I'm working on a model of achievement growth, in which I'd like to predict the slope by several time invariant covariates (like e.g., SES).
Now, I'm concerned if I should predict the slope by the intercept or rather should model the correlation between both factors (as seems to be standard to me).
Moreover, according to the effects of my covariates I'm a bit confused. If I correlate slope and intercept, nearly none of the paths from my covariates to the slope factor (slope on cov) reaches significance. But some of the same effects become significant when I model slope on intercept instead of slope with intercept.
I would have thought that this shouldn't make a difference.
Now, my questions are:
1. Which of the two alternatives (correlation vs. directed path) is correct and is this decision based on theoretical or statistical assumptions.
2. Why do different effects of the covariates result depending on the two alternatives?
How you specify your model should be guided by your research hypotheses.
In line with regular regression, when you use i as a covariate, the regression coefficients of s on x are partial coefficients conditioned on i. You should think about what this means in terms of the content of your variables.