Sampling dependence inherent in L2 studies is ignored

As Norouzian (Reference Norouzian2020) has discussed, the design of empirical L2 research has grown over time in complexity. Specifically, L2 research often tends to involve one or more of the following design aspects: (a) inclusion of multiple treatment groups along with a control/comparison group, (b) measurement over multiple occasions, and (c) measurement over multiple outcomes.

Take, for instance, Shintani and Ellis (Reference Shintani and Ellis2013). The study investigated the effect of direct corrective feedback (D) and the metalinguistic explanation (M) on ESL (English as a second language) writers’ implicit knowledge of the English indefinite article. To do so, the researchers adopted the design depicted in Figure 1.

Figure 1.

Design of Shintani and Ellis (Reference Shintani and Ellis2013).

As shown in Figure 1, Shintani and Ellis (Reference Shintani and Ellis2013) randomly assigned their classes (RC) to three groups (i.e., D, M, and control). Following an initial observation (O₁) of their participants’ knowledge of English indefinite article, they provided their intended treatments to each group. The two follow-up observations (O₂ and O₃) were made to measure the possible change in participants’ implicit knowledge of the English indefinite article over time in each group.

From this study, one can commonly obtain six effect-size estimates, in this case standardized mean differences (SMD; i.e., Hedges’s g). Namely, we can obtain two pre-feedback effect estimates at O₁, two immediate, post-feedback effect estimates at O₂, and finally two delayed, post-feedback effect estimates at O₃, all comparing the treatment groups with the same control group.

However, these effect-size estimates are likely dependent on (i.e., correlated with) each other due to the design aspects (A) and (B) adopted in Shintani and Ellis (Reference Shintani and Ellis2013). For instance, at each observation time, the effect-size estimates across the treatment groups are likely dependent due to being obtained by comparing the performance of each treatment group against that of the same control group (design aspect A). Being a pivot point, if the control group put up a superior performance, all of the treatments’ effects would become smaller. Conversely, if the control group put up a low performance, all treatments’ effects would become larger. Thus, treatments’ effects could vary together (i.e., be correlated) to a measurable degree as a function of their common control group’s performance.

Additionally, the effect-size estimates across the occasions for each treatment group are also likely dependent due to being obtained over repeated measurements (design aspect B). Specifically, because the same participants were measured over the three occasions (O₁, O₂, and O₃) in each group, their measurements across these occasions contain an estimable degree of correlation, likely more strongly so between the adjacent occasions than those further apart (Trikalinos & Olkin, Reference Trikalinos and Olkin2012). The same correlations carry forward to any statistic that is obtained from these measurements’ summaries such as effect-size estimates inducing dependence among them.

Indeed, Shintani and Ellis (Reference Shintani and Ellis2013) also measured their participants using an additional grammar correction test as a way to examine the participants’ explicit knowledge development on two occasions (O₁ and O₃). This would be considered a second outcome in the study. Thus, a new set of four effect-size estimates may be obtained from the same participants on this additional grammar correction test, if explicit knowledge development is of interest to the meta-analyst. If so, then this new set of four effect-size estimates on explicit knowledge development would be correlated with each other due to the design aspects A and B. But now these effect-size estimates on explicit knowledge development for each treatment group would also be correlated with the previous set of effect-size estimates on implicit knowledge development (design aspect C).

Because correlated effect-size estimates often stem from the use of the summary data from the same sample of participants, this form of inherent dependence in the effect-size estimates is sometimes referred to as sampling dependence (Stevens & Taylor, Reference Stevens and Taylor2009). Note that sampling dependence can occur in correlational L2 studies, too. For instance, if a correlational study examined the relationships of multiple subscales of L2 motivation with L2 achievement, the reported correlation coefficient estimates (i.e., effect-size estimates) for the subscales would likely be correlated with each other. This is because the same sample of participants has been measured on multiple subscales (design aspect C).

Classic methods of meta-analysis are ill-equipped to account for the sampling dependence inherent in effect-size estimates obtained from studies with one or more of the design aspects discussed above. Consequently, meta-analysts often find themselves making ad hoc decisions that oversimplify the nature of L2 studies (for a review of these decisions, see Hedges, Reference Hedges, Cooper, Hedges and Valentine2019). In addition, ignoring the sampling dependence threatens the validity of inferential conclusions regarding the summary effects and accurately estimating the magnitude of the heterogeneity in the underlying effects (Pustejovsky & Tipton, Reference Pustejovsky and Tipton2022).

