The 22nd SNU International Conference on Education Research (ICER)

Statistical Models for the Inference of Withinperson Relations in Longitudinal Design

Satoshi Usami
University of Tokyo
Email: usami_s@p.u-tokyo.ac.jp
HP: http://usami-lab.com/index-e.html/
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Overview of presentation• Inference of (within-person) reciprocal relation (Xt-1⇒Yt and/or Yt-1⇒Xt)
between variables is popular in observational studies of behavioral
science.
• Various longitudinal models have been developed in different contexts
and disciplines, and an issue of model choice is still under discussion.
• Researchers in psychology often use CLPM (or RI-CLPM) by SEM,
while other statistical models and estimation procedures are available.
In this presentation:(1) Introducing various statistical models for reciprocal relation andexplaining that (predetermined) RI-CLPM and DPM are safety options.
(2) Clarifying the potential problems of applying other models (LCM-SR,
LCS, GCLM).

(3) Briefly explaining an alternative estimation approach (Usami, 2022) toaccount for time-varying treatments/predictors and confounders.
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Outline• Introduction• Longitudinal models to examine reciprocal relation
CLPM RI-CLPM LCM-SR LCS GCLM DPM• Issue of time-varying confounders (Usami, 2022)• Summary and future research agenda

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Introduction• In educational and behavioral research, longitudinal data analysis has
been widely used to understand patterns, individual differences, andrelations of change in variables.
• One of the primary interests is the inference about reciprocal relationor causality between variables. RQ: How does change in one variable(𝑋𝑋𝑡𝑡−1 ) affect change in another variable (𝑌𝑌𝑡𝑡 )?
⇒Controlling lagged variables (𝑌𝑌𝑡𝑡−1 ) in dynamic process is a key.
• Also, researchers have shown strong interest in
disaggregating within-person relations from betweenperson differences (or unobserved heterogeneity).

RQ: How changes in a variable influence another for thesame person? (≠ group-level relation)⇒Controlling (time-invariant) latent factors is also a key.
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Introduction• In psychology, structural equation modeling (SEM)-based approachhas been popular for uncovering reciprocal relation in observation
studies, and a cross-lagged panel model (CLPM) has been a gold
standard.
• However, Hamaker et al (2015) criticizes the use of CLPM for theinference of within-person relation, and suggested a random-intercept
CLPM (RI-CLPM), which includes common factors called stable trait
factors (reaching more than 1600 citations on Google as of October2022).
• Increase of research that conducts secondary analysis and/or compares
estimation results through CLPM, RI-CLPM and other models (e.g.,
Orth et al., 2021).
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Introduction
Smoking intensity and exposure to smoking behavior through movie (Usami et al 2019)

Usami et al (2019)

Magnitude, sign and statistical significance of cross-lagged parameters(γ) estimates differ among models (Hamaker et al., 2015; Orth et al., 2021).
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Introduction• Various longitudinal models for to inferring within-person relation
have been developed in different contexts and disciplines.
• However, their relations have not been well recognized among
researchers and an issue regarding model choice is still underdiscussion.
• This lack of insight makes it difficult for researchers to select anappropriate statistical model when analyzing longitudinal data,
although estimation results can be largely influenced according to the
choice of model.
• Researchers in psychology often apply only CLPM or RI-CLPM,
while other statistical models and estimation procedures are available.
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Outline• Introduction• Longitudinal models to examine reciprocal relation
CLPM RI-CLPM LCM-SR LCS GCLM DPM• Issue of time-varying confounders (Usami, 2022)• Summary and future research agenda

