Online Appendices for: Modeling Latent Growth With Multiple Indicators: A Comparison of Three Approaches

Transcription

1 Online Appendices for: Modeling Latent Growth With Multiple Indicators: A Comparison of Three Approaches Jacob Bishop and Christian Geiser Utah State University David A. Cole Vanderbilt University Contents First-Order Autoregressive Structure Among Latent State Residuals 3 Mean and Covariance Structure in the SGM, GSGM, and ISGM 5 SGM Mean and Covariance Structure GSGM Mean and Covariance Structure ISGM Mean and Covariance Structure Autoregressive (AR Models Summary Proportionality Constraint 14 General to Specific Variance Ratio for the SGM General to Specific Variance Ratio for the GSGM General to Specific Variance Ratio for the ISGM Degrees of Freedom in the SGM, GSGM, and ISGM 16 Computing df Known Parameters Estimated Parameters and Degrees of Freedom Model Identification with Two Indicators per Time Point 1 Author Note. Jacob Bishop and Christian Geiser, Department of Psychology, Utah State University. David A. Cole, Department of Psychology and Human Development, Vanderbilt University. Correspondence concerning this article should be addressed to Jacob Bishop, 81 Old Main Hill Logan, UT jacob.bishop@usu.edu

2 MULTIPLE-INDICATOR GROWTH MODELS Variance Components and Coefficients 6 SGM GSGM ISGM Mplus Model Inputs for SGM, GSGM, and ISGM 9 Model 1a: SGM Model 1b: SGM, α, λ-mi Model a: GSGM Model b: GSGM, α, λ-mi Model c: GSGM, α, λ, γ-mi Model 3a: ISGM Model 3b: ISGM, γ-mi Monte-Carlo Simulation for the ISGM 46 Method Criteria for Evaluating the Performance of the Models Results Summary and Discussion References 65

3 MULTIPLE-INDICATOR GROWTH MODELS 3 Appendix A First-Order Autoregressive Structure Among Latent State Residual Variables Steyer and Schmitt (1994 were among the first to propose the inclusion of an autoregressive structure in LST models to capture stability beyond what can be accounted for by a common trait factor. Cole, Martin, and Steiger (5 examined the meaning of autoregressive structures in LST models in detail and noted that it is often appropriate to include autoregressive effects between latent state residual variables to model short-term stability or carry-over effects in these models. In a similar vein, Bollen and Curran (4 presented so-called autoregressive latent trajectory (ALT models that include autoregressive paths between observed variables in single-indicator LGC models. Murphy et al. (11 recently studied the meaning of autocorrelations in the SGM. A first-order autoregressive structure implies a time-dependence among adjacent latent state residuals, with an additional unpredictable error component δ t : ζ t = β t(t 1 ζ (t 1 + δ t, (A.1 where β t(t 1 is a real constant and the δ t variables have a mean of zero and are uncorrelated with each other as well as with all other latent and error variables in the model. Given that we assume a first-order autoregressive structure, the notation for the regression coefficient may be simplified by defining: β t β t(t 1. (A. We illustrate a first-order autoregressive structure graphically for the SGM in Figure A1A, the GSGM in Figure A1B, and the ISGM in Figure A1C. Including first-order autoregressive effects among the latent state residual variables relaxes the assumption that all of the across-time stability is entirely due to the trait and trait-change process. Instead, part of the (short-term stability can now be captured by the autoregressive effects, accounting for the fact that adjacent measurements are often more highly correlated than measurements that are further apart in time. In practical applications, it is often common to assume time-invariance of the autoregressive effect. This means that β t = β s = β is often assumed to hold for all s, t.

4 MULTIPLE-INDICATOR GROWTH MODELS 4 A ξ int E (ξ int 1 E (ξ lin ξ lin λ i1 λ j1 λ it λ jt α i1 α j1 α it α jt α is α js λ is λ js (t 1 λ it (t 1 λ jt (t 1 (s 1 λis (s 1 λjs (s 1 Y 11 Y i1 Y j1 Y 1t Y it Y jt Y 1s Y is Y js ε 11 ε i1 ε j1 ε 1t ε it ε jt ε 1s ε is ε js λ i1 λ j1 λ it λ jt λ is λ js ζ 1 β β t ζ t β (t+1 β s ζ s δ t δ s B ξ int E (ξ int 1 E (ξ lin ξ lin λ i1 λ j1 λ it λ jt α i1 α j1 α it α jt α is α js λ is λ js (t 1 λ it (t 1 λ jt (t 1 (s 1 λis (s 1 λjs (s 1 Y 11 Y i1 Y j1 Y 1t Y it Y jt Y 1s Y is Y js ε 11 ε i1 ε j1 ε 1t ε it ε jt ε 1s ε is ε js γ i1 γ j1 γ it γ jt γ is γ js ζ 1 β β t ζ t β (t+1 β s ζ s δ t δ s C ξ int1 E (ξ int1 ξ inti E (ξ inti ξ intj E ( ξ intj 1 E (ξ lini E (ξ lin1 E ( ξ linj t 1 t 1 ξ lin1 s 1 t 1 ξ lini s 1 ξ linj s 1 Y 11 Y i1 Y j1 Y 1t Y it Y jt Y 1s Y is Y js ε 11 ε i1 ε j1 ε 1t ε it ε jt ε 1s ε is ε js γ i1 γ j1 γ it γ jt γ is γ js ζ 1 β β t ζ t β (t+1 β s ζ s δ t Figure A1. LST Models with First-order Autoregressive Structure Among Latent State Residual Variables A: SGM Post-Transformation with a First-order Autoregressive Structure among Latent State Residual Factors. B: GSGM Post-transformation with a Firstorder Autoregressive State Residual Structure. C: ISGM Post-transformation with Firstorder Autoregressive State Residual Structure. δ s

5 MULTIPLE-INDICATOR GROWTH MODELS 5 Appendix B Mean and Covariance Structure in the SGM, GSGM, and ISGM SGM Mean and Covariance Structure The combined equations for the SGM are given in Equation (13. Recognizing that we set α 1t = and λ 1t = γ 1t = 1 to define the metric of the latent factors, it is sufficient to work with only the final line of these equations. This is given by: Y it = α it + λ it ξ int + λ it (t 1 ξ lin + λ it ζ t + ε it. (B.1 Mean Structure. We derive the mean structure by substituting Equation (B.1 into the expected value, and simplifying this expression according to the algebraic rules for expected values as follows. E (Y it = E (α it + λ it ξ int + λ it (t 1 ξ lin + λ it ζ t + ε it = E (α it }{{} +E (λ it ξ int + E (λ it (t 1 ξ lin + E (λ it ζ t }{{} α it = α it + λ it E (ξ int + λ it (t 1 E (ξ lin. + E (ε it }{{} Covariance Structure. First, we expand the covariance by substituting Equation (B.1 into the covariance operator for Y it and Y js : cov (Y it, Y js = cov ( αit + λ it ξ int + λ it (t 1 ξ lin + λ it ζ t + ε it, α js + λ js ξ int + λ js (s 1 ξ lin + λ js ζ s + ε js. Next, we expand using rules of covariance algebra (e.g., Kenny, This yields cov (Y it, Y js = cov (α it, λ js (ξ int + (s 1 ξ lin + λ js ζ s + ε js + }{{} cov (λ it (ξ int + (t 1 ξ lin + λ it ζ t + ε it, α js + }{{} ( λit (ξ cov int + (t 1 ξ lin + λ it ζ t + ε it, λ js (ξ int + (s 1 ξ lin + λ js ζ s + ε js = ( λit (ξ cov int + (t 1 ξ lin + λ it ζ t + ε it,. λ js (ξ int + (s 1 ξ lin + λ js ζ s + ε js The first two terms are zero because of the covariance rule cov (k, X = where k is a constant. Next, we decompose the covariance according to covariance addition rules. In this step, we separate the error terms ε it and ε js.

