Dynamic change meets mechanisms of change: Examining mediators in the latent change score framework

Abstract

Researchers in behavioral sciences are often interested in longitudinal behavior change outcomes and the mechanisms that influence changes in these outcomes over time. The statistical models that are typically implemented to address these research questions do not allow for investigation of mechanisms of dynamic change over time. However, latent change score models allow for dynamic change (not just linear or exponential change) over time and have flexibility in parameter constraints that other longitudinal models do not have. Developmental researchers also frequently utilize mediation analyses to investigate mechanisms of influence in longitudinal research implemented in path analytic or latent growth curve models. In this article, we provide three examples of how mediation can be tested in the latent change score framework by combining aspects of traditional mediation models with latent change score models of repeated measures outcomes (and mediators and predictors) with more than two timepoints. We also provide the Mplus syntax to complete these analyses and practical considerations of latent change score mediation (LCSM) models.

Keywords

Quantitative methods mediation latent difference latent change structural equation modeling longitudinal

Introduction

Researchers in developmental sciences often endeavor to investigate the mechanisms by which change in behavioral outcomes occurs, and many interventions based on developmental theory aim to change current behavior as a means to change a behavioral outcome over time. Mediation models allow for examination of the processes that influence the relation between predictor and outcome (MacKinnon, 2008). In addition, explicitly addressing change in an outcome requires longitudinal models that allow for a variety of experimental designs and covariates. Latent change score (LCS) models (McArdle, 2009) are statistical models that provide the flexibility to address dynamic change in addition to linear or exponential change. Specifically, LCS models are structural equation models that are parameterized to enable investigation of latent variables that represent change in an outcome. These models can also be extended to address developmental research questions involving mechanisms of change by incorporating mediators; such models are known as latent change score mediation (LCSM) models. In this article, we provide three illustrative examples of LCSM models to answer questions about dynamic change for more than two timepoints and discuss extensions of the LCS model that allow for further nuanced mediation analyses.

Dynamic Change Meets Mechanisms of Change

Mediation Analysis

Mediation models are used to examine how a predictor X influences an outcome Y through an intervening variable M, known as a mediator. One common use of mediation analyses is in the study of mechanisms influencing behavioral outcomes. For example, studies on gene–environment relations have posited that the relationship between children’s early life characteristics and the development of psychopathology operates through aspects of the parent–child relationship (i.e., evocative rGE; Knafo & Jaffee, 2013; Plomin et al., 1977). Thus, mediation analyses are needed to assess whether children’s early characteristics are related to change in psychopathology indirectly through changes in parenting practices.

One way this type of research question has been addressed is using single mediator models, in which single assessments of each of the X, M, and Y variables are modeled such that X is a predictor of both Y (c’ path) and M (a path), and M is a predictor of Y (b path). This simplest single mediator model can be described using the ordinary least squares (OLS) regression equations below using notation from MacKinnon (2008).

Y = i_{1} + c X + e_{1}

(1)

Y = i_{2} + c^{'} X + b M + e_{2}

(2)

M = i_{3} + a X + e_{3}

(3)

In single mediator models, the c path, or the total effect of X on Y, is the effect of the predictor on the outcome without including the mediator (equation (1)). The b path represents the effect of the mediator on the outcome holding the predictor constant, and the c’ path represents the effect of the predictor on the outcome holding the mediator constant (equation (2)). The a path represents the effect of the predictor on the mediator (equation (3)). For the more complex models described below, we will use similar notation; for example, we will also use b to denote the effect of M on Y, even if it is a dynamic effect. The product of the a and b paths ab is known as the indirect effect.

Tests of Mediation

There are several sets of guidelines for determining the occurrence of mediation (MacKinnon, 2008; MacKinnon et al., 2002). The most well-known approach is the causal steps approach (Baron & Kenny, 1986; Judd & Kenny, 1981), which requires significance of each of the c, a, and b paths. However, methodological research has shown that the causal steps approach has low statistical power compared with other tests, in part due to the requirement of a significant c path (Fritz et al., 2015; Fritz & MacKinnon, 2007; Kenny & Judd, 2014; MacKinnon, 2008; MacKinnon et al., 2002; O’Rourke & MacKinnon, 2015, 2018). The joint significance test is a causal steps approach that utilizes t or z statistics for the a and b path estimates and concurrently assesses whether each estimate is significantly different from zero but does not require the c path to be significant. Simulation research has shown that the joint significance test provides the best balance of statistical power with Type I error rates (MacKinnon et al., 2002).

Another way to test mediation is to construct confidence intervals for the product ab. The distribution of the product of a and b is nonnormal, so Monte Carlo methods such as bootstrapping must be used to construct asymmetric confidence intervals (MacKinnon et al., 2004; MacKinnon, Fritz, et al. 2007; Tofighi & MacKinnon, 2011). Recent research has indicated that for mediation with structural equation models, the percentile bootstrap method of constructing confidence intervals takes into account potential correlation between a and b and had an optimal balance of power and Type I error (Valente et al., 2016). In this article, we utilize these confidence intervals of the indirect effect ab in addition to the joint significance test.

Causal Assumptions of Mediation

Statistical mediation models make several assumptions that have important implications for causality. The first is the assumption of causal order, where it is assumed that the causal process is correctly specified such that X causes M, which in turn causes Y. This causal order assumption is necessary to many theories of human development (Lindenberger & Pötter, 1998; Selig & Preacher, 2009). The assumption of causal processes is what distinguishes mediation models from other analogous statistical models that do not assume causality, such as dynamical systems models. This causal order tenet also assumes no misspecification of causality (i.e., no reciprocal relations among the mediating variables, no unmeasured confounders, and no misspecification due to measurement error). This sub-assumption of no unmeasured confounders can be addressed for the X → M relation by randomizing the X variable; however, M usually cannot be randomized in mediation analyses, creating issues with causal interpretation for the M → Y relation. In addition, in many developmental studies, X is not randomized, creating further issues with causal interpretation for the relations of X → M and X → Y controlling for M.

