in ClusterRandomized Control Trials Sharon Wolf NYU Abu Dhabi Additional Insights Summer Training Institute June 15 2015 1 Outline Conceptual overview Analytic considerations PowerMinimum detectable differential effects ID: 550083
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Slide1
Individual-level Moderation in Cluster-Randomized Control Trials
Sharon WolfNYU Abu Dhabi Additional Insights Summer Training InstituteJune 15, 2015
1Slide2
OutlineConceptual overviewAnalytic considerations
Power/Minimum detectable differential effectsCross-level interactions in MLMCentering variablesRecommendations and tips
2Slide3
Conceptual OverviewWhen is the story in the subgroups?
3Slide4
Research Questions that Motivate Subgroup Analyses
Guide questions about how to target resources most efficiently:
How
widespread
are the effects of an intervention?
Is the intervention effective for a specific subgroup
?
Is the intervention effective for any subgroup?
Exploratory* versus confirmatory subgroup findings
4Slide5
Individual-level Moderation versus Cluster-level Moderation
Two examples from welfare reform in the United States and the different policy implications.
Michalopoulos
& Schwartz
(2000
) assessed two types of subgroups:
A range
of
person-level subgroups (e.g., education level
, prior employment experience, and risk of
depression).The nature of the program and program office practices.
5Slide6
Characteristics that Define Individual-level Subgroups
Characteristics believed to be related to the need for a particular intervention or the likelihood of benefiting from it.Demographic characteristics – e.g., gender, age, education levelRisk factors - past smoking, drug abuse, severity of disease, poverty statusCombinations
of characteristics – e.g., gender and age; cumulative levels of risk/risk index
6Slide7
Subgroups that are Exogenous versus Endogenous to the Intervention
Exogenous to the intervention: not affected by the intervention or correlated with its receipt (all pre-random assignment characteristics).
Endogenous
to the intervention: affected by the intervention or correlated with its receipt (e.g., dosage of the intervention). Valid
causal inferences much more
difficult.
Gambia: higher “dosage”
(i.e.,
higher attendance)
more learning?
Increased
attendance
could bring less advantaged students into the intervention
group, biasing the average treatment effect (ATE) downward
.
7Slide8
Exploratory versus Confirmatory Subgroup Analyses
Exploratory subgroup analysesProvide a basis for hypothesis-generationEssential step in the scientific method
Should be considered suggestive
Confirmatory subgroup analyses
Appropriate basis for testing hypotheses
Provide strong evidence if findings are: (a) consistent with existing findings, (b) large enough magnitude to be meaningful, (c) robust.
Bloom & Michalopoulos, 2010
8Slide9
Contextual considerationsInternal contextual considerationsFeatures of findings internal to a given study
E.g., pattern across all outcomes for a particular subgroup in a studyExternal contextual considerationsFeatures
of findings
external
to
a
given
study
E.g
., consistency with prior study findings
9Slide10
Subgroup analysis assess two questionsWhat is the impact of the program for each subgroup?
What are the relative impacts of the program across subgroups?
10Slide11
Power Calculations for Subgroup Analyses
Minimum detectable differential effects
11Slide12
Some considerations for power calculations of subgroups
1. Did the program work for a particular subgroup ? Assess impacts separately for this subgroup
Assess power to detect impacts for this subgroup
2. Were
the effects
different
for particular subgroups?
Assess impacts using a cross-level interaction
Assess power to detect a cross-level interaction
12Slide13
#1: Did the program work for a particular subgroup?Minimum Detectable Effect Size (MDES): the smallest true effect, in standard deviations of the outcome, that is detectable for a given level of power and statistical significance.
Accepted parameters:Power: 80% Statistical significance level: 0.0513Slide14
Establishing common notationρ = intraclass correlationδ
= MDESλ = non-centrality parameterJ = number of clustersn = number of units per cluster
14Slide15
Power in a 2-level CRT (J. Spybrook)Main effect:Main effect
with covariate:Cluster level Moderator:
Individual level
Moderator:Slide16
Considerations for power analysisThe
number of clusters (highest level units) is more important than the size of the cluster (lower level units) in reducing the MDES.
A higher
intra-cluster correlation
(ICC) increases the
MDES
(i.e., if
τ
00
is relatively large).
The proportion of variance in the outcome you can predict
with L1 and L2 variables (i.e.,
R
|X2 and R
|W
2
) reduces the
MDES.
16Slide17
Subgroup specific analysisMaintains a significant portion of power because the number of clusters (or L2 units) remains the same.