6 MULTIPLE-INDICATOR GROWTH MODELS 6 ( λit (ξ cov (Y it, Y js = cov int + (t 1 ξ lin + λ it ζ t + ε it,, λ js (ξ int + (s 1 ξ lin + λ js ζ s + ε js ( λit (ξ = cov int + (t 1 ξ lin + λ it ζ t + ε it, + λ js (ξ int + (s 1 ξ lin + λ js ζ s cov (λ it (ξ int + (t 1 ξ lin + λ it ζ t + ε it, ε js, ( λit (ξ = cov int + (t 1 ξ lin + λ it ζ t + ε it, + λ js (ξ int + (s 1 ξ lin + λ js ζ s cov (λ it (ξ int + (t 1 ξ lin, ε js + cov (λ it ζ t, ε js + cov (ε it, ε js, ( λit (ξ = cov int + (t 1 ξ lin + λ it ζ t, + λ js (ξ int + (s 1 ξ lin + λ js ζ s cov (ε it, λ js (ξ int + (s 1 ξ lin + cov (λ it (ξ int + (t 1 ξ lin, ε js + cov (ε it, λ js ζ s + cov (λ it ζ t, ε js + cov (ε it, ε js. The subsequent step focuses on expanding the first covariance term, the covariance between the latent trait and state residual components. cov (Y it, Y js = cov (λ it (ξ int + (t 1 ξ lin, λ js ζ s + cov (λ it (ξ int + (t 1 ξ lin, λ js (ξ int + (s 1 ξ lin + cov (λ it ζ t, λ js ζ s + cov (ε it, λ js (ξ int + (s 1 ξ lin + cov (λ it (ξ int + (t 1 ξ lin, ε js + cov (ε it, λ js ζ s + cov (λ it ζ t, ε js + cov (ε it, ε js. The second term in this expression, which represents the covariance among the latent trait variables is now expanded. At the same time, the constant terms λ it, λ js, (t 1, and (s 1 can be moved outside the covariance expression according to the rule cov (k X, Y = k cov (X, Y.

7 MULTIPLE-INDICATOR GROWTH MODELS 7 cov (Y it, Y js = λ it λ js cov ((ξ int + (t 1 ξ lin, ζ s + [(s 1 + (t 1] cov (ξ int, ξ lin + λ it λ js (t 1 (s 1 var (ξ lin + + var (ξ int λ it λ js cov (ζ t, ζ s + λ js cov (ε it, (ξ int + (s 1 ξ lin + λ it cov ((ξ int + (t 1 ξ lin, ε js + λ js cov (ε it, ζ s + λ it cov (ζ t, ε js + cov (ε it, ε js. Finally, this expression is simplified because for this model, the following restrictions apply cov (ξ it, ζ t cov (ξ it, ε it cov (ζ t, ε it cov (ζ t, ζ s = cov (ε it, ε js = The resulting general covariance equation is thus: cov (Y it, Y js = λ it λ js = for all i, j, s, t, (B. { var (ζt when t = s, otherwise, { var (εit when i = j, t = s, otherwise. [(s 1 + (t 1] cov (ξ int, ξ lin + (t 1 (s 1 var (ξ lin + var (ξ int λ it λ js cov (ζ t, ζ s + cov (ε it, ε js. + Note that the final two terms are zero except as given in Equations (B.3 and (B.4. GSGM Mean and Covariance Structure (B.3 (B.4 The combined equations for the GSGM are given in Equation (18. Recognizing that we set α 1t = and λ 1t = γ 1t = 1, it is sufficient to work with only the final line of these equations. Thus, the general equation for Y it is given by Y it = α it + λ it ξ int + λ it (t 1 ξ lin + γ it ζ t + ε it. (B.5 Mean Structure. We derive the mean structure by substituting Equation (B.5 into the expected value operator, and simplifying this expression according to the algebraic rules for expected values as follows.

8 MULTIPLE-INDICATOR GROWTH MODELS 8 E (Y it = E (α it + λ it ξ int + λ it (t 1 ξ lin + γ it ζ t + ε it = E (α it }{{} +E (λ it ξ int + E (λ it (t 1 ξ lin + E (γ it ζ t }{{} α it = α it + λ it E (ξ int + λ it (t 1 E (ξ lin. + E (ε it }{{} Note that this is equivalent to the mean structure for the SGM, since neither the latent state residual variable ζ t nor the corresponding loading γ it appears in the final simplified equation. Covariance Structure. We now derive the covariance structure in a similar way to the SGM. First, we expand the covariance by substituting the above for Y it and Y js : cov (Y it, Y js = cov ( αit + λ it ξ int + λ it (t 1 ξ lin + γ it ζ t + ε it, α js + λ js ξ int + λ js (s 1 ξ lin + γ js ζ s + ε js. Next, we expand according to the rules of covariance algebra. This yields cov (Y it, Y js = cov (α it, λ js (ξ int + (s 1 ξ lin + γ js ζ s + ε js + }{{} cov (λ it (ξ int + (t 1 ξ lin + γ it ζ t + ε it, α js + }{{} ( λit (ξ cov int + (t 1 ξ lin + γ it ζ t + ε it, λ js (ξ int + (s 1 ξ lin + γ js ζ s + ε js = ( λit (ξ cov int + (t 1 ξ lin + γ it ζ t + ε it,. λ js (ξ int + (s 1 ξ lin + γ js ζ s + ε js The first two terms are zero because of the covariance rule cov (k, X = where k is a constant. Next, we decompose the covariance according to the rules of covariance addition. In this step, we separate the error terms ε it and ε js.

9 MULTIPLE-INDICATOR GROWTH MODELS 9 ( λit (ξ cov (Y it, Y js = cov int + (t 1 ξ lin + γ it ζ t + ε it,, λ js (ξ int + (s 1 ξ lin + γ js ζ s + ε js ( λit (ξ = cov int + (t 1 ξ lin + γ it ζ t + ε it, + λ js (ξ int + (s 1 ξ lin + γ js ζ s cov (λ it (ξ int + (t 1 ξ lin + γ it ζ t + ε it, ε js, ( λit (ξ = cov int + (t 1 ξ lin + γ it ζ t + ε it, + λ js (ξ int + (s 1 ξ lin + γ js ζ s cov (λ it (ξ int + (t 1 ξ lin, ε js + cov (γ it ζ t, ε js + cov (ε it, ε js, ( λit (ξ = cov int + (t 1 ξ lin + γ it ζ t, + λ js (ξ int + (s 1 ξ lin + γ js ζ s cov (ε it, λ js (ξ int + (s 1 ξ lin + cov (λ it (ξ int + (t 1 ξ lin, ε js + cov (ε it, γ js ζ s + cov (γ it ζ t, ε js + cov (ε it, ε js. The subsequent step focuses on expanding the first covariance term, the covariance between the latent trait and state residual components. cov (Y it, Y js = cov (λ it (ξ int + (t 1 ξ lin, γ js ζ s + cov (λ it (ξ int + (t 1 ξ lin, λ js (ξ int + (s 1 ξ lin + cov (γ it ζ t, γ js ζ s + cov (ε it, λ js (ξ int + (s 1 ξ lin + cov (λ it (ξ int + (t 1 ξ lin, ε js + cov (ε it, γ js ζ s + cov (γ it ζ t, ε js + cov (ε it, ε js. The second term in this expression, which represents the covariance among the latent trait variables is now expanded. At the same time, the constant terms λ it, λ js, γ it, γ js, (t 1, and (s 1 can be moved outside the covariance expression according to the rule cov (k X, Y = k cov (X, Y.