Hierarchical dependence in underlying effects of L2 studies is ignored

That L2 studies can often produce multiple effect-size estimates due to the design aspects discussed in the previous section introduces sampling dependence among them prior to their meta-analysis. Another form of dependence may arise when we intend to envision a meta-analytic model for such effect-size estimates obtained from the literature.

Part of the job of a meta-analytic model is to statistically link the effect-size estimates we obtain from the literature to their underlying population effects (i.e., true effects). In its simplest form, when a group of L2 studies can each provide multiple effect-size estimates, then, by virtue of belonging to the same study, their underlying true effects are likely more similar in each study than those in other studies. This similarity amounts to a measurable degree of overlap (i.e., correlation) among the true effects in each study forming clusters of true effects across the studies. Because this second form of correlation occurs owing to the clustered nature of the underlying true effects envisioned for the meta-analysis of studies with complex research designs, it is sometimes referred to as hierarchical Footnote ² dependence (Stevens & Taylor, Reference Stevens and Taylor2009).

Classic methods of meta-analysis, which to date are the norm in the field, are unable to adopt a meta-analytic model for L2 studies that sufficiently allows for the hierarchical dependence among their underlying true effects. This inability has negative effect on the conclusion validity of our meta-analytic findings such as incorrect precision in average effect estimates, inflated Type I error rates, incorrectly narrow confidence intervals, and incomplete delineation among the sources of variation in the true effects (Matt & Cook, Reference Matt, Cook, Cooper, Hedges and Valentine2019).

Comparison across categories of substudy moderators may be invalid: The k situation

Perhaps one of the fundamental needs of L2 meta-analysts is to be able to compare meta-analytic findings across the theoretically distinct categories of their moderators. As Norouzian (Reference Norouzian2021) discussed, L2 moderators such as the type of treatment (e.g., direct feedback vs. meta-linguistic explanation) are often found at the substudy levels of L2 studies. That is to say, such moderators can vary within the same studies. For instance, L2 studies can each examine multiple types of treatment. Thus, comparing different types of treatment in a literature often entails obtaining multiple effect-size estimates from a combination of the same and different studies. This results in a situation where the number of effect-size estimates (usually denoted by “k” in L2 meta-analyses) obtained across the categories of a moderator (e.g., types of treatment) exceeds the number of the studies associated with those categories. For example, in a meta-analysis where 20 studies have employed direct feedback and/or indirect feedback, we may be able to obtain 20 effect-size estimates for direct feedback and 14 effect-size estimates for indirect feedback (i.e., 34 effect-size estimates from 20 studies). We refer to this situation as the “k situation.”

In the presence of a k situation, the effect-size estimates across the theoretically distinct categories of a moderator are not independent of each other.Footnote ³ For instance, effect-size estimates that make up the direct feedback category and those that constitute the meta-linguistic feedback category are not independent of each other. This is because some of these effect-size estimates have come from the same studies and carry along their sampling dependence. Placing these dependent effect-size estimates into seemingly independent categories (e.g., direct feedback vs. meta-linguistic explanation) and then comparing across them via traditional tests (e.g., Q _between test) to answer a research question (see the next section) may lead to invalid conclusions depending on the extent of the k situation.

Classic methods of meta-analysis are ill-suited to address the k situation. As noted above, this inability is, once again, rooted in the lack of a framework to account for the dependence among the effect-size estimates classified into seemingly independent categories (see Plonsky et al., Reference Plonsky, Hu, Sudina, Oswald, Mackey and Gass2023).

Findings may be biased

Meta-analysts are often interested in examining the relation between the effect of a phenomenon of interest (e.g., the effect of feedback on written accuracy) and a range of other substantively moderating variables (Shadish et al., Reference Shadish, Cook and Campbell2002). Cronbach’s (Reference Cronbach1982) UTOS paradigm is what is often invoked as an organizing framework for this purpose (Becker, Reference Becker, Benbow and Lubinski1996; Cook, Reference Cook, Albrecht and Otto1991; Matt & Cook, Reference Matt, Cook, Cooper, Hedges and Valentine2019). Under this framework, substantively moderating variables in the literature are divided into four dimensions, as applicable—namely, participants or more broadly units (U), treatments (T), outcomes (O), and settings (S). The meta-analyst then explores the association between instantiations of each dimension and the effect of a phenomenon of interest (Cook, Reference Cook1993, Reference Cook and Alkin2013).