The second assumption is that of temporal precedence in the measurement of variables, where the assumption is that X precedes M and M precedes Y in time. This second assumption, although distinct from the first, is also necessary for determining that the mediator is a causal mechanism; it is assumed that a period of time (no matter how small) has elapsed for the causal process to occur for each path. It is important to distinguish temporal precedence from the measurement timing of the variables. Again, mediation requires the assumption of causal ordering, so measurement timing alone is not sufficient to determine the causal processes of mediation (i.e., just because X precedes M in time does not necessarily mean that X causes M).

Longitudinal mediation models with multiple timepoints can often provide information or evidence on causal relations above and beyond mediation models where X, M, and Y are measured cross-sectionally. The choice of longitudinal mediation model should be based in the theory of change for each variable. In many studies, longitudinal mediation models answer questions about how levels of X predict levels of Y through levels of M. However, for situations in which researchers expect that change in (and not the level of) a variable is predicted by other variables, a more complex longitudinal model is needed that addresses hypotheses concerning changes in the X, M, or Y variables. The LCS models explained in more detail below can be parameterized to not only capture dynamic longitudinal processes but also include the components of traditional mediation models, resulting in the LCSM models presented in this article.

The LCS Framework

LCS models combine aspects of autoregressive and latent growth models (LGMs; Grimm et al., 2016; McArdle, 2009). The LCS model allows for investigation of within-person change between two or more timepoints as an outcome of interest by creating latent variables that represent the change in true scores between two times, t – 1 and t:

Δ y_{t i} = l y_{t i} - l y_{t - 1 i}

(4)

In equation (4), ∆y_ti is the LCS for person i at time t, expressed as the difference between $l y_{t i}$ (latent score y for person i at time t) and $l y_{t - 1 i}$ (latent score y for person i at time t – 1). These change scores can then serve as outcomes or predictors in models that address research questions about change between two timepoints or over a longer period of time. Generally, LCS models are best suited for continuous and normally distributed outcomes. LCS models can be conceptualized to yield parameters identical to LGMs by including certain model constraints (Serang et al., 2019). The advantage of LCS models, however, is that they do not need to be specified in this way, and parameters can be freed or constrained to answer a specific research question. In addition, it is important to note that the latent variables in LCSs are not latent variables in the traditional sense. Instead, they are “nodes” with single indicators that allow change scores to be parsed into constant and proportional parts. By design, many of the paths in the model are fixed to 1.0. This model parameterization allows us to interpret the LCSs as magnitude of change, but does not necessarily change interpretation over using the latent or observed variables themselves.

In LCS models, the initial true scores are individuals’ starting values in the longitudinal variable of interest, and they are represented by the intercept. There are also several change components in the LCS model. The first is constant change, also referred to as the slope, and it can be interpreted as such when change is linear and constant over time. The latent constant change variable can serve as a predictor or outcome in larger models. The second change component is the proportional change, which models acceleration (i.e., time-sequential change) over time. The proportional change component represents a major advantage of the LCS model as it enables examination of nonlinear change by examining how prior levels of a variable affect later change. Of note, the constant change of the LCS model is analogous to the slope in a LGM only when there is no proportional change component in the LCS model. Similarly, although LCS models that have both constant change and proportional change result in trajectories that, when plotted, capture similar information to a quadratic slope in an LGM, each change component captures different information about linear change and acceleration.

For example, consider a developmental research question regarding childhood antisocial behavior. The expectation is not that change in antisocial behavior will continue to increase in perpetuity, but rather that some participants might experience increases in antisocial behavior but that those increases will attenuate over time (Caspi & Moffitt, 1995). LCS models parameterized to examine longitudinal research questions involving such dynamic change are referred to as dual change score models because they include both constant change and proportional change components. In addition, when one variable is repeatedly measured and modeled over time using LCS models, the model can be referred to as univariate. Extensions of this model can include change in more than one variable over time (i.e., multivariate models). An advantage of the multivariate model is the ability for parameters from each univariate model to be included as outcome or as predictor, such that it is possible to include paths between the t – 1 latent levels of one variable and the latent change between t – 1 and t (i.e., the LCS) of another variable. In LCS models, this influence of prior latent levels of one variable on later latent change of another variable is known as the coupling parameter. Interpretation of the coupling parameter is similar to that of the proportional change described previously (i.e., attenuation or acceleration of change) with the exception that the coupling parameter provides an estimate of the degree to which the level of one variable at the previous measurement influences change in the trajectory of another variable at a subsequent measurement. The examples we provide in this article parameterize both univariate and multivariate models for mediation analyses.

As noted previously, LCS models provide flexibility beyond the LGMs commonly implemented in developmental research, including to test mediation hypotheses. In particular, unlike LGMs, which allow for relations between intercepts and slopes (e.g., in parallel process LGM models), the inclusion of the coupling parameter allows for investigation of prior levels of one variable on subsequent changes in the other (Grimm et al., 2012; Grimm et al., 2016). For example, developmental researchers may be interested in the extent to which children’s externalizing behavior elicits changes in their parents’ parenting behavior, which ultimately influences later substance use and risky behavior (cf. Sandler et al., 2015). Although LGM mediation models would enable a test of the overall slope of parenting behaviors as a mediator in this example, LCSM models present the advantage of predicting change in parenting between time t – 1 and t across time as an outcome and, therefore, as a mediator (e.g., a coupling path where prior levels of externalizing predicts change in parenting). In scenarios where these paths are expected to change in strength or direction over time, the paths can be freely modeled rather than constrained to be equal. This flexibility enables developmental researchers to more adequately match their analytic strategy with their substantive theories (Klopack & Wickrama, 2020). In the discussion, we note that extensions of multivariate LCSM models can be made to test additional mediation hypotheses regarding longitudinal change.

Mediation in the LCS Framework

As always, specific research questions should inform the choice of longitudinal mediation models. For example, mediation LGM models would be appropriate for researchers investigating mechanisms of change over the full course of an analysis (i.e., because the latent slope is the variable included in the mediation model). LCSM models are appropriate for research questions involving mechanisms of change in an outcome that attenuates or amplifies over time. Researchers have utilized these models with increasing frequency in recent years. Prior research using LCSM models has focused on change between only two timepoints and change that is completely nonoverlapping (i.e., change in X between Times 1 and 2, ΔX_1–2, as a predictor of change in M between Times 2 and 3, ΔM_2–3; Goldsmith et al., 2018; Selig & Preacher, 2009; Simone & Lockhart, 2019). These models investigated change between two timepoints for the X, M, or Y variable(s) in mediation analyses, but do not consider dynamic change as represented by more than two timepoints.

