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Appendix K - Details for Question B2

Power estimates

To control for costs, the study team powered the study to detect student outcomes only. The 

student outcome for confirmatory analysis in this study is fractions achievement measured by the 

Test of Understanding of Fractions (TUF). Teacher-level outcomes such as impacts on teacher 

knowledge are considered to be useful as exploratory analyses. Our preliminary power estimates1

for the student outcomes indicate that approximately 80 schools would be needed to achieve 

adequate power for detecting a true effect with an effect size of g = .12 for student outcomes. 

However, 84 schools will be recruited in the two states of Georgia and South Carolina for 

adequate statistical power in testing the effects of the math professional development 

intervention on student outcomes to handle anticipated attrition at the school level.

2 An even 

number of schools will be selected within each district to ensure operation of the matching 

strategy described below.

Power estimates for analyses of confirmatory student outcome (RQ1). 

This study is conceived as comprising three levels, with students (Level-1) nested within 

classrooms (Level-2), within schools (Level-3). School-pairs will be treated as fixed effects in 

the model. The study team will create school-pairs within each district using an optimal bipartite 

propensity score model, constructed from a vector of school-level variables including prior year 

third grade achievement on the state math assessment, % students eligible for free-reduced lunch, 

% of African American students, % of English language learners, and total school enrollment. 

One member of each pair will be randomly assigned to the treatment condition. 

Below is a list of assumptions and values used in conducting power analysis for the student 

outcome: 

  = .05 with a two-tailed test

 Power = .80

 2 = .05 (an estimate of teacher/classroom-level intra-class correlation)3

 3 = .10 (an estimate of school-level intra-class correlation)4

 

1

Power analyses were conducted using PowerUp! (Dong & Maynard, 2013) and cross-validated using 

Optimal Design software version 3.01 (Raudenbush et al., 2011). Equations were provided by Hedges and 

Rhoads (2009).

2

In a recent study of professional development for vocabulary instruction in 1st grade carried out by the 

PI of this proposal, the teacher attrition rate was low (5.2%) and was primarily due to the attrition of one 

school of 62.

3 Based on author estimates from a recent RCT implementation professional development in 62 schools 

(Gersten et al., manuscript in preparation).
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 R1

2

=.65 (proportion of variance at the student-level explained by student-level 

covariate)5

 R2

2

 =.70 (proportion of variance at the classroom-level explained by both school-level 

and student-level covariates)6

 R3

2

 =.75 (proportion of variance at the school-level explained by both school-level and 

student-level covariates)7

 J = 3 fourth grade teachers per school8

 n = 57 students per school on average (assuming 3 teachers/classrooms * 19 students 

per class after approximately 90% student participation rate [assuming 10% parent opt 

out] and subsequent 15% student attrition)9

Table G.1a below demonstrates the possible number of schools needed based on alternate 

assumptions about minimum detectable effect size (MDES). 

Table G.1a – Power analysis for RQ1: Student impact analysis

Student Math Achievement

MDES

.11 .12 .14

Estimated Number of Schools 100 80 60

In the prior literature review, in the studies that had at least substantively important findings and 

met WWC criteria, the smallest effect size observed was g =.20 on student outcomes on the state 

assessment for grades 5-8 (Sample McMeeking et al., 2012). This study will be powered to 

 

4 Given that Bloom, Richburg-Hayes, and Black (2007) reported a range of .05-.18 for low-income 

schools and a range of .05-.16 for low-achieving schools in third grade reading, .10 would seem to be a 

reasonable assumption for schools targeted for this study.

5 Assuming.65 or greater correlation between the prior year state assessment and post-test administration 

of the TUF. Bloom et al. (2007) reported school-level R

2 ranges of .48 to .83 for third grade math and 

from .47 to .73 for fifth grade math. They also reported student-level R

2

s ranging from .30 to .73 for third

grade reading. Student-level R

2

specifically in math was not reported.

6 Bloom et al. (2007) found school-level R2 ranging of .48 to .83 for third grade math and from .47 to .73 

for fifth grade math in school-level random assignment studies. Wijekumar, Hitchcock, Turner, Lei, and 

Peck (2009) reported .73 for classroom-level R

2

in a teacher random-assignment study using a math 

curriculum supplement.

