Why the binary latent growth model is not a special case of the ordinal latent growth model: Theoretical arguments and empirical evidence.

In the structural equation modeling framework, binary variable models are generally considered a special case of ordinal variable models, as both involve similar scale assignment processes. However, the scaling processes of the two model types differ, with these differences becoming increasingly pronounced in the context of latent growth models (LGMs). To define scale units, the two types of LGMs-specifically, one with ordinal variables and the other with binary variables-depend on different observed scale references, such as thresholds and standard deviations, which are derived from observed categorical variables. Applying distinct observed scale references to binary and ordinal LGMs results in systematic differences in the scale units of their corresponding latent response variables. Consequently, in binary LGMs, the transformed latent response variables used for model estimation may fail to accurately reflect the corresponding population information, and as a result, their parameter estimates are more likely to be systematically biased than those obtained from ordinal LGMs. This study investigates the impact of these differences on estimating ordinal and binary LGMs and underscores potential estimation concerns in binary LGMs from both theoretical and empirical perspectives.