How to Evaluate Causal Dominance Hypotheses in Lagged Effects Models

Figures & data

Figure 1. The bivariate CLPM for three waves and two observed variables; squares denote the observed variables, circles denote latent variables, triangles represent constants for the mean structure. The latent variables p and q are the grand-mean centered observed variables. Single-headed arrows indicate regression and double-headed arrows indicate correlations or (co)variances. The figure is based on Hamaker et al. (2015).

Figure 2. The bivariate RI-CLPM with the separation of within-person and between-person variation for three waves and two observed variables; squares denote the observed variables, circles denote latent variables, triangles represent constants for the mean structure. The variables $p^{*}$ and $q^{*}$ are the person-mean centered observed variables, and $\kappa$ and $\omega$ are random intercepts. Single-headed arrows indicate regression and double-headed arrows indicate correlations or (co)variances. This figure is based on Hamaker et al. (2015).

Table 1. Specifications in simulations for bivariate (RI-)CLPM with a total of 144 conditions (2 × 4 × 6 × 3).

Model	Number of waves	Sample size	Cross-lagged effects $(\beta ,\gamma )$
RI-CLPM and CLPM	3, 4, 5, 6	25, 50, 75, 100, 200, 300	(0.1, 0.2)
			(0.1, 0.4)
			(−0.07, −0.07)

Figure 3. An Illustration of how the three-wave data for the first participant (i = 1) is obtained from data generated from the VAR(1) model. This is repeated N times, that is, for each individual.

Figure 4. The (true) hypothesis rates when using the GORICA in a bivariate RI-CLPM for all simulation conditions, namely the number of participants (x-axis), the number of waves (different Colored lines), and the three different pairs of cross-lagged parameters ($\beta$ and $\gamma$): (a) ($\beta ,$ $\gamma$) = (0.1, 0.2), (b) ($\beta ,$ $\gamma$) = (0.1, 0.4), and (c) ($\beta ,$ $\gamma$) = (-0.07, -0.07). Note that in plots (a) and (b), $H_1$ is the true hypothesis, while in Plot (c) both $H_1:\left|\beta \right|<\left|\gamma \right|$ and $H_2:\left|\beta \right|>\left|\gamma \right|$ are true.

Figure 5. The tri-variate CLPM and tri-variate RI-CLPM; where squares denote the observed variables, circles denote latent variables and triangles represent constants for the mean structure. The latent variables p and q in Figure 5(a) are the grand-mean centered observed variables, while p, q, and r are the person-mean centered observed variables in Figure 5(b). Single-headed arrows indicate regression, double-headed arrows indicate correlations or (co)variances.

Table 2. Specifications in simulations for tri-variate (RI-)CLPM with a total of 36 conditions (2 × 1 × 6 × 3).

Model	Number of wave	Sample size	Cross-lagged effects (${\varphi }_{12},$ ${\varphi }_{21},$ ${\varphi }_{13},$ ${\varphi }_{31},$ ${\varphi }_{23},$ ${\varphi }_{32}$)
RI-CLPM and CLPM	3	25, 50, 75, 100, 200, 300	(0.1, 0.2, 0.1, 0.2, 0.1, 0.2)
			(0.1, 0.4, 0.1, 0.4, 0.1, 0.4)
			(−0.07, −0.07, −0.07, −0.07, −0.07, −0.07)

Figure 6. The (true) hypothesis rates when using the GORICA in a tri-variate RI-CLPM for all simulation conditions, namely the number of participants (x-axis) and the three different pairs of cross-lagged parameters (${\varphi }_{12},$ ${\varphi }_{21},$ ${\varphi }_{13},$ ${\varphi }_{31},$ ${\varphi }_{23},$ ${\varphi }_{32}$): (a) (${\varphi }_{12},$ ${\varphi }_{21},$ ${\varphi }_{13},$ ${\varphi }_{31},$ ${\varphi }_{23},$ ${\varphi }_{32}$)= (0.1, 0.2, 0.1, 0.2, 0.1, 0.2), (b) (${\varphi }_{12},$ ${\varphi }_{21},$ ${\varphi }_{13},$ ${\varphi }_{31},$ ${\varphi }_{23},$ ${\varphi }_{32}$)= (0.1, 0.4, 0.1, 0.4, 0.1, 0.4), and (c) (${\varphi }_{12},$ ${\varphi }_{21},$ ${\varphi }_{13},$ ${\varphi }_{31},$ ${\varphi }_{23},$ ${\varphi }_{32}$)= (−0.07, −0.07, −0.07, −0.07, −0.07, −0.07). Note that in plots (a) and (b), $H_3$ is the true hypothesis, while in Plot (c) both $H_3$ and its complement are true but $H_3$ is the most parsimonious one.

Figure 7. The number of times the incorrect hypotheses $H_4$ (Green) and $H_5$ (pink) are chosen in a RI-CLPM, for various participant numbers (x-axis) and two cross-lagged parameter specifications: (a) (${\varphi }_{12},$ ${\varphi }_{21},$ ${\varphi }_{13},$ ${\varphi }_{31},$ ${\varphi }_{23},$ ${\varphi }_{32}$)= (0.1, 0.2, 0.1, 0.2, 0.1, 0.2) and (b) (${\varphi }_{12},$ ${\varphi }_{21},$ ${\varphi }_{13},$ ${\varphi }_{31},$ ${\varphi }_{23},$ ${\varphi }_{32}$)= (0.1, 0.4, 0.1, 0.4, 0.1, 0.4).

Figure 8. Simplified graph of the RI-CLPM with standardized estimates for the autoregressive parameter estimates and the cross-lagged parameter estimates at time point t and $t+1(t=1,2)$ where the standardized cross-lagged paths are constrained to be the same across time.

Figure 9. Simplified graph of the RI-CLPM with standardized coefficients for the autoregressive parameter estimates and the cross-lagged parameter estimates at time point t and $t+1(t=1,2)$ in a wave-specific parameters model.

Table 3. GORICA values and weights for the hypotheses of interest ($H_{1a}$ and its complement) in a wave-independent parameters model.

	log-likelihood	penalty  term	GORICA values	GORICA weights
Hypothesis	($L_m$)	($P{T}_m$)	($\mathit{GORIC}{A}_m$)	($w_m$)
$H_{1a}$	34.238	12.5	−43.475	0.502
Complement	34.230	12.5	−43.460	0.498

Table 4. GORICA values and weights for the hypotheses of interest ($H_{1b}$ and its complement) in a wave-specific parameters model.

	Log-likelihood	penalty  term	GORICA values	GORICA weights
Hypothesis	($L_m$)	($P{T}_m$)	($\mathit{GORIC}{A}_m$)	($w_m$)
$H_{1b}$	8.410	2.977	−10.865	0.586
Complement	8.607	3.523	−10.168	0.414

Hamilton, J. (1994). Time series analysis. Princeton University Press.

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