess the fit of the model to our observed data. While both analyses indicated that the inclusion of informative priors improved model performance, the models without priors showed only a slight drop in performance that might not be practically significant. In other words, the priors were appropriately selected and were consistent with the data.

Cloze test data

Interaction effects of predictors

The primary focus of this analysis was to examine the interaction effects between different instructional conditions, test points, idiomaticity, and schools (Figure 6). The interaction between Condition and Test was credible for both the posttest and the delayed posttest, with the schematic feedback group demonstrating the biggest improvement in accuracy from pretest to posttest (β = 1.18, SE = 0.16, 95% CrI [0.88, 1.49]). This corresponds to a probability of approximately 0.76, suggesting that the schematic feedback group benefitted more than the traditional and corrective groups. Specifically, the improvement for the schematic feedback group was higher (β = 1.18) compared to the traditional (β = 0.92, SE = 0.16, 95% CrI [0.61, 1.22], probability ≈ 0.72) and corrective (β = 0.31, SE = 0.16, 95% CrI [–0.01, 0.62], probability ≈ 0.58). Similarly, the schematic feedback group demonstrated the most learning at the delayed posttest (β = 1.06, SE = 0.36, 95% CrI [0.37, 1.78], probability ≈ 0.74), suggesting the benefits of CL-informed instruction were sustained over time. While the traditional feedback group showed a sustained effect at the delayed posttest (β = 0.71, SE = 0.36, 95% CrI [0.01, 1.43]), the probability has dropped from ≈ 0.72 to ≈ 0.67 at the delayed posttest. Although there was no credible interaction found between Condition, Test, and Idiomaticity (β = 0.84, SE = 0.47, 95% CrI [–1.77, 0.07]), strong evidence was observed for the schematic feedback group at the delayed posttest (β = 1.78, SE = 0.74, 95% CrI [0.32, 3.20]) in a four-way interaction between Condition, Test, Idiomaticity, and School. For students in the lower-performing school, the schematic feedback was particularly effective in facilitating the learning of spatial polysemes (probability ≈ 0.86). However, the benefits of the schematic feedback took longer to unfold as such evidence was not observed at the posttest (β = 0.47, SE = 0.44, 95% CrI [–0.39, 1.34], probability ≈ 0.62). In summary, the interaction effects indicate that the schematic feedback group was the most effective overall, particularly in lower-performing schools and for spatial items in the long run. For detailed estimates of all parameters and their associated probabilities, please refer to supplementary material S3.

Figure 6. Forest plot of interaction effects in Cloze test. This plot displays the estimated interaction effects for different instructional conditions (Corrective, Schematic, Traditional) at the posttest and delayed posttest, including interactions with idiomaticity (Spatial, Non-spatial) and school (low performing vs. high performing). Each point represents the posterior median estimate, with the inner and outer intervals indicating the 66% and 95% credible intervals, respectively. The interactions illustrate how each instructional method impacts accuracy over time and in different contexts.

Figure 7. Forest plot of interaction effects in translation test. This plot displays the estimated interaction effects for different instructional conditions (Corrective, Schematic, Traditional) at the posttest, including interactions with idiomaticity (Spatial, Non-spatial) and school (Low performing, High performing). Each point represents the posterior median estimate, with the inner and outer intervals indicating the 66% and 95% credible intervals, respectively. The interactions illustrate how each instructional method impacts accuracy over time and in different contexts.

Translation test data

Discussion and conclusions

This replication study addressed the first research question regarding the role of cue focusing on preposition learning. Previous eCALL studies have suggested that practice and cue focusing promote L2 grammar learning (Presson, Davy, & MacWhinney, Reference Presson, Davy, MacWhinney and Schwieter2013; Presson, MacWhinney, & Tokowicz, Reference Presson, MacWhinney and Tokowicz2014), but none of those studies has examined the learning of prepositions. Similar to the original study, we did not find testing effect, and all treatment groups improved substantially upon training with the eCALL tutor. More importantly, with the added delayed posttest, we were able to observe internalization of accurate form–function mappings. As learners were repeatedly exposed to the polysemic contrasts, they were able to differentiate between the target and its highly competing cue.