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Related works• Usami,S. (2022). Within-person variability scores based causal inference: A two-stepestimation for joint effects of time-varying treatments. Psychometrika, in press
https://link.springer.com/content/pdf/10.1007/s11336-022-09879-1.pdf (open access)• Usami,S. (2021). On the differences between general cross-lagged panel model andrandom-intercept cross-lagged panel model: Interpretation of cross-lagged parametersand model choice. Structural Equation Modeling, 28, 331-344.
https://www.tandfonline.com/doi/pdf/10.1080/10705511.2020.1821690 (open access)• Usami,S., Todo,N., & Murayama,K. Modeling reciprocal effects in medical research:
Critical discussion on the current practices and potential alternative models. PLOS
(open access)ONE. 14(9): e0209133.
https://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0209133&type=printable• Usami,S., Murayama,K., & Hamaker,E.L. (2019). A unified framework of longitudinalmodels to examine reciprocal relations. Psychological Methods, 24, 637-657.
https://psycnet.apa.org/fulltext/2019-21491-001.pdf (open access)Coming soon (English version is under preparation)
・Usami,S. (in press). Statistical models for the inference of within-person relations:
A random intercept cross-lagged panel model and its interpretation. The Japanese
Journal of Developmental Psychology (written in Japanese).
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CLPM
𝑦𝑦𝑖𝑖𝑖𝑖 = α𝑦𝑦𝑦𝑦 + β𝑦𝑦𝑦𝑦 𝑦𝑦𝑖𝑖(𝑡𝑡−1) + γ𝑦𝑦𝑦𝑦 𝑥𝑥𝑖𝑖(𝑡𝑡−1) + 𝑑𝑑𝑦𝑦𝑦𝑦𝑡𝑡
𝑥𝑥𝑖𝑖𝑖𝑖 = α𝑥𝑥𝑡𝑡 + β𝑥𝑥𝑡𝑡 𝑥𝑥𝑖𝑖(𝑡𝑡−1) + γ𝑥𝑥𝑡𝑡 𝑦𝑦𝑖𝑖(𝑡𝑡−1) + 𝑑𝑑𝑥𝑥𝑖𝑖𝑖𝑖

𝑑𝑑𝑦𝑦𝑦𝑦𝑡𝑡~𝑁𝑁
𝑑𝑑𝑥𝑥𝑥𝑥𝑡𝑡ω2𝑦𝑦𝑦𝑦0,0ω𝑥𝑥𝑥𝑥𝑥𝑥

𝑖𝑖: individual 𝑡𝑡: time point 𝑥𝑥, 𝑦𝑦: observations 𝑑𝑑 : residuals
α: intercept β: autoregressive coefficient, γ: cross-lagged coefficientω𝑥𝑥𝑥𝑥𝑡𝑡
ω2𝑥𝑥𝑡𝑡Other formulation (use temporal group means µ instead of intercept α）.
∗
𝑦𝑦𝑖𝑖𝑖𝑖 = µ𝑦𝑦𝑦𝑦 + 𝑦𝑦𝑖𝑖𝑖𝑖∗
𝑥𝑥𝑖𝑖𝑖𝑖 = µ𝑥𝑥𝑡𝑡 + 𝑥𝑥𝑖𝑖𝑖𝑖∗∗∗
𝑦𝑦𝑖𝑖𝑖𝑖= β𝑦𝑦𝑦𝑦 𝑦𝑦𝑖𝑖(𝑡𝑡−1)+ γ𝑦𝑦𝑦𝑦 𝑥𝑥𝑖𝑖(𝑡𝑡−1)+ 𝑑𝑑𝑦𝑦𝑦𝑦𝑡𝑡∗∗∗
𝑥𝑥𝑖𝑖𝑖𝑖= β𝑥𝑥𝑡𝑡 𝑥𝑥𝑖𝑖(𝑡𝑡−1)+ γ𝑥𝑥𝑡𝑡 𝑦𝑦𝑖𝑖(𝑡𝑡−1)+ 𝑑𝑑𝑥𝑥𝑖𝑖𝑖𝑖µ: temporal group means𝑥𝑥 ∗ , 𝑦𝑦 ∗ ∶ temporal deviations

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CLPM

*Expression of this path diagram isbased on a unified framework of Usamiet al (2019)

