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# Statistical Mediation in Lifespan Developmental Analyses

## Summary and Keywords

Lifespan developmental research studies how individuals change throughout their lifetime and how intraindividual or interindividual change leads to future outcomes. Lifespan researchers are interested in how developmental processes unfold and how specific developmental pathways lead to an outcome. Developmental processes have been previously studied using developmental cascade models, concepts of equifinality and multifinality, and developmental interventions. Statistical mediation analysis also provides a framework for studying developmental processes and developmental pathways by identifying intermediate variables, known as mediators, that transmit the effect between early exposures and future outcomes. The role of statistical mediation in lifespan developmental research is either to explain how the developmental process unfolds, or to identify mediators that researchers can target in interventions so that individuals change developmental pathways. The statistical mediation model is inherently causal, so the relations between the exposures, mediators, and outcomes have to be correctly specified, and ruling out alternative explanations for the relations is of upmost importance.

The statistical mediation model can be extended to deal with longitudinal data. For example, the autoregressive mediation model can represent change through time by examining lagged relations in multiwave datasets. On the other hand, the multilevel mediation model can deal with the clustering of repeated measures within individuals to study intraindividual and interindividual change. Finally, the latent growth curve mediation model can represent the variability of linear and nonlinear trajectories for individuals in the variables in the mediation model through time. As a result, developmental researchers have access to a range of models that could describe the theory of change they want to study. Researchers are encouraged to consider mechanisms of change and to formulate mediation hypotheses about lifespan development.

# Conceptual Examples of Statistical Mediation in Lifespan Development

The field of developmental science studies change within a person, called intraindividual change, and also how change differs across people, called interindividual change. Longitudinal data are needed to investigate intra- and interindividual differences. Explanations for how people change largely depend on fundamental questions about the type of change (either constant or dynamic), if individuals can influence their own development, and the interactions between nature and nurture in human development (Kuther, 2017). Baltes and Nesselroade (1979) described five rationales for conducting longitudinal research: (1) identifying intraindividual stability or change, (2) identifying interindividual similarities or differences, (3) analysis of the relations between behavior change, (4) identifying the causes of intraindividual change, and (5) identifying the causes of interindividual change. Using the five rationales, researchers can formulate precise research questions about the nature of development (Grimm, Ram, & Estabrook, 2016) and answer questions about the causes and effects of intraindividual and interindividual developmental processes.

## Equifinality and Multifinality

Two important concepts in lifespan development relevant to mediating processes are multifinality and equifinality (Cicchetti & Rogosch, 1996; Masten & Cicchetti, 2010). Multifinality describes developmental pathways where the same situations or risk factors do not lead to the same outcomes. On the other hand, equifinality describes how different developmental pathways, situations, or risk factors can end in the same outcome (Cicchetti & Toth, 2009). As examples of multifinality, not all children with maladaptive parental attachment develop conduct disorder, and not all sexually abused children develop a mental disorder; the outcomes may depend on a certain environment or a genetic predisposition (Cicchetti & Toth, 2009). As an example of equifinality, there may be different developmental pathways to antisocial behavior, such as an authority conflict pathway, a covert pathway (i.e., from shoplifting to more serious crime), or an overt aggressive pathway (i.e., early fighting and aggression; Hinshaw & Lee, 2003). Therefore, the concepts of multifinality and equifinality suggest that investigating mediational hypotheses and processes might be more important than identifying single predictors of a disorder or an outcome. Equally important is to uncover the mechanisms that make individuals stay in or change developmental pathways, such as in the person-oriented approaches to statistical mediation (Bogat, von Eye, & Bergman, 2017; MacKinnon, 2008).

