Howard S. Bloom
Chief Social Scientist
MDRC
Prepared for the IES/NCER Summer Research Training Institute held at
Northwestern University on July 27, 2010.

 Sample size determinants
 Precision requirements
 Sample allocation
 Covariate adjustments
 Matching and blocking
 Subgroup analyses
 Generalizing findings for sites and blocks
 Using two-level data for three-level situations

The Basics

Statistical properties of grouprandomized impact estimators
Unbiased estimates
Y ij
= a
+B
0
T j
+e j
+ e ij
E(b
0
) = B
0
Less precise estimates
VAR( e ij
) = s
2
VAR(e j
) = t
2 r
= t
2 /( t
2 + s
2 )
GEM


n

r 
SE
C
b
0
SE
I
b
0

(for a given total number of individuals)
______________________________________
Intraclass Individuals per Group (n)
Correlation ( r) 10 50 500
0.01 1.04 1.22 2.48
0.05 1.20 1.86 5.09
0.10 1.38 2.43 7.13
_____________________________________

 Number of randomized groups (J)
 Number of individuals per randomized group (n)
 Proportion of groups randomized to program status (P)

 A minimum detectable effect (MDE) is the smallest true effect that has a “good chance” of being found to be statistically significant.
 We typically define an MDE as the smallest true effect that has 80 percent power for a two-tailed test of statistical significance at the 0.05 level.
 An MDE is reported in natural units whereas a minimum detectable effect size (MDES) is reported in units of standard deviations

Minimum Detectable Effect Sizes
For a Group-Randomized Design with r
= 0.05 and no Covariates
___________________________________
Randomized Individuals per Group (n)
Groups (J) 10 50 500
10 0.77 0.53 0.46
20 0.50 0.35 0.30
40 0.35 0.24 0.21
120 0.20 0.14 0.12
___________________________________

 It is extremely important to randomize an adequate number of groups.
 It is often far less important how many individuals per group you have.

Determining required precision






(speculative) (empirical)
_______________________________________________
Small = 0.2
s
Small = 0.15
s
Medium = 0.5
s
Medium = 0.45
s
Large = 0.8
s
Large = 0.90
s

Treatment:
13-17 versus 22-26 students per class
Effect sizes:
0.11
s to 0.22
s for reading and math
Findings are summarized from Nye, Barbara, Larry V. Hedges and Spyros
Konstantopoulos (1999) “The Long-Term Effects of Small Classes: A Five-
Year Follow-up of the Tennessee Class Size Experiment,” Educational
Evaluation and Policy Analysis , Vol. 21, No. 2: 127-142.

Reading
Grade Growth
Transition Effect Size
Math
Growth
Effect Size
----------------------------------------------------------------
K - 1 1.52 1.14
1 - 2 0.97 1.03
2 - 3 0.60 0.89
3 - 4 0.36 0.52
4 - 5 0.40 0.56
5 - 6 0.32 0.41
6 - 7 0.23 0.30
7 - 8 0.26 0.32
8 - 9 0.24 0.22
9 - 10 0.19 0.25
10 - 11 0.19 0.14
11 - 12 0.06 0.01
-------------------------------------------------------------------------------------------------
Based on work in progress using documentation on the national norming samples for the CAT5,
SAT9, Terra Nova CTBS, Gates MacGinitie (for reading only), MAT8, Terra Nova CAT, and SAT10.
95% confidence intervals range in reading from +/- .03 to .15 and in math from +/- .03 to .22

Performance gap between “average” (50 th percentile) and “weak” (10 th percentile) schools
Subject and grade District I District II District III District IV
Reading
Grade 3 0.31
0.18
0.16
0.43
Grade 5
Grade 7
Grade 10
0.41
.025
0.07
0.18
0.11
0.11
0.35
0.30
NA
0.31
NA
NA
Math
Grade 3
Grade 5
Grade 7
Grade 10
0.29
0.27
0.20
0.14
0.25
0.23
0.15
0.17
0.19
0.36
0.23
NA
0.41
0.26
NA
NA
Source: District I outcomes are based on ITBS scaled scores, District II on SAT 9 scaled scores, District
III on MAT NCE scores, and District IV on SAT 8 NCE scores.

Demographic performance gap in reading and math: Main NAEP scores
Subject and grade
Reading
Grade 4
Grade 8
Grade 12
Math
Grade 4
Grade 8
Grade 12
Black-
White
-0.83
-0.80
-0.67
-0.99
-1.04
-0.94
Hispanic-
White
-0.77
-0.76
-0.53
-0.85
-0.82
-0.68
Male-
Female
-0.18
-0.28
-0.44
0.08
0.04
0.09
Eligible-
Ineligible for free/reduced price lunch
-0.74
-0.66
-0.45
-0.85
-0.80
-0.72
Source: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics,
National Assessment of Educational Progress (NAEP), 2002 Reading Assessment and 2000 Mathematics Assessment.