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RI-CLPM (Hamaker et al., 2015)∗
𝑦𝑦𝑖𝑖𝑖𝑖 = µ𝑦𝑦𝑦𝑦 + 𝐼𝐼𝑦𝑦𝑦𝑦 + 𝑦𝑦𝑖𝑖𝑖𝑖∗
𝑥𝑥𝑖𝑖𝑖𝑖 = µ𝑥𝑥𝑡𝑡 + 𝐼𝐼𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑖𝑖𝑖𝑖∗∗∗
𝑦𝑦𝑖𝑖𝑖𝑖= β𝑦𝑦𝑦𝑦 𝑦𝑦𝑖𝑖(𝑡𝑡−1)+ γ𝑦𝑦𝑦𝑦 𝑥𝑥𝑖𝑖(𝑡𝑡−1)+ 𝑑𝑑𝑦𝑦𝑦𝑦𝑡𝑡∗∗∗
𝑥𝑥𝑖𝑖𝑖𝑖= β𝑥𝑥𝑡𝑡 𝑥𝑥𝑖𝑖(𝑡𝑡−1)+ γ𝑥𝑥𝑡𝑡 𝑦𝑦𝑖𝑖(𝑡𝑡−1)+ 𝑑𝑑𝑥𝑥𝑖𝑖𝑖𝑖µ𝑦𝑦𝑦𝑦 , µ𝑥𝑥𝑡𝑡 : temporal group means at 𝑡𝑡
𝐼𝐼𝑦𝑦𝑦𝑦 , 𝐼𝐼𝑥𝑥𝑖𝑖 : stable trait factors of 𝑖𝑖 𝐸𝐸 𝐼𝐼𝑦𝑦𝑦𝑦 = 𝐸𝐸 𝐼𝐼𝑥𝑥𝑥𝑥 = 0∗
𝑥𝑥𝑖𝑖𝑖𝑖, 𝑦𝑦𝑖𝑖𝑖𝑖∗ : temporal deviation from expected value of 𝑖𝑖 (µ𝑦𝑦𝑦𝑦 + 𝐼𝐼𝑦𝑦𝑦𝑦 , µ𝑥𝑥𝑥𝑥 + 𝐼𝐼𝑥𝑥𝑥𝑥 )∗⇒ 𝑥𝑥𝑖𝑖𝑖𝑖, 𝑦𝑦𝑖𝑖𝑖𝑖∗ represent within-person variability.

・Stable trait factors represent stable individual differences over time.

・Stable trait factors are assumed to be uncorrelated with within-person∗∗)=𝑐𝑐𝑐𝑐𝑐𝑐(𝐼𝐼𝑥𝑥𝑖𝑖 , 𝑦𝑦𝑖𝑖𝑖𝑖∗ )=𝑐𝑐𝑐𝑐𝑐𝑐(𝐼𝐼𝑥𝑥𝑖𝑖 , 𝑥𝑥𝑖𝑖𝑖𝑖)=0variability: 𝑐𝑐𝑐𝑐𝑐𝑐(𝐼𝐼𝑦𝑦𝑖𝑖 , 𝑦𝑦𝑖𝑖𝑖𝑖∗ )=𝑐𝑐𝑐𝑐𝑐𝑐(𝐼𝐼𝑦𝑦𝑦𝑦 , 𝑥𝑥𝑖𝑖𝑖𝑖
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CLPM

RI-CLPM*Expressions of diagrams are based on a unified framework of Usami et al (2019)

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RI-CLPM
・Data with T≧3 is required for model identification.
・RI-CLPM often shows better model fit than CLPM.
・Typically, estimates of β in the RI-CLPM become smaller than thatof CLPM. Estimates of γ would also differ between models.
・RI-CLPM is a possible option, but there are many other models.