## Interventions in Developmental Science

Mediating processes are important for lifespan development because they aid in understanding how development occurs. If the underlying mediating processes can be understood, then the most powerful aspect of mediation analysis can be applied. If the process by which early exposure affects later exposure is known, interventions could be designed to ameliorate or remove the negative effects of the early exposure or risk factors (Kurtines et al., 2008; MacKinnon, 2008). For example, childhood poverty adversely affects noncognitive abilities in children and it affects the child’s self-regulation strategies, coping abilities, and parent−child relations, which in turn are related to mental and physical health outcomes in the future. Consequently, if mediators are found for these relations, interventions could be designed to buffer the effects of poverty on those outcomes (Ramey & Ramey, 1998). For example, the Perry Preschool Program was a 2-year intervention for disadvantaged, 3- to 4-year-old, African American children. The program was comprised of morning school programs and afternoon teacher visits targeting raising IQ and motivation to learn. In a follow-up at age 40, children who participated in the intervention did not have higher IQs, but they had higher high school graduation rates, salaries, and home-ownership rates than children in the control group (Heckman, 2006). In another example, the Abecedarian program was an intervention for 4-month-old disadvantaged children, who received child care five days a week until kindergarten. The intervention targeted the development of cognitive, social, language, and gross motor skills. Results from the program indicated that participating children had higher IQ and higher noncognitive skills than those in the control group (Heckman, 2006). More early-life developmental interventions can be found in Reynolds, Wang, and Walber (2003). Another potential example is how conscientiousness is associated with length of life. Conscientious teenagers are more likely to smoke less, to wear seatbelts, and to engage in other healthy behaviors; and conscientious adults are more responsible, dependable, and achievement-oriented. So (in theory), interventions could be designed to increasing the mediator conscientiousness and thus influence length of life (Friedman, Kern, Hampson, & Duckworth, 2014). However, intervening on personality constructs might be difficult, so it might be more effective and efficient to target intervention toward changing specific health-related behaviors, rather than personality constructs like self-efficacy and social competence (English & Carstensen, 2014).

Overall, statistical mediation analysis provides a framework for testing hypotheses and investigating the mechanisms of change in how intrinsic and extrinsic influences affect human development, as well as how intrinsic and extrinsic influences cause intraindividual and interindividual change (MacKinnon, 2008; Selig & Preacher, 2009). Statistical mediation methods have undergone extensive development since the 1980s, and combining them with modern longitudinal modeling provides a comprehensive way to investigate mediating processes in developmental science. The mediating framework specifies both how an intervention changes the mediator and how the mediator is related to the outcome. The two aspects are reflected in statistical mediation analysis. Next, technical aspects of the statistical mediation model and its extensions to model longitudinal data are discussed.

# Statistical Aspects of Mediation Models

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Figure 1. Single mediator model.

Typically, researchers focus on the relation between two variables. Once the relation is observed, more elaborate theory is developed for how and why the two variables are related. The explanations are often in the form of mediating causal processes (Baron & Kenny, 1986; Judd & Kenny, 1981; Mackinnon, 2008; VanderWeele, 2015). This type of research represents an approach called mediation for explanation (MacKinnon, 2008), which originated in the work of Lazarsfeld (1955) and Kendall and Lazarsfeld (1950). Examples of the mediation for explanation approach include how mother−child reminiscing influences the child’s self-descriptive memory (Valentino et al., 2014), how selection strategies can compensate for cognitive loss (Baltes, 1997), and how spousal loss influences psychological well-being (Infurna, Gerstorf, & Ram, 2013). Another application of mediating variables is called mediation by design (MacKinnon, 2008), in which mediating variables are selected as targets for intervention because previous theory and empirical research indicate that the mediator is causally related to the outcome variable. Statistical mediation analysis applies for both types of research because it is used to identify and investigate mechanisms of change by considering intermediate variables that transmit the effects from the independent variable to the outcome (Baron & Kenny, 1986; MacKinnon, 2008). For example, consider the relationship between the Abecedarian program intervention [X] to IQ [Y], mediated by the development of cognitive skills [M]. For the single mediator model (see Figure 1), three equations are used to investigate mediating relations:

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In this case, c represents the total effect of independent variable X on the dependent variable Y, a is the effect of X on the mediator M, b is the effect of M on Y adjusted for X, and $c’$ is the effect of X on Y adjusted for M. The $i1$, $i2$, and $i3$ are the regression intercepts, and $e1$, $e2$, and $e3$ are the equation residuals. Also, $a^$, $b^,c^,$ and $c′^$ are sample estimators of a, b, c, and $c’$. Both the a-path and the b-path carry the information about mediation and are present in more complicated mediation models.