7 Bloom et al. (2007) found school-level R2 ranging of .48 to .83 for third grade math and from .47 to .73 

for fifth grade math in school-level random assignment studies. Wijekumar et al. (2009) reported .73 for 

classroom-level R

2

in a teacher random-assignment study using a math curriculum supplement.

8 Estimated from the Common Core of Data (CCD) database maintained by the National Center for 

Education Statistics (2013) using 2010/11 school-level enrollment data. Assuming a 20% non-response 

rate to teacher enrollment in the study. In a recent study of professional development for vocabulary 

instruction in 1st grade carried out by the PI of this proposal, the teacher attrition rate was low (5.2%). The 

student attrition rate was 7.2%. Twice that rate is assumed for the current power analysis.

9 On average 97 students were enrolled in grade 4 in each public elementary school across Georgia and 

South Carolina. Assuming an average default class-size of 25. 
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detect effect sizes at least as small as the lowest observed effect size from those previous studies. 

To be conservative, the study will include a minimum of 80 schools10 (combined treatment and 

control) to detect an effect size as small as g = 0.12. While small, an MDES of g = 0.12 

represents approximately one-third of the average annual growth from third to fourth grade 

(Bloom, Hill, Black, & Lipsey, 2008).

Given the limited research base on which to estimate both the ICCs and the R2values, there is 

some risk that these values will vary in practice from our estimates. Table G.1b examines a range 

of values for both of these parameters. For these estimates it is assumed that we will recruit 84

schools blocked into pairs within district, to handle possible attrition. It is assumed that on 

average 3 teachers will participate, and that on average 19 students (of 25) will participate from 

each classroom.

This exercise reveals that even under very conservative estimates of both the ICCs and R2 parameters the 

maximum MDES is .21. This effect size is still within the range of possible impacts from a professional 

development intervention.

 

10 The number of schools needed to detect an effect for this study is 80; however, we will over-recruit up to 

84 schools to account for possible school-level attrition. We expect that 82 schools will remain in the 

study after attrition.

Table G.1b - MDES with a range of R2and ICC assumptions

ICC

(2 ,3)*

R

2

(R1

2

, R2

2

, R3

2

)*

(.05, .10) (.10, .15) (.15, .20)

(.65, .70, .75) .12 .15 .17

(.60, .65, .70) .13 .16 .18

(.55, .60, .65) .14 .17 .20

(.50, .55, .60) .15 .18 .21

*Note: Values in the tables represent MDES given a sample of N=80 schools with a 50%/50% allocation.

R1

2

 = proportion of variance at the student-level explained by student-level covariate

R2

2

 = proportion of variance at the classroom-level explained by both school-level and student-level covariates

R3

2

 = proportion of variance at the school-level explained by both school-level and student-level covariates

2 = estimate of teacher/classroom-level intra-class correlation

3 = estimate of school-level intra-class correlation
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Power estimates for analyses of exploratory teacher outcomes (RQ2)

A power analysis was conducted to determine what MDES would be achieved at the teacher 

level, if the study were powered for student outcomes. See Table G.2 below. With a minimum 

sample of 80 schools11 (approximately 240 teachers), the MDES is estimated to be .33. Recent 

large-scale studies of math professional development at the seventh grade by Garet et al. (2010, 

2011) did not result in any significant impacts on total teacher knowledge scores for the primary 

analytic sample. However in subsequent exploratory analyses using a pooled sample of 

teachers12, they found significant impacts on the specialized knowledge of mathematics (Hill, 

Rowan, & Ball, 2005) needed for teaching (effect size 0.28; p<0.02). If the proposed study were 

to be powered based on these exploratory impacts from the Garet et al. (2011) study, to attain an 

MDES of .28, 110 schools would need to be recruited with 330 teachers, 30 schools more than 

the minimum sample of schools required for student outcomes. Therefore, no attempt was made 

to power this study for teacher outcomes, as these are not of primary interest to Alliance 

stakeholders and too expensive to target in the context of this RCT.

Table G.2 – Power analysis for RQ2: Teacher outcomes

Teacher Outcomes

MDES

.28a

.33

Estimated Number of Teachers

330 240

Estimated Number of Schools

(based on an assumption of 3 teachers per 

school)

110 80

a Size of significant impact reported in Garet et al. (2010, 2011) for a measure of teacher knowledge.