10 MULTIPLE-INDICATOR GROWTH MODELS 1 cov (Y it, Y js = λ it γ js cov (ξ int + (t 1 ξ lin, ζ s + [(s 1 + (t 1] cov (ξ int, ξ lin + λ it λ js (t 1 (s 1 var (ξ lin + + var (ξ int γ it γ js cov (ζ t, ζ s + λ js cov (ε it, (ξ int + (s 1 ξ lin + λ it cov ((ξ int + (t 1 ξ lin, ε js + γ js cov (ε it, ζ s + γ it cov (ζ t, ε js + cov (ε it, ε js. Finally, this expression is simplified because of the restrictions for this model, which are listed in Equations (B., (B.3, and (B.4. The resulting general covariance equation for the GSGM is thus: cov (Y it, Y js = λ it λ js [(s 1 + (t 1] cov (ξ int, ξ lin + (t 1 (s 1 var (ξ lin + var (ξ int γ it γ js cov (ζ t, ζ s + cov (ε it, ε js. + As before, the final two terms of this equation are zero except as given in Equations (B.3 and (B.4. ISGM Mean and Covariance Structure The combined equations for the ISGM are given in Equation (3. Recognizing that we set γ 1t = 1 to define the metric of each ζ t, it is sufficient to work with only the final line of these equations. Thus, the general equation for Y it is given by: Y it = α it + ξ inti + (t 1 ξ lini + γ it ζ t + ε it. (B.6 Mean Structure. We now derive the covariance structure in a similar way to the SGM and GSGM. First, Equation (B.6 is substituted into the expected value operator, and simplifying this expression according to the algebraic rules for expected values as follows. E (Y it = E (α it + ξ inti + (t 1 ξ lini + γ it ζ t + ε it = E (α it }{{} +E (ξ inti + E ((t 1 ξ lini + E (γ it ζ t }{{} α it = α it + E (ξ inti + (t 1 E (ξ lini. + E (ε it }{{}

11 MULTIPLE-INDICATOR GROWTH MODELS 11 Covariance Structure. The covariance structure for the ISGM is derived in a like manner by substituting Equation (B.6 for Y it and Y js in the covariance operator: cov (Y it, Y js = cov ( ξinti + (t 1 ξ lini + γ it ζ t + ε it, ξ intj + (s 1 ξ linj + γ js ζ s + ε js Next, we expand according to the covariance algebra rule cov (X, Y + Z = cov (X, Y + cov (X, Z. This is used to separate the error terms ε it and ε js. cov (Y it, Y js = ( (ξinti + (t 1 ξ lini + γ it ζ t + ε it, cov, (ξ intj + (s 1 ξ linj + γ js ζ s + ε js = ( (ξinti + (t 1 ξ lini + γ it ζ t + ε it, cov + intj + (s 1 ξ linj + γ js ζ s cov ((ξ inti + (t 1 ξ lini + γ it ζ t + ε it, ε js, ( (ξinti + (t 1 ξ lini + γ it ζ t + ε it, = cov + intj + (s 1 ξ linj + γ js ζ s cov (ξ inti + (t 1 ξ lini, ε js + cov (γ it ζ t, ε js + cov (ε it, ε js, ( (ξinti + (t 1 ξ lini + γ it ζ t, = cov + (ξ intj + (s 1 ξ linj + γ js ζ s ( cov ε it, (ξ intj + (s 1 ξ linj + cov ((ξ inti + (t 1 ξ lini, ε js + cov (ε it, γ js ζ s + cov (γ it ζ t, ε js + cov (ε it, ε js. The subsequent step focuses on expanding the first covariance term, the covariance between the latent trait and state residual components. cov (Y it, Y js = cov (ξ inti + (t 1 ξ lini, γ js ζ s + cov (ξ inti + (t 1 ξ lini, ξ intj + (s 1 ξ linj + cov (γ it ζ t, γ js ζ s + cov (ε it, ξ intj + (s 1 ξ linj + cov (ξ inti + (t 1 ξ lini, ε js + cov (ε it, γ js ζ s + cov (γ it ζ t, ε js + cov (ε it, ε js. The second term in this expression, which represents the covariance among the latent trait variables is now expanded. At the same time, the constant terms γ it, γ js, (t 1, and (s 1 can be moved outside the covariance expression according to the rule cov (k X, Y = k cov (X, Y..

12 MULTIPLE-INDICATOR GROWTH MODELS 1 cov (Y it, Y js = γ js cov (ξ inti + (t 1 ξ lini, ζ s + (t 1 cov (ξ intj, ξ lini + (s 1 cov (ξ inti, ξ linj + (t 1 (s 1 cov (ξ lini, ξ linj + cov (ξ inti, ξ intj γ it γ js cov (ζ t, ζ s + cov (ε it, ξ intj + (s 1 ξ linj + cov (ξ inti + (t 1 ξ lini, ε js + γ js cov (ε it, ζ s + γ it cov (ζ t, ε js + cov (ε it, ε js. + Finally, this expression is simplified because for this model, the following restrictions apply: cov (ξ it, ζ s = cov (ξ it, ε js = cov (ζ s, ε it =, for all i, j, s, and t. The resulting general covariance equation is thus: cov (Y it, Y js = (t 1 cov (ξ intj, ξ lini + (s 1 cov (ξ inti, ξ linj + (t 1 (s 1 cov (ξ lini, ξ linj cov (ξ inti, ξ intj γ it γ js cov (ζ t, ζ s + cov (ε it, ε js. + + As before, the final two terms of this equation are zero except as given in Equations (B.3 and (B.4. Autoregressive (AR Models In all three instances, the derivation of the autoregressive (AR model begins where the corresponding non-ar model left off. This is accomplished by substituting the quantity β t ζ t 1 + δ t wherever ζ t appears. Recall that β t(t 1 is a real constant and the δ t variables have a mean of zero and are uncorrelated with each other as well as with all other latent and error variables in the model. Thus, the mean structure for the AR-model is identical in each case to the mean structure for the non-ar model. It is not possible to explicitly re-write the equations for the covariance structure because the autoregressive substitution is recursive. Nevertheless, this substitution and expansion may be done for any specific case in which t and s are known. See Appendix E, where we illustrate this expansion for the specific case of two indicators, i = {1, } and three time points, t = {1,, 3}.

13 MULTIPLE-INDICATOR GROWTH MODELS 13 The restrictions for the AR model are slightly different than those of the non-ar model. For the AR model, the following restrictions apply: cov (ξ it, ζ s cov (ξ it, ε js cov (ζ t, ε js = for all i, j, s, t, (B.7 cov (δ t, ζ s cov (ξ it, δ s cov (ε it, δ s Summary cov (δ t, δ s = cov (ε it, ε js = var (ζ 1 when t = s = 1, var (δ t when t = s 1, otherwise, { var (εit when i = j, t = s, otherwise. (B.8 (B.9 The mean and covariance structure for the three models is summarized in Table B1. Table B1 Mean and Covariance Equations for Non-Autoregressive SGM, GSGM, and ISGM Models Mean and Covariance Structure SGM, non-ar Mean E (Y it = α it + λ ite (ξ int + λ it (t 1 E (ξ lin Covariance cov (Y it, Y js = λ it λ js [(s 1 + (t 1] cov (ξ int, ξ lin + λ it λ js (t 1 (s 1 var (ξ lin + λ it λ js var (ξ int + λ it λ js cov (ζ t, ζ s + cov (ε it, ε js GSGM, non-ar Mean E (Y it = α it + λ ite (ξ int + λ it (t 1 E (ξ lin Covariance cov (Y it, Y js = λ it λ js [(s 1 + (t 1] cov (ξ int, ξ lin + λ it λ js (t 1 (s 1 var (ξ lin + λ it λ js var (ξ int + γ it γ js cov (ζ t, ζ s + cov (ε it, ε js ISGM, non-ar Mean E (Y it = E (Y it = E (ξ inti + (t 1 E (ξ lini Covariance cov (Y it, Y js = (t 1 cov (ξ intj, ξ lini + (s 1 cov (ξ inti, ξ linj + (t 1 (s 1 cov (ξ lini, ξ linj + cov (ξ inti, ξ intj + γ it γ js cov (ζ t, ζ s + cov (ε it, ε js