The most commonly used LCSM model is a model similar to a piecewise LGM, where each LCS between time t – 1 and t is represented in the model (Selig & Preacher, 2009). As an example of this model, mediation would be represented as ΔX_1–2 predicting ΔM_2–3 (the a path) and ΔM_2–3 predicting ΔY_3–4 (the b path), as well as ΔX_1–2 predicting ΔY_3–4 (the c’ path). This conceptual model is a longitudinal extension of the simple mediation models widely represented in the psychology literature (see, for example, Figure 1). A similar model can be fit where loadings of observed variables on latent variables are fixed, such that an initial true score and LCSs from each time t – 1 to t are represented in the model, with unconstrained relations resulting in multiple a, b, and c’ paths (referred to in prior literature as a “modified” LCS model; Goldsmith et al., 2018b). The b path in these models can be lagged (ΔM_2–3 → ΔY_3–4 as is typically the case due to causal assumptions of temporal precedence) or contemporaneous (ΔM_2–3 → ΔY_2–3), which, as indicated in the discussion, has implications for the causal interpretation of mediation models.

Figure 1.

Example of Mediation Model as Conceptualized in Prior Latent Change Score Research.

Illustrative Examples

Extending prior research that has focused on LCSM models examining change between only two timepoints, we provide three examples that examine change between timepoints over five repeated measurements for various mediation variables. Data for these examples were simulated with SAS 9.4. Observed variables at single timepoints (i.e., X in Examples 1 and 2 and M in Example 2) were generated as random normal variables. Longitudinal variables (i.e., Y in Examples 1–3, M in Examples 2–3, and X in Example 3) were generated using matrix formulae with given univariate initial levels and slopes, and then related to one another using the fixed parameter values shown in Table 1 and their respective LCS equations (shown below for each example).

Table 1.

Sample Size and Parameters for Simulated Examples.

Parameter	True value
Parameter	Example 1(n = 530)	Example 2(n = 520)	Example 3(n = 530)
μ_x0	−	−	10
μ_sx	−	−	−0.5
β_x	−	−	0.02
$σ_{x 0}^{2}$	−	−	1
$σ_{s x}^{2}$	−	−	1
$σ_{e (x)}^{2}$	−	−	0.1
μ_m0	−	6	6
μ_sm	−	0.3	0.3
β_m	−	−0.04	0.04
$σ_{m 0}^{2}$	−	1	1
$σ_{e (m)}^{2}$	−	1	1
$σ_{sm}^{2}$	−	0.025	0.025
μ_y0	7	7	7
μ_sy	−0.5	−0.5	−0.5
β_y	0.1	0.1	0.1
$σ_{y 0}^{2}$	1	1	1
$σ_{s y}^{2}$	1	1	1
$σ_{e (y)}^{2}$	0.101	0.101	0.01
a	−0.4	−0.4	−0.07
b	−0.1	−0.130	−0.130
c’	0.1	0.1	0.0003

Table 1 shows the sample sizes and relevant parameter values used for data generation; correlations among initial score and constant change means for longitudinal variables were set to .5, and cross-sectional variables were generated as continuous normally distributed variables with mean of 0 and variance of 1. Sample sizes were chosen based on recent simulation work recommending minimum sample sizes for mediation with LCSs (Simone & Lockhart, 2019), and parameter values were chosen to produce realistic developmental trajectories in each variable. Table 1 provides the sample sizes and parameter values used for each model.

The Mplus (Muthén & Muthén, 1998–2017) code for each of the models presented in this article are included in Appendix 1. In the following example sections, we first provide an interpretation of the relevant LCS parameters (i.e., initial true score, constant change, proportional change, intercept-slope covariance), followed by interpretation of each path involved in mediation, as well as an example of how the mediation portions would be interpreted in the context of an applied developmental research question. We assessed mediation using the joint significance test as well as estimating the product of coefficients ab and its associated p value and 95% bootstrap confidence interval. In each of the examples, we constrained the constant change and proportional change parameters to be equal over time, resulting in single estimates of each parameter. However, certain research questions would warrant freeing these constraints, which we describe in the discussion.

Example 1: Single Predictor, Single Mediator, Longitudinal Outcome Model

An LCSM model with a single predictor, single mediator, and longitudinal outcome is provided in Example 1. Figure 2 provides a path diagram illustrating this model’s parameters.

Figure 2.

Path Diagram for the LCSM Model Presented in Example 1. Paths sharing labels are constrained to be equal. Unlabeled paths are constrained to be 1, except the paths marked by an asterisk. Variances and covariances are represented with double-headed arrows. Variances and covariances not shown are constrained to be 0.

This model can be expressed with the following system of equations:

M_{2} = a X_{1}

(5)

Δ y_{t i} = s_{y i} + β_{y} \times l y_{t - 1 i}

(6)

s_{y i} = b M_{2} + c^{'} X_{1} .

(7)

In this model, X at Time 1 is predicting M at Time 2 via the a path. The slope parameter s_yi represents constant change in Y for participant i, and β_y represents proportional change in Y relating prior level of latent Y to later change in latent Y. X at Time 1 is predicting constant change in Y via the c’ path, and M at Time 2 is predicting constant change in Y via the b path.

Results (including bootstrapped confidence intervals) from the LCSM model with a single predictor, single mediator, and longitudinal outcome are summarized in Table 2.

Table 2.

Unstandardized Estimates from the LCSM Model with Single X, Single M, and Longitudinal Y.