## Estimating Mediated Effects

Traditionally, researchers tested for mediation using Baron and Kenny’s (1986) four steps because Baron and Kenny wrote one of the first papers providing a clear step-by-step description for investigating mediation. The first step was to establish a significant relationship between X and Y (the c-path in Equation 1 is significant). The second step was that X should predict M (the a-path in Equation 2 is significant). The third step was that M should predict Y, adjusting for X (the b-path in Equation 3 is significant). Finally, the fourth step was that the relationship between X and Y should be nonsignificant after adjusting for M (Judd & Kenny, 1981) or should decrease in magnitude (the c’-path in Equation 3 is nonsignificant or less than the c-path in Equation 1). However, there were substantial problems with the mediation steps approach, among which were that a mediation relationship can exist even when there is not a significant c-path (MacKinnon, Krull, & Lockwood, 2000; O’Rourke & MacKinnon, 2015) and that the approach was severely underpowered compared to modern approaches (Fritz & MacKinnon, 2007).

The mediated effect, the influence of X on Y through M, is quantified by the product $ab^,$, and its statistical significance provides evidence in favor of M serving as a significant mediator between X and Y. A standard error $sab$ can be derived using Taylor-series approximation to test for statistical significance of the mediated effect $ab^$ (MacKinnon, 2008; MacKinnon & Dwyer, 1993; Sobel, 1986):

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(4)

The coefficients a and b are from Equation 2 and Equation 3, and the $sa2$ and $sb2$ are the squared standard errors of a and b. The ratio between $ab^$ and $sab$ yields a z-statistic, and it would be considered significant if the critical ratio is below −1.96 or above 1.96. However, the product of two normally distributed variables is not often normally distributed, which might make the statistical test inaccurate and lead to low statistical power and conservative conclusions regarding the significance of the mediated effect $ab^$. A lot of research in statistical mediation has focused on accurate methods to test for statistical significance of $ab^$ using the analytical distribution of the product of normal variables (MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002; Tofighi & MacKinnon, 2011), resampling approaches (MacKinnon, Lockwood, & Williams, 2004), tests of joint significance of $a^$ and $b^$ (MacKinnon et al., 2002; Valente, Gonzalez, Miočević, & MacKinnon, 2016), and Bayesian methods (Enders, Fairchild, & MacKinnon, 2013; Miočević, MacKinnon, & Levy, 2017; Yuan & MacKinnon, 2009). As of this writing, the recommended approaches to test for the significance of the mediated effect are to use percentile bootstrap confidence intervals or the distribution of the product asymmetric confidence intervals (MacKinnon, 2008). It is also recommended to accompany any significance test with an effect size, and an appropriate effect size for the mediated effect is $ab^$ divided by the standard deviation of the dependent variable Y (Miočević, O’Rourke, MacKinnon, & Brown 2017). Statistical mediation analyses can be estimated in a wide variety of programs, such as in SPSS, Stata, and SAS, and also in the structural equation modeling framework using statistical programs, such as R and Mplus. Programs like the PROCESS macro (Hayes, 2012) and Mplus (Muthén & Muthén, 1998–2017) have special capabilities for testing mediation hypotheses.

## Statistical Mediation Assumptions

Certain assumptions must hold to make accurate conclusions from the statistical mediation model (MacKinnon, 2008). Statistical mediation assumes that the functional form and temporal precedence among the three variables have been correctly specified. In the previous case, all of the relationships were linear, and X came before M, and Y came before X and M. In addition, it is assumed that no important variables have been left out of the model, such as variables that can confound relationships. Psychometrically sound measures and accurate timing in measurement of the variables are also assumed. Finally, four assumptions based on confounding (omitted causes among the variables) are needed to make causal statements about the mediated effect (Pearl, 2001; Valeri & VanderWeele, 2013):

1. 1. There are no unmeasured confounders in the relationship between X and Y.

2. 2. There are no unmeasured confounders in the relationship between M and Y.

3. 3. There are no unmeasured confounders in the relationship between the X and M.

4. 4. There are no measured or unmeasured confounders in the relationship between M and Y that have been affected by X.

