The Impact of Omitting Confounders in Parallel Process Latent Growth Curve Mediation Models: Three Sensitivity Analysis Approaches

Abstract Parallel process latent growth curve mediation models (PP-LGCMMs) are frequently used to longitudinally investigate the mediation effects of treatment on the level and change of outcome through the level and change of mediator. An important but often violated assumption in empirical PP-LGCMM analysis is the absence of omitted confounders of the relationships among treatment, mediator, and outcome. In this study, we analytically examined how omitting pretreatment confounders impacts the inference of mediation from the PP-LGCMM. Using the analytical results, we developed three sensitivity analysis approaches for the PP-LGCMM, including the frequentist, Bayesian, and Monte Carlo approaches. The three approaches help investigate different questions regarding the robustness of mediation results from the PP-LGCMM, and handle the uncertainty in the sensitivity parameters differently. Applications of the three sensitivity analyses are illustrated using a real-data example. A user-friendly Shiny web application is developed to conduct the sensitivity analyses.


Introduction
Mediation analysis (Baron & Kenny, 1986) is widely used in many disciplines to investigate the mechanisms underlying the effects of treatments on outcomes.Conventionally, mediation analysis is often conducted using single-timepoint (e.g., cross-sectional) data, where the treatment, mediator, and outcome are each measured at a single time point.Such single-time-point mediation analysis has several limitations, such as the inability to model the role of time (e.g., Maxwell & Cole, 2007;Mitchell & Maxwell, 2013).In recent years, longitudinal data have been increasingly used for mediation analysis.With longitudinal data, the role of time can be explicitly accounted for; furthermore, researchers can include intraindividual change as part of mediation mechanisms (MacKinnon & Fairchild, 2009;Selig & Preacher, 2009).
Different types of longitudinal mediation models have been proposed (Preacher, 2015).Among them, a popular type is the latent growth curve (LGC) mediation model (e.g., Bollen & Curran, 2006;Cheong et al., 2003;von Soest & Hagtvet, 2011).For example, LGC mediation models were adopted to test the mediation effects of a mobile-type intervention program on the level and change of depressive symptoms via the level and change of emotional self-awareness (Kauer et al., 2012), to evaluate whether the pathways from maternal parentification history to the level and change of maternal warm responsiveness were mediated by maternal infant development knowledge (Nuttall et al., 2015), and to investigate whether the level and change of English reading achievement mediated the pathways from English learners' proficiency in native language to the level and change of their achievement in science (Guglielmi, 2012).

The Parallel Process Latent Growth Curve Mediation Model (PP-LGCMM)
We consider the scenario where the treatment variable X i is time-invariant (e.g., intervention assignment), whereas the mediator M it and outcome Y it are both time-varying and are measured at time points t ¼ 1, :::, T for individuals i ¼ 1, :::, n: In this scenario, the PP-LGCMM (Cheong et al., 2003;MacKinnon, 2008) is frequently used to examine whether and how much the associations of treatment with the level and change of outcome are mediated by the level and change of mediator.
In the PP-LGCMM (Figure 1), repeated measures of the mediator and the outcome are viewed as manifest indicators of the latent mediator change trajectories and latent outcome change trajectories, respectively.To describe the within-person change trajectories, a widely used LGC model can be written as: (1) (2) Equation 1 is the mediator's LGC model, where I Mi and S Mi represent the latent mediator intercept and latent mediator slope for person i, respectively; Equation 2 presents the outcome's LGC model, where I Yi and S Yi denote person i' latent outcome intercept and latent outcome slope, respectively.In this article, we consider the LGC models with known slope loadings.That is, in Equation 1 (and Equation 2), the values of Time Mt (and Time Yt ) at all t ¼ 1, :::, T are known constants coded by researchers.For example, researchers could code the slope loadings as Time Mt ¼ t À 2 and Time Yt ¼ t À T, respectively.With such factor loadings, S Mi and S Yi represent the mediator's and outcome's linear rates of change, respectively; and I Mi and I Yi represent the mediator's level at the second time point (t ¼ 2) and the outcome's level at the last time point (t ¼ T), respectively.The within-person residuals e Mit and e Yit represent the deviations of person i's observed mediator and outcome scores at time point t from the person's latent mediator and outcome change trajectories, respectively.It is assumed that e Mit and e Yit have means 0 within each person (i.e., E½e Mit ji ¼ 0 and E½e Yit ji ¼ 0 for all t and all i).Let e i ¼ ðe Mi1 , :::, e MiT , e Yi1 , :::, e YiT Þ 0 contain the within-person residuals of the mediator and outcome of person i. e i is assumed to be independent across persons and follow a multivariate normal distribution with mean 0 and constant variancecovariance matrix H.We assume the within-person model in Equations 1-2 is correctly specified throughout.
With the PP-LGCMM, researchers are often interested in the mediation effects of treatment on the intercept and slope of outcome through the intercept and slope of mediator.To assess these mediation effects, the between-person model of the PP-LGCMM is often specified as Z is a vector of observed pretreatment covariates (collected at the baseline).The residual vector ðv IM , v SM Þ 0 of the mediator intercept and slope is modeled as independent of the residual vector ðv IY , v SY Þ 0 of the outcome intercept and slope.The two residual vectors ðv IM , v SM Þ 0 and ðv IY , v SY Þ 0 are assumed to be normally distributed with means 0 and covariance matrices W vM and W vY , respectively.The mediation paths include (1) a IM:X and a SM:X in Equations 3-4, path coefficients from treatment to the mediator intercept and slope in Figure 1, respectively (treatment-mediator relations; "a-paths"), and (2) b IY:IM , b IY:SM and b SY:IM , b SY:SM in Equations 5-6, path coefficients from mediator intercept and slope to the outcome intercept and slope in Figure 1, respectively (mediator-outcome relations; "b-paths").The direct paths from treatment to the outcome intercept and slope have coefficients b IY:X and b SY:X , respectively.In the PP-LGCMM, there are two mediator factors, I M and S M , and two outcome factors, I Y and S Y, leading to four mediation (i.e., indirect) pathways of interest.
In the real data example we will show later, participants were randomized to receive either a parenting intervention or a control condition; the mediation mechanisms by which the intervention affected the trajectories of children's lability/negativity (Y) via the level and/or change of maternal sensitive guidance (MÞ were investigated.The four mediation effects of substantive interest were the mediation effects of treatment (XÞ on the level (I Y) and on the change (S Y Þ of child lability/negativity via the level (I M ) and via the change (S M ) of maternal sensitive guidance.Using the product-of-coefficients approach (Baron & Kenny, 1986;Cheong et al., 2003;MacKinnon, 2008) We consider the PP-LGCMM in Equations 1-6 as the original model fitted in an empirical study, and refer to it as Model M0.To make inference of mediation effects from statistical mediation models, including the PP-LGCMM, several assumptions are needed (MacKinnon, 2008).A frequently discussed assumption is the no-omitted-confounders Besides the product-of-coefficients approach, mediation effects can be defined using approaches proposed in causal inference literature, which often involves the use of the potential outcomes framework (e.g., Pearl, 2001;Robins & Greenland, 1992;Valeri & VanderWeele, 2013).In this article, we consider the product-of-coefficients approach because it is widely used in empirical PP-LGCMM analysis in psychology and behavioral sciences and is relatively easy for substantive researchers to interpret.
assumption 2 (Judd & Kenny, 1981;MacKinnon, 2008).For Model M0, the no-omitted-confounders assumption requires that there should be no omitted pretreatment confounders of the relations between treatment and outcome intercept/slope, between treatment and mediator intercept/slope, and between mediator intercept/slope and outcome intercept/slope (MacKinnon, 2008;Tofighi et al., 2019).However, even when the treatment variable is a randomized assignment, meaning that the treatment-outcome and treatmentmediator relations would have no confounders in expectation, there could be unobserved pretreatment confounders of the mediator-outcome relations, because of the infeasibility to randomize the mediator.
In our real data example, even though the intervention assignment was randomized, the level and change of mothers' sensitive guidance were non-randomized, making the mediator-outcome relations subject to the influence of confounding variables.For instance, maternal depression was found to be related to children's emotional problems and mothers' negative behaviors during mother-child interactions (e.g., Goodman et al., 2011;Valentino et al., 2022).Thus, pretreatment maternal depression could be a potential confounder of the mediator-outcome relations in the PP-LGCMM, because (1) pretreatment maternal depression might influence posttreatment level (or change) of maternal sensitive guidance, and also influence posttreatment level (or change) of child lability/negativity, and (2) these influences could exist even conditional on the treatment and observed covariates.If data on pretreatment maternal depression were unobserved, the no-omitted-confounders assumption would be violated, and inference of the mediation effects from the PP-LGCMM (Model M0) could be misleading.
To handle potential violations of the assumption, sensitivity analysis can be conducted to examine the robustness of the inference from the original model.

