- Chapter 1: Introduction
- Chapter 2: Definition of a Mediator
- Chapter 3: Theoretical Framework
- Chapter 4: Statistical Power Formulas
- Chapter 5: Empirical Analysis
- Identifying Plausible R
^{2}Values - Additional Assumptions for the Statistical Power Calculations
- Empirical Results

- Identifying Plausible R
- Chapter 6: Summary and Conclusions
- Appendix A: Proof of Equation (18)
- References
- List of Tables
- List of Figures
- PDF & Related Info

The statistical power calculations were conducted using the following "real-world"
assumptions: (1) a two-tailed test, (2) a 5 percent significance level, (3) a balanced
allocation of schools to the treatment and control groups (*p* = 0.5), (4)
an average of 3 classrooms per school (*c* = 3 ), (5) an average of 23 students
per classroom, (6) data on student test score gains are available for 80 percent
of students in the sample (so that *m* = 18.2 ), and (7) data on mediating
outcomes are available for all teachers.

The statistical power calculations also required real-world assumptions on values for several additional parameters that enter the non-centrality parameter formulas, as discussed next.

*Reliability-Related Parameters (λ _{rel},
λ, and λ_{B})*. The reliability of a teacher practice
mediator, λ

The 0.2 and 0.5 values are in the range of plausible values for λ_{rel}
reported in Raudenbush et al. (2008) based on
an analysis of Classroom Assessment Scoring System (CLASS) data. Raudenbush et al.
(2008) estimated the measurement error variances
in (13) using the observed variation in instructional climate scores across raters
and time segments. The 0.2 to 0.5 reliability values are lower than those usually
reported for commonly-used classroom observation protocols. This is because the
reliability values found in the literature are typically based on the internal consistency
of item responses, and do not typ

Finally, for simplicity, the same parameter values are used for λ, λ_{rel},
and λ_{B} even though these parameters may differ in practice.

*The ratios ψ and ψ ^{Obs}*.
These parameters represent the extent to which mean mediator values vary across
schools, and enter the design effect formulas. As discussed, these parameters can
be obtained from

The *ICC* estimates for the mediators differ for the two studies. The *ICC*
estimates for the Reading Comprehension study are 0.21 for the interactive teaching
scale, 0.33 for the strategy instruction scale, 0.26 for the effective instruction
behavioral scale, and 0.20 for the classroom management scale. The *ICC*
estimates for the Teacher Induction study are 0.11 for the lesson content scale,
0.01 for the classroom culture scale, and 0.08 for the lesson implementation scale.

Due to this variation, a conservative mediator ICC value of 0.15 was assumed for
the analysis, which implies an estimate of about 0.5 for ψ. This 0.5 value
was also assumed for ψ_{Obs} (although ψ_{Obs}
and ψ may differ in practice).

*R _{MB,T}*

Two similar approaches were used for obtaining plausible values for β_{1,eff}^{2}.
First, a "rule-of-thumb" from the IV literature is that if the *F =t* =β̂_{1}^{2}
/*Vâr*(β̂_{1}) statistic from (2) is 10, then *T _{i}*
can be considered to be a strong instrument for

Based on these analyses, an *R _{MB,T}*

Finally, because *R _{MB,T}*