Causal mediation in developmental science: A primer

Abstract

Mediation has played a critical role in developmental theory and research. Yet, developmentalists rarely discuss the methodological challenges of establishing causality in mediation analysis or potential strategies to improve the identification of causal mediation effects. In this article, we discuss the potential outcomes framework from statistics as a means for highlighting several fundamental challenges of establishing causality in mediation analysis, including the difficulty of meeting the key assumption of sequential ignorability, even in experimental studies. We argue that this framework—which, although commonplace in other fields, has not yet been taken up in developmental science—can inform solutions to these challenges. Based on the framework, we offer a series of recommendations for improving causal inference in mediation analysis, including an overview of best practices in both study design and analysis, as well as resources for conducting analysis. In doing so, our overall objective in this article is to support the use of rigorous methods for understanding questions of mechanism in developmental science.

Keywords

Mediation causal mediation causal inference mechanisms of change

Developmental science has long been invested in understanding mediation, or the process by which one variable causally affects another via an intermediary mechanism (MacKinnon et al., 2007). Indeed, questions of mediation—including not just whether but how or why a stimulus produces an outcome—form the basis of most foundational theories of human development (e.g., Bandura, 1986; Bronfenbrenner, 1977; Piaget, 1969; Vygotsky, 1981). In addition to being useful in basic scientific inquiry, mediation is also increasingly utilized by applied psychologists to understand the mechanisms through which programs and interventions may influence children’s outcomes (Schindler et al., 2019).

Despite its conceptual and practical importance, testing mediation is fraught with methodological challenges. In recent years, researchers have leveraged a variety of methods to improve causal inference in child development research (e.g., Foster, 2010; Miller et al., 2016). Although the field has made tremendous progress overall, most discussions and advice in developmental science concerning mediation have focused on the logic of mediation and the correct estimation of standard errors (e.g., Baron & Kenny, 1986; Dearing & Hamilton, 2006; MacKinnon et al., 2007), whereas the challenges of identifying causal mediation effects have rarely been discussed in the field. In particular, although several threats to validity hamper causal interpretations in most mediation analyses, even when researchers employ longitudinal or experimental data (Maxwell & Cole, 2007; VanderWeele, 2015), research in developmental science often fails to explicitly recognize threats to internal validity or discuss strategies to address them. We argue that a better understanding of such threats and approaches to their mitigation is fundamental for advancing developmental theory and practice, including the design, refinement, and scaling of effective intervention programming.

In this article, we overview a general framework (and language) from statistics that makes explicit a key assumption underlying causality in mediation analysis: sequential ignorability. We argue that a clear understanding of sequential ignorability makes threats to validity more visible, thus motivating and facilitating the use of novel study designs and sensitivity checks that can strengthen the identification of causal mediation effects in developmental science. Rather than providing a detailed tutorial on how to conduct causal mediation according to the proposed framework, the current article seeks, for the first time in developmental science, to explicitly discuss the challenges of establishing causality in mediation analyses, to offer a brief overview of current best practices, and to refer readers to additional resources (including articles, books, and software) for further guidance. To enhance interpretation and relevance, throughout the manuscript we employ two hypothetical examples. The first includes an experimental evaluation of a fictitious, randomly assigned early childhood development (ECD) intervention that aims to increase parental engagement in stimulating activities (such as reading and playing) with children and, in turn, to support children’s cognitive development. The second includes an observational study exploring the effect of poverty on children’s cognitive development, mediated by parental engagement in stimulating activities.

Traditional Approaches to Estimating Mediation in Developmental Science

Mediation analysis is designed to examine how an antecedent variable, which we call the exposure variable (X), leads to or affects an outcome variable (Y) through a mediating variable (M) in a causal process $X \to M \to Y$ (MacKinnon et al., 2007). The exposure variable X can represent a treatment, intervention, or exposure to any stimulus or experience in a natural or experimental setting. The mediator M represents, in general, a biological, psychological, or social process. The outcome Y can represent any dependent variable, and in developmental science it typically represents a child skill or behavior. The main goal of mediation is to decompose the total effect of the exposure variable on the outcome variable into (1) an indirect or mediated effect via a potential mediator or mechanism and (2) a direct effect that represents all other possible mediators or explanations for the effect of X on Y (VanderWeele, 2016). In the case of our first hypothetical example, the indirect effect represents the effect of the ECD intervention (X) on children’s cognitive development (Y) via parental stimulation (M), whereas the direct effect represents the effect of the intervention on cognitive development through other unmeasured pathways, such as increments in the availability of stimulating materials at home. Similarly, in our second example, the indirect effect represents the effect of poverty (X) on children’s cognitive development (Y) via parental stimulation (M), and the direct effect represents the effect of poverty on cognitive development through all other pathways (e.g., parental depression and the availability of learning materials such as books and toys).