The assumptions that were used to estimate the above MDES for teacher outcomes (RQ2) are 

listed below: 

  =.05 

 Power = .80

 

11 The number of schools needed to detect an effect for this study is 80; however, we will over-recruit up 

to 84 schools to account for possible school-level attrition. We expect that 82 schools will remain in the 

study after attrition.

12 The pooled sample of teachers combined three groups of teachers: teachers who were in the first year 

impact analysis only, teachers who were in the second year impact analysis only, and teachers who were 

in both years of the impact analysis. Teachers who participated in both the first and second year were 

represented in the data-file twice, once for each year.
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  = .10 (an estimate of school-level intra-class correlation obtained from our earlier 

study of a professional development program)

 R

2

= .35 (based on an estimate pre-/post-test correlations of the Math Knowledge Test at 

both the teacher and school levels)

 n = 3 teachers per school 

Thus to recap, the study is being powered only to detect impacts at the student level. To detect a 

true effect with an effect size of .12 for student outcomes, a minimum of 80 schools will be 

needed. However, 84 schools will be recruited for adequate statistical power to handle possible 

attrition at the school level.13

Analytic Methods

Blocking and Random Assignment

After MOUs have been signed by district and school leaders and consent forms have been signed 

by participating teachers, schools will be randomly assigned to treatment and control conditions. 

An even number of schools will be selected within each district to ensure 1:1 matching. The 

study team will create school pairs within each district based on an optimal bipartite propensity

score model, constructed from a vector of school-level variables including prior year third grade 

achievement on the state math assessment, % students eligible for free-reduced lunch, % of 

African American students, % of English-language learners, and total school enrollment. The 

optmatch package (Hansen et. al., 2013) available for R will be used to conduct optimal 1:1 

matching within each district using the method pairmatch. One school from each pair will be 

randomly assigned to the DMI condition. To ensure the objectivity of the study, research staff 

not otherwise engaged with the study will perform the randomization. 

Specifications of the primary confirmatory impact analysis (RQ1)

Given that the data that will be used to address the research questions proposed for this study are 

of a nested nature (i.e., students nested within teachers and teachers nested within schools), 

hierarchical linear modeling (HLM) will be used to perform the main impact analyses. The HLM 

approach explicitly takes into account the nested data structure, and thus produces properlycomputed standard errors for the impact estimates (Raudenbush & Bryk, 2002).

For the confirmatory analysis (RQ1), a three-level model will be used to assess the impact of 

teacher professional development on student outcomes, with students at level 1, schools at level 

2, school-pairs at level 3. Given that no attempt is being made to generalize beyond the limited 

 

13 The number of schools needed to detect an effect for this study is 80; however, we will over-recruit up to 

84 schools to account for possible school-level attrition. We expect that 82 schools will remain in the 

study after attrition.
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number of districts included in this multi-site (i.e., school) randomized trial, the blocking 

variable (school-pairs) will be treated as fixed effects rather than random effects in the impact 

analyses. The intercept of the teacher-level model, which represents the average teacher outcome 

for a given school, is modeled as a random effect at the school level and as a function of school 

pair-specific intervention effects (captured by a set of professional development-by-pair 

interactions) and pair fixed effects (captured by a set of school-pair dummy indicators).

A measure of fourth grade student achievement in fractions (TUF), under development for this 

study by the Center for Improving Learning of Fractions (2014), will serve as the outcome for 

the confirmatory analysis. Prior-year state assessment scores and demographics will be included 

at the student level and a vector of teacher background characteristics as covariates at the teacher 

level, respectively, to improve the statistical precision of the impact estimates. 

Missing data

Following the most recent guidance from the WWC Procedures and Standards Handbook, v. 3.0 

(2013), which states that “the WWC prefers analyses conducted on actual, observed data” (p. 

16), only those students with a post-test score will be included in the analytic sample. However 

multiple imputation will be used to estimate missing pretest scores or values, so that the models 

may be estimated with a complete non-missing data matrix.

Multiple stochastic regression imputation will be used for missing pretest or covariate data to 

ensure that data is available for Intent-to-Treat analyses.14 Imputations of missing pretest or 

covariates will be conducted separately for both treatment and control units (i.e. students and 

teachers). No non-response weighting of individual cases will be done. 

The RQ1 model will be specified as follows:

Level 1 – Student level.