Impact of Omitting Confounders on Inference of Mediation
For single-mediator models, many sensitivity analysis methods have been proposed, often differing in the sensitivity parameters used to quantify omitted confounding effects.For example, sensitivity parameters can be the regression coefficients representing associations of an unobserved confounder with the mediator and outcome (e.g., Albert & Wang, 2015;Smith & VanderWeele, 2019;VanderWeele, 2010), the correlation between residuals of the mediator model and outcome model (e.g., Imai, Keele, & Tingley, 2010;Imai, Keele, & Yamamoto, 2010), or the weights calculated given the relations of an unobserved confounder with the mediator and outcome (Hong et al., 2018(Hong et al., , 2021)).
For multiple-mediator models, sensitivity analysis methods exist for models with two independent mediators, with multiple sequentially related mediators, or with other types of correlated mediators (e.g., Daniel et al., 2015;Imai & Yamamoto, 2013;Park et al., 2018Park et al., , 2022;;Park & Esterling, 2021).The original mediation models in these methods do not involve latent variables.For latent variable mediation models, particularly an LGCMM with a single-time-point manifest outcome and the latent mediator intercept and slope as two covarying mediator factors, Tofighi et al. (2019) proposed the correlated-augmented-model method and Kruger et al. (2022) developed related software for sensitivity analysis.In their method, the correlations of the outcome's residual with the mediator intercept's and mediator slope's residuals are the sensitivity parameters.
Many previous sensitivity analyses for mediation models adopted the frequentist approach, in which the sensitivity parameters are treated as fixed constants (Lash et al., 2009).However, uncertainty around the sensitivity parameters may exist.Incorporating such uncertainty is challenging with frequentist sensitivity analysis methods (e.g., Gustafson & Greenland, 2009).
In contrast, sensitivity parameters are treated as random variables with prior distributions from a Bayesian perspective (e.g., Greenland, 2005;Gustafson & McCandless, 2010;Harring et al., 2017).Uncertainty in the sensitivity parameters is allowed and can be accounted for conveniently via informative prior distributions.Sensitivity analysis approaches that use such prior distributions can include the Bayesian sensitivity analysis (BSA, e.g., Greenland, 2001;McCandless et al., 2007) and the Monte Carlo sensitivity analysis (MCSA, e.g., Phillips, 2003;Greenland, 2005).BSA approaches employ Bayesian estimation to obtain the adjusted point and interval estimates of focal parameters of interest.MCSA approaches obtain the adjusted point and interval estimates via iterative algorithms, in which the sensitivity parameters are drawn from their prior distributions, and then the focal parameters are drawn from the sampling distributions of their adjusted estimators (adjusted for the current drawn sensitivity parameter values).From either BSA or MCSA, the interval estimates account for uncertainty due to both random sampling and potential departures from the original assumptions (Greenland, 2005).
For sensitivity analysis of mediation models, uncertainty in the sensitivity parameters were considered in only two methods (to the best of our knowledge).Harring et al. (2017) proposed the phantom variable strategy with a random-parameter approach for assessing the influence of potential external misspecification (e.g., omitting confounders) on general structural equation models.Harring et al. 2 In addition to the no-omitted-confounders assumption, mediation inference from a statistical mediation model requires the model to have correct functional form specification and correct assumptions on the residual distributions, the causal ordering of treatment, mediator, and outcome, and measurement errors in observed variables (e.g., MacKinnon, 2008;Mackinnon et al., 2007).
(2017) illustrated the approach in a path analysis model, but did not focus on the inference of mediation effects.For mediation inference, McCandless and Somers (2019) developed an MCSA algorithm for the single-mediator model.
For the PP-LGCMM of interest, impacts of potential omitted confounders on the inference of the mediation effects have received limited attention; sensitivity analysis methods and tools are still unavailable for researchers to address concerns about potential uncontrolled confounding effects.

The Current Study
To reduce the research gap, the current study has two goals.First, we analytically examine how the omission of potential pretreatment confounders impacts the inferences of mediation effects from the PP-LGCMM.The treatment variable is considered as randomized and independent of pretreatment variables.Specifically, we construct a sensitivity analysis model using the phantom variable strategy (Harring et al., 2017).Then we analytically examine how the point estimates and test results of the mediation effects from the sensitivity analysis model differ from those yielded by the original model.
Second, we develop three sensitivity analysis approaches for PP-LGCMMs: the frequentist, BSA, and MCSA approaches.The frequentist approach presents researchers with information on when potential omitted confounders (i.e., what specific sensitivity parameter values) would reverse the original mediation inference (e.g., non-zero estimates to zero, statistical significance to nonsignificance).The BSA and MCSA help address the question on what the adjusted mediation inference would be, after incorporating researchers' knowledge and uncertainty about potential omitted confounders.
Applications of the three sensitivity analysis approaches will be illustrated using the real data example.Furthermore, we develop easy-to-use tools such as an R Shiny web application for implementing the proposed sensitivity analyses.
The rest of the article is organized as follows.We first conduct an analytical examination to investigate the impact of omitting confounders.Then, we develop the three sensitivity analysis approaches.Next, we illustrate the developed sensitivity analyses in the real data example.We end the article with discussing the implications and limitations of the current study and outlining future research directions.

Analytical Examination of the Impact of Omitting Confounders in the PP-LGCMM
In this section, we examine how the inferences of mediation effects in the PP-LGCMM (Model M0 in Equations 1-6) are influenced by the omission of a pretreatment confounder.We consider a randomized treatment variable and assume the omitted confounders are linearly related to the intercepts and slopes of the mediator and outcome.We build the sensitivity analysis model based on the phantom variable strategy in Harring et al. (2017) and the related latent augmented models in Tofighi and Kelley (2016) and Tofighi et al. (2019).Specifically, we augment the original model (Model M0) with a latent confounder C that may affect the mediator intercept I M and slope S M as well as the outcome intercept I Y and slope S Y .With the phantom variable strategy, C serves as a placeholder that accounts for all the remaining (uncontrolled) confounding effects after controlling for the observed covariates.As such, the latent C may represent the unique variation of one or more potentially omitted substantive confounders unexplained by observed covariates Z (see e.g., Liu & Wang, 2021;Tofighi et al., 2019;Tofighi & Kelley, 2016).
Specifically, the sensitivity analysis model (Model M1; depicted in Figure 2) is specified by the following equations.The superscript " Ã " is used for parameters and residuals in Model M1.The within-person equations are the same as those in the original model (i.e., Equations 1-2 of Model M0).In the between-person equations, the latent confounder C is included linearly and additively: In Model M1, C is independent of Z, as C is used to account for the remaining confounding effects after controlling for the observed covariates Z in the original model.The residual vector ðv Ã IY , v Ã SY Þ 0 of the outcome intercept and slope is assumed to be independent of the residual vector ðv Ã IM , v Ã SM Þ 0 of the mediator intercept and slope.In addition, ).To facilitate the analytical examination, we consider the maximum likelihood (ML) estimation.Without loss of generality, we specify C to have mean 0 and variance 1. Below we present the main findings.Technical details are available in the supplemental materials.
In terms of the point estimates, as the randomized X is independent of C, the point estimates of the effects of X on the mediator intercept and slope (the a-paths) are not influenced by the omission of C. The estimates from the original model equal their counterparts from the sensitivity analysis model; that is: However, the point estimates of the path coefficients from mediator intercept and slope to outcome intercept and slope Therefore, due to the omission of C, the mediation effect estimates (i.e., products of the corresponding a-path and bpath estimates) from the original model can be biased.