With over 88,000 citations in Google Scholar, the most common approach to quantifying mediation in developmental science is the product method (Baron & Kenny, 1986), which can be applied using ordinary least squares (OLS) or structural equation modeling (SEM). The product method employs two regression models to estimate direct and indirect effects (see also Figure 1, Panel A). In the first model, Y is regressed on X and M (Equation 1). In the second model, M is regressed on X (Equation 2). The direct effect is represented by the coefficient c in Equation 1 and reflects the association between X and Y while controlling for M. The indirect effect is calculated using the product of the coefficient for X in Equation 2 times the coefficient for M in Equation 1, that is, $a \times b$ .

Y_{i} = θ + c X_{i} + b M_{i} + ε_{i}

M_{i} = γ + a X_{i} + μ_{i}

Figure 1.

Mediation and Confounders.

Despite its popularity, the product method is often used to estimate how directly or indirectly variables are related to one another, without a clear articulation of the assumptions required to identify causal links (i.e., identifying assumptions; Keele et al., 2015; VanderWeele, 2015). We argue that a formal framework and language to think about the plausibility of causal mediation effects is fundamental to (1) better understand potential threats to the validity of mediation-related inferences, (2) design studies that minimize some of these threats, thus improving inferences, and (3) apply existing statistical tools (e.g., sensitivity checks and software) to improve mediation analysis in developmental science. In what follows, we review a general framework widely used in causal inference—the potential outcomes framework (POF; Rubin, 1974)—and explain its relevance for understanding and overcoming key challenges in mediation analysis within the developmental sciences.

A General Framework (and Language) for Causal Mediation

The POF (Rubin, 1974) offers a formal framework to think about causation. This framework was developed in the field of statistics; Neyman (see Neyman et al., 1990) introduced it in the context of experimental data, and Rubin (2005) extended it to define causal effects in both experimental and observational studies. Using the language of the POF, the effect of an exposure or treatment X (which can take the values of $x = 1$ if exposed and $x = 0$ if not exposed) on an outcome Y for an individual i is $Y_{i} (1) - Y_{i} (0)$ , or the difference between the value of the outcome if the individual experiences the exposure ( $Y_{i} (1)$ ) and the value of the outcome if the individual does not experience the exposure ( $Y_{i} (0)$ ). In the case of our first example, the effect of the ECD intervention for child i would therefore be represented by the difference between his or her cognitive development when receiving the intervention and his or her cognitive development had the intervention not taken place. In the second example, the effect of poverty for child i would be represented by the difference between child’s i cognitive development when exposed to poverty and child’s i cognitive development if the child had not been exposed to poverty.

In reality, these individual effects are not identifiable, as it is possible to observe only one potential outcome (i.e., one Y_i ) for each individual—the one that corresponds to his or her exposure status in real life (e.g., $Y_{i} (1) if X_{i} = 1$ ). For this reason, rather than examining the effect of an exposure for an individual, researchers typically seek to identify the average causal or treatment effect (ATE) over a population, as represented by the mean outcome difference between a group of exposed individuals and a counterfactual group of nonexposed individuals who represent what would have happened to the exposed individuals had exposure not taken place (Angrist & Pischke, 2011).

The POF allows researchers to identify the ATE under the assumption of ignorability. Ignorability—also known as no omitted variable bias, no selection bias, or exogeneity—implies that the model controls for all confounders in the association between X and Y. In other words, ignorability implies that conditional on a set of pre-exposure confounders, exposure is ignorable, which is similar to saying that it is as good as random, that there are no omitted confounders, or that the potential outcomes between exposed and unexposed individuals would have been the same had exposure not taken place (Angrist & Pischke, 2011). In the context of the POF, this assumption is unsurprising: Given that we only can observe one potential outcome for each person (the one that corresponds to his or her exposure status), the challenge is to infer the missing potential outcome (i.e., the counterfactual scenario) in a convincing way.