Through combining the two usage-based frameworks to address the second research question, Wong et al. (Reference Wong, Zhao and MacWhinney2018) demonstrated synergetic effect of CL and CM on L2 preposition learning. Interestingly, our findings aligned with the reanalysis of Wong et al.’s (Reference Wong, Zhao and MacWhinney2018) using Bayesian MEM, in which we observed a superiority effect of the CL-informed instruction in the cloze task at the posttest (β = 1.00, SE = 0.22, 95% CrI [0.57, 1.43], probability ≈ 0.73). While the authors of the original study observed a trend of the schematic feedback group demonstrating the most learning, the difference between conditions was not significant, as p > .05. This concurs with the reanalysis using the Bayesian approach as the traditional feedback group and the corrective feedback also demonstrated a comparable probability of ≈ 0.70 and ≈ 0.66 to improve respectively. Nevertheless, this replication study provides further evidence that the superiority effect of CL-informed instruction sustained over time whereas the improvement of the traditional instruction at the posttest did not. We believe that the participants were attuned to the conventional approach, in which rule-of-thumb definitions and examples were provided, prompting them to adopt the memorization strategy. However, without deeper conceptual stimulation, learners might fail to consolidate and internalize the form-meaning mappings in long run (Negueruela, Reference Negueruela2003). This might also explain why many L2 learners, even those with advanced proficiency, fail to fully acquire the English preposition system (Tyler, Reference Tyler2012).The translation data of the present study demonstrated a similar trend to that of the original study, in which the schematic feedback group demonstrated the highest probability (≈ 0.66) of making improvement, but it was not substantially different from those receiving traditional (≈ 0.59) or corrective feedback (≈ 0.62).

Unlike the cloze test, the positive instructional effects of the CL-informed instruction did not sustain over time. This finding did not concur with the findings of Wong (2023) where learners who received CL-informed instruction significantly outperformed the corrective feedback group at the delayed posttest. One possible reason for the substantial regression could be age related. The participants were junior secondary students; thus, the CL-inspired pedagogy might take more time to consolidate. Further, unlike sentence-level cloze task, translation tasks are uncommon for junior L2 learners, which might also have posed an additional cognitive challenge. Although we were unable to show sustainable learning effect at the production task as seen in previous research, varying learning outcomes were repeatedly found in different assessment tasks, thereby urging L2 researchers to incorporate multiple task types to better assess instructional effects and students’ learning trajectories.

For the learning of spatial and nonspatial polysemes, as in the original study, we observed no interaction effects between condition, time, and idiomaticity in the cloze task. However, with the newly added factor (School), the replication study found that CL-informed instruction benefits the learning of spatial polysemes among low-performing students. As for the translation task, learners from the traditional and corrective feedback groups did substantially better for spatial polysemes, whereas learners from the schematic feedback group showed the poorest performance. This observation is not consistent with previous findings, as the CL-informed instruction often shows better learning. Further, the findings on spatial and nonspatial polysemes did not replicate Wong’s (2023) findings where learners obtained higher accuracy for nonspatial polysemes in the receptive task (e.g., grammaticality judgement test). We observed a general trend that the three types of instruction benefited the learning of spatial polysemes, especially among lower-performing learners. Although many might think nonspatial polysemes pose a greater challenge due to their abstractness, spatial polysemes are much more versatile, increasing the difficulty for learners to choose in various contexts. For instance, the prepositions over, on, onto are all valid options for the sentence James put the blanket __ his bed. Therefore, when provided with options, learners might find nonspatial polysemes easier to recall than spatial polysemes as they are less abstract (Langacker, Reference Langacker2008). The substantial learning of spatial polysemes might indirectly suggest learners had not received sufficient help previously; therefore, the different instructions, despite variation, were useful in training spatial uses. Overall, the replication study provided further support for incorporating CL concepts into L2 preposition teaching and learning. With the Bayesian perspective, we gain a more comprehensive view of learning trajectories with increased flexibility, particularly on variable identification.

Future replication research

Our replication study marked an effort to apply Bayesian analysis to experimental research within applied CL. Subsequent replication endeavors should consider integrating prior information derived from the original research through the Bayesian lens, contrasting with the frequentist approach that tends to overlook insights from initial studies. In essence, researchers should prioritize the incorporation of informative priors in replication research to optimize the advantages inherent in adopting the Bayesian method. Moreover, posterior predictive checks should be performed for Bayesian analysis, as they are useful in assessing the validity of models, pinpointing any model inadequacy, quantifying lack of fit, and informing model improvement. In addition, when the original dataset is available, a replication study is advised to reanalyze the original data using the Bayesian approach, facilitating direct comparisons with new findings. Crucially, to foster robust replication research, it is essential for future studies to embrace open science principles, e.g., sharing complete sets of pedagogical intervention materials and collected data as well as the source codes for built models. Finally, future replication studies could explore how different proficiencies may lead to different degrees of uptake from various types of instruction (i.e., feedback types).