ES Results from Randomized Studies
Achievement Measure
Elementary School
Standardized test (Broad)
Standardized test (Narrow)
Specialized Topic/Test
Middle Schools
High Schools n
389
21
181
180
36
43
Mean
0.33
0.07
0.23
0.44
0.51
0.27

The ABCs of Sample Allocation

Balanced allocation
 maximizes precision for a given sample size;
 maximizes robustness to distributional assumptions.
Unbalanced allocation
 precision erodes slowly with imbalance for a given sample size
 imbalance can facilitate a larger sample
 Imbalance can facilitate randomization

 Equal variances: when the program does not affect the outcome variance.
 Unequal variances: when the program does affect the outcome variance.

MDES for equal variances without covariates
MDES

M
J

2
J r 
1
 n r
1
P ( 1

P )

1
P ( 1

P )

1
.
5 ( 1

.
5 )

2 .
00
1
.
6 ( 1

.
6 )

2 .
04
1
.
7 ( 1

.
7 )

2 .
18
1
.
8 ( 1

.
8 )

2 .
50
1
.
9 ( 1

.
9 )

3 .
33

Minimum Detectable Effect Size For Sample
Allocations Given Equal Variances
Allocation Example * Ratio to
Balanced
Allocation
0.5/0.5 0.54
s
0.6/0.4 0.55
s
0.7/0.3
0.8/0.2
0.9/0.1
0.59
s
0.68
s
0.91
s
1.00
1.02
1.09
1.25
1.67
*
________________________________________
Example is for n = 20, J = 10, r
= 0.05, a one-tail hypothesis test and no covariates .

Implications of unbalanced allocations with unequal variances
SE ( b
0
)
U
 s
2
P
J
P
 s
J
C
2
C
E ( se ( b
0
))
E
 s
2
P
J
C
 s
J
P
2
C

Implications Continued
The estimated standard error is unbiased


When the allocation is balanced
When the variances are equal
The estimated standard error is biased upward

When the larger sample has the larger variance
The estimated standard error is biased downward
 When the larger sample has the smaller variance


Don’t use the equal variance assumption for an unbalanced allocation with many degrees of freedom .
 Use a balanced allocation when there are few degrees of freedom .

References
Gail, Mitchell H., Steven D. Mark, Raymond J. Carroll,
Sylvan B. Green and David Pee (1996) “On Design
Considerations and Randomization-Based Inferences for Community Intervention Trials,”
Statistics in
Medicine 15: 1069 – 1092.
Bryk, Anthony S. and Stephen W. Raudenbush (1988)
“Heterogeneity of Variance in Experimental Studies:
A Challenge to Conventional Interpretations,”
Psychological Bulletin, 104(3): 396 – 404.

Using Covariates to Reduce
Sample Size

 Goal: Reduce the number of clusters randomized
 Approach: Reduce the standard error of the impact estimator by controlling for baseline covariates
 Alternative Covariates
 Individual-level
 Cluster-level
 Pretests
 Other characteristics

Impact Estimation with a Covariate y ij
 a  
0
T j
 
1 x ij
 e j
 e ij y ij
 a  
0
T j
 
1 x j
 e j
 e ij y ij
= the outcome for student i from school j
T j
X j
= 1 for treatment schools and 0 for control schools
= a covariate for school j x ij
= a covariate for student i from school j e j e ij
= a random error term for school j
= a random error term for student i from school j

Minimum Detectable Effect Size with a Covariate
MDES

M
J

K r
( 1

P ( 1

R
2
2
)
P ) J

( 1
 r
)( 1

P ( 1

R
P ) nJ
1
2
)
MDES = minimum detectable effect size
M
J-K
= a degrees-of-freedom multiplier 1
J = the total number of schools randomized n = the number of students in a grade per school
P = the proportion of schools randomized to treatment r = the unconditional intraclass correlation (without a covariate)
R
1
2 = the proportion of variance across individuals within schools (at level 1) predicted by the covariate
R
2
2 = the proportion of variance across schools (at level 2) predicted by the covariate
1 For 20 or more degrees of freedom M
J-K equals 2.8 for a two-tail test and 2.5 for a one-tail test with statistical power of 0.80 and statistical significance of 0.05