Parameter	Estimate (SE)	95% CI
Univariate dual change portion
μ_y0	6.987 (0.028)***	[6.933, 7.04]
μ_sy	−0.869 (0.153)***	[–1.172, −0.58]
β_y	0.151 (0.021)***	[0.111, 0.192]
$σ_{y 0}^{2}$	0.353 (0.024)***	[0.304, 0.398]
$σ_{s y}^{2}$	0.026 (0.004)***	[0.019, 0.036]
$σ_{y 0, s y}$	0.031 (0.011)**	[0.01, 0.051]
$σ_{e (y)}^{2}$	0.101 (0.003)***	[0.094, 0.108]
Bivariate portion
X → y₀	−0.014 (0.028)	[−0.069, 0.041]
M → y₀	−0.011 (0.029)	[−0.068, 0.045]
Mediation portion
a	−0.412 (0.038)***	[−0.488, −0.337]
b (M → s_y)	−0.095 (0.010)***	[−0.114, −0.077]
c’ (X → s_y)	0.122 (0.010)***	[0.102, 0.142]
ab (product of a and b)	0.039 (0.005)***	[0.03, 0.05]

Note. LCSM: latent change score mediation; 95% CI: bootstrap confidence interval; SE: standard error.

**p ⩽ .01. ***p ⩽ .001.

LCS Model Parameters

LCSs were included in this model for Y only, as Y was the only longitudinal variable in this example. The model-predicted mean initial true score for Y was 6.987. The mean constant change of Y was −0.869, indicating a negative trend in Y over time. The initial true score and constant change latent variables covaried significantly such that individuals with higher initial true scores on Y were also expected to have higher constant change in Y over time. In addition, the positive proportional change parameter (β = 0.151) indicated an overall amplification of the negative change in Y over time. Thus, predicted Y decreased over time and those decreases accelerated with each successive wave.

Mediation Parameters

The mediation portion of the model in Example 1 consisted of an a path equivalent to the single a path in traditional single mediator models. X and M were included as predictors of the slope of Y to represent the mediation b and c’ paths. X negatively predicted M (a = −0.412) and positively predicted change in Y over time (c’ = 0.122). M negatively predicted change in Y over time (b = −0.095). Using the joint significance test to determine the presence of mediation based on assessing the significance of the a and b paths separately, M was a significant mediator of the relationship between X and dynamic change in Y. The 95% percentile bootstrap confidence intervals of the product ab did not include zero, 95% CI = [0.03, 0.05], providing further evidence of mediation.

Contextual Substantive Illustration of the Mediation Parameters

To illustrate how this model could be implemented in a developmental research study, consider the following example: the effect of an early childhood temperamental trait, higher attention span, at a single timepoint may predict change in overt antisocial behavior over the course of adolescence through (i.e., mediated by) social preference, a social status indicator of how much children are liked by their peers, in middle childhood (e.g., a modified version of the environmental elicitation process tested in Berdan et al., 2008; Buil et al., 2017). Applied to this example, these simulated results would indicate that the relation between early childhood temperament and change in overt antisocial behavior over adolescence was mediated by social preference in middle childhood. Said differently, children scoring higher on high attention in early childhood were predicted to have lower social preference in middle childhood and a higher slope of antisocial behavior across adolescence. In turn, children with higher social preference in middle childhood were predicted to have a lower slope of antisocial behavior across adolescence.

Example 2: Single Predictor, Longitudinal Mediator, Longitudinal Outcome Model

Our second example is a LCSM model with a single predictor, longitudinal mediator, and longitudinal outcome. Figure 3 provides a path diagram illustrating this model’s parameters.

Figure 3.

Path Diagram for the LCSM Model Presented in Example 2. Paths sharing labels are constrained to be equal. Unlabeled paths are constrained to be 1, except the paths marked by an asterisk. Variances and covariances are represented with double-headed arrows. Variances and covariances not shown are constrained to be 0.

The following equations express the model in Figure 3:

Δ m_{t i} = s_{m i} + β_{m} \times l m_{t - 1 i}

(8)

s_{m i} = a X_{1}

(9)

Δ y_{t i} = s_{y i} + β_{y} \times l y_{t - 1 i} + b \times l m_{t - 1 i}

(10)

s_{y i} = c^{'} X_{1}

(11)

In this model, s_mi and s_yi represent constant change for participant i in M and Y, and β_m and β_y represent proportional change parameters for latent M and Y relating prior level to later change. X at Time 1 is predicting constant change in M via the a path and predicting constant change in Y via the c’ path (equations (9) and (11)). Prior latent level of M at time t – 1 is predicting later latent change in Y at time t via the constrained b path, which is a coupling parameter.

Coupling is used to represent the b path, which means that in this model the mediator is represented by the latent levels of M across time. A different parameterization could be specified such that the LCSs for M at t – 1 would predict later change in Y using this alternative for equation (10): $Δ y_{t i} = s_{y i} + β_{y} \times l y_{t - 1 i} + b \times Δ m_{t - 1 i}$ . Such a parameterization would not represent coupling (prior level influencing later change) but would examine prior change in M influencing later change in Y.

The results from this analytic example are summarized in Table 3.

Table 3.

Unstandardized Estimates from LCSM Model with Longitudinal M and Y.

Parameter	Estimate (SE)	95% CI
Multivariate dual change portion
Univariate information for M
μ_m0	6.041 (0.032)***	[5.977, 6.103]
μ_sm	0.382 (0.056)***	[0.273, 0.492]
β_m	−0.053 (0.009)***	[−0.071, −0.035]
$σ_{m 0}^{2}$	0.521 (0.033)***	[0.454, 0.583]
$σ_{s m}^{2}$	0.013 (0.001)***	[0.011, 0.017]
$σ_{m 0, s m}$	0.048 (0.006)***	[0.035, 0.06]
$σ_{e (m)}^{2}$	0.026 (0.001)***	[0.024, 0.027]
Univariate information for Y
μ_y0	7.04 (0.029)***	[6.982, 7.096]
μ_sy	−0.256 (0.165)	[−0.577, 0.064]
β_y	0.09 (0.012)***	[0.067, 0.114]
$σ_{y 0}^{2}$	0.38 (0.028)***	[0.324, 0.433]
$σ_{s y}^{2}$	0.053 (0.008)***	[0.039, 0.07]
$σ_{y 0, s y}$	0.079 (0.012)***	[0.055, 0.102]
$σ_{e (y)}^{2}$	0.101 (0.004)***	[0.094, 0.108]
Bivariate information
X → y₀	−0.012 (0.03)	[−0.07, 0.047]
X → m₀	0.025 (0.035)	[−0.044, 0.091]
σ_m0,y0	0.222 (0.023)***	[0.176, 0.266]
σ_sm,sy	0.017 (0.002)***	[0.012, 0.022]
σ_m0,sy	0.091 (0.016)***	[0.06, 0.123]
σ_y0,sm	0.035 (0.005)***	[0.026, 0.044]
Mediation portion
a (X → s_m)	−0.406 (0.008)***	[−0.42, −0.391]
b (δy [coupling M → Y], constrained to be equal over time)	−0.156 (0.02)***	[−0.196, −0.117]
c’ (X → s_y)	0.077 (0.016)***	[0.045, 0.107]
ab (product of a and b)	0.063 (0.008)***	[0.047, 0.08]