However, to validly estimate such substantive associations, an additional nuisance dimension needs to be accounted for as well. This additional nuisance dimension merely reflects the nonsubstantive and/or methodological (collectively denoted by M) differences across the studies selected for meta-analysis. As a result, the M dimension itself is not of theoretical interest or a target for meta-analysis. Rather, this dimension acts as a regulator to ensure obtaining unbiased substantive associations (Aloe et al., Reference Aloe, Thompson, Reed, Albers, Shlonsky and Mildon2020; Becker, Reference Becker, Sampson, Dozier, Osborn and Lenz2017; Becker & Aloe, Reference Becker and Aloe2008; Cheung & Slavin, Reference Cheung and Slavin2016).

For example, L2 studies in a meta-analysis sample might likely differ in several substantive instantiations of UTOS dimensions (denoted by lower-case letters) such as in (1) the characteristics of participants (u’s, e.g., their proficiency level), (2) the types of treatment (t’s, e.g., types of corrective feedback), (3) the types of outcome (o’s, e.g., types of linguistic features targeted), and (4) the context of the study (s’s, e.g., language institute vs. university). However, the same L2 studies might also differ in ways that merely reflect their nonsubstantive and/or methodological differences. For instance, whereas some studies might have been able to randomly form their study groups (a true experiment), others might have been unable to do so purely by chance, for example, due to logistical constraints (i.e., a quasi-experiment). Or for example, some studies might have taken longer to complete and others have taken less time to complete (i.e., difference in study length). Such nonsubstantive and/or methodological differences need to be always accounted for in the background so the substantive association between UTOS dimensions and the effect of the phenomenon of interest can be validly assessed in the foreground (Berlin & Antman, Reference Berlin and Antman1992; Tipton et al., Reference Tipton, Pustejovsky and Ahmadi2019).

Classic methods of meta-analysis are not readily capable of accommodating both the nonsubstantive and/or methodological confounds along with the substantive moderators. This incapability is due to the classic methods’ fragmented approach toward moderators in a meta-analysis. That is, classic methods can commonly take one moderator at a time. This limitation is evident when, for example, we see tables presenting the results of numerous isolated moderator analyses in an L2 meta-analysis disregarding the connections between the substantive features of interest and the methodological choices that produced them. Consequently, classic methods of meta-analysis may not provide a valid picture regarding the association between the UTOS moderators (e.g., proficiency) and the effect of the phenomenon of interest (e.g., teacher feedback on written accuracy). In addition to biasing these associations, such a shortcoming in classic methods of meta-analysis can also bias the amount of heterogeneity in the effects explained by the UTOS moderators and, therefore, that which cannot be explained by the UTOS moderators.

Multivariate, multilevel meta-analysis

The flexibility of regression analyses over the commonly used ANOVA techniques is well established in L2 research (Plonsky & Ghanbar, Reference Plonsky and Ghanbar2018; Plonsky & Oswald, Reference Plonsky and Oswald2017). Also well documented in L2 methodological literature is the fact that multilevel models (MLMs) add new capabilities to the regression analyses by accounting for the various types of dependence in the data (Cunnings, Reference Cunnings2012; Gries, Reference Gries2021). Multivariate, multilevel meta-analysis (3M) is a flexible approach to meta-analysis that builds on the capabilities of MLMs in the context of meta-analysis.

In its simplest formulation, 3M allows for examining the association between the average size of the effect (dependent variable) of an L2 phenomenon of interest (e.g., effect of teacher feedback on written accuracy) and a set of (M)UTOS moderators (independent variables) taking into account two important facts. First, studies could produce multiple dependent effect-size estimates prior to their meta-analysis (sampling dependence). Second, their corresponding underlying true effects are likely more similar in each study than those in other studies forming clusters across the studies (hierarchical dependence). The application of these two features in 3M is conceptually explained in the next two sections.

Hierarchical dependence in 3M

In theory, 3M is capable of envisioning many potential levels of hierarchical dependence for the underlying true effects of our effect-size estimates. For instance, one assumption to envision might be that the true effects of an L2 phenomenon of interest (e.g., effect of corrective feedback on written accuracy) are more similar in each study than those in other studies (study as a level). But when, for example, treatment groups in each study are tested on multiple occasions, as in Shintani and Ellis (Reference Shintani and Ellis2013) discussed earlier, one might also reason that the true effects for each treatment group in a study are likely more similar than those for other treatment groups in that study (treatment group as a level).