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LCM-SR (latent curve model with structured residuals; Curran et al, 2013)∗
𝑦𝑦𝑖𝑖𝑖𝑖 = 𝐼𝐼𝑦𝑦𝑦𝑦 + (𝑡𝑡 − 1)𝑆𝑆𝑦𝑦𝑦𝑦 + 𝑦𝑦𝑖𝑖𝑖𝑖∗
𝑥𝑥𝑖𝑖𝑖𝑖 = 𝐼𝐼𝑥𝑥𝑖𝑖 + (𝑡𝑡 − 1)𝑆𝑆𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑖𝑖𝑖𝑖∗∗∗
𝑦𝑦𝑖𝑖𝑖𝑖= β𝑦𝑦𝑦𝑦 𝑦𝑦𝑖𝑖(𝑡𝑡−1)+ γ𝑦𝑦𝑦𝑦 𝑥𝑥𝑖𝑖(𝑡𝑡−1)+ 𝑑𝑑𝑦𝑦𝑦𝑦𝑡𝑡∗∗∗
𝑥𝑥𝑖𝑖𝑖𝑖= β𝑥𝑥𝑡𝑡 𝑥𝑥𝑖𝑖(𝑡𝑡−1)+ γ𝑥𝑥𝑡𝑡 𝑦𝑦𝑖𝑖(𝑡𝑡−1)+ 𝑑𝑑𝑥𝑥𝑖𝑖𝑖𝑖

・Growth trajectory (mean structure) is modeled with growth factors(intercept factor 𝐼𝐼 and slope factor 𝑆𝑆). ⇒ group mean µ is not included.
・Growth factors further decompose within-person variability of RI∗∗
CLPM into 𝑡𝑡 − 1 𝑆𝑆𝑦𝑦𝑦𝑦 and 𝑦𝑦𝑖𝑖𝑖𝑖(or 𝑡𝑡 − 1 𝑆𝑆𝑥𝑥𝑖𝑖 and 𝑥𝑥𝑖𝑖𝑖𝑖).

・𝑆𝑆 indicates individual differences of changes as (linear) trend. If thisindividual differences in growth trajectories are considered a criticalcomponent of reciprocal relations, γ would be biased.
⇒The problem of overadjustment.
⇒”Throwing the baby out with the bathwater” (Usami et al., 2019).
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LCM-SR

RI-CLPM*Expressions of diagrams are based on a unified framework of Usami et al (2019)

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LCS (latent change score model; McArdle & Hamagami, 2001)
𝑦𝑦𝑖𝑖𝑖𝑖 = 𝑓𝑓𝑦𝑦𝑖𝑖𝑖𝑖 + ε𝑦𝑦𝑦𝑦𝑦𝑦 𝑓𝑓𝑦𝑦𝑦𝑦𝑦𝑦 = 𝐴𝐴𝑦𝑦𝑦𝑦 + β𝑦𝑦 𝑓𝑓𝑦𝑦𝑦𝑦(𝑡𝑡−1) + γ𝑦𝑦 𝑓𝑓𝑥𝑥𝑖𝑖(𝑡𝑡−1) + 𝑑𝑑𝑦𝑦𝑦𝑦𝑡𝑡
𝑥𝑥𝑖𝑖𝑖𝑖 = 𝑓𝑓𝑥𝑥𝑖𝑖𝑖𝑖 + ε𝑥𝑥𝑖𝑖𝑖𝑖 𝑓𝑓𝑥𝑥𝑖𝑖𝑖𝑖 = 𝐴𝐴𝑥𝑥𝑥𝑥 + β𝑥𝑥 𝑓𝑓𝑥𝑥𝑥𝑥(𝑡𝑡−1) + γ𝑥𝑥 𝑓𝑓𝑦𝑦𝑦𝑦(𝑡𝑡−1) + 𝑑𝑑𝑥𝑥𝑖𝑖𝑖𝑖

・f: true (or common factor) score ε: measurement error (or unique factor)

・𝐴𝐴: accumulating factor (Usami et al., 2019).
⇒has non-zero mean and is allowed to be correlated with 𝑓𝑓.
⇒is included in the lagged regression and it has both direct and indirect
effects on observations, while stable trait factor has only direct effect.
・Time-invariant β and γ parameters are usually assumed. Estimates are
influenced by linear transformation of observations if time-varying
parameters are assumed.

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LCS (latent change score model)
・Two primary analytic purposes (i.e., inferring reciprocal relations
between variables, and describing growth trajectories/mean structures)are intertwined.
Common potential problem in LCM-SR and LCS:
・Misspecification in growth trajectories/mean structure through latent
factors (S or A) could lead to biased estimates of γ , while the growthtrajectory can take on any shape in (RI-)CLPM.