14 MULTIPLE-INDICATOR GROWTH MODELS 14 Appendix C Proportionality Constraint The proportionality constraint refers to an implicit constraint on the ratio of general to specific variance that is present in the SGM, but not the GSGM or ISGM. The ratio of general to specific variance for the first time point (t = 1 was given in the main text, while this ratio for the general case (for an arbitrary number of time points is given below. Note that even for the general case, only the ratio for the SGM is subject to the so-called proportionality constraint. General to Specific Variance Ratio for the SGM We begin with the decomposition of variance for an observed variable, var (Y it, in the SGM, which is given by (the general mean and covariance equations for all three models are derived in Appendix B: var (Y it = λ it var (ξ int +(t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin +λ it var (ζ t +var (ε it. (C.1 For any two indicators Y it and Y jt, at the same time point t, the variance ratio is constrained to be identical: λ it var (ξ int + (t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin λ it var (ζ t = λ jt var (ξ int + (t 1 λ jt var (ξ lin + (t 1 λ jt cov (ξ int, ξ lin λ jt var (ζ t = var (ξ int + (t 1 var (ξ lin + (t 1 cov (ξ int, ξ lin. var (ζ t (C. General to Specific Variance Ratio for the GSGM In the GSGM, the variance decomposition is given by: var (Y it = λ it var (ξ int +(t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin +γ it var (ζ t +var (ε it, (C.3 and the ratio of general to specific variance is given by λ it var (ξ int + (t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin γ it var (ζ t λ jt var (ξ int + (t 1 λ jt var (ξ lin + (t 1 λ jt cov (ξ int, ξ lin γ jt var (ζ t for indicator i, for indicator j.(c.4 The ratio of these variances is not constrained.

15 MULTIPLE-INDICATOR GROWTH MODELS 15 General to Specific Variance Ratio for the ISGM We now consider the general variance equation for the ISGM: var (Y it = var (ξ inti + (t 1 var (ξ lini + (t 1 cov (ξ inti, ξ lini + γ it var (ζ t + var (ε it, (C.5 which leads to the following ratio of general to specific variance: var (ξ inti + (t 1 var (ξ lini + (t 1 cov (ξ inti, ξ lini γit var (ζ. (C.6 t Obviously, the ISGM also does not impose a proportionality constraint because in general, the state residual loadings γ it in the denominator can be estimated freely for all indicators 1. 1 Note that there are specific identification conditions in certain designs that may preclude the state residual (γ loadings from being freely estimated in the GSGM and ISGM. For example, designs with only indicators per time point would not allow the state residual (γ loadings to be freely estimated unless (1 a significant autoregresive process is present and estimated for the state residual factors or ( each state residual factor is significantly related to at least one external variable.

16 MULTIPLE-INDICATOR GROWTH MODELS 16 Appendix D Degrees of Freedom in the SGM, GSGM, and ISGM The degrees of freedom (df of a model is an important consideration when choosing or comparing models; df is important when choosing a model because insufficient df may result in model non-identification; df is also important when comparing models since it provides the context within which fit statistics are interpreted (for more detail see e.g., Walker, 194. For simplicity, we compare the df for the restricted case in which latent state residuals are assumed to be uncorrelated, and all conditions of MI are met for all models (i.e., that all loadings and intercepts are time-invariant. The df for this situation represented as an algebraic function of the input parameters (m and n are shown in general in Table D3, and for specific numeric values (of m and n in Table D1. The derivation of the formulas in Table D3 is based on the known and estimated parameters given in Table D. The technical details for this derivation are given below. Examining these tables reveals that for a given set of variables the SGM has the most df, followed by the GSGM, and finally the ISGM. Computing df The degrees of freedom (df for each model are computed as: df = #known #estimated, (D.1 where #known is the number of known parameters (means, variances, and non-redundant covariances of the observed variables Y it and #estimated is the number of free parameters estimated in the model. Known Parameters The total number of known parameters is equal to the number of uniquely identified non-redundant elements in the measurement covariance matrix (including both variances Table D1 Degrees of Freedom for SGM, GSGM, and ISGM Models SGM GSGM ISGM m n n n a 5 a 43 a 65 a 4 a 18 a 36 a 58 a Note. SGM = second-order growth model; GSGM = Generalized second-order growth model; ISGM = indicator-specific growth model; m = number of indicators; n = number of time points. Here, we assume that measurement invariance across time holds for all intercepts and factor loadings. a The two indicator GSGM and ISGM are subject to additional constraints needed for model identification.

17 MULTIPLE-INDICATOR GROWTH MODELS 17 Table D Number of Estimated Model Parameters in Different Multi-indicator Linear Growth Models Model Estimated Parameters SGM GSGM ISGM Intercept and Growth Factor Means E (ξ inti, E (ξ lini m Intercept and Growth Factor Variances var (ξ inti, var (ξ lini m Error Variances var (ε it mn mn mn State Residual Variances var (ζ t or var (δ t n n n Intercept and Growth Factor Covariances cov (ξ inti, ξ intj, cov (ξ inti, ξ linj, cov (ξ lini, ξ linj Intercepts 1 1 m (m 1 α it n (m 1 if α it α is (m 1 if α it = α is Factor Loadings λ it n (m 1 if λ it λ is (m 1 if λ it = λ is Autoregressive Effects β t State Residual Factor Loadings γ it if cov (ζ t, ζ s = 1 if ζ t = β t ζ (t 1 + δ t and β t = β s (n 1 if ζ t = β t ζ (t 1 + δ it and β t β s n (m 1 if γ it γ is (m 1 if γ it = γ is Note. SGM = second-order growth model; GSGM = Generalized second-order growth model; ISGM = indicator-specific growth model; m = number of indicators; n = number of time points.

18 MULTIPLE-INDICATOR GROWTH MODELS 18 Table D3 Algebraic Degrees of Freedom for SGM, GSGM, and ISGM Models Model #known #estimated df SGM GSGM ISGM (mn + 3mn (mn + 3mn (mn + 3mn m + mn + n + 3 mn + 3m + n + m + 4m + mn + n 1 m n + mn 4m n 6 m n + mn 6m n 4 m n 4m + mn 8m n + Note. #known = number of known means, variances, and covariances; #estimated = number of estimated parameters; SGM = second-order growth model; GSGM = Generalized second-order growth model; ISGM = indicator-specific growth model; m = number of indicators; n = number of time points. Here, we assume that all latent state residual factors are uncorrelated and that measurement invariance across time holds for all intercepts and factor loadings. and covariances plus the total number of expected values that can be computed from the measurements (see, e.g., Bollen, In the present context, the number of known parameters is #known = (mn + 3mn. Estimated Parameters and Degrees of Freedom In contrast to the number of known parameters, the number of parameters estimated varies across models. The number of degrees of freedom is then calculated by subtracting the number of estimated parameters from the number of known parameters, as given in Equation (D.1. For simplicity, we assume linear growth, and that all intercepts (α it, trait loadings (λ it, and state residual loadings (γ it are time-invariant. We also assume that the state residual factors are all uncorrelated (no autoregressive structure. For less restrictive models, or models with other forms of growth, more parameters will be estimated. SGM. #estimated = }{{} E (ξ int E (ξ lin + }{{} var (ξ int var (ξ lin }{{} 1 + (m 1 + (m 1 }{{}}{{} cov(ξ int,ξ lin α i λ i = 4 + (m 1 + m n + n + 1 = m + mn + n + 3 4m + mn + n + 6 = + m }{{ n } + }{{} n + var(ε it var(ζ t