Note. LCSM: latent change score mediation; 95% CI: bootstrap confidence interval; SE: standard error.

***p ⩽ .001.

LCS Model Parameters

Example 2 expands upon Example 1 by replacing the single mediator with a longitudinal mediator, so the results for this model include LCSs and related parameters for both M and Y. Starting with the mediator, the mean initial true score for M was 6.041. A positive linear trend in M was expected over time, as the mean constant change of M was 0.382. The covariance between the initial true score and constant change for M was positive and significant, such that individuals with higher initial true scores on M were also expected to have greater constant change in M over time. Regarding the proportional change parameter, there was an overall attenuation of the positive change in M over time (β_m = −0.053). Taking the constant change and proportional change together, these results show that M was predicted to increase over time and those increases attenuated with each successive wave.

Regarding the outcome, the mean initial true score for Y was 7.04; the mean constant change of Y was −0.256, indicating a negative but nonsignificant trend in Y over time. As was also the case with M, individuals with higher initial true scores on Y were also expected to have higher constant change in Y over time given the positive and significant covariance between the initial true score and constant change for Y. The positive proportional change for Y (β_y = 0.09) indicated that the negative change in Y amplified over time. Thus, Y was predicted to decrease over time and those increases were larger with each successive wave.

Mediation Parameters

Example 2 included longitudinal measurement of both M and Y, which presents the opportunity to address research questions that involve dynamic change in a mediator, inherently changing the interpretation of the mediation b path. That is, in Example 2, M is longitudinal and also a predictor of Y, enabling the use of the coupling parameter as the b path. These results show that X (at its single measurement) negatively predicted the slope of M (a = −0.406) and positively predicted the slope of Y (c’ = 0.077). Coupling from M to Y was significant, with higher prior (i.e., t – 1) levels of M negatively predicting later growth in Y (b = −0.156). According to the joint significance test, mediation was again present in this model. This means that the single X variable significantly predicted dynamic change in M, and prior levels of M significantly predicted later dynamic change in Y. The 95% percentile bootstrap confidence interval of ab did not include zero, 95% CI = [0.047, 0.080], supporting the conclusion that mediation is present.

Contextual Substantive Illustration of the Mediation Parameters

Continuing with the substantive example from Model 1, Model 2 could be implemented in a study in which a single measurement of early childhood temperament (high attention) may predict change in social preference across middle childhood, and prior levels of social preference may, in turn, predict change in antisocial behavior. In that case, results from the mediation portion of this LCSM model from simulated data would indicate that high attention in early childhood predicted a lower slope of social preference and higher slope of antisocial behavior. In addition, coupling between social preference and antisocial behavior over time was present such that higher levels of social preference at one timepoint predicted less change in antisocial behavior at the next timepoint. Thus, these results would indicate that mediation was present in the relation between high attention in early childhood and change in antisocial behavior through change in social preference.

Example 3: Longitudinal Predictor, Longitudinal Mediator, Longitudinal Outcome Model

Our third LCSM is the most complex model presented in this article, with a longitudinal predictor, longitudinal mediator, and longitudinal outcome. Figure 4 provides a path diagram illustrating this model’s parameters.

Figure 4.

Sample Path Diagram for the LCSM Model Presented in Example 3. Paths sharing labels are constrained to be equal. Unlabeled paths are constrained to be 1. Variances and covariances are represented with double-headed arrows. Variances and covariances not listed are constrained to be 0.

The following system of equations express the model in Figure 4:

Δ x_{t i} = s_{x i} + β_{x} \times l x_{t - 1 i}

(12)

Δ m_{t i} = s_{m i} + β_{m} \times l m_{t - 1 i} + a \times l x_{t - 1 i}

(13)

Δ y_{t i} = s_{y i} + β_{y} \times l y_{t - 1 i} + b \times l m_{t - 1 i} + c^{'} \times l x_{t - 1 i}

(14)

Equation (12) is a dual change score model for X, including a constant change component for participant i, s_xi, and proportional change component β_x. In equations (13) and (14), s_mi and s_yi represent constant change in M and Y for participant i, and β_m and β_y represent proportional change for latent M and Y. The constrained a path is a coupling parameter relating prior latent level of X at time t – 1 to later latent change in M at time t. The constrained b path is a coupling parameter relating prior latent level of M to later latent change in Y. Finally, the constrained c’ path is a coupling parameter relating prior latent level of X to later latent change in Y. In this model, coupling is used to represent the b path, and therefore the mediator is represented by the latent levels of M across time. As mentioned in Example 2, an alternative parameterization could be specified such that the LCSs for M would predict later change in Y.

The results for this analytic model are summarized in Table 4.

Table 4.

Unstandardized Estimates from the LCSM Model with Longitudinal X, M, and Y.

