When outcome change over time occurs in a nonlinear fashion, nonlinear growth models should be considered. The LGM allows for the flexibility of factor loadings being associated with the slope growth factor to fit various nonlinear growth trajectories. The commonly used approaches to specify nonlinear growth trajectory include, but arenot limited to, polynomial function, orthogonal polynomial function, piecewise polynomials, and free time scores.

A polynomial function is often used to specify nonlinear outcome change. A polynomial time function can be constructed by introducing quadratic, cubic, or higher order functions of time scores. The order of the polynomial function to choose depends on patterns of change in the outcome over time. Theoretically, the orders of polynomial function can go as high as (T − 1) where T is the total number of observation time points. However, the higher the polynomial order, the more difficult it is for the model to fit and the more difficult it is to interpret the results. Here our examples were confined to a quadratic function (i.e., the second order of the polynomial time function) even though higher polynomial terms might be justified in some instances. A quadratic growth means one departure (either curve up or curve down) from linearity. With the regular quadratic change assumption, an additional latent slope growth factor with appropriate time scores can be added to represent the quadratic term (Muthén and Muthén, 1998–2010). ...

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