Questions Addressed Empirically about the
Predictive Power of Covariates
 School-level vs. student-level pretests
 Earlier vs. later follow-up years
 Reading vs. math
 Elementary vs. middle vs. high school
 All schools vs. low-income schools vs. low-performing schools

Empirical Analysis
 Estimate r
, R
2
2 and R
1
2 from data on thousands of students from hundreds of schools, during multiple years at five urban school districts
 Summarize these estimates for reading and math in grades
3, 5, 8 and 10
 Compute implications for minimum detectable effect sizes

Estimated Parameters for Reading with a School-level Pretest Lagged One Year
___________________________________________________________________
School District
___________________________________________________________
A B C D E
___________________________________________________________________
Grade 3 r
0.20 0.15 0.19 0.22 0.16
R
2
2 0.31 0.77 0.74 0.51 0.75
Grade 5 r
0.25 0.15 0.20 NA 0.12
R
2
2
Grade 8 r
R
2
2
Grade 10 r
0.33 0.50 0.81 NA 0.70
0.18 NA 0.23 NA NA
0.77 NA 0.91 NA NA
0.15 NA 0.29 NA NA
R
2
2 0.93 NA 0.95 NA NA
____________________________________________________________________

Minimum Detectable Effect Sizes for Reading with a School-Level
Pretest (Y
-1
) or a Student-Level Pretest (y
-1
) Lagged One Year
________________________________________________________
Grade 3 Grade 5 Grade 8 Grade 10
________________________________________________________
20 schools randomized
No covariate 0.57 0.56 0.61 0.62
Y
-1 y
-1
0.37 0.38 0.24 0.16
0.38 0.40 0.28 0.15
40 schools randomized
No covariate 0.39 0.38 0.42 0.42
Y
-1 y
-1
0.26 0.26 0.17 0.11
0.26 0.27 0.19 0.10
60 schools randomized
No covariate 0.32 0.31 0.34 0.34
Y
-1 y
-1
0.21 0.21 0.13 0.09
0.21 0.22 0.15 0.08
________________________________________________________

Key Findings



Using a pretest improves precision dramatically.
This improvement increases appreciably from elementary school to middle school to high school because R
2
2 increases.
School-level pretests produce as much precision as do student-level pretests.
 The effect of a pretest declines somewhat as the time between it and the post-test increases.
 Adding a second pretest increases precision slightly.
 Using a pretest for a different subject increases precision substantially.
 Narrowing the sample to schools that are similar to each other does not improve precision beyond that achieved by a pretest.

Source
Bloom, Howard S., Lashawn Richburg-Hayes and Alison
Rebeck Black (2007) “
Using Covariates to Improve
Precision for Studies that Randomize Schools to
Evaluate Educational Interventions” Educational
Evaluation and Policy Analysis , 29(1): 30 – 59.

A Tail of Two Tradeoffs
(“It was the best of techniques. It was the worst of techniques.”
Who the dickens said that?)

Why match pairs?
 for face validity
 for precision
How to match pairs?
 rank order clusters by covariate
 pair clusters in rank-ordered list
 randomize clusters in each pair

 When the gain in predictive power outweighs the loss of degrees of freedom
 Degrees of freedom
 J - 2 without pairing
 J/2 - 1 with pairing

Deriving the Minimum Required
Predictive Power of Pairing
Without pairing
MDE ( b
0
)
GR
 M
J

2
SE ( b
0
)
GR
With pairing
MDE ( b
0
)
GR
 M
J / 2

1
1

R
2
SE ( b
0
)
GR
Breakeven R 2
R
2 m in

1
 M
2
M
J
2
J

2
/ 2

1

The Minimum Required
Predictive Power of Pairing
Randomized Required Predictive
Clusters (J) Power (R min
2 ) *
6 0.52
8 0.35
10
20
30
* For a two-tail test.
0.26
0.11
0.07

 Blocking for face validity vs. blocking for precision
 Treating blocks as fixed effects vs.random effects
 Defining blocks using baseline information

Subgroup Analyses #1:
When to Emphasize Them

 Confirmatory: Draw conclusions about the program’s effectiveness if results are
 Consistent with theory and contextual factors
 Statistically significant and large
 And subgroup was pre-specified
 Exploratory: Develop hypotheses for further study
45

 Before the analysis, state that conclusions about the program will be based in part on findings for this set of subgroups
 Pre-specification can be based on
 Theory
 Prior evidence
 Policy relevance
46

 When should we discuss subgroup findings?
 Depends on
 Whether significant differences in impacts across subgroups
 Might depend on whether impacts for the full sample are statistically significant
47