19 MULTIPLE-INDICATOR GROWTH MODELS 19 Thus, the degrees of freedom for the SGM are given by GSGM. df = #known #estimated ( = (mn + 3mn 4m + mn + n + 6 = m n + 3mn 4m mn n 6 = m n + mn 4m n 6 #estimated = }{{} E (ξ int E (ξ lin + }{{} var (ξ int var (ξ lin + }{{} mn + }{{} n + var(ε it var(ζ t }{{} 1 + (m 1 + (m 1 + (m 1 }{{}}{{}}{{} cov(ξ int,ξ lin α i λ i γ i = (m 1 + m n + n + 1 = 4 + 3m 3 + m n + n + 1 = mn + 3m + n + mn + 6m + n + 4 = Thus, the degrees of freedom for the GSGM are given by ISGM. #unknown = }{{} m E (ξ inti E (ξ lini df = #known #estimated ( = (mn + 3mn m n + 6m + n + 4 = m n + 3mn mn 6m n 4 = m n + mn 6m n 4 + m }{{} var (ξ inti var (ξ lini + m n }{{} var(ε it + n }{{} var(ζ t ( + = (m + m m + m + m n + n + m 1 = m + 4m + mn + n 1 m m } {{ } cov(ξ it,ξ js + (m 1 }{{} γ i

20 MULTIPLE-INDICATOR GROWTH MODELS Thus, the degrees of freedom for the ISGM are given by df = #known #estimated = (mn + 3mn ( m + 4m + mn + n 1 ( = (mn + 3mn 4m + 8m + mn + n = m n 4m + mn 8m n +. Note that for models with autoregressive parameters (β t as well as partial or complete non-invariance of measurement parameters (α it, λ it, γ it, models with fewer df would result.

21 MULTIPLE-INDICATOR GROWTH MODELS 1 Appendix E Model Identification with Two Indicators per Time Point To demonstrate the limitations of a growth model with only two indicators per time point, it is sufficient to consider only the case with three time-points. That is, i = {1, }, t = {1,, 3}. The covariance matrix for this case, in terms of Y it for all i, t is shown in Table E1. Table E1 Covariance Matrix for Indicators and 3 Time-points j = 1 s = 1 var (Y 11 i = 1 i = i = 1 i = i = 1 i = t = 1 t = 1 t = t = t = 3 t = 3 j = s = 1 cov (Y 11, Y 1 var (Y 1 j = 1 s = cov (Y 11, Y 1 cov (Y 1, Y 1 var (Y 1 j = s = cov (Y 11, Y cov (Y 1, Y cov (Y 1, Y var (Y j = 1 s = 3 cov (Y 11, Y 13 cov (Y 1, Y 13 cov (Y 1, Y 13 cov (Y, Y 13 var (Y 13 j = s = 3 cov (Y 11, Y 3 cov (Y 1, Y 3 cov (Y 1, Y 3 cov (Y, Y 3 cov (Y 13, Y 3 var (Y 3 To show the general case, we first write out the complete model for the GSGM. This is accomplished by starting with the generic covariance relation for the autoregressive GSGM given in Table B1. The substitution of values for the indices into the general covariance equation is very straightforward. However, as noted in Appendix B, β t ζ t 1 + δ t must be recursively substituted for ζ t (if t > 1 or s > 1. Then, the resulting covariance relation must be expanded and simplified. We demonstrate this for all except the trivial instance where t = s = 1. Note that the order in which the values (t, s appear is not important (cov (ζ t, ζ s = cov (ζ s, ζ t. (1, = (, 1 (1, 3 = (3, 1 (, cov (ζ, ζ 1 = cov (β ζ 1 + δ, ζ 1 = cov (β ζ 1, ζ 1 + cov (δ, ζ 1 = β var (ζ 1 cov (ζ 1, ζ 3 = cov (ζ 1, β 3 [β ζ 1 + δ ] + δ 3 = cov (ζ 1, β 3 β ζ 1 + β 3 δ + δ 3 = β 3 β var (ζ 1 cov (ζ, ζ = cov (β ζ 1 + δ, β ζ 1 + δ = cov (β ζ 1 + δ, β ζ 1 + cov (β ζ 1 + δ t, δ = cov (β ζ 1, β ζ 1 + cov (δ, β ζ 1 + }{{} cov (β ζ 1, δ + cov (δ, δ }{{} = β var (ζ 1 + var (δ

22 MULTIPLE-INDICATOR GROWTH MODELS (, 3 = (3, (3, 3 cov (ζ, ζ 3 = cov (β ζ 1 + δ, β 3 [β ζ 1 + δ ] + δ 3 = cov (β ζ 1, β 3 [β ζ 1 + δ ] + δ 3 + cov (δ, β 3 [β 1 ζ 1 + δ ] + δ 3 = β 3 β cov (ζ 1, ζ 1 + cov (δ, β 3 β ζ 1 + β 3 δ + = β 3 β var (ζ 1 + β 3 var (δ, δ cov (ζ 3, ζ 3 = cov (β 3 [β ζ 1 + δ ] + δ 3, β 3 [β ζ 1 + δ ] + δ 3 = cov (β 3 [β ζ 1 + δ ], β 3 [β ζ 1 + δ ] + cov (β 3 [β ζ 1 + δ ], δ 3 + }{{} cov (δ 3, δ 3 = cov (β 3 β ζ 1 + β 3 δ, β 3 β ζ 1 + β 3 δ + cov (β 3 β ζ 1, β 3 β ζ 1 + cov (β 3 β ζ 1, β 3 δ + }{{} cov (β 3 δ, β 3 δ + var (δ 3 = β 3β var (ζ 1 + β 3 var (δ + var (δ 3 The results from substituting all possible combinations for the specific values of i and t into this covariance equation, and simplifying these according to the model restrictions given in Equations (B.7-(B.9 are shown in Table E3. Similarly, the results of this substitution for the case of uncorrelated state residual variables, simplified according to the restrictions given in Equations (B.-(B.4, are shown in Table E3. We assume MI and uncorrelated state residual variables in order to examine when the model is not identified. The identification problem comes from the state residual (ζ t parameters. This is evident from the fact that the latent state residual variable for the first time point, ζ 1, appears in exactly three equations from Table E3: var (Y 11 = var (ξ int + var (ζ 1 + var (ε 11 cov (Y 11, Y 1 = λ 1 var (ξ int + γ 1 var (ζ 1 var (Y 1 = λ 1 var (ξ int + γ 1 var (ζ 1 + var (ε 1 These are also the only equations in which the parameter γ 1 appears. In these equations, var (Y 11, var (Y 1, and cov (Y 11, Y 1 are known, while var (ξ int and λ 1 can be identified from the other model equations. The parameters γ 1, var (ζ 1, var (ε 11 and var (ε 1 are not known. These parameters do not appear elsewhere and thus cannot be estimated from other information. Without formal proof, we note that the situation remains unchanged regardless of the number of time-points present. This leads to the conclusion that