Parameter	Estimate (SE)	95% CI
Multivariate dual change portion
Univariate information for X
μ_x0	10.027 (0.031)***	[9.967, 10.088]
μ_sx	−0.435 (0.216)*	[−0.872, −0.025]
β_x	0.014 (0.023)	[−0.029, 0.059]
$σ_{x 0}^{2}$	0.447 (0.032)***	[0.387, 0.511]
$σ_{s x}^{2}$	0.013 (0.003)***	[0.008, 0.02]
σ_x0,sx	0.037 (0.012)**	[0.012, 0.06]
$σ_{e (x)}^{2}$	0.096 (0.003)***	[0.09, 0.103]
Univariate information for M
μ_m0	6.02 (0.025)***	[5.971, 6.071]
μ_sm	0.465 (0.116)***	[0.239, 0.693]
β_m	0.039 (0.019)*	[0.004, 0.078]
$σ_{m 0}^{2}$	0.333 (0.021)***	[0.293, 0.375]
$σ_{s m}^{2}$	0.049 (0.005)***	[0.041, 0.06]
σ_m0,sm	0.064 (0.008)***	[0.049, 0.081]
$σ_{e (m)}^{2}$	0.025 (0.001)***	[0.023, 0.027]
Univariate information for Y
μ_y0	7.005 (0.022)***	[6.961, 7.049]
μ_sy	−0.419 (0.155)**	[−0.725, −0.112]
β_y	0.122 (0.012)***	[0.098, 0.146]
$σ_{y 0}^{2}$	0.263 (0.017)***	[0.232, 0.295]
$σ_{s y}^{2}$	0.003 (0.001)*	[0.002, 0.007]
σ_y0,sy	0.008 (0.003)**	[0.002, 0.013]
$σ_{e (y)}^{2}$	0.01 (<0.001)***	[0.01, 0.011]
Multivariate information
σ_x0,m0	0.183 (0.02)***	[0.143, 0.223]
σ_x0,y0	0.163 (0.018)***	[0.129, 0.2]
σ_m0,y0	0.151 (0.015)***	[0.121, 0.181]
σ_sx,sm	0.013 (0.003)***	[0.008, 0.018]
σ_sx,sy	0.003 (0.001)*	[0.001, 0.006]
σ_sm,sy	0.004 (0.002)*	[0, 0.007]
σ_x0,sm	0.074 (0.011)***	[0.054, 0.095]
σ_x0,sy	0.025 (0.011)*	[0.004, 0.046]
σ_m0,sx	0.033 (0.007)***	[0.019, 0.047]
σ_m0,sy	0.005 (0.005)	[−0.005, 0.015]
σ_y0,sx	0.031 (0.006)***	[0.019, 0.043]
σ_y0,sm	0.058 (0.007)***	[0.045, 0.072]
Mediation portion
a (δm [coupling X → M], constrained to be equal over time)	−0.088 (0.016)***	[−0.119, −0.057]
b (δy [coupling M → Y], constrained to be equal over time)	−0.092 (0.014)***	[−0.119, −0.064]
c’ (coupling X → Y, constrained to be equal over time)	−0.046 (0.027)	[−0.1, 0.009]
ab (product of a and b)	0.008 (0.002)***	[0.005, 0.012]

Note. LCSM: latent change score mediation; 95% CI: bootstrap confidence interval; SE: standard error.

p ⩽ .05. **p ⩽ .01. ***p ⩽ .001.

LCS Model Parameters

Because Example 3 includes the longitudinal measurement of X, M, and Y, each have corresponding LCS parameters. The initial predictor X exhibited a negative trend over time (i.e., the mean initial true score for X was 10.027, and the mean constant change of X was −0.435). The initial true score and constant change covaried significantly such that individuals with higher initial true scores on X were also expected to have higher constant change in X over time. In addition, the positive proportional change parameter (β_x = 0.014) indicated an overall acceleration (though nonsignificant) of the negative change in X over time. That is, X decreased over time and those decreases accelerated with each successive wave.

The mean initial true score for M was 6.02. The mean constant change of M was 0.465, indicating a positive trend in M over time. As was the case with X, the initial true score and constant change of M covaried significantly; individuals with higher initial true scores on M were also expected to have higher constant change in M over time. In addition, the positive proportional change parameter (β_m = 0.039) indicated an overall acceleration of the positive change in Y over time. Therefore, M was predicted to increase over time, and those increases accelerated with each successive wave.

For Y, the mean initial true score was 7.005, and the mean constant change was −0.419, indicating a negative trend in Y over time. Again, the covariance between the initial true score and constant change was positive and significant; individuals with higher initial true scores on Y were also expected to have higher constant change in Y. The positive proportional change parameter (β_y = 0.122) taken with the negative slope indicated that Y decreased over time, and those decreases accelerated with each successive wave.

Mediation Parameters

Given the longitudinal measurement of each of the variables involved in the mediation portion of the model presented in Example 3, coupling was used to represent each of the mediation paths: a, b, and c’. Results showed that prior levels of X negatively predicted later growth in M (i.e., coupling from X to M; a = −0.088), and they did not significantly predict additional acceleration or attenuation of change in Y over time (i.e., coupling from X to Y; c’ = −0.046). Prior levels of M significantly predicted later decline in Y (i.e., coupling from M to Y; b = −0.092). That is, prior levels of X significantly predicted later dynamic change in M, and prior levels of M predicted later dynamic change in Y; thus, mediation was again present in this model using the joint significance test. The 95% percentile bootstrap confidence interval of the product of coupling parameters ab also did not include zero, 95% CI = [0.005, 0.012], indicating presence of mediation and supporting results of the joint significance test.

Contextual Substantive Illustration of the Mediation Parameters

Model 3 might be implemented, for example, in a developmental study of college students in which prior levels of heavy episodic drinking predicts changes in academic performance, and prior levels of academic performance predicts later change in psychological distress (e.g., an extension of the cross-sectional associations tested by Tembo and colleagues, 2017). The results from this model would demonstrate that college students’ heavy episodic drinking was negatively coupled with academic performance over time (i.e., higher prior levels of heavy episodic drinking at a given timepoint predicted less growth in academic performance to the next timepoint) and was not coupled with psychological distress. Higher academic performance was negatively coupled with psychological distress, such that higher academic performance at the prior timepoint was associated with attenuation in psychological distress at the next timepoint. Thus, these results would indicate that although heavy episodic drinking did not appear to directly influence change in psychological distress over time, support was found for mediation such that heavy episodic drinking was a risk factor for attenuated growth in academic performance and, in turn, students with higher academic performance demonstrated attenuated increases in psychological distress.