Magnitudes of the biases depend on the confounding strengths of C (quantified by b
) and the variation and covariation of I M and S M unexplained by the treatment and observed covariates Z (quantified by /12 vM , /11 vM , and /22 vM from the original model).For example, Equation 13 implies that holding everything else constant, the bias in the original model's estimate for the path from I M to I Y (i.e., bIY:IM ) tends to be larger if b Ã IY:C is larger; the bias would also be larger if a Ã IM:C /11 vM is larger.Furthermore, due to the correlation between the intercept and slope of mediator, the estimates of the mediation effects via mediator intercept (or slope) from the original model can be different from those in the sensitivity analysis model, even when confounder C does not affect the mediator intercept (or slope).Take the mediation effect of treatment on I Y via I M as an example.When a Ã IM:C ¼ 0 (i.e., the confounder does not influence the mediator intercept), the estimate âIM:X bIY:IM from the original model can differ from the confounding-adjusted estimate âÃ IM:X bÃ IY:IM from the sensitivity analysis model, and the difference is âIM:X /12 vM Â b Ã IY:C a Ã SM:C : Even when a Ã IM:C ¼ 0, this difference can still exist if the mediator intercept and slope have a non-zero residual covariance estimate from the original model (i.e., /12 vM 6 ¼ 0).Besides the point estimates, the omission of confounder C can impact the test results of the mediation effects.To analytically demonstrate the impact, we consider the joint significance test (Mackinnon et al., 2002), 4 in which a mediation effect is statistically significant if the a-path and b-path are both statistically significant.For testing a path coefficient, we consider the Wald z-test, which is commonly produced by SEM software.
Take the mediation effect of treatment on I Y via I M as an example.For the a-path, the standard error estimate from the sensitivity analysis model is the same as that from the original model, b Together with Equation 11 for the point estimate (â Ã IM:X ¼ âIM:X ), the zstatistic and the associated p-value for testing the a-path is thus unaffected if C is omitted.
However, for the b-path, the standard error estimate from the sensitivity analysis model can be expressed as: where b seð bIY:IM Þ is the standard error estimate of the b-path estimate from the original model.Equation 17shows that where ŴvM is the estimated residual covariance matrix of ðv IM , v SM Þ 0 yielded by the original model.
The joint significance test can be used because the analytical results in Yuan and Bentler (2006) suggest that for the PP-LGCMM with known slope loadings (i.e., Time Mt and Time Yt are known), the ML estimates of the a-paths âÃ IM:X and âÃ SM:X from Model M1 (or the counterparts from Model M0) are asymptotically independent of the ML estimates of the b-paths, bÃ IY:IM , bÃ IY:SM , bÃ SY:IM , and bÃ SY:SM from Model M1 (or the counterparts from Model M0).
for the b-path, the standard error estimates from the original vs. sensitivity analysis models can differ.Due to the biased point estimate (Equation 13) and altered standard error estimate, the z-statistic of the b-path from the original model, the sensitivity analysis model.Hence, with the joint significance test, the test result of the mediation effect a IM:X b IY:IM in the original model could be reversed (from significance to nonsignificance or vice versa) when accounting for confounder C i .
To summarize, the analytical examination demonstrates that even when the treatment X i is randomized, the point estimates and test results of mediation effects from the original PP-LGCMM (Model M0) could be misleading, if a pretreatment confounder of the mediator-outcome relations were omitted.We therefore emphasize the importance of the no-omitted-confounders assumption for mediation inference with PP-LGCMMs.

Proposed Sensitivity Analysis Approaches
We propose three sensitivity analysis approaches-the frequentist (FSA), Bayesian (BSA), and Monte Carlo (MCSA) approaches-for evaluating the robustness of mediation inferences from PP-LGCMMs.For each approach, we describe the key ideas and procedures.Technical details are available in the supplemental materials.At the end of this section, we discuss similarities and differences of the three approaches.
We consider two types of confounding sensitivity parameters: (1) the confounder path coefficients a Ã IM:C , a Ã SM:C , b Ã IY:C , and b Ã SY:C in Equations 7-10 of the sensitivity analysis model, representing the associations of potential omitted confounder C with the latent intercepts and slopes of the mediator and outcome; and (2) the zero-order confounder correlations of C with the latent mediator intercept, mediator slope, outcome intercept, and outcome slope, denoted by r Ã IM:C , r Ã SM:C , r Ã IY:C , and r Ã SY:C , respectively.Under the sensitivity analysis model, a given set of confounder path coefficients can be transformed to the corresponding set of confounder correlations and vice versa.
The sensitivity analysis model cannot be identified 5 using information in the observed data only.To estimate the sensitivity analysis model, researchers need to provide information regarding the sensitivity parameters based on substantive knowledge.

Frequentist Sensitivity Analysis (FSA)
In the FSA, researchers provide fixed values for the sensitivity parameters.The adjusted mediation inferences are then obtained by fitting the sensitivity analysis model (e.g., via software Mplus) or using our analytical results .To study what specific fixed sensitivity parameter values would reverse an inference, we used the analytical results to obtain specific confounder correlation values that (1) make the point estimate of a mediation effect be zero, and/or (2) reverse the test result of the mediation effect (from statistical significance to nonsignificance or vice versa), after adjusting for the omitted confounder C with the sensitivity analysis model.In addition, we determine the admissible ranges of the confounder correlations, within which the covariance matrix of C, I M , S M , I Y, S Y, X, and Z implied by the sensitivity analysis model is positive definite.
We present the FSA results via sensitivity plots.From the plots, researchers can evaluate the robustness of the original mediation inferences conveniently, by assessing the plausibility of the highlighted sensitivity parameter values that would yield zero estimates or reversed testing results of the mediation effects given substantive knowledge.

Bayesian Sensitivity Analysis (BSA)
In the BSA, researchers can conveniently incorporate uncertainty about plausible strengths of the potential omitted confounder C to obtain the adjusted mediation inferences.To make a fair comparison in the results from the original and sensitivity analysis models, both models are estimated using Bayesian methods in the BSA, as described below.
In the Bayesian framework, parameters are treated as random quantities that have distributions.For the original PP-LGCMM (Model M0), parameters include h M0 ¼ fa, b, W vM , W vY , H}, where a and b contain all the a and b path coefficients in the between-person models (Equations 3-6).Their prior distributions can be non-informative or vague.Frequently used vague priors include a normal distribution with mean 0 and large variance such as Nð0, 10 10 Þ for a path coefficient, inverse-gamma distribution IGð10 À3 , 10 À3 Þ for a residual variance parameter, and inverse-Wishart distribution IWðI, p þ 1Þ for a p-dimensional residual covariance matrix.
For the sensitivity analysis PP-LGCMM (Model M1), the vector of parameters has two components: h Ã M0 contains the counterparts of the original model parameters in the sensitivity analysis model, and the confounder path coefficients (sensitivity parameters).Following Harring et al. (2017), the mean and variance of the latent confounder C are set as 0 and 1, respectively, and hence are not included as part of model parameters.The priors for h Ã M0 in the sensitivity analysis model are the same as those for h M0 in the original model.Particularly, the BSA use proper priors (e.g., proper inverse-Wishart priors) for the residual covariance matrices (W Ã vM and W Ã vY in Model M1 and their counterparts in Model M0) to avoid inadmissible (i.e., not positive definite) residual covariance matrices estimates.
For the confounder path coefficients in h Ã C , their priors are informative priors that researchers specifies to reflect prior information regarding the plausible strengths of In the SEM literature, model identification is about whether it is possible to find unique estimates of the model parameters (Bollen, 1989).Under the sensitivity analysis model (Model M1), the maximum likelihood of the observed data can be obtained with non-unique values of Model M1's parameters.Thus, with observed data only, Model M1 is not identified if there are no additional information (e.g., informative priors) about the model parameters.
omitted confounder C. Uncertainty in the confounder path coefficients is also accounted for through the prior specification.For instance, researchers may specify the prior for a confounder path coefficient as a normal distribution that centers at its most likely value, with a prior variance reflecting one's uncertainty about the probable values that this coefficient may take.A smaller prior variance generally indicates a lower level of uncertainty.
The impact of omitting confounder C on the mediation inferences is evaluated by comparing the posterior distributions of mediation effects from the two models.For example, comparing the posterior means from the two models can tell how robust the magnitudes of the original mediation effect estimates are to omitting the confounder.The robustness of the mediation test results can be assessed by comparing the posterior percentile intervals from the two models.If the interval estimate of a mediation effect from the original model did not cover 0 but that from the sensitivity analysis model did, the test result of the mediation effect is sensitive to the omitted confounder C.
Our BSA has connections with the simulated-confounder sensitivity analysis proposed by Qin and Yang (2022) for single-mediator models.Similar to their method, the BSA simulates a value for the latent C at each MCMC iteration from its conditional distribution given the parameters and other variables in the sensitivity analysis model.However, the simulated-confounder method by Qin and Yang (2022) is performed in the frequentist framework and uses the sensitivity parameters as fixed constants.In contrast, the BSA is done in the Bayesian framework, where (i) the sensitivity parameters are random quantities with uncertainty specified by their prior distributions, and (ii) the prior distributions are integrated with observed data following the Bayes' theorem to obtain the posterior distributions of the mediation effects in the sensitivity analysis model.