In our first hypothetical example, if the ECD intervention was randomized, we would expect the treatment and control groups to be equivalent in all ways except for their exposure to the intervention, mitigating concerns about violating the ignorability assumption. We may have more concerns about ignorability, however, in observational studies where the exposure of interest was self-selected or not randomly assigned. In our second study about poverty, for example, there may be fundamental but unmeasurable differences between the experiences of lower versus higher income children (e.g., community-level factors) that may explain disparities in children’s cognitive development, thus confounding the “effect” of poverty. Developmental researchers have become increasingly interested in designing studies to satisfy the ignorability assumption as a means for establishing credible causal links between exposures and outcomes. Some examples of methods that help to support ignorability are randomized controlled trials and, in observational studies, instrumental variables, difference-in-difference, regression discontinuity, matching methods, and fixed-effects models (Duncan et al., 2004; Foster, 2010; Gennetian et al., 2008; McCartney et al., 2006; Miller et al., 2016).

It is possible to extend the POF to reflect causal mediation effects (Imai, Keele, & Tingley, 2010). In the framework, $M_{i} (1)$ represents the potential mediator value had exposure taken place, and $M_{i} (0)$ otherwise. $Y_{i} (x, m)$ represents the resulting outcome when exposure status X_i equals x (where x is 1 if exposure occurred and 0 otherwise) and the mediator M_i equals value m. Consequently, the individual causal total effect will be defined by $Y_{i} (1, M (1)) - Y_{i} (0, M (0))$ , that is, the difference between the outcome variable caused by the value of the mediator had exposure taken place $Y_{i} (1, M (1))$ and the outcome variable caused by the value of the mediator had exposure not taken place $Y_{i} (0, M (0))$ .

As noted earlier, total effects in mediation analysis can be decomposed into direct and indirect effects. Using the language of the POF, the individual causal indirect (or mediation) effect is $Y_{i} (x, M_{i} (1)) - Y_{i} (x, M_{i} (0))$ . In other words, according to the POF, the fundamental question that causal mediation seeks to answer is: What is the difference between the value of the outcome variable that would occur had the mediator taken the value that would be realized under exposure ( $M_{i} (1)$ ) versus the value of the outcome variable that would occur had the mediator taken the value that would be realized had exposure not taken place $(M_{i} (0)$ ), holding exposure status constant at x? (Imai, Keele, & Tingley, 2010). If exposure does not have any effect on the mediator, then $M_{i} (1) = M_{i} (0)$ . In our first hypothetical example, $Y_{i} (1, M_{i} (1))$ therefore represents the cognitive development for child i caused by the level of parental stimulation that arises after participating in the ECD intervention, whereas $Y_{i} (1, M_{i} (0))$ represents a counterfactual scenario, that is, the cognitive development for child i, who participates in the ECD intervention, but considering a level of parental stimulation that would have arisen had the ECD intervention not taken place. In the second example, $Y_{i} (1, M_{i} (1))$ represents the cognitive development for child i caused by the level of parental stimulation produced by exposure to poverty, whereas $Y_{i} (1, M_{i} (0))$ represents the cognitive development for child i, who experiences poverty, but considering a level of parental stimulation that would have arisen without exposure to poverty.

Finally, the individual causal direct effect is $Y_{i} (1, M_{i} (x)) - Y_{i} (0, M_{i} (x))$ , or the difference in the outcome variable for individual i under the condition of exposure versus non-exposure given all other possible mediating variables. In our first example, $Y_{i} (1, M_{i} (x))$ represents the direct effect of the ECD intervention on child i’s cognitive development via all other mechanisms (e.g., increments in the availability of learning materials at home), holding parental stimulation constant at the level that would have arisen had the ECD intervention not taken place. Similarly, in the second example $Y_{i} (1, M_{i} (x))$ represents the direct effect of poverty on child i’s cognitive development via all other pathways (e.g., differences in parental depression and availability of learning materials at home), holding parental stimulation constant at the level that would have arisen in the absence of poverty.

Just as in the traditional POF, it is impossible to identify individual-level mediation effects, as only one potential outcome can be observed for each person in a given data set (the one that corresponds to the actual exposure status, e.g., $Y_{i} (1, M_{i} (x)) if X_{i} = 1$ ). As such, researchers who use the POF to understand mediation focus on identifying the average causal mediation effects (ACME) and the average causal direct effects (ADE) averaging over a population. In this case, the ATE will be the sum of the ACME and ADE (Keele et al., 2015). The POF also makes it visible that in mediation the main challenge is to infer the missing potential mediator and outcome (i.e., counterfactual scenario, which is not observable) in a convincing way, given that it is only possible to observe the potential mediator and outcome that corresponds to each individual’s actual exposure status.