In practice, however, (a) limitations in the number of studies and their effect-size estimates and/or (b) having possibly several (M)UTOS moderators at various hierarchical levels (e.g., study level and treatment level) of the literature can curtail one’s ability to encode the many potential levels of hierarchical dependence into a 3M model (Hedges, Reference Hedges, Cooper, Hedges and Valentine2019). For most purposes in L2 research, a general form of hierarchical dependence might simply acknowledge that the underlying true effects of our effect-size estimates are likely more similar in each study as a whole than those in other studies. The product of such an acknowledgement is that we can obtain the random variation in studies’ average true effects (study-level variation) and the random variation in any given study’s own true effects (effect-level variation). We say “any given study” because the effect-level variation is somewhat averaged across all studies, thus representing such variation in any given study. Figure 2 symbolically depicts these two sources of random variation with the first study at the top figuratively used as a “given study.”

Figure 2.

Two sources of random variation captured by 3M.

Importantly, these two sources of variation are called random because they account for the variation in the underlying true effects of an L2 phenomenon of interest that could not be systematically attributed to any of the (M)UTOS moderators used in a 3M model (e.g., variation in magnitude of true effects beyond that attributable to proficiency level differences). This formulation reinforces the belief that despite potentially systematic reasons (i.e., [M]UTOS moderators) for variation in the magnitude of true effects for an L2 phenomenon of interest, in the end studies as a whole and their constituent effects could each potentially have their own unique but random differences captured by these two sources of random variation.

For purposes of describing such a model, it is customary to refer to the studies’ average true effects as Level 1 (highest level), the underlying true effects of the studies’ effect-size estimates as Level 2 (middle level), and the studies’ own effect-size estimates as obtained from the literature as Level 3 (lowest level). Thus, this would be an example of a three-level, 3M model.

Applying 3M to a complex literature

To promote the use of 3M in L2 research, we apply it to one of the most polemic areas in instructed second language acquisition (ISLA) research: the effectiveness of written corrective feedback (WCF) in improving L2 writers’ accuracy. Because of the nature of the so-called WCF debate, a large and often complex body of studies has contributed to the WCF literature. For substantive purposes, Brown, Liu, & Norouzian (Reference Brown, Liu and Norouzian2023) extensively reviewed the details of and meta-analyzed a large set of WCF studies. For the methodological purposes of this article, we randomly selected 35 studies from the full set of the WCF studies in Brown et al.

Design complexities of WCF studies

The WCF literature includes studies with a variety of design features. From those features, four in particular are worth mentioning here. First, WCF studies usually employ multiple treatment groups (group) receiving different types of WCF all compared against a control/comparison group. Second, the studies often involve multiple measurements of the same participants (time) to assess the durability of WCF’s effect on learners’ written accuracy. Third, they can also measure multiple outcomes to study WCF’s effect on multiple types of linguistic errors (err_type). Finally, given the debates over the treatment of the control group’s members (e.g., whether they should only engage in writing practice or receive feedback on content) and its subsequent influence on the results of WCF studies (Truscott, Reference Truscott1996, Reference Truscott2004), occasionally studies (Van Beuningen et al., Reference Van Beuningen, De Jong and Kuiken2012) tend to use more than one type of control group (cont_type) against which all WCF-receiving groups are compared. Figure 3 examines, classifies, and visually displays these design features for our selected WCF studies (to reproduce Figure 3, see: https://github.com/rnorouzian/i/blob/master/1.r).

Figure 3.

Design complexity in WCF literature.

In Figure 3, each panel represents a unique combination of the design features found in the WCF literature. Changes in color denote multiplicity in the design feature titled above each column (e.g., group, time). For instance, we see that all the studies in the WCF literature have measured their participants on more than one occasion (pre- and posttests). This is because across all the panels, time always changes in color. In contrast, in panel F, there are 11 studies that have used only one control group, one treatment group, and one error type. The numbers in each box denote the identifiers (e.g., 0 for cont_type = the control group only engaging in writing practice) or indices (1 for group = the first treatment group in the study) used by the meta-analyst to code these design features in each study in the coding sheet.