Monte Carlo Sensitivity Analysis (MCSA)
In the MCSA, uncertainty in the sensitivity parameters can also be conveniently incorporated.Unlike the BSA, both the original and sensitivity analysis models are estimated using frequentist methods in the MCSA.This can be an advantage of the MCSA over the BSA if the original analysis is done in a frequentist framework, because changing estimation frameworks for the original model (from frequentist to Bayesian) might change the results from the original model (Harring et al., 2017).Specifically, the MCSA approach is performed in the following steps: (Step 1) Specify a joint informative prior for the confounder path coefficients, and randomly draw K (e.g., 10,000) sets of confounder path coefficients from the specified prior.
(Step 2) For the k-th (k ¼ 1, :::, K) set of confounder path coefficients, check whether it is within the admissible ranges for the covariance matrix of (C, I M , S M , I Y, S Y, X, Z 0 ) 0 in the sensitivity analysis model to be positive definite.If not, discard this set of confounder path coefficients following recommendations in the literature (Lash, Fox, Fink, et al., 2009;MacLehose & Gustafson, 2012) In Step 2a, we obtain the mediation path estimates and their covariance matrix estimate from the sensitivity analysis model using our analytical results described in the previous section.Alternatively, researchers may obtain the estimates by fitting the sensitivity analysis model in SEM software, given the k-th set of confounder path coefficients.In this case, nonconvergence may occur, especially if the sample sizes are small, the variances of the latent intercepts/slopes are low, or the confounder path coefficients are near the boundaries of the admissible ranges (e.g., De Jonckere & Rosseel, 2022; Tofighi et al., 2019;Yuan & Chan, 2008).Step 2b is done to incorporate the sampling error in the estimates in Step 2a.

Comparing the Frequentist, Bayesian, and Monte Carlo Approaches
Table 1 summarizes the similarities and differences in the developed FSA, BSA, and MCSA approaches.The three approaches share the same overall goal: to evaluate the robustness of the original mediation inference from the PP-LGCMM.They differ in how the robustness is evaluated.Specifically, the FSA obtains the sensitivity parameter values needed to make the mediation inference be reversed after adjusting for the potential omitted confounder; whereas the BSA and MCSA obtain the adjusted mediation inference.Consequently, in the FSA, researchers would be less concerned about the robustness if the needed sensitivity parameter values were less plausible given researchers' substantive knowledge; in the BSA or MCSA, the robustness could be less concerned if minor differences existed between the original and the adjusted results (e.g., no difference in statistical significance).
For sensitivity parameter specifications, when researchers have "a fairly well-defined idea" (Harring et al., 2017, p. 622) about omitted confounding strengths, the FSA can be conducted.When researchers are uncertain in the sensitivity parameter values, the BSA and/or MCSA can be conducted by using informative priors for the sensitivity parameters.Regarding the estimation frameworks, frequentist estimation methods are conventionally used, and Bayesian estimation methods are increasingly considered as complements/alternatives to frequentist estimation.Comparisons and suggestions for choosing between frequentist and Bayesian estimation are provided in the literature (e.g., Gelman et al., 2013;Lee & Song, 2012;Smid et al., 2020;Zhang et al., 2007).Considerations often include reliance on asymptotic theories, usage of priors in inference, interpretation of inference results, handling of certain practical issues such as missing data or small samples (see e.g., Ashby, 2006, van de Schoot et al., 2017, Rupp et al., 2004 for more details).
To carry out the sensitivity analysis, we have developed R functions and a user-friendly Shiny web application (https:// github.com/xliu12/PP-LGCMM_sensitivity_analysis;see the supplemental materials for instructions on running the application).
In sum, the three sensitivity analysis approaches are not exclusive and can provide complementary information.As such, researchers can use one or more of the approaches to evaluate the robustness of PP-LGCMM analyses.

A Real-Data Example
We use longitudinal data (sample size n ¼ 123) from Speidel et al. (2020) to illustrate the application of the proposed sensitivity analysis methods.The treatment X i is a randomized assignment to the Reminiscing and Emotion Training (RET) intervention or control condition, where 60 participants were assigned to the intervention condition and 63 were assigned to the control condition.M i1 , M i2 , and M i3 are maternal sensitive guidance measured at baseline (T1), 2 months later (post-intervention, T2), and 6 months later (T3), respectively.Y i1 , Y i2 , and Y i3 are child lability/negativity at T1, T2, and T3, respectively.Following Speidel et al.
(2020), we set the loadings of the mediator slope (Time Mt ) at T1, T2, and T3 as À0.33, 0, and À0.04, respectively; and those of the outcome slope (Time Yt ) as À1, À0.19, and 0, respectively. 6Child age at T1 was used as the observed covariate Z i for predicting the outcome's latent intercept and slope.The within-person residuals of the mediator (or outcome) were specified to be uncorrelated and have homogeneous variance r 2 eM (or r 2 eY ) over time.Figure 3 depicts the original PP-LGCMM (i.e., Model M0) in this real-data illustration.Inference of the four mediation effects of interest-the mediation effects of intervention (X) on child lability/negativity's latent level at T3 (I Y) and latent change from T1 to T3 (S Y) via maternal sensitive guidance's latent level at T2 (I M ) and latent change from T1 to T3 (S M )-requires the no-omitted-confounders assumption.As illustrated in the introduction, pretreatment maternal depressive symptoms at baseline could be a potential omitted confounder for the mediator-outcome relations.Based on the child psychopathology literature, we expect that severer maternal depressive symptoms at baseline may relate to severer posttreatment children emotional problems such as lability/negativity (e.g., a review in Goodman et al., 2011), and might also relate to lower posttreatment maternal sensitivity guidance (e.g., Crespo et al., 2017;Lunkenheimer, et al., 2021;Valentino et al., 2022).Even conditional on the observed pretreatment covariates (baseline child age) and treatment, estimates of the four mediation effects from Model M0 could be confounded by pretreatment maternal In Speidel et al. (2020), the linear LGC model did not have adequate fit for the mediator.Thus, they freely estimated the mediator slope loading at T3, with those at T2 and T1 fixed respectively at 0 and À0.33 (i.e., the 2-month interval between T1 and T2 divided by the 6-month total study duration).For the outcome, the linear LGC model did not have adequate fit.Thus, they freely estimated the outcome slope loading at T2, with those at T3 and T1 fixed respectively at 0 and À1.In the current study, the slope loadings of mediator and outcome from Speidel et al. ( 2020) are used in both the original and sensitivity analysis models to illustrate the developed sensitivity analyses.
depressive symptoms (potential omitted confounder C in this illustration).
For illustrative purposes, we apply the FSA, BSA, and MCSA, to evaluate the robustness of the mediation inference from the original model (Model M0).In the following subsections, we first consider frequentist inference and illustrate the FSA and MCSA.We then consider Bayesian inference and demonstrate the BSA.