Yijk =  0jk +  1jk * (PreState)ijk + eijk

Where:

 Yijk is the outcome for student i for teacher j in school k

 (PreState)ijk is the prior year 3rd grade state assessment scores in mathematics for each 

student

  0jk is the average outcome of students with teacher j in school k

 

14 This will be accomplished using the mi procedures incorporated in STATA version 12.1 (StataCorp LP, 2013).
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  1jk is the association between pretest state assessment scores and student outcomes on 

the TUF 

 eijk is a random error associated with student i with teacher j in school k; eijk ~ N (0, σ2).

Level 2 - Teacher level. 

 0jk = β00k + β01k*(X)jk + r0jk

1jk = β10k

Where:

 00kis the average outcome of teachers in school j; 

 X is a vector of teacher background characteristics, including teaching experience, and 

credentials, grand-mean centered;

 01kis the relationship between a vector of teacher background characteristics and the 

outcome in school k; and

 r0jk is a random error associated with teacher i in school j, rijk ~ N (0, σ2).

At Level 3 - School level.

β00k = d k

N

d

d

*(PAIR )

1

 000



 + d k

N

d

d

*(PD*PAIR )

1

 001



 + u00k 

β01k = 010

β10k = 100

Where:

 PD is the indicator variable for the intervention condition (1 = PD school, 0= control 

school)

 PAIRk, k=1, 2, ..., N, are dummy indicator variables representing the N schools in the 

sample

 PD * PAIRd are N intervention-by-pair interaction terms

 δk, k = 1, 2, ..., N, represents the N pair intercepts

 λk, k = 1, 2, ..., N, represents the effect of PD for each of the N pairs

 100 is the fixed effect representing the average student pretest across all schools 

 u0j is a random error associated with teacher j on teacher average outcome; u0j ~ N (0, 

τ00). 

The three-level model for estimating student impacts will generate a pair-specific intervention 

impact (001d) for each of the N pairs. The overall treatment effect will be computed as the 
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weighted average of pair-specific treatment effects, weighting each pair by the N of the treatment 

school.

In addition to the statistical significance of the intervention’s effects, the magnitude of the effects 

will also be gauged by computing the effect sizes (standardized mean difference, or Hedges’ g) 

associated with the impact estimates. Effects will be interpreted with an effect size equal to or 

greater than 0.25 as a “substantively important” effect (WWC Procedures and Standards 

Handbook, v. 3.0, 2013). 

Specifications of the exploratory impact analyses (RQ2)

For exploratory analyses of DMI’s impact on teacher outcomes, a two-level HLM model will be 

used with teachers at level 1 and schools at level 2. Given that the school pairs do not represent a 

larger population of schools, school-pairs will be treated as fixed effects rather than random 

effects in the impact analyses. The intercept of the teacher-level model, which represents the 

average teacher outcome for a given school, is modeled as a random effect at the school level 

and as a function of pair-specific intervention effects (captured by a set of professional 

development-by-pair interactions) and pair fixed effects (captured by a set of pair dummy 

indicators). The specification of the model is as follows:

Level 1 - Teacher level. 

 Yij= β0j + β1j*(X)ij + rij

Where: 

 Yij is the outcome measure of teacher knowledge for teacher i in school j; 

 β0j is the average outcome of teachers in school j; 

 X is a vector of teacher background characteristics, including teaching experience, and 

credentials, grand-mean centered;

 Β1j is the relationship between a vector of teacher background characteristics and teacher 

knowledge [MKT] and the outcome in school j; and

 rij is a random error associated with teacher i in school j, rij ~ N (0, σ2).

Level 2 - School level.

β0j = d j

d

d

*(PAIR )

10

1

 00



 + d j

d

d

*(PD*PAIR )

10

1

 01



 + u0j 

β1j = 10
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Where:

 PAIRd , d = 1, 2, ..., Nd, are dummy indicators representing the N pairs; 

 00d, d = 1, 2, ..., Nd, represents the average teacher outcome in comparison schools in 

pair d; 

 PD* PAIRd,, d = 1, 2, ... Nd, are a set of PD-by-pair interaction terms (PD = 1 for 

intervention schools, and 0 for comparison schools);

 01d, d = 1, 2, ... Nd, represents the difference in average teacher outcomes between 

intervention school and comparison school (i.e., intervention effect) in pair d;

 10 is the average relationship between teacher background characteristics and the 

outcome across all schools; and

 u0j is a random error associated with school j on school average teacher outcome; u0j ~ N 

(0, τ00).