Consequently, the POF allows us to formalize a key assumption to identifying causal mediation effects, namely sequential ignorability (Imai, Keele, & Tingley, 2010). Sequential ignorability implies two ignorability assumptions in a row: first, controlling for pre-exposure confounders, exposure is assumed to be ignorable (i.e., as good as random or independent of potential outcomes and potential mediators); second, the mediator is assumed to be ignorable given exposure status and pre-exposure confounders (i.e., mediator is independent of potential outcomes) or, in other words, the model controls for all confounders in the relation between mediator and outcome. (Figure 1, Panel B, presents an observed confounder Z which can be included as a covariate in the model. Panel C presents an unobserved confounder U in the association between mediator and outcome that violates sequential ignorability.)

It is possible to satisfy the first part of the sequential ignorability assumption by randomizing the exposure variable within an experimental design or exploiting quasi-experimental approaches that developmental researchers already use (Foster, 2010; Miller et al., 2016). The second part of sequential ignorability, however, is a very strong assumption in that it requires controlling for all pre-exposure confounders in the relation between mediator and outcome, which likely does not occur even when exposure is randomized (VanderWeele, 2015). In our first example, even if the ECD program was successfully randomized, the second part of the ignorability assumption would be violated if an unobserved confounder such as parental stress or neighborhood crime simultaneously affects both parental stimulation and child development. In the second example, the sequential ignorability assumption is even harder to meet, as selection effects, or endogeneity bias, are a concern at both stages (i.e., poverty to parental stimulation and parental stimulation to child cognition). The use of quasi-experimental approaches (e.g., fixed-effects and regression discontinuity) could mitigate to some extent concerns about nonrandom selection into poverty, but it will likely not reduce concerns about potential confounders in the link between parental stimulation and children’s cognitive development.

Best Practices for Addressing Challenges in Causal Mediation

The general framework presented in this manuscript has important implications for developmental researchers who seek to improve their mediation analysis. In particular, the POF makes visible the key assumption to identify causal mediation effects (i.e., sequential ignorability), making it possible to think about and apply sensitivity checks and invest in study designs that make sequential ignorability more plausible. In this section, we highlight different approaches for mitigating challenges in mediation analysis.

Study Design

As noted above, experimental and quasi-experimental study designs rarely guarantee sequential ignorability (Imai et al., 2011). To further improve causality in mediation analysis, researchers can follow additional steps when designing studies to improve the plausibility of causal mediation claims. First, the use of longitudinal data with rich covariates, including pre-exposure measures of outcomes and mediators, guarantees temporal precedence and allows researchers to control for multiple confounders in the relation between the outcome and mediator, relaxing to some extent the sequential ignorability assumption. Second, a strong theoretical model can allow researchers to identify potential mediators and confounders in the causal chain, informing data collection and model design. Finally, quasi-experimental methods that exploit natural variation or that employ principal stratification or matching methods can aid researchers in checking balance in observed characteristics and reducing, to some extent, threats to the identifying assumption (Jo et al., 2011).

Alternative experimental approaches where the mediator can be directly or indirectly manipulated also offer promise (Imai et al., 2013). One such approach is a parallel design, where each individual is randomly assigned to either (a) random exposure only or (b) both random exposure and random mediator. Another approach is a crossover design, where each individual participates in two sequential experiments: First, exposure is randomly assigned, and the researcher observes the resulting values of the mediator and outcome variables. Then, exposure status is reversed, and the mediator is fixed to the value observed in the first experiment. A main limitation of these designs is that it is often difficult to manipulate mediators perfectly, particularly if mediators relate to psychological processes (Imai et al., 2013).

General Estimation Method for Mediation

Imai, Keele, and Tingley (2010) have proposed a general estimation method that makes more salient the rationale of and assumptions behind the POF, aiming to predict the (not observed) potential mediator and outcome. Moreover, the method can be used in parametric and nonparametric models, thereby removing linearity assumptions common to SEM and OLS methods. Linearity assumptions are not tenable in much developmental research, including cases where researchers use binary outcome variables (e.g., the child is developmentally on track or not). As such, removing such assumption aids causal identification.