・ALT (autoregressive trajectory model; e.g., Curran & Bollen, 2001) has
similar potential problems as LCS.

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RI-CLPM

LCS

indirect effectdirect effect*Expressions of diagrams are based on a unified framework of Usami et al (2019)

GCLM (general cross-lagged panel model; Zyphur et al., 2020ab)
𝑦𝑦𝑖𝑖𝑖𝑖 = α𝑦𝑦𝑡𝑡 + λ𝑦𝑦𝑦𝑦 𝐵𝐵𝑦𝑦𝑦𝑦 + β𝑦𝑦𝑡𝑡 𝑦𝑦𝑖𝑖(𝑡𝑡−1) + γ𝑦𝑦𝑡𝑡 𝑥𝑥𝑖𝑖(𝑡𝑡−1) + δ𝑦𝑦𝑦𝑦 𝑑𝑑𝑦𝑦𝑖𝑖(𝑡𝑡−1) + ζ𝑦𝑦𝑦𝑦 𝑑𝑑𝑥𝑥𝑖𝑖(𝑡𝑡−1) + 𝑑𝑑𝑦𝑦𝑦𝑦𝑡𝑡
𝑥𝑥𝑖𝑖𝑖𝑖 = α𝑥𝑥𝑡𝑡 + λ𝑥𝑥𝑡𝑡 𝐵𝐵𝑥𝑥𝑖𝑖 + β𝑥𝑥𝑡𝑡 𝑥𝑥𝑖𝑖(𝑡𝑡−1) + γ𝑥𝑥𝑡𝑡 𝑦𝑦𝑖𝑖(𝑡𝑡−1) + δ𝑥𝑥𝑡𝑡 𝑑𝑑𝑥𝑥𝑖𝑖(𝑡𝑡−1) + ζ𝑥𝑥𝑡𝑡 𝑑𝑑𝑦𝑦𝑖𝑖(𝑡𝑡−1) + 𝑑𝑑𝑥𝑥𝑖𝑖𝑖𝑖

・Moving average (MA) terms (δ𝑦𝑦𝑦𝑦 𝑑𝑑𝑦𝑦𝑖𝑖(𝑡𝑡−1) + ζ𝑦𝑦𝑦𝑦 𝑑𝑑𝑥𝑥𝑖𝑖(𝑡𝑡−1) ) are included.

・Accumulating factors with time-varying loadings λ𝑡𝑡 are included.

・If time-varying loadings λ𝑡𝑡 are assumed, like LCM-SR, “throwing
the baby out with the bathwater” problem might occur.
・Exact meanings of MA terms (or residuals) are obscure in general,
while including these can be useful for prediction.

・These aspects make interpretation of cross-lagged parameters much
more difficult, and it can also lead to biased estimates of γ because of
possible overadjustment (Usami, 2021).
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DPM (dynamic panel model)
𝑦𝑦𝑖𝑖𝑖𝑖 = α𝑦𝑦𝑡𝑡 + 𝐴𝐴𝑦𝑦𝑦𝑦 + β𝑦𝑦 𝑦𝑦𝑖𝑖(𝑡𝑡−1) + γ𝑦𝑦 𝑥𝑥𝑖𝑖(𝑡𝑡−1) + 𝑑𝑑𝑦𝑦𝑦𝑦𝑦𝑦
𝑥𝑥𝑖𝑖𝑖𝑖 = α𝑥𝑥𝑡𝑡 + 𝐴𝐴𝑥𝑥𝑥𝑥 + β𝑥𝑥 𝑥𝑥𝑖𝑖(𝑡𝑡−1) + γ𝑥𝑥 𝑦𝑦𝑖𝑖(𝑡𝑡−1) + 𝑑𝑑𝑥𝑥𝑥𝑥𝑥𝑥

・Accumulating factor (usually called as unit effect) 𝐴𝐴 has zero mean.

・In econometrics, usually time-invariant β and γ are assumed andunidirectional relation (X⇒Y) is considered. Other terms (e.g., trend) might be
included.