When independent variable X is random assignment to groups, such as in an intervention study, assumptions 1 and 3 are met in expectation. Therefore, one can assume that the relation between X and Y and the relation between X and M are causal. However, in the field of developmental science, it is unlikely that the X variable is randomized, which renders assumptions 1 and 3 questionable. Even in randomized studies, assumptions 2 and 4 are not met because M is not randomized, so there is no way to test if the relation between M and Y is causal. These untestable assumptions can be studied with sensitivity analyses (Cox, Kisbu-Sakarya, Miočević, & MacKinnon, 2013; MacKinnon & Pirlott, 2015; MacKinnon, Valente, & Wurpts, in press). Sensitivity analyses presume a hypothetical situation where there are unmeasured confounders of relations in the model. Sensitivity analyses estimate how large the relationship between the variables in the model and the confounders would have to be in order for the mediated effect to be nonsignificant (Mauro, 1990). Developmental researchers could then hypothesize about the unmeasured variables that cause both M and Y with a large enough effect to make the mediated effect nonsignificant. For example, social norms may be a confounder of an observed relation between self-control and drug abuse. Sensitivity analysis would investigate how large the effect of social norms on both self-control and drug abuse would have to be for the mediation relations to be nonsignificant.

## Extensions to Multiple Mediators

The statistical mediation model can be expanded to include multiple predictor X variables and multiple M mediators. These models are particularly useful when researchers need to control for pretest measures of the variables in the mediation model. In the case of multiple predictors X, the predictors are included in the same regressions that predict M and Y, so there would be aj and cj coefficients for j predictors. For the case of multiple mediators M, each mediator would have its own regression equation, and all of the mediators would be included simultaneously in the regression equation that predicts outcome Y. Therefore, there would be ap and bp coefficients for p mediators. Although a model with multiple predictors and multiple mediators indicates the unique influence of each predictor and mediator on the outcome, relations between individual predictors and individual mediators can also be studied separately. Specific mediated effects, unique for each mediator, can be defined by the product of a single a and a single b. Adding all of the specific mediated effects leads to a total mediated effect. Also, researchers can use structural equation modeling to simultaneously estimate models with multiple mediators and multiple outcomes (Bollen, 1989; MacKinnon, 2008). Finally, multiple mediator models can also be useful in conceptualizing equifinality, multifinality, and developmental cascades. Equifinality would be a multiple mediator model (different processes) with a common outcome, and multifinality would be conceptualized as a single mediator model (common process) with multiple outcomes. Developmental cascades would be a multiple mediator model where the mediators are the sequencing risk factors predicting an outcome (a pathology).

## Role of Mediation and Lifespan Analyses

The role of statistical mediation analysis in lifespan developmental studies is to explain how longitudinal processes unfold over time (Krull, Cheong, Fritz, & MacKinnon, 2017). Mediation analysis of longitudinal data allows researchers to go beyond investigating mediating hypotheses about interindividual change to study hypotheses about intraindividual change. In this way, statistical mediation can help researchers investigate what contextual, personality, or developmental factors influence lifespan processes, and in turn how lifespan processes influence future behavior.

Stronger evidence for a mediating process is obtained with longitudinal data common in lifespan research (MacKinnon, 2008). The statistical mediation model assumes temporal precedence of the variables in the model, where X comes before M, and X and M come before Y. With longitudinal data, pretest measures of X, M, and Y can be modeled in several ways. Data structures where the cause and effect are repeatedly measured allow for more accurate examination of how a cause affects a mediator that affects an outcome. It is unlikely that variables exert simultaneous effects on each other, so time has to pass for the variables to exert their effects on others (Gollob & Reichardt, 1991). For cross-sectional data, it is reasonable to believe that the randomized intervention variable X comes before M and Y. However, for the M to Y relation or in nonrandomized studies, it is difficult to establish the order of the variables. Longitudinal data can help examine temporal relations between X, M, and Y by looking at lagged relations between the variables over time. For example, researchers can investigate how variable X at time 1 affects M at time 2, and in turn how those affect Y at time 3 (Cheong, MacKinnon, & Khoo, 2003; Cole & Maxwell, 2003; MacKinnon, 2008). Another benefit of longitudinal data is that the data can be used to disentangle within-person effects and between-person effects. With cross-sectional data, one can study only how individuals differ from each other, but with repeated measures, researchers can study intraindividual change. Consequently, repeated measures can also help rule out the influence of extraneous variables and alternative explanations in bivariate relationships. With repeated measures data, participants serve as their own control because controlling for measures at a previous timepoint can help rule out stable influences or confounders.