23 MULTIPLE-INDICATOR GROWTH MODELS 3 Table E Variance and covariance equations for the GSGM-AR with two indicators and three time points. var (Y 11 = var (ξ int + γ 11 var (ζ 1 + var (ε 11 cov (Y 11, Y 1 = λ 1 var (ξ int + γ 1 var (ζ 1 cov (Y 11, Y 1 = cov (ξ int, ξ lin + var (ξ int + β var (ζ 1 cov (Y 11, Y = λ [cov (ξ int, ξ lin + var (ξ int ] + γ β var (ζ 1 cov (Y 11, Y 13 = [ cov (ξ int, ξ lin + var (ξ int ] + β 3 β var (ζ 1 cov (Y 11, Y 3 = λ 3 [ cov (ξ int, ξ lin + var (ξ int ] + γ 3 β 3 β var (ζ 1 var (Y 1 = λ 1 var (ξ int + λ 1 var (ζ 1 + var (ε 1 cov (Y 1, Y 1 = λ 1 [cov (ξ int, ξ lin + var (ξ int ] + γ 1 β var (ζ 1 cov (Y 1, Y = λ 1 λ [cov (ξ int, ξ lin + var (ξ int ] + γ 1 γ β var (ζ 1 cov (Y 1, Y 13 = λ 1 [ cov (ξ int, ξ lin + var (ξ int ] + γ 1 β 3 β var (ζ 1 cov (Y 1, Y 3 = λ 1 λ 3 [ cov (ξ int, ξ lin + var (ξ int ] + γ 1 γ 3 β 3 β var (ζ 1 var (Y 1 = cov (ξ int, ξ lin + var (ξ lin + var (ξ int + β var (ζ 1 + var (δ + var (ε 1 cov (Y 1, Y = λ [ cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] + [ γ β var (ζ 1 + var (δ ] cov (Y 1, Y 13 = [3 cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] + [ β3 β var (ζ 1 + β 3 var (δ ] cov (Y 1, Y 3 = λ 3 [3 cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] + [ γ 3 β3 β var (ζ 1 + β 3 var (δ ] var (Y = λ [ cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] + [ β var (ζ 1 + var (δ ] + var (ε γ cov (Y, Y 13 = λ [3 cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] + [ γ β3 β var (ζ 1 + β 3 var (δ ] cov (Y, Y 3 = λ λ 3 [3 cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] + [ γ γ 3 β3 β var (ζ 1 + β 3 var (δ ] var (Y 13 = 4 cov (ξ int, ξ lin + 4 var (ξ lin + var (ξ int + β 3 β var (ζ 1 + β 3 var (δ + var (δ 3 + var (ε 13 cov (Y 13, Y 3 = λ 3 [4 cov (ξ int, ξ lin + 4 var (ξ lin + var (ξ int ] + [ γ 3 β 3 β var (ζ 1 + β3 var (δ + var (δ 3 ] var (Y 3 = λ 3 [4 cov (ξ int, ξ lin + 4 var (ξ lin + var (ξ int ] + [ β 3 β var (ζ 1 + β3 var (δ + var (δ 3 ] + var (ε 3 γ 3

24 MULTIPLE-INDICATOR GROWTH MODELS 4 Table E3 Variance and covariance equations for the GSGM with two indicators and three time points. var (Y 11 = var (ξ int + var (ζ 1 + var (ε 11 cov (Y 11, Y 1 = λ 1 var (ξ int + γ 1 var (ζ 1 cov (Y 11, Y 1 = cov (ξ int, ξ lin + var (ξ int cov (Y 11, Y = λ [cov (ξ int, ξ lin + var (ξ int ] cov (Y 11, Y 13 = cov (ξ int, ξ lin + var (ξ int cov (Y 11, Y 3 = λ 3 [ cov (ξ int, ξ lin + var (ξ int ] var (Y 1 = λ 1 var (ξ int + γ 1 var (ζ 1 + var (ε 1 cov (Y 1, Y 1 = λ 1 [cov (ξ int, ξ lin + var (ξ int ] cov (Y 1, Y = λ 1 λ [cov (ξ int, ξ lin + var (ξ int ] cov (Y 1, Y 13 = λ 1 [ cov (ξ int, ξ lin + var (ξ int ] cov (Y 1, Y 3 = λ 1 λ 3 [ cov (ξ int, ξ lin + var (ξ int ] var (Y 1 = [ cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] + var (ζ + var (ε 1 cov (Y 1, Y = λ [ cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] + γ var (ζ cov (Y 1, Y 13 = 3 cov (ξ int, ξ lin + var (ξ lin + var (ξ int cov (Y 1, Y 3 = λ 3 [3 cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] var (Y = λ [ cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] + γ var (ζ + var (ε cov (Y, Y 13 = λ [3 cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] cov (Y, Y 3 = λ λ 3 [3 cov (ξ int, ξ lin + var (ξ lin + var (ξ int ] var (Y 13 = 4 cov (ξ int, ξ lin + 4 var (ξ lin + var (ξ int + var (ζ 3 + var (ε 13 cov (Y 13, Y 3 = λ 3 [4 cov (ξ int, ξ lin + 4 var (ξ lin + var (ξ int ] + γ 3 var (ζ 3 var (Y 3 = λ 3 [4 cov (ξ int, ξ lin + 4 var (ξ lin + var (ξ int ] + γ 3 var (ζ 3 + var (ε 3

25 MULTIPLE-INDICATOR GROWTH MODELS 5 the GSGM is not generally identified with only two indicators at each time-point, unless further constraints are imposed or effects are added to the model. If the parameters γ t are known, the remaining unknown parameters in these equations can be estimated. This can be accomplished by fixing the state residual loadings γ t to some value (i.e., γ 1t = γ t = 1. Alternatively, the SGM, which we have shown to be a special case of the GSGM in which γ it = λ it can be used to adequately constrain this model, since the λ it parameters can be estimated from other model information. Another case in which the model information is sufficient for the parameters γ t and var (ζ t to be estimated occurs when there is a significant autoregressive structure. The γ t loadings in this case are related by the β t parameters. This can be seen by comparing the covariance relations in Table E to those in Table E3, paying particular attention to the parameters containing the terms containing ζ 1 and δ t.

26 MULTIPLE-INDICATOR GROWTH MODELS 6 Appendix F Variance Components and Coefficients In each of the three growth models, the observed variance can be partitioned into variance due to the growth process, variance due to state residual variability, and measurement error variance as shown in Figure F1. Observed Variance var (Y it Trait & Growth Variance State Residual Variance Error Variance Latent State Variance var (τ it var (ε it Figure F1. Variance Decomposition in the SGM, GSGM, and ISGM Based on the variance decomposition in each model (see Table F1, several coefficients can be defined (Eid, Courvoisier, & Lischetzke, 1: Coefficient of reliability: Rel (Y it = Coefficient of consistency: Con (τ it = Coefficient of occasion-specificity: Table F1 Variance Decomposition OSpec (τ it = Model Variance Component Equation SGM GSGM ISGM State Variance Observed Variance = var (τ it var (Y it. Trait & Growth Variance. State Variance State Residual Variance. State Variance Trait & Growth Variance λ it var (ξ int + (t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin State Residual Variance Error Variance λ it var (ζ t 1 var (ε it Trait & Growth Variance λ it var (ξ int + (t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin State Residual Variance γ it var (ζ t Error Variance var (ε it Trait & Growth Variance var (ξ inti + (t 1 var (ξ lini + (t 1 cov (ξ inti, ξ lini State Residual Variance γ it var (ζ t Error Variance var (ε it

