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Technical Methods Report: Do Typical RCTs of Education Interventions Have Sufficient Statistical Power for Linking Impacts on Teacher Practice and Student Achievement Outcomes?

NCEE 2009-4065
October 2009

Appendix A: Proof of Equation (18)

The asymptotic variance of ŷ1,OLSb ME can be approximated as follows:

asymptotic variance

where all terms were defined in the main text. Thus, the associated non-centrality parameter is:

non-centrality parameter

where the last equality holds using the definition of λ. Note that the partial squared correlation, Ry,M|T2 can be expressed as follows:

partial squared correlation

where Ry,T2 is the squared population correlation between yijk and Ti. Thus, solving (A.3) for (1 -RM,T2) and inserting this expression into (A.2) yields:

solution of A.3 and a.2

Finally, (18) can be obtained from (A.4) using the two relations: (1) σ2y2(1 -Ry,MObs,T2), where Ry,MObs,T2 is the total regression R2 value; and (2) (1 -Ry,MObs,T2) =(1 -Ry,T2)(1 -λRy,M|T2).