The addition of repeated measures in longitudinal datasets requires more elaborate mediation models than for the cross-sectional model previously described. For repeated measures data, each case is not independent because data from the same participant may be more similar than data from two different participants. Ignoring this dependency might lead to inaccurate research conclusions and high Type I error rates, so longitudinal models accommodate the dependency by explicitly modeling within- and between-person effects. Three frameworks for analyzing longitudinal data are the autoregressive framework, the multilevel framework, and the latent growth curve framework, and these frameworks can be extended to test mediation hypotheses. A technical treatment for advanced longitudinal mediation models is beyond the scope of this article, but important concepts are discussed to show how statistical mediation can be examined in data with repeated measures (for more on longitudinal models, see Bollen & Curran, 2006, Grimm et al., 2016, Krull et al., 2017, and MacKinnon, 2008).

## Autoregressive Mediation Models

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Figure 2. Autoregressive mediation model with contemporaneous effects.

The most common type of longitudinal mediation model is the autoregressive model (Cole & Maxwell, 2003; MacKinnon, 1994, 2008; Maxwell & Cole, 2007). The autoregressive model expands the single mediator model to include repeated measures and autoregressive relations where the same variable at previous waves predicts the current wave. Mediated effects are represented by the different possible relations of X, M, and Y over time. Three waves of data are usually needed to specify a full autoregressive mediator model, where X at time 1 predicts M at time 2, which in turn predicts Y at time 3. However, contemporaneous relations between the variables can also be specified. Equations 5 to 10 define a three-wave, autoregressive mediation model with contemporaneous and longitudinal mediated effects (see Figure 2).

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where Xt, Mt, and Yt are the predictor, mediator, and outcome measured at time t, and the s coefficient is the stability coefficient, which is constrained to be the same for each autoregressive relation for each variable (MacKinnon, 2008). The longitudinal mediated effects are those where a and b come from the relations across waves (as in lagged relations), while contemporaneous mediated effects are those where a and b come from the same wave. Although longitudinal mediated effects meet the temporal order assumption and are of most interest, detecting mediated effects might depend on the time of measurement between waves. Therefore, there might be times where contemporaneous mediated effects could be more accurate and closer to the true mediating relationship than longitudinal mediated effects (MacKinnon, 2008). Cross-lagged paths could also be included in the model, where the outcome in a previous wave predicts the mediator in the next wave (such as a path from Y1 to M2). Although cross-lagged relationships violate temporal order, cross-lagged relationships between mediator and outcome might be reasonable if both were associated in a previous, unmeasured time (MacKinnon, 2008). Chi-square difference tests can be used to determine if stability coefficients should be constrained to be the same or if cross-lagged relationships or contemporaneous relationships are needed in the model. Although the autoregressive model meets the temporal precedence assumption of mediation, one of its limitations is that it focuses on interindividual differences and does not explicitly model intraindividual variability (MacKinnon, 2008; Selig & Preacher, 2009).

An example of the autoregressive model in developmental science is Pardini, Loeber, and Stouthamer-Loeber’s (2005) study on the relationship between parent and peer influence on adolescents’ belief about delinquent behavior. Parent−child relationship quality, peer deviant behavior, and the adolescent’s deviant beliefs were modeled as latent variables and had autoregressive relationships across six waves of data, from sixth to eleventh grade. Although statistical mediation was not directly tested in the study, the temporal precedence between the variables allows researchers to test the hypothesis. Researchers could test if the relationship between parent−child relationship quality in early adolescence and the adolescent deviant beliefs in eleventh grade are mediated by the peer deviant behavior (treating it as a measure of peer influence and association). Researchers can also test how the mediating relationship changes across the intermediate waves of data.

## Multilevel Modeling and Statistical Mediation

Variables used in lifespan analyses might be clustered, as with participants providing repeated measures. For these situations, repeated observations within participants might be more similar than data from two different participants. Ignoring the clustering in data structures could have detrimental effects on regression analysis because standard errors are smaller and test statistics are larger than normal, leading to high Type I error rates and possibly false-positive conclusions about regression analyses. To overcome these limitations, multilevel modeling could be used to accommodate the clustered structure of datasets.

