1. Developmental Models: Latent Growth Models Brad Verhulst & Lindon Eaves

2. Two Broad Categories of Developmental Models Autoregressive Models: – The things that happened yesterday affect what happens today, which affect what happens tomorrow – Simplex Models Growth Models – Latent parameters are estimated for the level (stable) and the change over time (dynamic) components of the traits

3. The Univariate Simplex Model (in singletons) Y1Y1 ε 1 ε1ε1 Y2Y2 ε 1 ε2ε2 Y3Y3 ε 1 ε3ε3 Y4Y4 ε 1 ε4ε4 With a Univariate Simplex Model, the Y t causes Y t+1 and is caused by Y t-1 Note that the disturbance terms are uncorrelated This is also called an AR1 model as the “lag” is only 1 time point

4. Latent Growth Model C1C1 L1L1 Q1Q1 X1X1 X2X2 X4X4 X3X3 Δ 1 δ1δ1 Δ 1 δ2δ2 Δ 1 δ3δ3 Δ 1 δ4δ4 1 0 1 4 9 1 1 1 1 1 2 0 3 ψ 11 ψ 33 ψ 22 ψ 12 ψ 13 ψ 23 μCμC μLμL μQμQ Note that the means for the latent variables are being estimated within the model

5. Identification of Mean Structures SEMs with Mean Structures must be identified both at the level of the Mean and at the level of the Covariance. – You can only estimate each mean once If your model is unidentified at either the mean or the covariance level, your model is unidentified – An overidentified covariance structure will not help identify the mean structure and vice versa.

6. Mean Structures in Factor Models ξ1ξ1 Y1Y1 Y2Y2 ψ1ψ1 Y2Y2 μ1μ1 μ2μ2 μ3μ3 ψ2ψ2 ψ3ψ3 1 VFVF λ1λ1 λ2λ2 λ3λ3 μFμF 1 You must choose one of the other, as both ΛE(ξ) and Μ are not simultaneously identified

7. Latent Growth Models (LGM) Latent Growth Models are (probably) the most common SEM with mean structures in a single sample. Data requirements for LGM: 1.Dependent Variables measured over time 2.Scores have the same units and measure the same thing across time Measurement Invariance can be assumed 3.Data are time structured (tested at the same intervals) The intervals do not have to be equal – 6 months, 9 months, 12 months, 18 months

8. C1C1 L1L1 Q1Q1 X 12 X 13 X 15 X 14 A 1 E 1 e1e1 A 1 a2a2 E 1 e2e2 A 1 a3a3 E 1 e3e3 A 1 a4a4 E 1 e4e4 1 0 1 4 9 1 1 1 1 1 2 03 a1a1 Growth Model for Two Twins A 1 E 1 ecec acac A 1 E 1 eqeq aqaq A 1 E 1 elel alal C2C2 L2L2 Q2Q2 X 12 X 13 X 15 X 14 A 1 E 1 e1e1 A 1 a2a2 E 1 e2e2 A 1 a3a3 E 1 e3e3 A 1 a4a4 E 1 e4e4 0 1 4 9 1 1 1 1 1 2 03 a1a1 A 1 E 1 ecec acac A 1 E 1 eqeq aqaq A 1 E 1 elel alal

9. Intuitive Understanding of the LGM Each time point is represented as an indicator of all of the latent growth parameters – The constant is analogous to the constant in a linear regression model. – All of the (unstandardized) loadings are fixed to 1. The linear effect is analogous to the regression of the observations on time (with loadings of 0, 1, …, t) – If latent slope loadings are set to -2,1,0,1,2,…,t, then the intercept will be at the third measurement occasion The quadratic increase is analogous to a non-linear effect of time on the observed variables and is interpreted in a very similar way to the linear effect. Cubic, Quartic and higher order time effects – Be cautious in your interpretation as they may only be relevant in your sample. – Interpretations of high order non-linear effects are difficult.

