### Power Calculations for Latent Growth Modeling

(the 2nd version)

**Introduction**

A method first proposed by Satorra and Saris (1985) is utilized to compute the
power estimates and applied, here, to the growth modeling framework developed
by Muthen and Curran (1997).

The key technique in determining
the power associated with the likelihood ratio test is based on the fact that
(1) the distribution of the test statistic under the alternative hypothesis
approaches a noncentral chi-square distribution;
(2) the noncentrality parameter
can be approximated by using existing
statistical software R.

**A Latent Growth Model**

This is a two-level latent growth model which consists of a measurement model
and a structural model with two latent variables denoted by _{1}, representing *initial status*, and _{2}, representing *rate of change*.

__A Measurement Model:__
Modeling components of growth restricted to linear, which are the same for an individual(*i*) in both intervention and control groups with repeated measures (*j*=1,...p).

.

where the unique variances (variances of ), are assumed to be the same across time and correlations between components of are zero.

__A Structural Model:__
Modeling intervention and control differences on the latent growth variable's
distributions.

.

Under normality, it is equivalent to
.

where _{2.1} is the residual variance after
regressing _{2} on _{1}.
Click here for the joint distribution of _{1} and
_{2}.

**Four Tests**

**Case 1:** Mean change in growth rates differs by the intervention group.

**Case 2:** Variance in growth rates differs by the intervention group.

**Case 3:** Mean and variance in growth rates differ by the intervention group.

**Case 4:** Mean, variance in growth rates and interaction with baseline level differ by the intervention group.

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*Last updated: 04/18/98*