27 MULTIPLE-INDICATOR GROWTH MODELS 7 Note that whereas the reliability coefficient is defined for the observed variables Y it, the Con and OSpec coefficients are defined for the true score variables τ it, so as to partition the total systematic variance into growth versus state residual variance. Below we show the specific calculations for each model. SGM Total Variance: var (Y it = λ it var (ξ int +(t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin +λ it var (ζ t +var (ε it. Reliability Coefficient: Rel (Y it = λ it var (ξ int + (t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin + λ it var (ζ t. var (Y it Consistency Coefficient: Con (τ t = var (ξ int + (t 1 var (ξ lin + (t 1 cov (ξ int, ξ lin, var (τ t = Occasion-specificity Coefficient: GSGM OSpec (τ t = var (ζ t var (τ t, Total Variance: = var (ξ int + (t 1 var (ξ lin + (t 1 cov (ξ int, ξ lin var (ξ int + (t 1 var (ξ lin + (t 1 cov (ξ int, ξ lin + var (ζ t. var (ζ t var (ξ int + (t 1 var (ξ lin + (t 1 cov (ξ int, ξ lin + var (ζ t. var (Y it = λ it var (ξ int +(t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin +γ it var (ζ t +var (ε it. Reliability Coefficient: Rel (Y it = λ it var (ξ int + (t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin + γit var (ζ t. var (Y it Consistency Coefficient: Con (τ it = λ it var (ξ int + (t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin, var (τ it = λ it var (ξ int + (t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin λ it var (ξ int + (t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin + γ it var (ζ t. Occasion-specificity Coefficient: OSpec (τ t = γ it var (ζ t var (τ it, = γ it var (ζ t λ it var (ξ int + (t 1 λ it var (ξ lin + (t 1 λ it cov (ξ int, ξ lin + γ it var (ζ t.

28 MULTIPLE-INDICATOR GROWTH MODELS 8 ISGM Total Variance: var (Y it = var (ξ inti + (t 1 var (ξ lini + (t 1 cov (ξ inti, ξ lini + γ it var (ζ t + var (ε it. Reliability Coefficient: Rel (Y it = var (ξ int i + (t 1 var (ξ lini + (t 1 cov (ξ inti, ξ lini + γit var (ζ t. var (Y it Consistency Coefficient: Con (τ it = var (ξ int i + (t 1 var (ξ lini + (t 1 cov (ξ inti, ξ lini, var (τ it = Occasion-specificity Coefficient: var (ξ inti + (t 1 var (ξ lini + (t 1 cov (ξ inti, ξ lini var (ξ inti + (t 1 var (ξ lini + (t 1 cov (ξ inti, ξ lini + γ it var (ζ t. OSpec (τ it = γ it var (ζ t var (ξ inti + (t 1 var (ξ lini + (t 1 cov (ξ inti, ξ lini + γ it var (ζ t. Note that for the SGM, the consistency coefficient and its complement (OSpec are dependent only upon τ t. Thus, this coefficient is the same for all indicators measured at the same time point in this model. However, for both the GSGM and ISGM, Con and OSpec are indicator-specific (dependent on τ it because there is not a common latent state factor for each time-point in these two models. Additional coefficients can be defined in the case of an autoregressive structure among the state residuals as shown in Eid et al. (1.

29 MULTIPLE-INDICATOR GROWTH MODELS 9 Appendix G Mplus Model Inputs for SGM, GSGM, and ISGM The order and names used for the following model specifications match the specification in Table 3 (of the main paper. The data file used for this analysis is provided for practice purposes only. It is available for download at psycarticles/supplemental/met18/met18_supp.html. Publication of this data in any form, including results from analyzing this data is expressly forbidden. For requests to use this data for publication or other purposes, please contact Dr. David Cole at david.cole@vanderbilt.edu. Model 1a: SGM filename: 1a_SGM_LinearGrowth_AlphaLamInv.inp TITLE: Second-Order Multiple-Indicator Growth Model. Linear Growth. First-order representation. (after Schmid-Leiman Transformation. DATA: file = Anxiety_Data_3_Indicators_4_Waves.dat; VARIABLE: names = grpid y11 y1 y31 y1 y y3 y13 y3 y33 y14 y4 y34; usevariables = y11 y1 y31 y1 y y3 y13 y3 y33 y14 y4 y34; missing = all(-99; MODEL:! Common xi. This is the trait intercept.! Set the first trait loading (lambda to 1.! Estimate the other two.! Measurement Invariance on lambda not assumed. xi_int by y11@1 (lambda11 y1 (lambda1 y31 (lambda31 y1@1 (lambda1 y (lambda y3 (lambda3 y13@1 (lambda13 y3 (lambda3 y33 (lambda33 y14@1 (lambda14 y4 (lambda4

30 MULTIPLE-INDICATOR GROWTH MODELS 3 y34 (lambda34;! Estimate the mean of the trait factor.! (zero by default in Mplus. [xi_int*];! Variance of the trait factor (xi_int. xi_int (xi_intv;! Growth Model. This is the trait slope.! Wave 1 is missing, because lambda would be multiplied by.! Multiply by 1*lambda for the second wave, by *lambda for! the third, and 3*lambda for the fourth. xi_lin by y1@1 (lambda1 y (lambda y3 (lambda3 y13@ (lambda13d y3 (lambda3d y33 (lambda33d y14@3 (lambda14t y4 (lambda4t y34 (lambda34t;! Estimate the mean of the growth factor.! (zero by default in Mplus. [xi_lin*];! Variance of the growth factor (xi_lin. xi_lin (xi_linv;! These loadings must be the same as the trait! loadings given above...a constraint of the SGM. zeta1 by y11@1 (lambda11 y1 (lambda1 y31 (lambda31; zeta by y1@1 (lambda1 y (lambda y3 (lambda3; zeta3 by y13@1 (lambda13 y3 (lambda3 y33 (lambda33; zeta4 by y14@1 (lambda14 y4 (lambda4 y34 (lambda34;! Set intercepts of state residual factors to zero.! (would otherwise be estimated as the default in Mplus [zeta1-zeta4@];! Delta error variances on each zeta. zeta1 (zeta1var; zeta (zetavar;

31 MULTIPLE-INDICATOR GROWTH MODELS 31 zeta3 (zeta3var; zeta4 (zeta4var;! Constant intercepts (alpha_it.! Set the intercept on the first indicator! (alpha1 to, and estimate the other! two (alpha, alpha3.! This assumes alpha-mi. [y1*] (alpha1; [y31*] (alpha31; [y*] (alpha; [y3*] (alpha3; [y3*] (alpha3; [y33*] (alpha33; [y4*] (alpha4; [y34*] (alpha34;! Estimate the Error Variances. y11 (ev11; y1 (ev1; y31 (ev31; y1 (ev1; y (ev; y3 (ev3; y13 (ev13; y3 (ev3; y33 (ev33; y14 (ev14; y4 (ev4; y34 (ev34;! Non-admissible correlations. xi_int with zeta1-zeta4@; xi_lin with zeta1-zeta4@; zeta1 with zeta-zeta4@; zeta with zeta3-zeta4@; zeta3 with zeta4@; MODEL CONSTRAINT:! lambdaid is double lambda i. lambda3d = lambda3*; lambda33d = lambda33*;! lambdait is triple lambda i. lambda4t = lambda4*3; lambda34t = lambda34*3; OUTPUT: sampstat stdyx; Model 1b: SGM, α, λ-mi filename: 1b_SGM_LinearGrowth_AlphaLamInv.inp TITLE: Second-Order Multiple-Indicator Growth Model. Measurement invariance assumed on alpha and lambda. Linear Growth. First-order representation. (after Schmid-Leiman Transformation. DATA: file = Anxiety_Data_3_Indicators_4_Waves.dat; VARIABLE: names = grpid y11 y1 y31 y1 y y3 y13 y3 y33