Multilevel modeling can be conceptualized as fitting a series of regression models per level in the data (Raudenbush & Bryk, 2002; Singer & Willett, 2003). Consider the case of observations nested within people, where observations are the first or lowest level of analysis and comparisons across people are the second level of analysis. The first level of analysis (L1) describes intraindividual differences and is estimated by the regression equation,

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(11)

where Yij is the outcome, Xij is the predictor, and eij is the residual, all for observation i for person j. The predictor X could either be repeated measures, such as daily measures of physical activity or mental health, or a timing variable, as is found in developmental research. The β0j and β1j are the intercept and slope of each person j, respectively. Conceptually, L1 applies a regression equation to observations per person. The second level (L2) models the variability of β0j and β1j (intercept and slope) across individuals, such as:

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where γ00 is the overall intercept for the data and u0j is the intercept deviation of person j from the overall intercept for the data. The γ10 is the overall slope for the data and u1j is the slope deviation of person j from the overall slope for the data. In this case, β0j and β1j are the same as in Equation 11 and are treated as outcome variables. Conceptually, L2 describes how the starting point and the rate of change differ across people. Both L1 and L2 equations can be combined into a single regression equation by substituting L1 coefficients β0j and β1j with the L2 equations, such as,

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where γ00 and γ10 are the fixed effects and represent the overall model, and u0j and u1j are the random effects and their variance describes the group deviations from an overall model. Both u0j and u1j are assumed to be multivariate normally distributed, with a mean of zero and variance-covariance matrix $τ$. Finally, the multilevel framework can also be used to model nested data, such as individuals nested within communities, students nested within classrooms, or siblings nested within a family. Therefore, each community, classroom, or family would have an estimated intercept and slope, and the multilevel model estimates the variability of the intercepts and slopes across communities, classrooms, or families.

Statistical mediation can be estimated for repeated measures data by extending the statistical mediation equations to a multilevel model (Krull & MacKinnon, 2001). Therefore, researchers can estimate mediated effects when either X or M is measured at multiple levels. A convenient notation to represent the levels of measurement for the mediation variables is to indicate its level in the sequential order, as in the mediation model in the following structure X$→$ M $→$ Y. For example, a 2$→$1$→$1 model indicates that predictor X is measured at L2, mediator M is measured at L1, and outcome Y is measured at L1 (see Figure 3). Conceptually, each measured variable appears in the L1 or L2 equations depending on their level. The a coefficient, describing differences across people, is estimated with the equations:

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The equations for the b coefficient, describing differences within people, are:

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In another example, the a coefficient in the 2 $→$ 2 $→$ 1 model is estimated with the equation:

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The b coefficient in the 2 $→$ 2 $→$ 1 model is estimated with the equations:

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Figure 3. Multilevel mediation model.

The estimate for the mediated effect is the product ab, which combines information across levels, and it is tested either using the Sobel standard error in Equation 4 or bootstrap confidence intervals (Krull et al., 2017). It is important to note that when there are 1$→$1 links in the model (L1 variables predicting each other), such as in the first example, the random intercepts were estimated but the random slopes were treated as fixed. If the random slopes were estimated, it would mean that the mediated effect also varies randomly across L2 units. This is particularly relevant in the 1$→$1$→$1 model, where all variables are measured at the lowest level. The estimation of the mediated effect in the 1$→$1$→$1 is complex because coefficients a and b might covary, so the product ab might not be an accurate representation of the mediated effect. More details are included in Bauer, Preacher, and Gil (2006), Krull et al. (2017), and MacKinnon (2008) about the estimation and complications of the 1$→$1$→$1 model. A final limitation is that links from lower-order variables to higher-order variables, such as 1$→$2, cannot be handled in the multilevel framework, but multilevel structural equation modeling might be able to estimate these relationships (Krull et al., 2017; Preacher, Zyphur, & Zhang, 2010).

In an example of mediation in the multilevel framework, physical activity is an important factor for life satisfaction, but it is unclear how physical activity impacts life satisfaction and if the relationship differs across the lifespan (Maher, Pincus, Ram, & Conroy, 2015). Older adults might have a stronger relationship between physical activity and life satisfaction because physical activity might represent health maintenance and the possibility of pursuing relevant goals. This relationship might not hold for emerging and young adults because they have not suffered health declines related to age. Maher and colleagues (2015) used multilevel mediation to investigate between- and within-person differences in the relation between physical activity and life satisfaction and if the relation was mediated by physical and mental health. Researchers concluded that the relationship between life satisfaction and age was curvilinear, decreasing in emerging and young adulthood (persons 18 to 35 years old), increasing in middle adulthood (35 to 64 years), and then decreasing in older adulthood (more than 65 years), and that age moderated the effect of physical activity and life satisfaction. A 2$→$2$→$1 model was estimated for between-person differences in middle or older adults. Results suggested that usual mental and physical health mediated the effect of usual physical activity and usual life satisfaction. This mediating relationship was not present for younger adults. On the other hand, a 1$→$1$→$1 model was estimated for the within-person differences across all age groups. Results suggested that daily mental and physical health mediated the effect of daily physical activity and daily life satisfaction. In other words, when participants engage in more exercise than normal, they report better physical and mental health and in turn report higher life satisfaction.