Discussion

In the present article, we provided three examples of mediation in the LCS framework, which can be used to address behavioral research questions regarding mechanisms of dynamic change in an outcome. We illustrated novel parameterizations of LCSM models to show how change in behavioral variables can be included as predictors, mediators, and outcomes, interpretations of the models’ parameters, and provided the Mplus code to conduct these analyses. As developmental researchers continue to pursue longitudinal research involving mediation, the LCSM models described in this article and its extensions will be useful in examining mechanisms of dynamic change.

Conceptualizations of Mediation Paths and Indirect Effects

The LCSM models presented in this article show three parameterizations of mediation in the LCS framework wherein the mediation paths either involve slopes of the outcomes or paths are constrained to be equal over time. Although the parameterizations presented here are likely to be appropriate in applied research settings, they do not represent an exhaustive list of the possibilities for assessing mediation in the LCS framework. This is one main advantage of using LCSM models to examine longitudinal mediation. In the models presented, the coupling a, b, and c’ paths could be freely estimated instead of constrained to be equal over time; for example, coupling paths representing peer influence on antisocial behavior may be expected to weaken in strength over time. The resulting model would then provide multiple estimates of the mediation path; the utility of such estimates would depend on the research question but could easily be accommodated by removing the constraints in the code provided in this article.

In addition, the models presented in Examples 1 and 2 use a more parsimonious approach to include a, b, and c’ paths when X is measured at a single timepoint by including each path as X predicting the slope of M or Y. An alternative, but less parsimonious model, would be one where X predicts the latent change in M or Y for the mediation paths. The models presented here are more readily interpretable from a conceptual and practical standpoint, in that they assess whether X predicts constant change in M or Y, and they provide information on the proportion of variance in constant change that is accounted for by X. However, if the research question dictated that X would predict change in M or Y differentially over time (i.e., paths should be freely estimated over time instead of being constrained to be equal), it would be appropriate to fit the less parsimonious model without constraints to allow for those coupling paths to vary over time.

Furthermore, for each pair of longitudinal variables, we use coupling to assess bivariate dynamic change over time, but other specifications could be used to examine the influence of change in one variable on change in another. Goldsmith and colleagues (2018b) provide an example of a modified LCSM model, conceptualized such that the specific indirect paths all ultimately arrive at the final measurement of Y. In doing so, they also included paths for multiple mediators, as well as LCSs that were measured contemporaneously (i.e., ∆X_1-2 → ∆M_1-2). Models presented by Selig and Preacher (2009) provide all potential indirect effects in an LCSM model that investigated change between two timepoints for each of the X, M, and Y variables (i.e., ∆X_1-2 → ∆M_2-3 → ∆Y_3-4). This model inherently included the simplest mediation model (X₁ → M₂ → Y₃), but it also included all possible combinations of levels and change between two timepoints as each of the variables in the mediation model. Furthermore, models could include parameters to assess influence of latent level in one variable on latent level in another (i.e., lx₁ → lm₁) in addition to coupling paths to the LCSs. To summarize the issue of estimation of indirect effects in LCSM models, a wide variety of specific indirect effects could be included in the model when the mediation paths are estimated freely and not constrained to be equal over time, and mediation paths with longitudinal variables do not necessarily have to be conceptualized as coupling paths. Thus, researchers should carefully consider which parameterization most appropriately answers their research questions regarding mechanisms of dynamic change.

Causality

One issue that is not unique to LCSM models but deserves particular attention here is the issue of causality. In the following paragraphs, we describe issues of temporal precedence for longitudinal mediation generally and the LCSM model specifically, and address the issue of confounding for M and nonrandomized X in the LCSM model.

Temporal Precedence

Mediation implies that there is a temporal ordering of model effects (MacKinnon, 2008). However, mediation models are often applied to cross-sectional data where X, M, and Y are examined at single timepoints (e.g., X₁ → M₁ → Y₁). Such models may fail to appropriately model the temporal effects. To illustrate the potential problem, take for example a model in which researchers have longitudinal measurement of their X, M, and Y measures, each at Waves 1 through 3. When each of the estimated paths represents the relation between one measure at time t – 1 with another at time t (e.g., X₁ → M₂ → Y₃), the assumption of temporal precedence is met and does not hinder the causal interpretation of these paths (when other causal assumptions are also met). Longitudinal mediation models where some or all of the model variables are measured repeatedly over time do not necessarily suffer from this problem (Selig & Preacher, 2009). As mentioned in the previous section, however, LCSM models can be specified with the mediation paths represented by coupling parameters. When the coupling parameter is lagged such that change in one variable between times t – 1 and t is predicted by levels of the other variable at time t – 1, the process being predicted is effectively temporally preceded by the predictor at t – 1, again avoiding problems regarding the assumption of temporal precedence.

However, there are unique circumstances in LCSM models that can be problematic for causal interpretation. LCSM models could, for example, be specified so that the paths represent contemporaneous change (e.g., b path where ΔM_2–3 → ΔY_2–3) or where the process is effectively being predicted by a variable measured at a later timepoint (e.g., b path where M₃ → ΔY_2–3). In each of these cases, the assumption of temporal precedence is not met. Researchers should be aware that this issue arises in decisions about specifications and make decisions regarding which estimates are more appropriate, giving consideration to both their substantive research question and the statistical implications. To this point, though conventional wisdom regarding mediation requires that M temporally precede Y (MacKinnon, Fairchild, & Fritz, 2007) in certain circumstances with longitudinal mediation, contemporaneous paths may be theoretically justifiable or provide better model fit, although this is more common in clinical trials research with randomized X where the first measurement of the mediator occurs closer in time to (or at the same time as) the outcome (Goldsmith et al., 2018a, 2018b; Wang et al., 2009). Thus, developmental researchers studying relations where the true processes are lagged may more commonly use lagged mediation measurements with temporal precedence for X, M, and Y. As with all mediation models, it is essential that researchers should always choose mediation models to match their theory.