The above model estimates a set of pair-specific intervention effect (01d). The overall 

intervention effect will be computed as the weighted average of pair-specific treatment effects, 

weighting each pair by the N of the treatment school.15 Again impact results will also be reported 

as a Hedges’ g and discussed in terms of statistical significance and discussed relative to the 

WWC’s benchmark for substantive importance (g ≥ 0.25).

Attrition, non-response, cross-overs, and participation

An intent-to-treat approach will be used for data collection, which means outcome data from all 

teachers and students in the original randomized study sample will be collected to the fullest 

extent possible. 

Following the most recent guidance from the WWC Procedures and Standards Handbook, v. 3.0 

(2013), which states that “the WWC prefers analyses conducted on actual, observed data” (p. 

16), non-responders at post-test will be treated as attrition cases and not included in the analytic 

sample. 

Included in the analytic sample (for RQ1) will be only those students with a complete and valid 

post-test. For the teacher knowledge outcome (RQ2) only teachers with knowledge test scores 

would be considered in the analytic sample (though students of teachers missing knowledge 

scores would be included in the analytic sample for RQ1 if the students have non-missing 

scores).

 

15 The sample size-weighted average effect will be similar to a simple average effect if the sample size or 

the intervention effect is similar across the districts.
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For missing pretest scores and covariate values, however, multiple imputation will be used so 

that a complete data matrix is available for analysis. Multiple imputation will be conducted 

separately for both treatment and control groups.16

Attrition rates will be calculated by overall sample and by treatment group. These findings will 

be compared to the WWC Standards for attrition that are current at the time of the final impact 

analysis. To examine differences in pretest scores and demographic characteristics, t-tests and 

chi-square tests will be used, comparing the sample of teachers and students at the time of 

randomization and the final analytic sample. These comparisons will be conducted by treatment 

group. 

Crossovers

As a general rule, once students have been assigned to a condition (as a function of being 

initially enrolled in a school randomly assigned to that condition) they will remain in that 

condition for analytic purposes. Specifically, students who move during the study from a 

treatment classroom to a control classroom (in another school) will still be treated as if they were 

in the treatment condition for analytic purposes. Students who move during the study from a 

control classroom to a treatment classroom (in another school) will be treated as if they were in 

the control condition for analytic purposes. Students who move to a non-participating classroom 

within a participating school will be considered to be in the same condition as originally assigned 

and assessed. School-level MOUs will be written to include specification of these procedures.

All crossovers will be documented throughout the study and reported to provide context for the 

impact results. Students who move from a study school to a non-study school will not be tested 

and will be treated as attriting. This is because there would not be an MOU in place for testing 

students in schools outside of the study.

It is possible (though unlikely) that a teacher during the year may move to another school, either 

a school in another condition, or a non-participating school. Such teachers will still be assessed 

on teacher knowledge outcomes and treated analytically in the condition they were originally 

assigned. 

Participation analysis

Two types of teacher participation will be examined. The first will be a comparison of teachers 

in study schools who signed consent forms for participation in the study versus those who did 

not. The second will be a comparison of teachers who completed 7 or more of the 8 DMI 

sessions versus those who signed consent forms but completed fewer than 7 sessions. These 

 

16 Multiple imputation will be accomplished through the use of the mi procedures incorporated in STATA 

version 12.1 (StataCorp LP, 2013) using 20 imputations. Imputations of missing pretest or covariates will 

be conducted separately for both treatment and control units (i.e. students and teachers). No weighting of 

individual cases will be done.
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groups will be compared using t-tests and chi-square tests on teacher demographics and teacher 

knowledge outcomes. For the first question, teacher demographics will be derived from school 

records for all teachers in participating schools.

For students, a waiver of active consent was granted by the IRB. However, should a substantial 

number of parents opt-out after receiving information of their school’s participation in the study

demographics will be compared for those students whose parents allowed participation versus

the demographics of all students in study schools. These school-wide demographics will be 

derived from school records.

The RQ1 impact analysis will be conducted as a sensitivity analysis, using only the subsample of 

teachers (and their students) who completed 7 or more DMI sessions. This is analogous to a 

treatment-on-treated (TOT) analysis, where completion of 7 or more DMI sessions is considered 

receiving intended treatment.
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