The only statistical difference between the subexperiment and the full experiment is the number of L1 units per cluster.17Slide18
Example using OD software18Slide19
#2: Did the program work differently for different subgroups?Minimum Detectable Effect Size Differences (MDESD): the smallest true effect of the
difference in program impacts for two subgroups, in standard deviations of the outcome, that is detectable for a given level of power and statistical significance.Accepted parameters:Power: 80%
Statistical significance level: 0.05
19Slide20
Power in a 2-level CRT (J. Spybrook)Main effect:
Main effect with covariate:Cluster level Moderator:
Individual level
Moderator:Slide21
Considerations for power analysisWithin-level variance becomes increasingly important. Implications include:
The number of cases per cluster (lower
level units)
become more important for increasing power.
The
intra-cluster
correlation
(ICC)
becomes less significant in affecting power (though still important).
The
proportion of variance in the outcome you can predict with L1 variables (not L2; i.e
.,
R
|X2) increases power.
21Slide22
Cross-level interactions
Assessing individual-level moderation in cluster-randomized trials using multi-level models22Slide23
Testing relationships using multi-level modeling
Lower level direct effects. Does a L1 predictor X (e.g., student gender) have a relationship with the L1 outcome variable Y (e.g., student reading
)?
Cross-level direct effects.
Does a L2 predictor (e.g.,
school treatment status
) have a relationship with an L1 outcome variable Y (e.g.,
student reading
)?
Cross-level interaction effects. Does the nature or strength of the relationship between a L1 variable (e.g., gender) and the outcome (e.g.,
reading
) change as a function of a higher-level variable (e.g.,
school treatment status)?
23Slide24
Model of Program Impacts
Level 1
Level 2
Y
ij
= outcome for individual
i
in cluster j
T
j
=
1
for program-group members,
0
for control-group
γ
00
= mean outcome for the control group
γ
01
= true program impact
r
ij
= error component for individual
i
from cluster j
u
0j
= error component for cluster j
24Slide25
Combined equation
25Slide26
Adding in a cross-level interactionLevel 2 (e.g., school treatment status
) and Level 1 (e.g., student gender) variables interacting to produce an effect on the outcome (e.g., student reading scores).
In terms of your impact estimation equation:
(a) Add Level 1 predictor (moderator).
(b) Expand Level 2 model to include a fixed slope (
1
).
(c) Add a level 2 predictor (treatment status) to the slope.
26Slide27
Model of Moderated Program Impacts
Level 1
Level 2
Added L1 predictor (moderator)
Expand L2 slope
Add L2 predictor to the slope
27Slide28
Model of Moderated Program Impacts
Level 1
Level 2
γ
00
= mean outcome for the control group
γ
01
= estimated program impact for
M
ij
=0
γ
10
= main effect for the moderating variable,
T
j
=0
γ
11
= moderated effect (i.e., interaction)
Coefficient for cross-level interaction
28Slide29
Combined equation
29Slide30
How to graph a cross-level interaction effect“Simple Regression Equation”: Calculate the expected values of Y
ij under different conditions of Tj and MijFor continuous moderators, plot at values of one standard deviation below the mean, the mean, and one standard deviation above the mean for
M
.
I
t may also be useful to choose additional values that may be informative in specific contexts.
30Slide31
Simple Regression EquationE(Yij
| Mij ,Tj) =
γ
00
+
γ
01
(
T
j
)+ γ10(Mij )+ γ11(
M
ij
)(Tj)
Under control conditions:
E
(
Y
ij
|
M
ij
,Tj = 0) = γ00 + γ10(Mij)Under treatment conditions:E
(
Y
ij
|
M
ij
,
T
j
= 1) =
γ
00
+
γ
01
+
γ
10
(
M
ij
)
+
γ
11
(
M
ij
)
31Slide32
BREAKIf we need it32Slide33
Centering Variables in Multi-level Models
Implications for interpreting effect estimates and detecting impact variation
33Slide34
Decisions about centering depend on your data and your research questionHow do you want to interpret the intercept in your model? The coefficients?Example: School diversity/cultural awareness program
H1: Improved sense of belonging for minority students (L1 moderator).
H2:
Improved
sense
of belonging for minority
students in less diverse schools (L1 & L2 moderators).
The distribution of the moderator variable across clusters needs to be considered.
