Extensions of simple linear Growth Curve models
This chapter introduced varies of growth curve models by extending the simple linear growth curve models. It extends the simple model in three ways:
1. more complex level-1 error structures
Other level-1 error structures: a) error may depend on other variables. For example, error is a function of age. b) Error variances at each time point are allowed to be different. c) autocorrelations may exist among errors for each person
2. piecewise linear growth models
When there may be nonlinear growth suggested by data, one option is to break up the curvilinear growth trajectories into separate linear components. (This option allows us to compare the growth curves in different periods.)
3. time-varying covariates / quadratic curve
Other covariates except for time.
Quadratic relationship between time and outcomes.
Comparing hierarchical, multivariate repeated-measures, and structural equation models
This chapter explains several population terms in multilevel modeling and similarity/ differences/ relations among them.
MRM model : requires specification of main effects and interactions
requires balanced “time-structured” data (i.e., the number and spacing of time points to be invariant across persons)
Hard to extend to third level.
Hierarchical model : does not require specification of main effects and interactions
does not require balanced “time-structured” data
Easy to build the higher levels
SEM model : can test a wide range of covariance structures (autocorrelations)
Requires balanced “time-structured” data
Multi-level models for binary data
This chapter introduces how to model the binary data using hierarchical generalized linear models. General idea is simple: to use link functions.
Think the linear model as a special case of the HGLM. (with the link function f(x)=x)
For binary data, the link function is logit function (f(x)=log(phi/(1-phi))), where phi is the probability of a correct answer.
Phi can be modeled using Rasch or 2PL or other IRT models.
For a Rasch model, the level-1 model is
Phi_it=1/(1+exp{-beta_it})
And the level-2 model is
Beta_i=a0+a1x_i+u_it