The general estimation method comprises five steps. First, the researcher fits two regression models: (a) the mediator is regressed on the exposure variable and pre-exposure confounders, and (b) the outcome is regressed on the mediator, exposure variable, and pre-exposure confounders. These regression models can be linear or nonlinear (e.g., logit or Poisson). Second, using the estimates from the fitted mediator model (Model a), the researcher generates two sets of predicted values for the mediator for each observation, one corresponding to the mediator value under exposure and one for non-exposure conditions. Third, using the coefficients for fitted outcome model (Model b), the researcher predicts potential outcomes corresponding to the exposure condition and using the mediator value for the exposure conditions predicted in Step 2, and potential outcomes corresponding to the non-exposure condition and using its corresponding mediator value. The ACME will be the average difference between the predicted outcome under the exposure and non-exposure conditions. Fourth, the researcher repeats the process N times (typically 5,000–10,000, although there are not clear guidelines on minimum numbers of resamples necessary; Imai, Keele, & Tingley, 2010; Keele et al., 2015) to obtain uncertainty estimates or confidence intervals using bootstrapping. Finally, hypothesis testing is conducted using the uncertainty estimates to assess statistical significance.

Sensitivity Analysis

Given that sequential ignorability is a very strong assumption, it is of cardinal importance to quantify the sensitivity of the results from mediation analyses to potential confounders, or the degree to which identifying assumptions must be violated for a given conclusion to be invalid. Outside of mediation analysis, developmentalists have proposed sensitivity analyses, such as the coefficient of proportionality, to assess the impact that unobserved confounders would need to have to nullify a result (Dearing & Zachrisson, 2019).

Other fields, such as epidemiology, have proposed similar approaches to assess the validity of causal mediation claims (VanderWeele, 2015). Considering the general estimation method proposed by Imai, Keele, and Tingley (2010), the error terms for the outcome and mediator models should be independent (r = 0) if sequential ignorability holds (as these error terms should be random and not explained by an omitted variable). As such, the correlation between both error terms can serve as a sensitivity parameter, where the higher the correlation, the less likely sequential ignorability holds (Keele et al., 2015). Imai, Keele, and Yamamoto (2010) also proposed a sensitivity check similar to the coefficient of proportionality. This sensitivity check assesses the extent to which an omitted variable would change the coefficient of determination (R ²) of the mediation model, by assessing how much the R ² varies when controlling for observed characteristics that may act as confounders. In this test, relative changes in the R ² (in models ignoring observed confounders relative to models controlling by these confounders) serve as a sensitivity parameter. Even though we encourage the use of these approaches, they only allow researchers to examine the influence of pre-exposure confounders, not post-exposure confounders that may be influenced by exposure itself (Imai, Keele, & Tingley, 2010). Moreover, there is no established threshold for the sensitivity parameters, which makes their interpretation largely subjective (Keele et al., 2015).

Resources for Developmentalists

Resources for causal mediation analysis have evolved rapidly over the last decade and are increasingly available within multiple statistical software packages. Table 1 provides a brief list of resources for developmentalists interested in using causal mediation methods in their own work, including available statistical functions to apply the general estimation method for mediation, sensitivity analysis, and other estimation methods for causal mediation effects, along with references for detailed descriptions, tutorials, and examples. Developmental researchers interested in learning more about causal mediation approaches more generally can also reference work by Imai et al. (2010), Imai et al. (2011), VanderWeele (2015, 2016), and other references included in this manuscript. We argue that the wide-scale use of these resources can make the application of best practices in causal mediation more common in developmental science.

Table 1.

Sample Resources for Causal Mediation Analysis.

	Suggested statistical functions with tutorials and examples
	Reference	Statistical function
General estimation method for mediation
Stata	-Hicks and Tingley (2011)	medeff ^b
R	-Tingley et al. (2014)	mediate^b
Sensitivity analysis
Stata	-Hicks and Tingley (2011)	medsens^b
R	-Tingley et al. (2014)	medsens^b
Other estimation methods for causal mediation effects^a
Stata	-VanderWeele (2015, p. 40)	paramed
Stata	-Valente et al. (2020) -Discacciati et al. (2018)	Med4way
R	-Steen et al. (2017) -Valente et al. (2020)	medflex
Mplus	-Muthén and Asparouhov (2015) -Valente et al. (2020)	MODEL INDIRECT
SAS	-Valente et al. (2020) -Valeri and VanderWeele (2013) -VanderWeele (2015, p. 35)	mediation
SAS	-SAS Institute (2018)	PROC CAUSALMED
SPSS	-Valente et al. (2020) -Valeri and VanderWeele (2013) -VanderWeele (2015, p. 38)	Mediation

Note. ^a Other estimation methods are not based on the general estimation method described above (i.e., Imai et al., 2010). ^bStatistical function/command can be installed with package mediation.