Inspecting Figure 3 a bit further, we also realize that the use of multiple control groups (cont_type) is very infrequent in our WCF study pool, occurring in only two studies (Panel A). Conversely, having multiple treatment groups is a quite common design feature in the WCF studies appearing in twenty studies (Panels A, B, and C). Additionally, studying WCF’s effect on multiple error types (outcomes) occurs in a handful of studies (Panels B and D). Other design features can also be examined in a similar manner. However, notice that all studies have invariably produced multiple effect-size estimates, as indicated by the multiple row indices (effect; row-wise location of each effect-size estimate in the data) in each panel.

In short, these design complexities may give rise to (a) having several effect-size estimates from each WCF study, (b) the fact that these estimates are likely dependent on one another in a variety of ways in each study, and potentially, and (c) the existence of both systematic and random variation (i.e., heterogeneity) in the true effects of WCF on written accuracy both at the study and the effect levels. From these three consequences, the last one differs in nature from the other ones in that it involves using a range of (M)UTOS moderators backed by theory and/or prior research. Specifically, to explore the degree to which the heterogeneity in the magnitude of the true effects of WCF might be due to the features systematically distinguishing the studies as a whole and/or finer elements within the studies (e.g., participants) from each other, a number of candidate (M)UTOS moderators may be considered. A selection of these moderators is provided in the next section.

The (M)UTOS moderators

In theory, there may be a large number of (M)UTOS moderators that can potentially account for the systematically explainable heterogeneity in the magnitude of the true effects of WCF. In practice, however, a smaller set may be found and consistently coded in the WCF literature along each dimension of (M)UTOS. For the purposes of this study, Table 1 provides 10 (M)UTOS moderators, two instantiations of each (M)UTOS dimension. Table 1 also includes the general as well as the coding definitions of these moderators (for a fuller list of potential moderators see Brown et al., Reference Brown, Liu and Norouzian2023).

Table 1.

Coding scheme for the selected moderators

Although space limitation prevents us from applying 3M to all the moderators listed in Table 1, we try to offer a road map by applying the 3M to a substantive moderator (i.e., proficiency) and examine the critical decision points encountered along the way. A description of the data set used for this purpose is presented in the next section.

The wcf data set

The data set used in this article is called wcf. In addition to the (M)UTOS moderators described in Table 1, the wcf data set includes 245 effect-size estimates in the standardized mean difference (SMD) unit (Hedges’s g), which is a common metric used in applied linguistics research. As shown in Table 2, all studies contributed more than one effect-size estimate to the database such that, on average, each WCF study provided seven effect-size estimates.

Table 2.

Distribution summary of effect-size estimates across studies

These effect-size estimates are all corrected for a systematic bias inherent in them, especially when obtained from small-sized group comparisons. As is customary, we refer to these bias-corrected effect-size estimates as Hedges’s g and denote them by g in the wcf data set.

Accompanying these effect-size estimates are their sampling variances, denoted by var in the wcf data set. Sampling variances are simply the squared version of the estimates’ standard errors (Viechtbauer, Reference Viechtbauer2007). They denote how dispersed, in the variance metric, the distribution of effect-size estimates might be, if the exact same group comparisons used to obtain the effect-size estimates were to be replicated infinitely many times. Although these sampling variances indicate how much each effect-size estimate in a study might vary in its replication sampling, they convey no information regarding how different estimates in that study might covary with one another likely due to the design aspects discussed earlier (see Sampling dependence inherent in L2 studies is ignored).

The wcf data set also uses the name of the study authors as the identifier for each study. Thus, variable study provides a unique name for each study. Needing to be unique to each study, if the same authors contributed more than one study to the study pool, their names for each study were distinguished from one another by appropriate suffixes (e.g., Author A vs. Author B).

Similar to studies, each effect-size estimate is associated with a numeric identifier. This variable, which is called effect in the wcf data set, simply denotes the row-wise location of the effect-size estimates. For example, the first effect-size estimate in the first row gets a value of 1, the second one gets a value of 2, and so on. Therefore, such numbers have no real numeric value. We will need this variable to be able to distinguish the effect-level variation from the study-level variation in the true effects (see preceding text).

Together with all the (M)UTOS moderators shown in Table 1, the wcf data set is built into the R programs introduced in the next section. Thus, readers will automatically have access to the data set, once they install the programs in the next section.