Frequentist: Original Analysis, FSA, and MCSA
We fitted the original model using ML estimation in Mplus.The 95% confidence intervals for the mediation effects were obtained using the Monte Carlo confidence interval method (Selig & Preacher, 2008;Tofighi & MacKinnon, 2011) with 1000 draws from the sampling distribution of the mediation path estimates yielded from the original model.The results are listed in Table 2 (under "Original analysis").
From the original model, the treatment-mediator relations a IM:X and a SM:X were both statistically significant, indicating that compared to those assigned to the control condition, participants assigned to the intervention condition had significantly higher maternal sensitive guidance level at T2 (I M ) and significantly greater change in maternal sensitive guidance from T1 to T3 (S M ).Regarding the mediator-outcome relations, b IY:IM was statistically significant with a negative point estimate, indicating higher maternal sensitive guidance level at T2 was associated with lower child lability/negativity level at T3.The other three b-paths, b SY:IM , b IY:SM , and b SY:SM , were statistically nonsignificant.Based on the joint significance test and the 95% Monte Carlo confidence interval, maternal sensitive guidance level at T2 was a statistically significant mediator for the effect of intervention on child lability/negativity level at T3 (i.e., significant mediation effect of X on I Y via I M ).The other three mediation effects were not statistically significant.In addition, baseline child age was not a significant covariate for the outcome intercept and slope ( bIY:Z and bSY:Z were À0.016 and 0.095, with p-value 0.975 and 0.834, respectively).

Application of the FSA
The FSA provides a sensitivity plot for each of the four mediation effects (Figures 4-7).In the plots, the sensitivity parameters are the four confounder correlations r Ã IM:C , r Ã SM:C , r Ã IY:C , and r Ã SY:C : To display the four confounder correlations in a two-dimensional sensitivity plot, two correlations are fixed at specified values, one correlation has varying values displayed on the x-axis, and the other correlation has values solved analytically and displayed on the y-axis.
For example, for the mediation effect of X on I Y via I M , the pair of confounder correlations of C with S M and S Y (i.e., r Ã SM:C and r Ã SY:C ) are fixed at specified values because they are not involved in the mediation effect under sensitivity assessment.The confounder correlation of C with I M (r Ã IM:C ), is varied from À1.0 to 1.0 and displayed on the xaxis.Then, given the three confounder correlations (r Ã SM:C , r  signs of the confounder correlations were informed by empirical studies (e.g., Valentino et al., 2022).The magnitude was chosen to be a medium-sized zero-order correlation (i.e., 0.3 ;Cohen, 1988).Figure 4 displays the sensitivity plot for this mediation effect.Similar procedures were applied to obtain the sensitivity plots for the other three mediation effects (Figures 5-7).In the supplemental materials, we display the sensitivity plots where the unvaried confounder correlations were specified to large or small sizes (i.e., À0.5, À0.1 for r Ã IM:C and r Ã SM:C ; 0.5, 0.1 for r Ã IY:C and r Ã SY:C ).The findings from these plots are consistent and thus are not presented in the main text.
In the sensitivity plots, the solid lines show the admissible ranges of the confounder correlations under the sensitivity analysis model.The dashed lines show the confounder correlations that would make the point estimates of the mediation effects zero.The shaded areas show the confounder correlations with which the test results of the  Note.Under "Original analysis".The 95% confidence intervals for the mediation effects were obtained via the Monte Carlo confidence interval method.The 95% confidence intervals for the mediation paths were obtained based on their standard error estimates and the asymtotic normality feature of maximum likelihood estimates.Under "MCSA," the 95% percentile intervals were obtained from the MCSA procedure with K ¼ 10,000 sets of confounder path coefficients drawn from the specified priors.100% of the draws were within the admissible ranges.The intervals that do not cover 0 are in bold.mediation effects would be reversed, if there are any.To evaluate the robustness of the original mediation inferences, researchers would gauge to what extent it is plausible that the confounder correlations fall in the shaded areas or dashed lines within the admissible ranges.For example, the mediation effect estimate of treatment on I Y via I M was negative and statistically significant from the original analysis.In its sensitivity plot (Figure 4), the dashed lines within the admissible ranges appeared far from the origin, indicating that relatively large confounder correlations are needed to completely explain the non-zero estimate from the original analysis.Thus, if such large confounder correlations are not plausible, researchers could be confident in that the point estimate of the mediation effect is non-zero.In addition, the red and orange areas show the combinations of the confounder correlations that would make the mediation test result become nonsignificant in the sensitivity analysis model.Confounder correlations in the orange areas would also make the estimated b-path (i.e., the path from I M to I Y) , and subsequently the estimated mediation effect, oppose the direction of the original estimate.For this example, the red and orange areas within the admissible ranges appeared small and not close to the origin.If such large confounder correlations do not seem plausible in the substantive research context, researchers could be more confident in the robustness of the significant mediation test result to potential omitted confounders of the mediator-outcome relations.
The mediation effect estimate of treatment on S Y via I M was nonsignificant and thus the direction of the effect was inconclusive from the original analysis.The sensitivity plot for this mediation effect is displayed in Figure 5 with varying sensitivity parameters r Ã IM:C (x-axis) and r Ã SY:C (y-axis) and fixed r Ã IY:C ¼ 0:3 and r Ã SM:C ¼ À0:3: In this plot, the green areas show the combinations of confounder correlations that would make the mediation effect become significant after adjusting for confounder C with the sensitivity analysis model.The green areas within the admissible ranges were not small.Depending on how plausible the combinations of confounder correlations in the green areas are given one's substantive knowledge, the nonsignificant mediation test result from the original analysis can be sensitive to the omission of the confounder C.
The sensitivity plots for the mediation effects of treatment on I Y and S Y through S M (Figures 6 and 7) have no shaded areas.This indicates that the nonsignificant test results of the two mediation effects are robust to all the combinations of confounder correlations displayed in the sensitivity plots.The robustness of the statistical nonsignificance is consistent with the large sampling errors in the estimates of the two mediation effects.Both mediation effects had wide 95% intervals in the original analysis, and their associated b-paths had large p-values (0.810 for b IY:SM , 0.995 for b SY:SM ).

Application of the MCSA
In the MCSA, we specify the joint informative prior for the confounder path coefficients (sensitivity parameters) to be independent normal distributions: The prior means were specified to correspond to medium-sized confounder correlations (i.e., As in the application of FSA, the signs of these values were informed by previous empirical studies.For the prior standard deviations, they were specified such that the 95% prior intervals of the corresponding confounder correlations (i.e., r Ã IM:C and r Ã SM:C ) range from À0.5 (large; Cohen, 1988) to À0.1 (small), and the 95% prior intervals of the corresponding confounder correlations (i.e., r Ã IY:C and r Ã SY:C ) range from 0.4 (moderately large) to 0.1 (small).From the specified informative prior distributions, K ¼ 10 4 sets of confounder path coefficients were drawn.100% of the draws were within the admissible range limits.Table 2 displays the results (under "MCSA").
The test results of the mediation effects (i.e., whether the 95% MCSA intervals covered 0) remained the same as the original analysis.Thus, the MCSA suggests that the mediation test results were not sensitive to omitting potential confounder C with the specified informative priors.
For the point estimates, the MCSA estimates of the mediation effects of X on outcome intercept and slope via I M were in the same direction as the original estimates but had smaller magnitudes.This indicates that the directions of the estimated mediation effects via I M are robust, and that the magnitudes of the estimates would be reduced after accounting for C with the specified informative prior.The reductions in the point estimates of the two mediation effects are consistent with the frequentist sensitivity plots in Figures 4 and 5 (and those in the supplemental materials).From the FSA plots, negative confounder correlations r Ã IM:C and r Ã SM:C , coupled with positive r Ã IY:C and r Ã SY:C can make estimates of the mediation effects via I M become zero after adjusting for the potential confounder.In the MCSA, the 95% prior intervals of the confounder path coefficients were indeed corresponding to negative r Ã IM:C and r Ã SM:C and positive r Ã IY:C and r Ã SY:C : For the two mediation effects via S Mi , both the 95% intervals from the MCSA and the 95% confidence intervals from the original analysis were wide, indicating large uncertainty.Thus, although their point estimates from the MCSA and the original model differed in magnitude and/or direction, the differences are inconclusive because of such large uncertainty.