Subgroup Analyses #2:
Creating Subgroups

Defining Features
 Creating subgroups in terms of:
 Program characteristics
 Randomized group characteristics
 Individual characteristics

Defining Subgroups by
Program Characteristics
 Based only on program features that were randomized
 Thus one cannot use implementation quality

Defining Subgroups by Characteristics
Of Randomized Groups
 Types of impacts
 Net impacts
 Differential impacts
 Internal validity
 only use pre-existing characteristics
 Precision
 Net impact estimates are limited by reduced number of randomized groups
 Differential impact estimates are triply limited (and often need four times as many randomized groups)

Defining Subgroups by
Characteristics of Individuals
 Types of impacts
 Net impacts
 Differential impacts
 Internal validity
 Only use pre-existing characteristics
 Only use subgroups with sample members from all randomized groups
 Precision
 For net impacts : can be almost as good as for full sample
 For differential impacts: can be even better than for full sample

Program
Group
Differential Impacts by Gender
Boys Girls
Control
Group
Boys Girls
Boys Girls Boys Girls
Boys Girls Boys Girls
Boys Girls
Y
PB
- Y
PG
Boys Girls
Y
CB
- Y
CG

Generalizing Results from
Multiple Sites and Blocks

Fixed vs. Random Effects Inference:
A Vexing Issue
 Known vs. unknown populations
 Broader vs. narrower inferences
 Weaker vs. stronger precision
 Few vs. many sites or blocks

 Implicitly through a pooled regression
 Explicitly based on
 Number of schools
 Number of students
 Explicitly based on precision
 Fixed effects
 Random effects
 Bottom line: the question addressed is what counts

Using Two-Level Data for Three-
Level Situations

The Issue
 General Question: What happens when you design a study with randomized groups that comprise three levels based on data which do not account explicitly for the middle level?
 Specific Example: What happens when you design a study that randomizes schools (with students clustered in classrooms in schools) based on data for students clustered in schools?

3-level vs. 2-level Variance Components
Outcomes
Expressive vocab-spring
School
19.84
3-Level Model
Class
32.45
Student
306.18
Variance Components
Total
358.48
Stanford 9 Total Math Scaled Score 1273.15
1424.69
School
38.15
131.39
2-Level Model
Student
321.11
1293.24
Total
359.26
1424.63
Stanford 9 Total Reading Scaled Score 1581.86
1849.56
181.77
1666.48
1848.25
Sources: The Chicago Literacy Initiative: Making Better Early Readers study (CLIMBERs) database and the School Breakfast Pilot Project (SBPP) database.

3-level vs. 2-level MDES for Original Sample
Outcomes
Expressive vocab-spring
3-Level Model
Unconditional
0.482
Conditional
0.386
MDES
2-Level Model
Unconditional
0.495
Conditional
0.311
Stanford 9 Total Math Scaled Score 0.259
0.184
0.259
0.184
Stanford 9 Total Reading Scaled Score 0.261
0.148
0.264
0.150
Sources: The Chicago Literacy Initiative: Making Better Early Readers study (CLIMBERs) database and the School Breakfast Pilot Project (SBPP) database.

Further References
Bloom, Howard S. (2005) “Randomizing Groups to Evaluate Place-Based
Programs,” in Howard S. Bloom, editor, Learning More From Social
Experiments: Evolving Analytic Approaches (New York: Russell Sage
Foundation).
Bloom, Howard S., Lashawn Richburg-Hayes and Alison Rebeck Black (2005)
“Using Covariates to Improve Precision: Empirical Guidance for Studies that
Randomize Schools to Measure the Impacts of Educational Interventions” (New
York: MDRC).
Donner, Allan and Neil Klar (2000) Cluster Randomization Trials in Health
Research (London: Arnold).
Hedges, Larry V. and Eric C. Hedberg (2006) “Intraclass Correlation Values for
Planning Group Randomized Trials in Education” (Chicago: Northwestern
University).
Murray, David M. (1998) Design and Analysis of Group-Randomized Trials (New
York: Oxford University Press).
Raudenbush, Stephen W., Andres Martinez and Jessaca Spybrook (2005) “Strategies for Improving Precision in Group-Randomized Experiments” (University of
Chicago).
Raudenbush, Stephen W. (1997) “Statistical Analysis and Optimal Design for
Cluster Randomized Trials” Psychological Methods , 2(2): 173 – 185.
Schochet, Peter Z. (2005) “Statistical Power for Random Assignment Evaluations of
Education Programs,” (Princeton, NJ: Mathematica Policy Research).