Growth Trajectories

What the page reads

MODEL RESULTS, and nothing else is needed. A growth factor is printed there with the | symbol and with the time score of every occasion beneath it, so the time scores are read from the output itself rather than from the input, and a time score that was estimated rather than fixed is read with the value it took. Beneath them come the covariance between the factors under WITH, the factor means under Means, the factor variances under Variances and the residual variance of every occasion under Residual Variances. When the factors are regressed on predictors, Mplus prints intercepts and residual variances in place of means and variances, and the page says so and reports the trajectory of a case whose predictors are all zero.

The trajectory

In a linear growth model, let ${\lambda }_t$ be the time score attached to occasion t in the Mplus growth statement. The outcome is

where ${\xi }_I$ is the level at the point whose time score is zero, ${\xi }_S$ is change per unit of the chosen time metric, and ${\epsilon }_t$ is the occasion-specific residual (Geiser, 2021, Section 3.5.2.2). Taking the expectation gives the mean implied at that occasion. Writing the equation with ${\lambda }_t$, rather than with the occasion number, matters when observations are unequally spaced or one or more time scores are freely estimated. A quadratic or cubic model adds further growth factors with the corresponding powers or time-score loadings, and the page follows the loadings actually printed in the file.

The variance at every occasion, and the part the growth factors account for

For the linear model the variance of the outcome decomposes as

and the share of it the growth factors account for is the first three terms over the whole (Geiser, 2021, Section 3.5.2.3). The table gives each term separately, since the two factors and their covariance move the variance in different directions: with a negative covariance between level and change the variance can fall before it rises, and the term that produces that is then visible on its own. The same share is the R-square Mplus prints for that variable in the standardized solution, and when the file carries it the two are reported side by side.

The trajectory written about another occasion

The time score of an occasion enters the equation above only as a multiplier of the change factor, so writing the trajectory about another occasion is a substitution. With the time score c taken as the origin, the equation becomes

so that the factor for the level becomes ${\xi }_1+c{\xi }_S$. Its mean, its variance and its covariance with the change factor are those of that sum, which the ordinary rules for the mean and the variance of a linear combination give as $E\left({\xi }_1\right)+cE\left({\xi }_S\right)$, $\mathrm{Var}\left({\xi }_1\right)+2c\mathrm{Cov}+c^2\mathrm{Var}\left({\xi }_S\right)$ and $\mathrm{Cov}+c\mathrm{Var}\left({\xi }_S\right)$. With a quadratic model the same substitution puts ${\xi }_1+c{\xi }_S+c^2{\xi }_Q$ in the first place and ${\xi }_S+2c{\xi }_Q$ in the second, and the page applies the transformation the powers of the time scores call for.

The quantity this bears on is the correlation between level and change. It is not a property of the process: it is the correlation at the occasion whose time score is zero, and moving that point moves it, sometimes from a clear negative value to a clear positive one over the span of the study. The table is there so that the correlation is read at the occasion the question is about, and reported with that occasion named.

Unsupported inputs

A file whose MODEL RESULTS carries no factor written with |, since there is then no growth model to read. The output of a multiple-group run and of a two-level run, which print one set of factors per group or per level. A run whose messages put it among those whose results are not to be interpreted. The substitution of the origin is left out, with the reason given, when the time scores are not the successive powers of one time variable, which is the case with a piecewise specification. A linear model with a freely estimated time score is not in that class: its time scores are read with the values they took and the substitution applies to them as it does to fixed ones.

Self-test

Append ?selftest=1 to check time-score parsing, implied means and variances, variance decomposition, recentering of time, quadratic trajectories, and unsupported inputs.

• Geiser, C. (2021). Longitudinal structural equation modeling with Mplus: A latent state–trait perspective. Guilford Press.
• Grimm, K. J., Ram, N., & Estabrook, R. (2017). Growth modeling: Structural equation and multilevel modeling approaches. Guilford Press.
• Wickrama, K. A. S., Lee, T. K., O'Neal, C. W., & Lorenz, F. O. (2022). Higher-order growth curves and mixture modeling with Mplus: A practical guide (2nd ed.). Routledge. https://doi.org/10.4324/9781003158769
• Muthén, L. K., & Muthén, B. O. (1998–2017). Mplus user's guide (8th ed.). Muthén & Muthén. https://www.statmodel.com/ugexcerpts.shtml