Bayesian: Original Model and BSA
In the Bayesian estimation of the original model (Model M0), we specified weakly informative priors for the residual covariance matrices of the latent intercepts and slopes of the mediator and outcome (i.e., W vM and W vY , respectively), and non-informative priors for the other parameters.The use of the weakly informative priors is due to the small sample size and number of time points in this example.When using the frequently used vague priors (e.g., IW½I, 3), the Bayesian estimation of the original model (Model M0) did not converge based on the potential scale reduction factor (PSRF > 1.10, indicating non-convergence; [Gelman & Rubin, 1992]) even after 100,000 MCMC iterations.Previous studies have shown that using weakly informative priors such as data-dependent priors can be helpful for mitigating small sample size issues and obtaining satisfactory inference results (e.g., S.-Y. Lee & Song, 2004;McNeish, 2016b).Thus, following McNeish (2016b), we specified the prior for the residual covariance matrix W vM (or W vY ) as an Inverse-Wishart distribution with 2 degrees of freedom and the scale matrix being its ML estimate multiplied by this degrees of freedom. 7These same priors were also used in the Bayesian estimation of the sensitivity analysis model (Model M1).
For the confounder path coefficients in Model M1, the prior distributions were the same as those specified in the MCSA.For both models (models M0 and M1), Bayesian estimation was performed using the Gibbs sampler algorithm in Mplus (Asparouhov & Muth en, 2010), where two chains were run and each chain had 40,000 iterations with the first half being the burn-in period.The PSRF values for both models were around 1.00, indicating the convergence of MCMC.Results from the original model with Bayesian estimation and the BSA results are displayed in Table 3.
We noticed that the original model with frequentist vs. Bayesian estimation had the same test results while numerically different point and interval estimates of the mediation effects.The numerical differences from different estimation frameworks are not unexpected, especially with smaller samples (e.g., McNeish, 2016a;Smid et al., 2020).As we noted earlier, such differences can make the results from BSA not comparable with those from the original frequentist analysis.Thus, we compare the BSA results with those from the original model with Bayesian estimation.
The original model with Bayesian estimation and BSA both yielded significant test results (i.e., 95% posterior intervals not covering 0) for the mediation effect of X on outcome intercept I Y via mediator intercept I M , and nonsignificant results for the other three mediation effects (i.e., the mediation effects of X on S Y via I M , on I Y via S M , and on S Y via S M ).This shows that the mediation test results are robust to the omission of potential confounder C whose confounder path coefficient values are distributed as the specified informative priors.
In terms of the point estimates (i.e., posterior means), the BSA estimates of the mediation effects on the outcome intercept and slope via I M had the same signs as those from the original model, although the magnitudes were smaller.This pattern is consistent with the MCSA results.For the two mediation effects via S M , the BSA estimates and the Bayesian estimates from the original model differed in signs and magnitudes; nonetheless, their 95% posterior intervals were wide, making the differences inconclusive.
An additional observation is that for the a-paths a IM:X and a SM:X , the estimates and 95% posterior intervals from the original model were close to those from the BSA.This is consistent with the analytical examination that omitting C does not affect the inferences of the a-paths in the PP-LGCMM with randomized treatment assignment X.

Summary of the Sensitivity Analysis Results
Through the applications of the three proposed sensitivity analyses, the statistical significance of the mediation effect of intervention (X) on child liability/negativity at T3 (I Y) via maternal sensitive guidance at T2 (I M ) appeared robust.The statistical nonsignificance of the mediation effects on the level and change of child liability/negativity (I Y and S Y) via the change in maternal sensitive guidance from T1 to T3 (S M ) also appeared robust.The statistical nonsignificance of the mediation effect on the change in child liability/negativity from T1 to T3 (S Y) via maternal sensitive guidance at T2 (I M ) were robust in the BSA and MCSA, but somewhat less robust than the other three mediation effects in the FSA.Overall, the mediation inferences from the original PP-LGCMM analysis were not highly sensitive to the omission of potential betweenperson confounders of the mediator-outcome relations.
Our applications are for illustrative purposes.In practice, the setups of sensitivity analyses (e.g., specifying plausible values or informative priors for sensitivity parameters), and interpretation of the results (e.g., criteria of robustness, plausibility of confounder correlations), should be based on the substantive research context.

PP-
LGCMMs are commonly used in empirical longitudinal mediation studies to estimate and test the mediation effects of a treatment variable on the level and change of an outcome through the level and change of a mediator.A frequently discussed threat to the validity of empirical mediation inference is the violation of the no-omitted-confounders assumption, which has attracted methodological attention in recent years (e.g., Cox et al., 2013;Lee et al., 2021;Qin & Yang, 2022).Despite the popularity of PP-LGCMMs, limited research has been done to examine the impact of violating the no-omitted-confounders assumption on the inferences of mediation effects in PP-LGCMMs.Furthermore, sensitivity analysis methods for assessing the robustness of the mediation inferences from PP-LGCMMs have not been previously developed.
In the current study, we analytically examined how the point estimates and test results of the mediation effects in the PP-LGCMM are impacted by the omission of a pretreatment confounder that can affect the mediator intercept and slope as well as outcome intercept and slope.Furthermore, using the analytical results, we extended the ideas in the previous sensitivity analysis literature (e.g., Cox et al., 2013;Harring et al., 2017;Liu & Wang, 2021;McCandless & Somers, 2019;Tofighi et al., 2019) to develop three easy-touse sensitivity analysis approaches for PP-LGCMMs-the FSA, BSA, and MCSA approaches (see Table 1).With the FSA, researchers obtain sensitivity plots to investigate the specific correlation values of a potential omitted confounder with mediator's and outcome's intercepts and slopes that would (1) make the point estimate of the mediation effect zero, and/or (2) reverse the mediation test result, after adjusting for the omitted confounder in the sensitivity analysis model.The plots also display the ranges of admissible confounder correlations to make the between-person covariance matrix of the sensitivity analysis model positive definite.From the plots, researchers can assess the robustness of both the magnitudes of mediation effect estimates and the mediation test results conveniently, by using substantive knowledge to evaluate the plausibility of confounder correlations that are admissible and fall in the highlighted areas.
The BSA and MCSA examine what the adjusted mediation inferences would be, after adjusting for potential omitted confounders and incorporating one's uncertainty about the strengths of potential omitted confounders.Such uncertainty is quantified by the informative prior distributions specified for the sensitivity parameters.Thus, the inference from the BSA or MCSA is based on the distributions of the adjusted mediation effect estimators in which both the random errors in the observed data and the uncertainty in strengths of potential omitted confounders are taken into account.This differs from the FSA, in which the adjusted mediation inference is conditional on specific (fixed) strengths of potential omitted confounders.Comparing the BSA and MCSA, the two approaches differ in the estimation frameworks used to estimate the sensitivity analysis model (Bayesian vs. frequentist).Therefore, to assess the sensitivity of the original mediation inferences obtained using frequentist estimation, the MCSA may be more suitable than the BSA, as the results from BSA may reflect not only the impact of omitting potential confounders but also the influence of changing estimation frameworks.

Notes on Using the Developed Sensitivity Analysis Approaches
The developed sensitivity analyses require two types of information: information about the original PP-LGCMM analysis and information about confounding sensitivity parameters.
For the first type, the BSA requires the raw data and prior distributions used for Bayesian estimation of the original model.The FSA and MCSA can be conducted without the raw data, and use the sample means and sample covariance matrix of the observed variables in the original model instead.
Information on the confounding sensitivity parameters generally would come from previous studies and researchers' knowledge in the substantive context.The sensitivity analysis literature has discussion on sources of information about sensitivity parameters (e.g., Lash et al., 2009Lash et al., , 2014)).For example, prior elicitation (e.g., Garthwaite et al. 2005;Stefan et al., 2022) can be used to translate expert knowledge into plausible values or informative prior distributions of the sensitivity parameters.Researchers may also specify the sensitivity parameters based on the relationships of the mediator intercept/slope and outcome intercept/slope with observed confounders in the study (e.g., the range of their path coefficients or correlations; Qin & Wang, in press).Alternatively, educated guesses based on the substantive literature of similar topics can inform the sensitivity parameter specification.In the realdata example, the signs of the confounder correlations were based on information from relevant substantive studies; and the magnitudes were based on typical correlation values in psychological research (Cohen, 1988;Meyer et al., 2001).
Note that not all sensitivity parameter values are admissible; some conflict with the original data and should be avoided (Lash et al., 2014).In our case, the confounding sensitivity parameters outside the admissible ranges would render the covariance matrix of the potential confounder, latent intercepts and slopes of mediator and outcome, treatment variable, and observed baseline covariates not positive definite (which could result in negative residual variances of the latent intercepts/slopes in the sensitivity analysis model).In the developed methods, the FSA visualizes the admissible ranges to help avoid drawing conclusions based on inadmissible values; the MCSA discards any inadmissible confounding sensitivity parameter values.For the BSA, similar to other Bayesian analysis, it addresses the admissibility issue by specifying the priors of the residual covariance matrices to allow only positive definite matrices (Can et al., 2015;Martin & McDonald, 1975).To facilitate the specification, we provided R functions for checking whether a set of confounder correlation values is within the admissible range and for transforming between confounder path coefficients and confounder correlations (also available in the web application).
On reporting the sensitivity analysis results, we suggest that the specifications of the sensitivity parameters (fixed values in the FSA and prior distributions in the BSA or MCSA) and reasons for the specifications should be clearly described.Moreover, for the BSA, the prior specifications for parameters in the original model should be provided and justified.Finally, researchers can discuss potential avenues for future research based on the sensitivity analysis results (e.g., Lash et al., 2014;Rosenbaum, 2004;Rosenbaum & Rubin, 2022).For example, if the FSA shows that the test result of a mediation effect would be reversed with confounder correlations of moderate sizes, or if the BSA or MCSA suggests that the inferences of a mediation effect would be meaningfully changed with the specified priors of confounding sensitivity parameters, researchers may consider collecting more information on pretreatment covariates that may relate to the mediator intercept/slope and outcome intercept/slope in future research (e.g., a validation study; Lash et al., 2009), to further investigate the mediation effects in the PP-LGCMM with controlling for more observed covariates.