Detecting outliers in 3M models

Before finalizing our 3M model, a critically important step is to ensure that the model is not negatively affected by outlying estimates of effect size. A useful approach to do so is to examine how each effect-size estimate interactively influences a 3M model (Viechtbauer, Reference Viechtbauer, Schmid, Stijnen and White2021). Interactive outliers are those that from multiple perspectives are considered outlying. Three such perspectives include an effect estimate (1) having a standardized deleted residual that exceeds ± 1.96 (such estimates do not fit the 3M model well), (2) exerting a large influence on the results once deleted (an estimate’s Cook’s distance exceeding the upper limit of a box plot of all estimates’ Cook’s distances), and (3) leveraging considerable influence on its own predicted true effect value given the meta-analytic weight and the moderator values attached to it (an estimate’s Hat value exceeding the upper limit of a box plot of all estimates’ Hat values).

The idea behind detecting interactive outliers is to find effect-size estimates that simultaneously meet the three perspectives discussed above. For our m_prof model, we can achieve this goal by running the following in R with the results shown in Figure 4:

Figure 4.

Detection of interactive outliers.

If detected, interactive outliers are indicated by their row-wise location in the Figure generated by the R function interactive_outlier. In our case, out of 245 estimates of effect size obtained from the WCF literature, only one effect-size estimate (not an entire study) on row 116 is identified as an interactive outlier. The interactive_outlier function automatically removes the interactive outliers and provides the new data set:

Because this new data set is smaller than the original one due to the removal of one interactively outlying effect-size estimate, we also need to redefine our sampling dependence structure initially defined in R code (2) with this new data set:

Finally, we can update our original 3M model (m_prof) with this new data set (wcf_new) and the new sampling dependence structure (V_new):

Exploring results from a 3M model

At its core, a 3M model is a multilevel regression model (see multivariate, multilevel meta-analysis). This means that the output will largely resemble that from multilevel regression analyses. However, this output is often perceived to be difficult to interpret for researchers in applied linguistics (Plonsky & Oswald, Reference Plonsky and Oswald2017) as well as those in other fields of study (Fox & Weisberg, Reference Fox and Weisberg2018). The interpretations become particularly subtle as moderators in the model, including the categorical and the continuous ones, grow in number.

Additionally, recall that in addition to the moderators involved in our research question (i.e., how does proficiency influence the effect of WCF on written accuracy over time?), other variables (i.e., M moderators) in the 3M simply act as bias-controlling agents. Such bias-controlling agents, as noted previously, do not necessarily merit an interpretation and may be adjusted for in the background (Berlin & Antman, Reference Berlin and Antman1992; Tipton et al., Reference Tipton, Pustejovsky and Ahmadi2019).

The R function post_rma() helps us obtain a piece of output that is focused on our research question, keeping the M moderators adjusted for in the background. For our research question, the function may be used as follows:

In (11), m_prof_new represents our fitted 3M model in R code (9). The ~prof*time part indicates the moderators involved in our research question. These primary moderators are also the ones defined in our 3M model in R code (3). The output from post_rma is displayed in Table 3.

Table 3.

3M results for the influence of proficiency on WCF effect over time

The column titled Mean represents the meta-analytic average effects found for each proficiency level at each measurement time in the WCF literature, SE represents the standard errors for those average effects, and Lower and Upper represent the 95% confidence intervals for the estimates of Mean. The column t is the t value resulting from testing the null hypothesis that each Mean is in reality nonexistent (here 0). Thus, the p values indicate whether we can reject such null hypotheses using some decision rule for statistical significance (e.g., p < .05).

Similar to a primary study, it would be helpful to visualize the patterns represented in Table 3 for our meta-analytic study. The R function plot_rma() is designed for this purpose. We can direct the function to plot the proficiency groups’ Mean effects over time (prof~time):

Such visuals have other benefits as well. For instance, looking at Figure 5, we realize that in the WCF literature, low-proficiency learners are likely quite understudied relative to intermediate and advanced learners. This is evident in that the confidence intervals for the beginners are much wider than are those for the other proficiency levels. Additionally, not only have WCF studies largely focused on intermediate learners, this proficiency level has been more longitudinally studied than others have. This is seen in the confidence intervals for the intermediate learners, which are much narrower on the second posttest than those for the other proficiency levels.

Figure 5.

Average meta-analytic effects for proficiency levels over time.

Despite these benefits, it is often difficult to compare the groups in any precise manner by solely looking at visualizations such as Figure 5. This is why it is necessary to conduct post-3M comparisons to better explore the findings of the study.