32 MULTIPLE-INDICATOR GROWTH MODELS 3 y14 y4 y34; usevariables = y11 y1 y31 y1 y y3 y13 y3 y33 y14 y4 y34; missing = all(-99; MODEL:! Common xi. This is the trait intercept.! Set the first trait loading (lambda to 1.! Estimate the other two.! This assumes Measurement Invariance on lambda.! Do this if the lambdas have to be the same! for a given test across testing occasions. xi_int by y11@1 y1 y31 (lambda1-lambda3 y1@1 y y3 (lambda1-lambda3 y13@1 y3 y33 (lambda1-lambda3 y14@1 y4 y34 (lambda1-lambda3;! Estimate the mean of the trait factor.! (zero by default in Mplus. [xi_int*];! Variance of the trait factor (xi_int. xi_int (xi_intv;! Growth Model. This is the trait slope.! Wave 1 is missing, because lambda would be multiplied by.! Multiply by 1*lambda for the second wave, by *lambda for! the third, and 3*lambda for the fourth. xi_lin by y1@1 (lambda1 y (lambda y3 (lambda3 y13@ (lambda1d y3 (lambdad y33 (lambda3d y14@3 (lambda1t y4 (lambdat y34 (lambda3t;! Estimate the mean of the growth factor.! (zero by default in Mplus. [xi_lin*];! Variance of the growth factor (xi_lin. xi_lin (xi_linv;! These loadings must be the same as the trait! loadings given above...a constraint of the SGM. zeta1 by y11@1 y1 y31 (lambda1-lambda3; zeta by y1@1 y y3 (lambda1-lambda3; zeta3 by y13@1 y3 y33 (lambda1-lambda3;

33 MULTIPLE-INDICATOR GROWTH MODELS 33 zeta4 by y4 y34 (lambda1-lambda3;! Set intercepts of state residual factors to zero.! (would otherwise be estimated as the default in Mplus Delta error variances on each zeta. zeta1 (zeta1var; zeta (zetavar; zeta3 (zeta3var; zeta4 (zeta4var;! Constant intercepts (alpha_it.! Set the intercept on the first indicator! (alpha1 to, and estimate the other! two (alpha, alpha3.! This assumes alpha-mi. [y1*] (alpha; [y31*] (alpha3; [y*] (alpha; [y3*] (alpha3; [y3*] (alpha; [y33*] (alpha3; [y4*] (alpha; [y34*] (alpha3;! Estimate the Error Variances. y11 (ev11; y1 (ev1; y31 (ev31; y1 (ev1; y (ev; y3 (ev3; y13 (ev13; y3 (ev3; y33 (ev33; y14 (ev14; y4 (ev4; y34 (ev34;! Non-admissible correlations. xi_int with zeta1-zeta4@; xi_lin with zeta1-zeta4@; zeta1 with zeta-zeta4@; zeta with zeta3-zeta4@; zeta3 with zeta4@; MODEL CONSTRAINT:! lambdaid is double lambda i. lambdad = lambda*; lambda3d = lambda3*;! lambdait is triple lambda i. lambdat = lambda*3; lambda3t = lambda3*3; OUTPUT: sampstat stdyx; Model a: GSGM filename: a_gsgm_lineargrowth.inp TITLE: Generalized Second-Order Multiple-Indicator Growth Model. Linear Growth. First-order model representation.

34 MULTIPLE-INDICATOR GROWTH MODELS 34 (after Schmid-Leiman Transformation. DATA: file = Anxiety_Data_3_Indicators_4_Waves.dat; VARIABLE: names = grpid y11 y1 y31 y1 y y3 y13 y3 y33 y14 y4 y34; usevariables = y11 y1 y31 y1 y y3 y13 y3 y33 y14 y4 y34; missing = all(-99; MODEL:! Common xi. This is the trait intercept.! Set the first trait loading (lambda to 1.! Estimate the other two.! Measurement Invariance on lambda not assumed. xi_int by y11@1 (lambda11 y1 (lambda1 y31 (lambda31 y1@1 (lambda1 y (lambda y3 (lambda3 y13@1 (lambda13 y3 (lambda3 y33 (lambda33 y14@1 (lambda14 y4 (lambda4 y34 (lambda34;! Estimate the mean of the trait factor.! (zero by default in Mplus. [xi_int*];! Variance of the trait factor (xi_int. xi_int (xi_intv;! Growth Model. This is the trait slope.! Wave 1 is missing, because lambda would be multiplied by.! Multiply by 1*lambda for the second wave, by *lambda for! the third, and 3*lambda for the fourth. xi_lin by y1@1 (lambda1 y (lambda y3 (lambda3 y13@ (lambda13d y3 (lambda3d y33 (lambda33d

35 MULTIPLE-INDICATOR GROWTH MODELS 35 (lambda14t y4 (lambda4t y34 (lambda34t;! Estimate the mean of the growth factor.! (zero by default in Mplus. [xi_lin*];! Variance of the growth factor (xi_lin. xi_lin (xi_linv;! These loadings are NOT the same as the trait! loadings given above. This is the difference! between the GSGM and the SGM.! Do not assume measurement invariance of the state residual! factor loadings (gamma_it=gamma_is. zeta1 by (gamma11 y1 (gamma1 y31 (gamma31; zeta by (gamma1 y (gamma y3 (gamma3; zeta3 by y13@1 (gamma13 y3 (gamma3 y33 (gamma33; zeta4 by y14@1 (gamma14 y4 (gamma4 y34 (gamma34;! Set intercepts of state residual factors to zero.! (would otherwise be estimated as the default in Mplus [zeta1-zeta4@];! Delta error variances on each zeta. zeta1 (zeta1var; zeta (zetavar; zeta3 (zeta3var; zeta4 (zeta4var;! Constant intercepts (alpha_it.! Set the intercept on the first indicator! (alpha1 to, and estimate the other! two (alpha, alpha3.! This assumes alpha-mi. [y11@]; [y1*] (alpha1; [y31*] (alpha31; [y1@]; [y*] (alpha; [y3*] (alpha3; [y13@]; [y3*] (alpha3; [y33*] (alpha33; [y14@]; [y4*] (alpha4; [y34*] (alpha34;! Estimate the Error Variances. y11 (ev11; y1 (ev1; y31 (ev31;

36 MULTIPLE-INDICATOR GROWTH MODELS 36 y1 (ev1; y (ev; y3 (ev3; y13 (ev13; y3 (ev3; y33 (ev33; y14 (ev14; y4 (ev4; y34 (ev34;! Non-admissible correlations. xi_int with zeta1-zeta4@; xi_lin with zeta1-zeta4@; zeta1 with zeta-zeta4@; zeta with zeta3-zeta4@; zeta3 with zeta4@; MODEL CONSTRAINT:! lambdaid is double lambda i. lambda3d = lambda3*; lambda33d = lambda33*;! lambdait is triple lambda i. lambda4t = lambda4*3; lambda34t = lambda34*3; OUTPUT: sampstat stdyx; Model b: GSGM, α, λ-mi filename: b_gsgm_lineargrowth_alphalaminv.inp TITLE: Generalized Second-Order Multiple-Indicator Growth Model. Measurement invariance assumed on alpha and lambda. Linear Growth. First-order model representation. (after Schmid-Leiman Transformation. DATA: file = Anxiety_Data_3_Indicators_4_Waves.dat; VARIABLE: names = grpid y11 y1 y31 y1 y y3 y13 y3 y33 y14 y4 y34; usevariables = y11 y1 y31 y1 y y3 y13 y3 y33 y14 y4 y34; missing = all(-99; MODEL:! Common xi. This is the trait intercept.! Set the first trait loading (lambda to 1.! Estimate the other two.! This assumes Measurement Invariance on lambda.! Do this if the lambdas have to be the same! for a given test across testing occasions.