## Latent Growth Curve Modeling and Statistical Mediation

In longitudinal studies, individuals are measured on multiple occasions, so a typical research question is to investigate the average rate of change across all the individuals and how individual trajectories vary. Latent growth curve modeling (LGM) uses structural equation modeling to analyze growth over time by introducing latent variables to represent the intercept and slope of a growth process (Grimm et al., 2016), thus allowing the analysis of intraindividual change in mediation. The growth parameters are the mean and the variance of each latent variable and they represent the average and individual differences in the expected value at the initial measurement point (intercept) and the growth rate (slope), respectively (Krull et al., 2017). For any given response Y[t] at time t,

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In other words, the intercept and slope factors represent means ($α$n and $α$s) and individual variations ($ζ$ni and $ζ$si) from the means. Each measurement occasion is used as an indicator of the intercept and slope factors, but instead of estimating factor loadings, each of the factor loadings is fixed to the shape of the growth process in the data (Krull et al., 2017). For a linear growth process, the factor loadings for the intercept factor are set to 1, and the factor loadings for the slope factor are fixed to represent the time that has passed since the first measurement occasion. Specifically, if participants were measured four times, such as at baseline, year 1, year 3, and year 5, then the factor loadings on the slope factor would be constrained to 0, 1, 3, and 5, respectively. Researchers can represent higher-order processes, such as quadratic growth, by introducing another growth factor and fixing the factor loadings to the squared values of those from the linear model (such as 0, 1, 9, and 25).

Statistical mediation can be evaluated using LGM when either X, M, or Y is measured across three or more occasions (Cheong et al., 2003). In this case, the mediated effect emphasizes within-person differences and is related to changes in the X, M, or Y trajectories. LGM provides a more flexible framework for modeling individual differences than autoregressive models or difference scores because it explicitly models the growth trajectories and it can accommodate nonlinear growth. The first step in the LGM approach to mediation is to model the growth processes for X, M, and Y individually. Researchers can investigate growth trajectories by plotting their datasets and identifying the best model to describe the trajectories. Then, growth parameters, such as mean and variances of intercept and slope factors, are estimated per variable. In other words, there is an intercept mean, intercept variance, slope mean, and slope variance for X (IX, SX), M (IM, SM), and Y (IY, SY). The mean estimates of IX, IM, and IY represent the level of X, M, and Y at the first measurement occasion, and the mean estimates of SX, SM, and SY represent the average growth rate per time unit for X, M, and Y, respectively. The variance of each growth parameter estimate represents the individual differences in initial level and growth rate. Finally, the mediation relations are specified between the growth parameters for X, M, and Y, such that the initial level and growth rate of M and Y are predicted by the initial level and growth rate of X and M. Equations 25 to 28 describe the LGM mediation model (see Figure 4):

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In this case, i1,i2,i3, and i4 describe regression intercepts and e1i, e2i, e3i, and e4i are equation residuals for person i. Equations 25 and 27 describe that the M and Y initial levels are functions of the X and M initial levels, and these relations are often not of interest. It is most interesting to investigate the relations between the growth rate of a variable as a function of growth rates of other variables. Equation 26 describes that the growth rate of M (SM) is a function of the initial level of X (IX) and the growth rate of X (SX). Equation 28 describes that the growth rate of Y (SY) is a function of the initial level of X (IX), the growth rate of X (SX), the initial level of M (IM), and the growth rate of M (SM). The mediated effect of most interest is the influence of the growth rate of X (SX) on the growth rate of Y (SY) through the growth rate of M (SM), which is the product of coefficients a and b. In this case, a and b model intraindividual change. One can test for significance of ab using the Sobel standard error in Equation 4.

Click to view larger

Figure 4. Latent growth curve mediation model.