Notes on the PP-LGCMM
We focused on the no-omitted-confounders assumption, which is an important assumption frequently concerned in mediation analysis, due to the infeasibility to randomize mediators.In addition to this assumption, several aspects of the PP-LGCMM are worth noting.First, the choices of time points at which the slope loadings of mediator and outcome (i.e., Time Mt and Time Yt ) are coded as zero (i.e., how the time variables are centered) require careful consideration (Preacher, 2015;Selig & Preacher, 2009).These choices influence interpretations of the intercepts of mediator and outcome and thus interpretations and inference of the mediation effects.In practice, the slope loadings should be consistent with the temporal ordering in the hypothesized mediation mechanism.Specifically, I M should represent the mediator's level at a time point before or concurrent with the outcome level represented in I Y and the outcome change represented in S Y; and the outcome level represented in I Y should be after or concurrent with the mediator change represented in S M (Soest & Hagtvet, 2011).
Second, the changes represented in the mediator slope S M and outcome slope S Y in the PP-LGCMM happen concurrently.Concurrent effects are meaningful to study and interpret in longitudinal research (Goldsmith et al., 2018;Singer et al., 2003).However, constructing lagged relations from S M to S Y can strengthen inferences about causal relations.Therefore, when five or more time points are available, researchers may consider modeling the mediator slope and outcome slope in piecewise growth models to better account for the temporal ordering (e.g., O'Laughlin et al., 2018;Selig & Preacher, 2009).
Third, the LGC models for the mediator and outcome (Equations 1-2) should be specified and evaluated separately before incorporating them into the PP-LGCMM (Cheong et al., 2003;MacKinnon, 2008).LGC model specification would involve specifying the slope loadings and residual covariance structure based on the hypothesized trajectory shape and model fit.Inappropriate specification of the mediator's and/or outcome's LGC models could lead to inaccurate inferences of parameters related to the latent intercepts and slopes (e.g., Kwok et al., 2007;Sivo et al., 2005).For more detailed discussions about considerations involved in conducting PP-LGCMM analysis, see Cheong et al. (2003), MacKinnon (2008), Soest and Hagtvet (2011), among others.
Our sensitivity analysis approaches are extendable to assess the sensitivity of mediation inference with more complex LGC models.For example, the original model could be extended, where the LGCs for mediator and/or outcome are the piecewise growth models or have additional latent factors (e.g., a factor capturing quadratic change).Then, the sensitivity analysis model could be extended accordingly, where the sensitivity parameters could be the associations of a phantom confounder with latent factors of the extended LGCs for mediator and outcome.To extend the FSA, researchers could solve for values of the sensitivity parameters to reverse the mediation inference (analytically or extending certain algorithms e.g., Fisk et al., 2023;Leite et al., 2021).An extended BSA approach could be using Bayesian estimation for the extended sensitivity analysis model, with informative priors specified for the sensitivity parameters.The MCSA could also be extended, by obtaining the confounding-adjusted sampling distribution of the mediation path estimates from the extended sensitivity analysis model (analytically or via SEM software) with the specified prior distributions for the sensitivity parameters.

Limitations and Future Directions
The current study has limitations and can be extended in the future.First, we considered the parametric modeling approach to mediation analysis, with which the mediation effects are defined using the product-of-coefficients approach (Baron & Kenny, 1986).However, this approach requires that the parametric model is correctly specified.For the PP-LGCMM (Model M0), it assumes linearity and no interaction among observed covariates, treatment, and mediator intercept and slope, which may not hold.Besides the parametric modeling approach, causal inference approaches to mediation analysis have been proposed, with which various types of causal mediation effects (e.g., natural indirect effect, interventional indirect effect) have been defined using the potential outcomes framework (e.g., Imai, Keele, & Tingley, 2010;Loh et al., 2022;Robins & Greenland, 1992;VanderWeele, 2015;Vansteelandt & Daniel, 2017).Particularly, to handle the post-treatment confounding issue 8 posed by the mediator intercept and slope's mutual dependence, researchers have extended the interventional indirect effects to the PP-LGCMM (Liu, 2022).Obtaining valid inferences of the interventional indirect effects also requires the no-omitted-confounders assumption.It could be an important future direction to develop sensitivity analysis methods for assessing the robustness of causal mediation effect inferences (e.g., the interventional indirect effects) in PP-LGCMMs.
Furthermore, the sensitivity analysis model we constructed assumes the potential omitted confounder is linearly related to the intercepts and slopes of mediator and outcome.Such linearity assumption may be violated if the omitted confounder interacts with treatment, mediator intercept, and/or mediator slope.In the mediation analysis literature, there are derived bias formulas and the related sensitivity analysis techniques that do not make assumptions about the functional forms of the relations between unmeasured confounders and observed variables (Ding & Vanderweele, 2016;Smith & VanderWeele, 2019;VanderWeele, 2010).How to extend the previous work to develop sensitivity analysis methods for the PP-LGCMM that have relaxed assumptions about the potential omitted confounders is a worthwhile future direction.
Second, we assume the treatment variable is randomized.Nonetheless, randomization of treatment is not always feasible.With non-randomized treatment assignment, the relations of treatment with mediator intercept/slope and with outcome intercept/slope can be confounded due to omitted confounders.For univariate LGC mediation models, Tofighi (2021) considered non-randomized treatment assignment and developed sensitivity analysis methods to deal with potential omitted confounders.However, to our knowledge, sensitivity analysis approaches for PP-LGCMMs with nonrandomized treatment assignment are not available and await future development.
The current study could also be extended in other directions.In addition to omitting confounders, the mediation effect estimates from the PP-LGCMM may be biased due to other reasons.For example, the observed variables may contain measurement errors, misclassifications, and/or have non-ignorable missingness.In the sensitivity analysis literature, methods have been developed to assess the consequences of the potential co-occurrence of omitted confounders and one or more of these bias sources for treatment effect estimation for regression models and the single-mediator model (e.g., Greenland, 2005;Lash & Silliman, 2000;Liu & Wang, 2021).Extending the previous studies to develop sensitivity analysis methods that account for multiple potential bias sources in PP-LGCMMs may lead to useful additions to researchers' toolbox.
Other longitudinal mediation models, such as the latent difference score, state space, continuous time, dynamic models, exist and have been applied in empirical longitudinal mediation analysis (e.g., Gu et al., 2014;Huang & Yuan, 2017;O'Laughlin et al., 2018;Preacher, 2015).Similar to the PP-LGCMM, obtaining valid inferences of mediation effects from these models requires certain no-omitted-confounders assumptions that are often challenging to satisfy in practice.Therefore, future research should investigate how In causal mediation analysis, the natural indirect effect of a treatment on an outcome via a mediator cannot be causally identified, when there exists a posttreatment confounder(s) of the mediator-outcome relation, either omitted or not, that is itself affected by the treatment (e.g., VanderWeele & Vansteelandt, 2009;VanderWeele & Vansteelandt, 2014).This issue is different from the omitted-pretreatment-confounders issue considered in our current study.The omitted-pretreatment-confounders issue can be potentially addressed by collecting data on and adjusting for all (or a sufficiently rich set of) pretreatment confounding variables in the analysis.The post-treatment confounding issue, however, is the issue with the existence of such posttreatment confounders, irrespective of whether they are omitted from or included in the analysis.
the omission of potential confounders would impact the mediation inferences from these models and develop sensitivity analysis tools for these models.
In conclusion, by examining the impact of omitting pretreatment confounders on inferences of the mediation effects in PP-LGCMMs, the current study emphasizes the importance of the no-omitted-confounders assumption and offers researchers useful sensitivity analysis tools for addressing potential violations of this assumption in PP-LGCMMs.