Conclusion

Mediation analysis is fundamental to understanding developmental theories, questions, and processes. However, developmental science has long overlooked important methodological challenges to establishing credible causal mediation effects, in both experimental and observational studies. Methodological advances in other fields offer exciting opportunities for developmentalists to improve causal inference in mediation analysis. Even though the key assumption to identify causal mediation effects is very strong and unsatisfiable even in the context of a randomized experiment, a rich collection of covariates, novel experimental designs, and sensitivity analyses can offer evidence to evaluate the plausibility of causal claims. When broadly applied, the framework for mediation reviewed in this article can allow developmental researchers to explore mediating processes in a rigorous way to better inform both theory and practice.

Footnotes

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Jorge Cuartas

References

Angrist

J. D.

Pischke

J.-S.

(2008). Mostly Harmless Econometrics: An Empiricist’s Companion. Princeton: Princeton University Press.

Angrist

Pischke

J.-S.

(2011). In Angrist

Pischke

J.-S.

(Eds.), Mostly harmless econometrics: An empiricist’s companion (Vol. 38, pp. 281–282).

Bandura

(1986). Social foundations of thought and action: A social cognitive theory. Prentice Hall.

Baron

R. M.

Kenny

D. A.

(1986). The moderator–mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51(6), 1173–1182. https://doi.org/10.1037/0022-3514.51.6.1173

Bronfenbrenner

(1977). Toward an experimental ecology of human development. American Psychologist, 32(7), 513–531. https://dx-doi-org.web.bisu.edu.cn/10.1037/0003-066X.32.7.513

Dearing

Hamilton

L. C.

(2006). V. Contemporary advances and classic advice for analyzing mediating and moderating variables. Monographs of the Society for Research in Child Development, 71(3), 88–104. https://doi.org/10.1111/j.1540-5834.2006.00406.x

Dearing

Zachrisson

H. D.

(2019). Taking selection seriously in correlational studies of child development: A call for sensitivity analyses. Child Development Perspectives, 13(4), 267–273. https://doi.org/10.1111/cdep.12343

Discacciati

Bellavia

Lee

J. J.

Mazumdar

Valeri

(2018). Med4way: A Stata command to investigate mediating and interactive mechanisms using the four-way effect decomposition. International Journal of Epidemiology, 48(1), 15–20. https://doi.org/10.1093/ije/dyy236

Duncan

G. J.

Magnuson

K. A.

Ludwig

(2004). The endogeneity problem in developmental studies. Research in Human Development, 1(1–2), 59–80. https://doi.org/10.1080/15427609.2004.9683330

10.

Foster

E. M.

(2010). Causal inference and developmental psychology. Developmental Psychology, 46(6), 1454–1480. https://doi.org/10.1037/a0020204

11.

Gennetian

L. A.

Magnuson

Morris

P. A.

(2008). From statistical associations to causation: What developmentalists can learn from instrumental variables techniques coupled with experimental data. Developmental Psychology, 44(2), 381–394. https://doi.org/10.1037/0012-1649.44.2.381

12.

Hicks

Tingley

(2011). Causal mediation analysis. Stata Journal, 11(4), 605–619. http://www.stata-journal.com/article.html?article=st0243

13.

Imai

Stuart

E. A.

(2011). Commentary: Using potential outcomes to understand causal mediation analysis. Multivariate Behavioral Research, 46(5), 861–873. https://doi.org/10.1080/00273171.2011.606743

14.

Imai

Keele

Tingley

(2010). A general approach to causal mediation analysis. Psychological Methods, 15(4), 309–334. https://doi.org/10.1037/a0020761

15.

Imai

Keele

Yamamoto

(2010). Identification, inference and sensitivity analysis for causal mediation effects. Statistical Science, 25(1), 51–71. https://doi.org/10.1214/10-STS321

16.

Imai

Tingley

Yamamoto

(2013). Experimental designs for identifying causal mechanisms. Journal of the Royal Statistical Society: Series A (Statistics in Society), 176(1), 5–32. www-jstor-org.web.bisu.edu.cn/stable/23355175

17.