Post-3M comparisons

Although Table 3 and Figure 5 provide useful information about how WCF has worked for each proficiency level at a given measurement time, knowing precisely how WCF’s effect compares within and across the proficiency levels requires making post-3M comparisons. Some of these comparisons may be simple (e.g., Prior to receiving WCF, how do the proficiency groups differ at the pretesting occasion?). Others may be more complex (e.g., How does WCF’s effect for proficiency groups differ from pre- to the final posttest?).

For simple comparisons such as comparing groups at the pretesting time, a pairwise comparison involving the first three Means (include=1:3) in Table 3 results, which focus on the proficiency groups’ average effects at the pretesting occasion, is all that is needed. To do so, we may use the pairwise functionality of post_rma:

As can be seen in Table 4, no statistically significant differences are found across the proficiency groups at the pretesting occasion (i.e., prior to receiving WCF). This finding may, however, be taken with a grain of salt. This is so because the CIs for the differences are fairly wide, indicating a potential inability to detect such differences, when some could in reality exist (i.e., Type II error in decision making).

Table 4.

Difference between proficiency groups at the pretest

With this Type II error possibility, comparing the proficiency groups at any posttesting occasion (e.g., final posttest) without considering their pretesting status may provide an invalid picture regarding how WCF has worked for different proficiency groups at that point. Consistent with this view, we can instead compare the gains each proficiency group has made from the pretest to any posttesting occasion of interest. This is precisely what our second comparison (i.e., How does WCF’s effect for proficiency groups differ from pre- to the final posttest?) is targeting. To address our second comparison, we can define the long-term gain for each proficiency group between pre- and second posttest as follows:

Our guiding plan then is to compare each pair of proficiency groups on such a gain, which will require testing the following contrasts:

Such customized contrasts, however, may not be readily tested by statistical software. Instead, we will need to locate the Means from Table 3 to form the gains for each proficiency group and then contrast them with one another. For example, for contrast (15) we can easily see that the difference in the gains entails the following mean locations from Table 3:

The above, in words, reads

Contrast (18) can be simplified by applying the − sign between its parentheses:

Similarly, for contrasts (16) and (17), we will, respectively, obtain the following Mean locations from Table 3:

Having determined the location of the Means in Table 3 for contrasts (15) through (17), we can test them using contrast_rma. To do so, we will supply the function with the Table 3 results (post_prof) and the Mean locations indicated in (20) through (22) for our customized contrasts:

The results are shown in Table 5. Based on these results, long-term gains for the proficiency groups in the WCF literature are not statistically different from one another. Once again, however, the fairly wide CIs for these differences prevent us from placing full trust in these results. We, as indicated previously, know that the bulk of WCF research has been focused on intermediate learners, with the other two proficiency groups remaining largely understudied. And this is the underlying reason we tend to see a fair amount of uncertainty in our results.

Table 5.

Difference in long-term gains across proficiency levels

Note, however, that our customized contrasts focused on the difference in gains (not gains themselves) from WCF across the proficiency groups to determine whether WCF might have differentially benefited the proficiency groups. Thus, the lack of a statistically significant finding should not mean that WCF has not produced any benefit for each proficiency group per se. Indeed, we can easily obtain the long-term gains from WCF per equation (14) for each proficiency group using contrast_rma. To do so, we can again supply contrast_rma with the relevant Mean locations for each proficiency group’s long-term gain from Table 3 (e.g., 8 − 2 for beginners’ gain):

The results in Table 6 show that the long-term gain in written accuracy from WCF is statistically significant for intermediate learners and so with a reasonable level of certainty, 95% CI [.356, .729]. For other understudied proficiency groups, however, the high level of uncertainty, indicated by wide confidence intervals, prevents a precise estimate of the long-term gains in written accuracy.

Table 6.

Long-term gains in writing accuracy across proficiency levels

Exploring variation (heterogeneity) in 3M

Besides exploring WCF’s mean effects for various proficiency levels, our 3M model allows us to examine how much of the total variation (i.e., sum of σ study- and σ effect-level variations) in the underlying WCF effects is systematically due to proficiency adjusting for the M moderators. Stated differently, we can ask the question: to what extent the variation in the magnitude of underlying WCF effects is because they come from different proficiency levels controlling for the methodological differences in the WCF studies? To answer this, it is customary to compare our existing model with one that has no (M)UTOS moderator in it. This procedure is akin to obtaining the R ² in regression analyses. To achieve this, we can use R2_rma():

Table 7 displays the results. As can be seen, our model (m_prof_new) has accounted for 5.2% of the total variation (in SD unit; σ total) in underlying WCF effects given its moderators. These moderators also tend to collectively be statistically significant (p < .05). Clearly, the rest of the variation in the underlying WCF effects remains unexplained by proficiency, perhaps a reason for the meta-analyst to explore other moderators in subsequent analyses.