10. Interpretation of the Latent Growth Factors Residuals: The variance in the phenotype that is not explained by the latent growth structure Factor Loadings: The same as you would interpret loadings in any factor model (but they are typically not interpreted) Factor Means: The average effect of the intercept/linear/quadratic in the population (more on this next) Factor Covariance: Random Effects of the Latent Growth Parameters (more on this too)

11. Means of the Latent Parameters μ C : Mean of the latent Constant – The average level of the latent phenotype when the linear effect is zero μ L : Mean of the latent Linear slope – The average increase (decrease) over time μ Q : Mean of the latent Quadratic slope – The average quadratic effect over time

12. Variances of the Latent Growth Parameters ψ ii : Variance of the latent growth parameters – Dispersion of the values around the latent parameter – Large variances indicate more dispersion Large variance on a Latent slope may indicate that the average parameter increase but some of the latent trajectories may be negative Typically the variance of higher order parameters are smaller than the variances of lower order parameters – ψ 11 > ψ 22 > ψ 33

13. Covariances between the Latent Growth Parameters ψ ij : Covariance of the latent growth parameters – Generally expressed in terms of correlations – Important to keep in mind what the absolute variance in the constituent growth parameters are E.G. if the variance of the linear increase is really small, the correlation may be very large as an artifact of the variance ψ 12 : Covariance between the intercept and the linear increase – ψ 12 > 0: the higher an individual starts, the faster they increase – ψ 12 < 0: the higher an individual starts, the slower they increase

14. Presenting LGC Results The basic formula for the average effect of the growth parameters is very similar to the simple regression equation: These expected values can easily be plotted against time. It is also possible to include either the standard errors of the parameters or the variances of the growth parameters in the graphs. Some people like to include the raw observations also.

15. Example Presentation of the LGC Results

16. Alternative Specifications Instead of fixing the loadings to 0, 1, 2, 3, if we fix the loadings to -3, -1, 1, 3, it will reduced the (non-essential) multicolinearity Importantly, the model fit will not change! So what correlations are the right ones?

17. Caveat If the starting point is zero, then it might be best to pick a value in the middle of the range for the intercept, or generate an orthogonal set of contrasts – Just because you started your study when people were 8, 14, 22, doesn’t mean that 8, 14 or 22 are meaningful starting points. If zero is a meaningful then it might be a good idea to keep that value as 0. – Critical Event – Treatment (with pretests and follow-up tests)

18. LGM on Latent Factors η1η1 Y1Y1 Y2Y2 λ 11 δδ 1 λ 21 Y2Y2 δ λ 31 1 1 δ1δ1 δ2δ2 δ3δ3 ζ 1 ζ1ζ1 η2η2 Y4Y4 X5X5 λ 42 δδ 1 λ 52 X6X6 δ λ 62 1 1 δ4δ4 δ5δ5 δ6δ6 ζ 1 ζ2ζ2 η3η3 X7X7 X8X8 λ 73 δδ 1 λ 83 X9X9 δ λ 94 1 1 δ7δ7 δ8δ8 δ9δ9 ζ 1 ζ3ζ3 η4η4 X 10 X 11 δδ 1 λ 10,5 X 12 δ λ 11,5 1 1 δ 10 δ 11 δ 12 ζ 1 ζ4ζ4 λ 12,5 ξCξC ξCξC ξCξC φ 11 φ 33 φ 22 φ 12 φ 13 φ 23 1 μCμC μLμL μQμQ

19. Autoregressive LGM η1η1 Y1Y1 Y2Y2 λ 11 δδ 1 λ 21 Y2Y2 δ λ 31 1 1 δ1δ1 δ2δ2 δ3δ3 ζ 1 ζ1ζ1 η2η2 Y4Y4 X5X5 λ 42 δδ 1 λ 52 X6X6 δ λ 62 1 1 δ4δ4 δ5δ5 δ6δ6 ζ 1 ζ2ζ2 η3η3 X7X7 X8X8 λ 73 δδ 1 λ 83 X9X9 δ λ 94 1 1 δ7δ7 δ8δ8 δ9δ9 ζ 1 ζ3ζ3 η4η4 X 10 X 11 δδ 1 λ 10,5 X 12 δ λ 11,5 1 1 δ 10 δ 11 δ 12 ζ 1 ζ4ζ4 λ 12,5 β1β1 β2β2 β3β3 ξCξC ξCξC ξCξC φ 11 φ 33 φ 22 φ 12 φ 13 φ 23 1 μCμC μLμL μQμQ