Researchers should be aware of the following considerations before using LGM for statistical mediation and developmental processes. Model convergence problems may arise when factor variances are very small, and this is commonly observed for slope variances. Solutions to nonconvergence are to constrain factor variances to zero, but mediated effects are challenging to test through that constrained factor because it does not have variance. Furthermore, the relations among the growth parameters should be interpreted with caution if all of the measures of X, M, and Y have been collected concurrently. A potential solution is to use a two-stage piecewise model (Raudenbush & Bryk, 2002), so that early and later phases of the development process (along with their growth parameters) are analyzed separately (Cheong et al., 2003). In this case, growth parameters in the early phases can affect growth parameters in the later phases (Krull et al., 2017). For example, linear growth in self-control during childhood could predict linear growth in drug abuse during adolescence. Finally, the mediated effect in the LGM framework for repeated measures can be estimated in the multilevel framework; with some constraints, these frameworks can give essentially the same answers. Details on the similarities between the multilevel and LGM frameworks are provided elsewhere (Krull et al., 2017).

# Conclusion and Future Directions

This article describes the statistical mediation model in developmental lifespan analyses. There are many examples of mediating processes in lifespan developmental research, both as explanations of developmental processes and also as targets for the development and evaluation of interventions. This article provides examples of mediating relationships in the areas of childhood poverty, substance abuse, and developmental psychopathology, along with discussion of important developmental concepts, such as multifinality and equifinality and developmental cascade models.

Future directions in the area of mediation in developmental science are to continue to apply mediation models to understand the mechanisms of lifespan change. However, longitudinal studies that can track individuals through their entire lifespan are scarce and expensive. Accelerated longitudinal designs, where different cohorts of participants are followed with some time measurement overlap (Bell, 1953; Little, Preacher, Selig, & Card, 2007), can help researchers model longitudinal relationships for an extended amount of time. However, there have not been many studies combining statistical mediation and accelerated longitudinal designs (see Birkett, Newcomb, & Mustanski, 2015, for an example). Another approach is to combine information from different longitudinal studies to understand lifespan processes. Currently, there are initiatives for creating longitudinal and aging study databases (Erten-Lyons et al., 2012) to pool developmental information and to conduct analyses through data harmonization or integrative data analysis (Friedman et al., 2014; Hofer & Piccinin, 2009, 2010). Future research should also continue to investigate the influence of the time metric in mediation results (Grimm, Ram, & Estabrook, 2016), along with studying mediation models with a continuous time metric, such as differential equation models (MacKinnon, 2008). On the other hand, future directions for the estimation of statistical mediation in longitudinal data include the influence of time-varying confounding on conclusions about the mediated effect. The models presented in this article assume that the effect of any confounding variable is stable through time (such as, the effect of poverty on academic achievement is constant through time), but current methods could give biased results if the confounding variable changes over time (such as, if the effect of poverty and academic achievement changes over time; VanderWeele, Hawkley, Thisted, & Cacioppo, 2011; VanderWeele & Tchetgen, 2017). Marginal structural models could be used to get unbiased results for mediation. Finally, the models assume that the measured variables are psychometrically sound. Therefore, the impact of measurement bias (construct changes over time) and measurement error (unreliable measures) in longitudinal mediation models should be considered and further studied (Gonzalez, Valente, & MacKinnon, 2017).

Overall, the ubiquity of longitudinal data makes the application of longitudinal mediation models ideal for developmental research, but, to date, mediation models have not been widely applied. There are several options for these models (MacKinnon, 2008), which mostly reflect how the dependency in measures is modeled, such as in autoregressive relationships, multilevel modeling, or latent growth curve modeling. Researchers must consider the following decisions in the application of longitudinal mediation models. First, theory of change has to be considered (MacKinnon, 2008; Selig & Preacher, 2009). Second, there are options regarding the type of model for change in X, M, and Y, such as whether it is linear or nonlinear. Third are questions about the type of relation for how X affects M, and M affects Y. Linear, threshold, cascading, and other nonlinear models may reflect the causal relations among these variables. Fourth, there are many different possible mediation effects, including whether effects are rapid or may emerge after many years. Finally, the types of relations may differ at different time points. Statistical mediation models can be developed for all these options, but the empirical results and theory for the effects may not yet be present in lifespan research. As a result, statistical application of modern mediation models will help with this research, but it must all rest on the theory for how X affects M, which affects Y.

# Acknowledgments

This research was supported in part by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1311230 and the National Institute on Drug Abuse under Grant No. R37 DA09757.

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