Stuart

E. A.

MacKinnon

D. P.

Vinokur

A. D.

(2011). The use of propensity scores in mediation analysis. Multivariate Behavioral Research, 46(3), 425–452. https://doi.org/10.1080/00273171.2011.576624

18.

Keele

Tingley

Yamamoto

(2015). Identifying mechanisms behind policy interventions via causal mediation analysis. Journal of Policy Analysis and Management, 34(4), 937–963. https://doi.org/10.1002/pam.21853

19.

MacKinnon

D. P.

Fairchild

A. J.

Fritz

M. S.

(2007). Mediation analysis. Annual Review of Psychology, 58(1), 593–614. https://doi.org/10.1146/annurev.psych.58.110405.085542

20.

Maxwell

S. E.

Cole

D. A.

(2007). Bias in cross-sectional analyses of longitudinal mediation. Psychological Methods, 12(1), 23–44. https://doi.org/10.1037/1082-989X.12.1.23

21.

McCartney

Bub

K. L.

Burchinal

M. R.

(2006). VI. Selection, detection, and reflection. Monographs of the Society for Research in Child Development, 71(3), 105–126. https://doi.org/10.1111/j.1540-5834.2006.00407.x

22.

Miller

Henry

Votruba-Drzal

(2016). Strengthening causal inference in developmental research. Child Development Perspectives, 10(4), 275–280. https://doi.org/10.1111/cdep.12202

23.

Muthén

Asparouhov

(2015). Causal effects in mediation modeling: An introduction with applications to latent variables. Structural Equation Modeling: A Multidisciplinary Journal, 22(1), 12–23. https://doi.org/10.1080/10705511.2014.935843

24.

Neyman

Dabrowska

D. M.

Speed

T. P.

(1990). On the application of probability theory to agricultural experiments. Essay on principles. Section 9. Statistical Science, 5(4), 465–472. www-jstor-org.web.bisu.edu.cn/stable/2245382

25.

Piaget

(1969). The psychology of the child. Basic Books.

26.

Rubin

D. B.

(1974). Estimating causal effects of treatments in randomized and nonrandomized studies. Journal of Educational Psychology, 66(5), 688–701. https://doi.org/10.1037/h0037350

27.

Rubin

D. B.

(2005). Causal inference using potential outcomes. Journal of the American Statistical Association, 100(469), 322–331. https://doi.org/10.1198/016214504000001880

28.

29.

Schindler

H. S.

McCoy

D. C.

Fisher

P. A.

Shonkoff

J. P.

(2019). A historical look at theories of change in early childhood education research. Early Childhood Research Quarterly, 48, 146–154. https://doi.org/10.1016/j.ecresq.2019.03.004

30.

Steen

Loeys

Moerkerke

Vansteelandt

(2017). Medflex: An R package for flexible mediation analysis using natural effect models. Journal of Statistical Software, 1(11), https://doi.org/10.18637/jss.v076.i11

31.

Tingley

Yamamoto

Hirose

Keele

Imai

(2014). Mediation: R package for causal mediation analysis. Journal of Statistical Software, 59(5), 38. https://doi.org/10.18637/jss.v059.i05

32.

Valente

M. J.

Rijnhart

J. J. M.

Smyth

H. L.

Muniz

F. B.

MacKinnon

D. P.

(2020). Causal Mediation Programs in R, Mplus, SAS, SPSS, and Stata. Structural Equation Modeling: A Multidisciplinary Journal, 27(6), 975–984. https://doi.org/10.1080/10705511.2020.1777133

33.

Valeri

VanderWeele

T. J.

(2013). Mediation analysis allowing for exposure–mediator interactions and causal interpretation: Theoretical assumptions and implementation with SAS and SPSS macros. Psychological Methods, 18(2), 137–150. https://doi.org/10.1037/a0031034

34.

VanderWeele

T. J.

(2015). Explanation in causal inference: Methods for mediation and interaction. Oxford University Press.

35.

VanderWeele

T. J.

(2016). Mediation analysis: A practitioner’s guide. Annual Review of Public Health, 37(1), 17–32. https://doi.org/10.1146/annurev-publhealth-032315-021402

36.

Vygotsky

L. S.

(1981). Mind in society: The development of higher psychological processes (Nachdr., Ed.). Harvard University Press.