Abstract: Latent growth models (LGMs) are a powerful tool for analyzing longitudinal data, and have attracted the attention of scholars in psychology and other social science disciplines. For a latent variable measured by multiple indicators, we can establish both a univariate LGM (also called first-order LGM) based on composite scores and a latent variable LGM (also called second-order LGM) based on indicators. The two model types are special cases of the first-order and second-order factor models respectively. In either case, we need to scale the factors, that is, to specify their origin and unit. Under the condition of strong measurement invariance across time, the estimation of growth parameters in second-order LGMs depends on the scaling method of factors/latent variables. There are three scaling methods: the scaled-indicator method (also called the marker-variable identification method), the effect-coding method (also called the effect-coding identification method), and the latent-standardization method.
The existing latent-standardization method depends on the reliability of the scaled-indicator or the composite scores at the first time point. In this paper, we propose an operable latent-standardization method with two steps. In the first step, a CFA with strong measurement invariance is conducted by fixing the mean and variance of the latent variable at the first time point to 0 and 1 respectively. In the second step, estimated loadings in the first step are employed to establish the second-order LGM. If the standardization is based on the scaled-indicator method, the loading of the scaled-indicator is fixed to that obtained in the first step, and the intercept of the scaled-indicator is fixed to the sample mean of the scaled-indicator at the first time point. If the standardization is based on the effect-coding method, the sum of loadings is constrained to the sum of loadings obtained in the first step, and the sum of intercepts is constrained to the sum of the sample mean of all indicators at the first time point. We also propose a first-order LGM standardization procedure based on the composite scores. First, we standardize the composite scores at the first time point, and make the same linear transformation of the composite scores at the other time points. Then we establish the first-order LGM, which is comparable with the second-order LGM scaled by the latent-standardization method.
The scaling methods of second-order LGMs and their comparable first-order LGMs are systematically summarized. The comparability is illustrated by modeling the empirical data of a Moral Evasion Questionnaire. For the scaled-indicator method, second-order LGMs and their comparable first-order LGMs are rather different in parameter estimates (especially when the reliability of the scale-indicator is low). For the effect-coding method, second-order LGMs and their comparable first-order LGMs are relatively close in parameter estimates. When the latent variable at the first time point is standardized, the mean of the intercept-factor of the first-order LGM is close to 0 and not statistically significant; so is the mean of the intercept-factor of the second-order LGM through the effect-coding method, but those through two scaled-indicator methods are statistically significant and different from each other.
According to our research results, the effect-coding method is recommended to scale and standardize the second-order LGMs, then comparable first-order LGMs are those based on the composite scores and their standardized models. For either the first-order or second-order LGM, the standardized results obtained by modeling composite total scores and composite mean scores are identical.

Key words: latent growth model, composite score, scaling method, scaled-indicator, effect-coding, latent-standardization