・Interestingly, if stable trait factor and within-person variability are allowed tobe correlated in RI-CLPM (called predetermined RI-CLPM) and timeinvariant parameters are assumed for β , γ and residual (co)variances, in thissituation estimates of γ would be the same between predetermined RI-CLPM
and DPM (Andersen, 2021).

・Model choice to infer within-person relation: DPM, (predetermined) RI-CLPM,
STARTS (ε is included in the RI-CLPM). But see Usami et al (2019) for more discussion.
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predetermined RI-CLPM

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Unified Framework （Usami et al., 2019）
・To clarify the conceptual and mathematical similarities anddifferences among models:
𝑦𝑦𝑖𝑖𝑖𝑖 = 𝑓𝑓𝑦𝑦𝑖𝑖𝑖𝑖 + ε𝑦𝑦𝑖𝑖𝑖𝑖
𝑥𝑥𝑖𝑖𝑖𝑖 = 𝑓𝑓𝑥𝑥𝑖𝑖𝑖𝑖 + ε𝑥𝑥𝑖𝑖𝑖𝑖∗∗
𝑓𝑓𝑦𝑦𝑖𝑖𝑖𝑖 = [µ𝑦𝑦𝑦𝑦 +{𝐼𝐼𝑦𝑦𝑖𝑖 + 𝑡𝑡 − 1 𝑆𝑆𝑦𝑦𝑖𝑖 ]𝑓𝑓𝑦𝑦𝑦𝑦𝑦𝑦, 𝑓𝑓𝑥𝑥𝑖𝑖𝑖𝑖 = [µ𝑥𝑥𝑡𝑡 +{𝐼𝐼𝑥𝑥𝑖𝑖 + 𝑡𝑡 − 1 𝑆𝑆𝑥𝑥𝑖𝑖 ]𝑓𝑓𝑥𝑥𝑖𝑖𝑖𝑖∗∗∗
𝑓𝑓𝑦𝑦𝑦𝑦𝑦𝑦= {𝐴𝐴𝑦𝑦𝑖𝑖 + 𝑡𝑡 − 1 𝐵𝐵𝑦𝑦𝑦𝑦 } + β𝑦𝑦𝑡𝑡 𝑓𝑓𝑦𝑦𝑖𝑖(𝑡𝑡−1)+ γ𝑦𝑦𝑦𝑦 𝑓𝑓𝑥𝑥𝑖𝑖(𝑡𝑡−1)+ 𝑑𝑑𝑦𝑦𝑦𝑦𝑡𝑡∗∗∗
𝑓𝑓𝑥𝑥𝑖𝑖𝑖𝑖= {𝐴𝐴𝑥𝑥𝑖𝑖 + 𝑡𝑡 − 1 𝐵𝐵𝑥𝑥𝑖𝑖 } + β𝑥𝑥𝑡𝑡 𝑓𝑓𝑥𝑥𝑥𝑥(𝑡𝑡−1)+ γ𝑥𝑥𝑡𝑡 𝑓𝑓𝑦𝑦𝑦𝑦(𝑡𝑡−1)+ 𝑑𝑑𝑥𝑥𝑖𝑖𝑖𝑖

Three key aspects:
・Whether measurement error ε is included.
・Whether temporal group mean µ is included (or growthtrajectory is modeled) for mean structure.
・Whether stable trait factor (𝐼𝐼) or growth factors (𝐼𝐼 and/or 𝑆𝑆), or
accumulating factors (𝐴𝐴 and/or 𝐵𝐵) are included.
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Usami et al (2019)

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Conceptual diagrams（Usami et al., 2019）
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Outline• Introduction• Longitudinal models to examine reciprocal relation
CLPM RI-CLPM LCM-SR LCS GCLM DPM• Issue of time-varying confounders (Usami, 2022)• Summary and future research agenda

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Time-varying confounders• Models described so far do not account for time-varying
confounders. However, within-person relation is different fromcausal relation.
• In SEM approach, one needs to correctly specify how timevarying observed confounders are functionally related totreatments/predictors and outcomes at the within-person level.
• The RI-CLPM as SEM (or, covariance structure analysis) has
obvious drawback in that it generally demands linear regressions
for latent variables (i.e., within-person variability/temporaldeviations) .