Table 7.

Variation in underlying WCF effects accounted for by the fitted 3M

Using variation (heterogeneity) for evidence-based recommendations

Understanding the magnitude and sources of variation in the underlying WCF effects also provides a window into the distribution of effects in that domain of research (Mathur & VanderWeele, Reference Mathur and VanderWeele2019). Supplementing this information with the Mean effects (Table 3) or the effects from our main comparisons (e.g., gains in Table 6) provides a highly useful means of offering evidence-based recommendations to the audience of our meta-analysis. For example, we could predict how likely it would be, on average, for L2 learners to positively benefit (i.e., gain) from teachers’ written feedback given their proficiency levels. To do so, we can feed Table 6 results (gain_prof) for proficiency groups’ long-term gains (gain=TRUE) into function prob_rma() with the results shown in Table 8:

Table 8.

Prediction for WCF benefitting learners given their proficiency

Based on Table 8, intermediate learners are, on average, ~81% (column Probability) likely to positively benefit from teachers’ written feedback in the long run. But a positive gain could include any amount of gain even as small as to be considered practically insignificant. Thus, we can define a minimum threshold for that gain to be practically more meaningful. For instance, we could predict how likely it would be for learners, on average, to gain at least as large as 0.2 on the effect-size scale from teachers’ written feedback given their proficiency levels. To do so, we can provide prob_rma with our target effect of .20 with the results shown in Table 9:

Table 9.

Prediction for WCF benefitting learners by a target effect given their proficiency

Now, we can see that the likelihood of benefitting from teachers’ written feedback for intermediate learners, on average, decreases from ~81% to ~71% in the long run (i.e., from pre- to the second posttest controlling for the differences in the studies’ lengths). This decrease in likelihood is, of course, expected given that we have set a higher expectation for learning gains from teachers’ written feedback. Notice also that in both Tables 8 and 9, our ability to make such easy-to-understand evidence-based recommendations depends, among other things, on the amount of the evidence supporting these benefits. For instance, because beginners and advanced learners are understudied in the WCF literature, the minimum (column Min) and maximum (column Max) bounds on how much they, on average, can benefit from teachers’ written feedback in the long-run are quite far apart. As a result, evidence-based recommendations in this respect may be made with caution (e.g., perhaps using or emphasizing the Min probabilities).

Sensitivity of 3M to sampling dependence

Recall from our discussion on sampling dependence (see Sampling dependence in 3M) that the medium-high correlation (r = .60) assumed for effect-size estimates in each study was more of a guesstimate. As such, we may want to subject this value to a sensitivity analysis. This means that we would want to assume a range of different correlations for effect-size estimates in each study to see how robust the result of the 3M model (m_prof_new) will be to the change in such a guesstimate.

To achieve this goal, we can use the sense_rma() function in R, which uses our initial Table 3 results (post_prof) as well as the name of the variable in our data denoting the sampling variances (var):

Table 10 shows the results of our sensitivity analysis under five different degrees of correlation assumed for effect-size estimates in each study. The last column assesses the variability (in SD unit) in the results under all these degrees of correlation. Although there is no preset cut point, almost all mean effects (except for beginner post2) are close to one another, resulting in their variability (column SD) being fairly small.

Table 10.

Sensitivity of 3M results to different amounts of sampling dependence

Note. The asterisk denotes the r value used in the current model.

For the total variation (in SD unit) in true WCF effects (last row), we do see a bit of sensitivity. Specifically, as r becomes larger, the total variation decreases in magnitude at a faster rate than that in other terms in Table 10. Ultimately, the point is to acknowledge the fact that our addressing the dependencies in individual studies, at least in some ways, is not and perhaps will never be perfect. Rather, it to some extent is sensitive to the degree of correlation assumed for the evidence collected from our complex studies. However, these small sensitivities, in the end, often outweigh a complete ignorance of the dependent nature of the evidence collected from such studies.

The steps illustrated above are important in describing the role of proficiency as a critical variable in WCF effectiveness. But they also provide a road map for exploring all other moderators listed in Table 1. Also, due to space limitation other moderators are not demonstrated here, so the interested readers are encouraged to implement the steps described in the present article for other moderators given the open-data-and-materials nature of the present article.