Funding
Lijuan Wang and Zhiyong Zhang are grateful for the financial support from IES grant R305D210023 during the study.Lijuan Wang is also grateful for the financial support from NIH grant R01HD091235 during the study.
, 1 the mediation effects of treatment X on I Y via I M and via S M are a IM:X b IY:IM and a SM:X b IY:SM , respectively.These effects quantify how much the treatment indirectly influences the outcome level through influencing the mediator level and mediator change, respectively.The mediation effects of X on S Y via I M and via S M are a IM:X b SY:IM and a SM:X b SY:SM , respectively, representing to what extent the treatment effect on the change in outcome is transmitted by the mediator level and mediator change, respectively.Inference about these mediation effects can promote understanding of pathways from treatments (e.g., interventions, antecedents of interest) to outcomes, particularly whether the mediator's level or change plays a role in the pathways.

Figure 1 .
Figure 1.A parallel process latent growth curve mediation model (PP-LGCMM).We consider this model as the original model, Model M0.The observed pretreatment covariates Z are not depicted for simplicity.

Figure 2 .
Figure 2. Diagram of the sensitivity analysis model, Model M1.The observed pretreatment covariates Z are not displayed for simplicity.The unobserved confounder C is represented by the dashed circle.The effects of C on the intercept I M and slope S M of the mediator and on the intercept I Y and slope S Y of the outcome are represented by the dashed arrows.

(
the b-paths) are influenced by the omission of C. Let ŴÀ1 vM be the inverse of the estimated residual covariance matrix of the mediator intercept and slope from the original model, with /12 vM denoting the off-diagonal element and /11 vM and /22 vM denoting the first and second diagonal elements, respectively. 3The estimates of the b-paths from the sensitivity analysis model with adjusting for C (i.e., bÃ IY:IM , bÃ IY:SM , bÃ SY:IM , and bÃ SY:SM from Model M1) and the estimates yielded by the original model (i.e., bIY:IM , bIY:SM , bSY:IM , and bSY:SM from Model M0) have the following relationships: bÃ Ã SY:C , and r Ã IM:C ), the values of the confounder correlation of C with I Y (r Ã IY:C ) are solved based on the Equations 13-17 to reverse the original inference of this mediation effect and for the admissible ranges.The results on r Ã IY:C are shown on the y-axis of the sensitivity plot.In this example, r Ã SM:C and r Ã SY:C -correlations of C (e.g., baseline maternal depressive symptoms) with maternal sensitive guidance slope and child lability/negativity slope-are À0.3 and 0.3, respectively.The

Figure 3 .
Figure 3.The PP-LGCMM used in the original analysis (i.e., Model M0) in the real-data example.The hypothesized mediation mechanism is that the effects of RET intervention (X) on the latent level of child lability/negativity at T3 (I Y, Lability/Negativity Intercept) and latent change in child lability/negativity from T1 to T3 (S Y, Lability/Negativity Slope) are mediated via the latent level of maternal sensitive guidance at T2 (I M , Maternal Sensitive Guidance Intercept) and latent change in maternal sensitive guidance from T1 and T3 (S M , Maternal Sensitive Guidance Slope).The slope loadings were specified following Speidel et al. (2020).MSG: maternal sensitive guidance; L/N: child lability/negativity; RET: reminiscing and emotion training; PP-LGCMM: parallel process latent growth curve mediation model.

Figure 4 .
Figure 4. Sensitivity plot for the mediation effect of treatment X on the outcome intercept I Y via the mediator intercept I M from the FSA in the real-data example.The confounder correlations r Ã IM:C and r Ã IY:C of C with I M and I Y are varied from À1.0 to 1.0 and displayed on the x-and y-axes ("rCIm" and "rCIy"), respectively, with the other two confounder correlations r Ã SM:C and r Ã SY:C fixed (shown in the panel names "rCSm" and "rCSy," respectively).FSA: frequentist sensitivity analysis.

Figure 5 .
Figure 5. Sensitivity plot for the mediation effect of treatment X on the outcome slope S Y via the mediator intercept I M from FSA in the real-data example.The confounder correlations r Ã IM:C and r Ã SY:C of C with I M and S Y are varied from À1.0 to 1.0 and displayed on the x-and y-axes ("rCIm" and "rCSy"), respectively, with the other two confounder correlations r Ã SM:C and r Ã IY:C fixed (shown in the panel names "rCSm" and "rCIy," respectively).FSA: frequentist sensitivity analysis.

Figure 6 .
Figure 6.Sensitivity plot for the mediation effect of treatment X on the outcome intercept I Y via the mediator slope S M from the FSA in the real-data example.The confounder correlations r Ã SM:C and r Ã IY:C of C with S M and I Y are varied from À1.0 to 1.0 and displayed on the x-and y-axes ("rCSm" and "rCIy"), respectively, with the other two confounder correlations r Ã IM:C and r Ã SY:C fixed (shown in the panel names "rCIm" and "rCSy," respectively).FSA: frequentist sensitivity analysis.

Figure 7 .
Figure 7. Sensitivity plot for the mediation effect of treatment X on the outcome slope S Y via the mediator slope S M from the FSA in the real-data example.The confounder correlations r Ã SM:C and r Ã SY:C of C with S M and S Y are varied from À1.0 to 1.0 and displayed on the x-and y-axes ("rCSm" and "rCSy"), respectively, with the other two confounder correlations r Ã IM:C and r Ã IY:C fixed (shown in the panel names "rCIm" and "rCIy," respectively).FSA: frequentist sensitivity analysis. 8 . If it is, perform (2a) and (2b): (2a) Given the k-th set of confounder path coefficients, obtain the adjusted mediation path estimates from the sensitiv-, b Ã ðkÞ IY:IM , b Ã ðkÞ IY:SM , b Ã ðkÞ SY:IM , b Ã ðkÞ SY:SM Þ 0 from the sampling distribution Nð d medpaths Ã ðkÞ , ŜÃ ðkÞ medpaths Þ in (2a).Calculate the mediation effects as a Ã ðkÞ IM:X b Ã ðkÞ IY:IM , a Ã ðkÞ IM:X b Ã ðkÞ SY:IM , a Ã ðkÞ SM:X b Ã ðkÞ IY:SM , and a Ã ðkÞ SM:X b Ã ðkÞ SY:SM : (Step 3) Compute the mean, lower and upper 2.5% quantiles for each mediation effect based on the K sets adjusted results fa Ã ðkÞ IM:X b Ã ðkÞ IY:IM , a Ã ðkÞ IM:X b Ã ðkÞ SY:IM , a Ã ðkÞ SM:X b Ã ðkÞ IY:SM , a Ã ðkÞ SM:X b Ã ðkÞ SY:SM g k¼1, :::, K yielded from Step 2, to obtain the MCSA point estimates and 95% percentile intervals of the mediation effects.

Table 1 .
Comparing the three developed sensitivity analysis approaches for the PP-LGCMM: the frequentist (FSA), Bayesian (BSA), and Monte Carlo (MCSA) approaches.Focal parameters Mediation effects of treatment X on outcome intercept I Y and slope S Y via mediator intercept I M or slope S M a Given a set of fixed values of the sensitivity parameters, researchers can also obtain the adjusted mediation inference results, by fitting the sensitivity analysis model via software or using the analytical results.PP-LGCMM: parallel process latent growth curve mediation model.6

Table 2 .
Results from the original analysis and the Monte Carlo sensitivity analysis (MCSA) in the real-data example.

Table 3 .
Results from the original model with Bayesian estimation and the Bayesian sensitivity analysis (BSA) in the real-data example.Note.Estimate: Posterior mean.95% Posterior Interval: 95% percentile interval of the posterior distribution.Intervals that do not cover 0 are in bold.