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Epidemiological approach fortime-varying treatments and confounders• Potential outcome approach such as marginal structural models(MSMs; Robins, 1999; Robins, Hernan, & Brumback, 2000) has been popularin epidemiology to account for time-varying treatments.
• Although the number of application has been relatively
infrequent, structural nested models (SNMs: Robins, 1989, 1994) with
G-estimation are in principle better tailored for dealing with
failure of the usual assumptions of no unobserved confounders
or sequential ignorability (Robins, 1999; Robins & Hernan, 2009;
Vansteelandt & Joffe, 2014).

• MSMs and SNMs use observations in the model and they do not
explicitly include latent factors (e.g., stable traits). But estimation
procedures used in MSMs and SNMs can be utilized to infer
reciprocal relations at the within-person level.
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Two step estimation (Usami, 2022)• Assuming data generating process like RI-CLPM (stable trait factor).
1: Within-person variability score of each individual is predicted foreach variable (outcome 𝑌𝑌, treatment 𝐴𝐴, confounder 𝐿𝐿) through SEM.
2: Causal parameters (or cross-lagged parameters) are then estimatedby MSMs or SNMs, using calculated within-person variability scores.
Usami, 2022)

𝑌𝑌: outcome
𝐴𝐴: treatment(𝑋𝑋)𝐿𝐿: observed
confounderOct, 21 ICER 2022 Satoshi Usami

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Two step estimation (Usami, 2022)
・MSM/SNM is more flexible than the RI-CLPM in modeling how
observed confounders are functionally related to outcomes andtreatments/predictors.
・Notably, estimator of SNM through estimating equation has doubly
robust property: consistent estimator of (joint) treatment effect if either
outcome model or treatment model can be correctly specified.
・SNMs can allow one to directly model interactions/moderations
effects of treatments/predictors A with observed confounders L.
・ Through simulation it was shown that the proposed method canrecover causal parameters well and that causal estimates might beseverely biased if one does not properly account for stable traits.
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Outline• Introduction• Longitudinal models to examine reciprocal relation
CLPM RI-CLPM LCM-SR LCS GCLM DPM• Issue of time-varying confounders (Usami, 2022)• Summary and future research agenda

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Summary• Various longitudinal models for examining reciprocal relations at thewithin-person level exist, and an issue regarding model choice is still
under discussion.
• Researchers in psychology often use CLPM (or RI-CLPM) by SEM,
while other statistical models and estimation procedures are available.
Potential problems of existing models for reciprocal relation:• Overadjustment by including time-varying latent factors S or B (LCS
and GCLM).
• Estimation results of within-person relation (γ) could be biased if
growth trajectory/mean structure is misspecified (LCS and LCM-SR).
• Interpretative difficulty of γ because of inclusion of MA terms(GCLM).
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Conclusion and future research agenda
・(predetermined) RI-CLPM and DPM seem to be better options for theinference of reciprocal relation at the within-person level, and comparing
estimation results from these models could be especially useful.
・From the view of causal inference, SNM with G-estimation as used in
Usami (2022) could be a potential estimation option to account for timevarying confounders in inferring (within-person) reciprocal relation.
Future research:・Comparison between DPM and (predetermined) RI-CLPM in timevarying parameters setting.
・Extension of Usami (2022):
Additional simulation research/ Including measurement errors/ Assuming
data generating process as DPM/ Extension to other research questions (e.g.,
mediation effects).
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Reference• Andersen, H.K. (2021). Equivalent approaches to dealing with unobserved heterogeneity incross-lagged panel models? Investigating the benefits and drawbacks of the latent curvemodel with structured residuals and the random intercept cross-lagged panel model.
Psychological Methods. https://doi.org/10.1037/met0000285• Curran, P.J., & Bollen, K.A. (2001). The best of both worlds: Combining autoregressive andlatent curve models. In L.M. Collins & A. G. Sayer (Eds.), New methods for the analysis ofchange (pp. 105–136). Washington, DC: American Psychological Association.
• Curran, P.J., Howard, A.L., Bainter, S.A., Lane, S.T., & McGinley, J.S. (2013). The
separation of between-person and within-person components of individual change overtime: A latent curve model with structured residuals. Journal of Consulting and ClinicalPsychology, 82, 879–894.
• Hamaker, E.L., Kuiper, R.M., Grasman, R.P.P.P. (2015). A critique of the cross-lagged panel
model. Psychological. Methods, 20, 102–116.
• McArdle, J.J., & Hamagami, F. (2001). Latent difference score structural models for linear
dynamic analyses with incomplete longitudinal data. In L. Collins & A. Sayer (Eds.), New
methods for the analysis of change (pp. 137–175). Washington, DC: American
Psychological Association.
• Orth, U.D., Clark, A.M., Donnellan, B., & Robins, R. W. (2021). Testing prospective effects
in longitudinal research: Comparing seven competing cross-lagged models. Journal of
Personality and Social Psychology. 120, 1013–1034.
Oct, 21 ICER 2022 Satoshi Usami