Confounding of Mediation Paths

Another issue related to causality is the implication of randomizing X in a mediation model, which applies to the LCSM model as well. The gold standard for studying mechanisms of change is to have a mediation model where X is randomized, for example, in a randomized controlled trial (RCT). When X is randomized, the assumption can be made that there are no unmeasured confounders of the X → M relation (i.e., the a path; MacKinnon, 2008). However, developmental researchers often have questions involving mediators where X is not a randomized variable or where randomization is not possible or ethical. When X is not randomized to conditions, this may violate the mediation assumption of no unmeasured confounders, and may introduce bias into the estimates due to unmeasured variables influencing the mediation relations. The same issue of unmeasured confounders is a forefront issue in mediation for the mediator, M, which is typically not randomized and is assumed to cause Y. Although confounding of X can more commonly be addressed by randomizing the X variable, randomizing the mediator is not often possible, and unmeasured confounders could potentially bias the M → Y relation (i.e., the b path).

Several approaches are available to address the potential confounding issues in simple mediation models stemming from nonrandomized X or M, mainly coming from the causal inference literature (MacKinnon, 2008; MacKinnon & Pirlott, 2015). However, in certain longitudinal mediation models, the possible influence of unmeasured confounders can be addressed by including correlated residuals (i.e., correlated measurement errors) between X and M and between M and Y (Goldsmith et al., 2018a, 2018b). Goldsmith et al. (2018a) recommended including contemporaneous residual covariances (e.g., residual covariance of X with M both at time t instead of lagged timepoints).

As with other longitudinal mediation models, we were able to include correlated residuals between M → Y in Example 2 where both M and Y were longitudinal, and between X → M and M → Y in Example 3 when all variables were longitudinal, partially addressing the issue of unmeasured confounders for longitudinal mediation. However, this method cannot be used for unmeasured confounders in LCSM models such as Example 1, where only one variable is longitudinal and therefore residual covariances are not present. Such models may use methods to deal with confounding that were developed for simple mediation models (MacKinnon, 2008; MacKinnon & Pirlott, 2015).

Time Measurement and LCSM Models

As with all mediation models, it is assumed that the mediation variables are measured with time metrics that are meaningful to the constructs being measured. Recent research has shown that measurement of time metric has important implications for interpreting bivariate LCS models, particularly showing that bivariate coupling estimates are biased when an incorrect time metric is assumed, and measurement lag and initial starting point are ignored (O’Rourke et al., 2021). This research has implications for the LCSM model as well, particularly the fully longitudinal model shown in Example 3. For results to be unbiased, the assumption must be made that all mediation variables exist on the same measurement timescale. If one variable were to change at a significantly faster or slower rate, or if one variable contained unmodeled lags, model results could demonstrate false mediation, or fail to detect a real mediated effect. An additional consideration of timing is the measurement choice of single versus repeated measures of the mediator. When the mediator is measured at a single timepoint, it is assumed that the mediator has no underlying temporal process; when repeated measures are used, an underlying temporal process is assumed.

Finally, time metric consideration should also inform the choice of mediation model itself. For example, LCSM models do not require specification of functional form of change. If researchers have strong theory about the specific functional form of change of each variable, a continuous time mediation model may be more appropriate (Deboeck & Preacher, 2016).

Limitations and Future Research

There are several important limitations to consider regarding this study. The pristine condition of simulated data is unlikely to be found in real longitudinal data, in which researchers encounter issues including attrition or missingness and non-normal (e.g., count or binary) mediators or outcomes. Additional research on missingness in LCS models is needed to more clearly understand best practices for accounting for missing data. Furthermore, one of the assumptions of the LCS model is that variables are continuous and follow a multivariate normal distribution, so these models are not appropriate for non-normal longitudinal outcomes, a circumstance which also applies to the LCSM model. There are many additional parameterizations of LCSM models that answer specific research questions, some of which were mentioned earlier in the discussion; although a full accounting of all of these examples are beyond the scope of this article, it should be noted that these extensions are available. Researchers may also be interested in questions that involve heterogeneity of the mediated effect (e.g., gender differences), which would warrant solutions including multigroup LCSM models.

One limitation regarding longitudinal mediation models in general is the lack of interpretable mediation effect sizes. Most mediation effect sizes have been adapted only for classical single-variable mediation, without feasible extensions to longitudinal models. For example, adjusted R² (Lachowicz et al., 2018) is based on calculations of variance in Y for a linear regression mediation model without extensions to models with repeated measures for Y. Partially and fully standardized mediation effect sizes perform well in terms of power and Type I error (Miočević et al., 2018), but the interpretation of these effect sizes is not meaningful in latent change models where the mediation paths are in the metric of the LCSs (i.e., not the original metric of Y). The ratio and proportion mediation effect sizes tend to perform poorly (Miočević et al., 2018) and κ² (a ratio of the estimated mediated effect to a maximum possible mediated effect; Preacher & Kelley, 2011) has been shown to be non-monotonic (Wen & Fan, 2015) and has not been extended for use with longitudinal models. MedES (Kraemer, 2014) requires a binary X variable and is not useful for repeated measures of continuous X.

Finally, limited work has been conducted on power analyses for LCSM models. Simone and Lockhart (2019) provided sample size guidelines for models that include combinations of single measurements of X, M, and/or Y and LCSs between only two timepoints. Additional research is needed regarding power and sample size planning for LCSM models in which X, M, and/or Y are measured over more than two timepoints and when coupling paths are implemented as mediation paths.

Conclusion

In summary, mediation can be assessed in the LCS framework whenever one or more of the X, M, or Y variables are longitudinal and researchers are interested in questions concerning dynamic change. Previous research using LCSM models has focused on change between two timepoints. The present article provided example implementations of an LCSM model over a longer period of time, with combinations of cross-sectional and longitudinal mediation variables. In the discussion, we highlight important considerations related to LCSM models, namely that each data structure requires consideration of the multiple possible constructions of mediated effects, temporal ordering of variables, and the causal assumptions of mediation. LCSM models provide a modern and flexible method to investigate mechanisms of dynamic change over time, with many options for adjustment to nuanced research questions in the developmental literature.

Footnotes

Appendix 1 Acknowledgements

The authors are extremely grateful to Kevin J. Grimm (ASU) for his insightful feedback on an earlier version of this article.

Compliance with Ethical Standards

The analyses were conducted on simulated data and did not involve human participants or animals.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

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

ORCID iD

Chanler D. Hilley

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