34/36

Reference• Robins, J. M. (1989). The analysis of randomized and non-randomized AIDS treatment
trials using a new approach to causal inference in longitudinal studies. In L. Sechrest, H.
Freeman, & A. Mulley (Eds.), Health service research methodology: A focus on AIDS (pp.
113–159). U.S. Public Health Service, National Center for Health Services Research.
• Robins, J. M. (1994). Correcting for non-compliance in randomized trials using structural
nested mean models. Communications in Statistics-Theory and Methods, 23, 2379–2412• Robins, J.M. (1999). Marginal structural models versus structural nested models as tools forcausal inference. Epidemiology, 116, 95-134.
• Robins, J.M., Hernán, M.A., & Brumback, B. (2000). Marginal structural models and causal
inference in epidemiology. Epidemiology, 11, 550-560.
• Robins, J.M., & Hernán, M.A. (2009). Estimation of the causal effects of time-varying
exposures. In G. Fitzmaurice et al. (Eds.), Handbooks of modern statistical methods:
Longitudinal data analysis (pp. 553-599). Boca Raton: CRC Press.
• Usami,S. (2021). On the differences between general cross-lagged panel model and randomintercept cross-lagged panel model: Interpretation of cross-lagged parameters and model
choice. Structural Equation Modeling, 28, 331-344.
https://www.tandfonline.com/doi/pdf/10.1080/10705511.2020.1821690• Usami,S. (2022). Within-person variability scores based causal inference: A two-stepestimation for joint effects of time-varying treatments. Psychometrika, in press.
https://link.springer.com/content/pdf/10.1007/s11336-022-09879-1.pdf• Usami,S., Murayama,K., & Hamaker,E.L. (2019). A unified framework of longitudinalmodels to examine reciprocal relations. Psychological Methods, 24, 637-657.
https://psycnet.apa.org/fulltext/2019-21491-001.pdf
Oct, 21 ICER 2022 Satoshi Usami

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Reference• Vansteelandt, S., & Joffe, M. (2014). Structural nested models and g-estimation: The
partially realized promise. Statistical Science, 29, 707–731.
• Zyphur, M.J., Allison, P.D., Tay, L., Voelkle, M.C., Preacher, K.J., Zhang, Z., Hamaker, E.L.,
Shamsollahi, A., Pierides, D.C., Koval, P., & Diener, E. (2020a). From data to causes I:
Building a general cross-lagged panel model (GCLM). Organizational Research Methods, 23,
651–687.
• Zyphur, M.J., Voelkle, M.C., Tay, L., Allison, P.D., Preacher, K.J., Zhang, Z., Hamaker, E.L.,
Shamsollahi, A., Pierides, D.C., Koval, P., & Diener, E. (2020b). From data to causes II:
Comparing approaches to panel data analysis. Organizational Research Methods, 23, 688–716.

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