34Slide35
Two options to center in multi-level modelsCGM = Centering at the grand meanDeviations calculated from the sample mean for all individualsCGM L1 with all individuals; L2 with all clusters
CWC = Centering within clustersaka, group-mean centeringDeviations calculated around the mean of the cluster j to which case i belongs
35Slide36
Y
(outcome)
X
(predictor)
The distribution of M is highly variable across clusters.
36
Cluster 1
Cluster 2
Cluster 3Slide37
Y
(outcome)
CGM
M
37
X
(predictor)Slide38
Y
(outcome)
X
(predictor)
CGM
M
38Slide39
Centering at the Grand Mean (CGM)Does not affect the rank order of scores on the variable. The complex, multilevel association between the L1 and L2 variables is unaffected.Yields scores that are correlated with variables at both levels of the hierarchy. (This is a critical differences with CWC.)
Produces an interaction coefficient (γ11) that is a weighted combination of the within- and between-cluster regression coefficients.
39Slide40
Y
(outcome)
X
(predictor)
CWC
M
1
M
2
M
3
40
The distribution of M is highly concentrated within clusters.Slide41
Y
(outcome)
X
(predictor)
CWC
M
1
M
2
M
3
41Slide42
Centering within Cluster (CWC)Affects the rank order of scores of variables within the sample.Produces scores that are uncorrelated with Level 2 variables (because the mean for all L2 variables is zero).
Produces an interaction coefficient (γ11) that is an unbiased estimate of the Level 1 associationγ11 is a pure estimate of the cross-level interaction, no longer confounded with the Level 2 interaction.
42Slide43
Y
(outcome)
X
(predictor)
43
The distribution of M is even across clusters.Slide44
Y
(outcome)
X
(predictor)
CGM
M
44Slide45
Y
(outcome)
X
(predictor)
CWC
M1
M2
M3
45Slide46
Main TakeawaysCentering will affect estimates more if the predictor variable is not evenly distributed across clusters.Cross-level interaction term using CGM will provide a coefficient estimate that is a mix of the L1 and L2 effects.
Cross-level interaction term using CWC will provide a pure estimate of the L1 relationship.Decisions on how to center depend on your data and your research question (!!).
46Slide47
Example
Predictor: Treatment status (L2)Individual level moderator: Student age (L1) (continuous)Outcome: Reading score
Some options on how to center the data and what it means for interpreting your moderated effect…
47Slide48
Option 1: No centering
Level 1
Level 2
γ
00
is the average school mean reading score for the control group when age=0
γ
10
is the composite
of
the relationship of within
school age-reading
scores and
between-school age reading
scores
γ
11
is the composite
of
the interaction between treatment and within
school age-reading
scores and treatment and between-school
age reading
scores.
48Slide49
Option 1: CGM age
Level 1
Level 2
γ
00
is the average school mean reading
score for
schools
for the control group.
γ
10
is the composite of the relationship of within school age-reading scores and between-school age reading
scores.
γ
11
is
still the
composite of the interaction between treatment and within school age-reading scores and treatment and between-school age reading scores.
49Slide50
Option 3: CWC age at L1, CGM age at L2
50
Level 1
Level 2
γ
00
is the average school mean reading
score across
the
schools for the control group.
γ
10
is the average change in school mean reading
score for
a 1 unit increase in school mean age across schools (between school age relationship)
γ
11
is the composite of the interaction between treatment and within school age-reading scores and treatment and between-school age reading scores.
Slide51
Option 4: CWC age at L1, CGM age at L2 interacted with treatment status51
Level 1
Level 2
γ
00
is the average school mean reading fluency across the
schools for the control group.
γ
10
is the average change in school mean reading fluency for a 1 unit increase in school mean age across schools (between school age relationship
).
γ
03
is the moderated relationship between treatment and between school age reading scores.
γ
11
is the
moderated relationship between
treatment and within school age-reading
scores.
Slide52
General Recommendations and Tips
52Slide53
Multiple Hypothesis T
estingDistortions to statistical inferences
can
occur when multiple related hypothesis tests
are conducted.
Suggested approaches:
Explicitly
distinguish between exploratory and confirmatory
findings
Minimize the number of confirmatory hypothesis tests conducted by a given study.
Create an omnibus hypothesis test about the intervention’s effects that considers all outcome measures and subgroups together. (e.g., composite measure of individual outcomes).
Consider family-wise error correction (reduces statistical
power
considerably).
53Slide54
Practical tips and recommendations
Calculate ρ for all levels.Determine your research question and relevant approach to assessing subgroup affects.
Calculate the power needed to detect a subgroup effect (either for a particular subgroup, or for a cross-level interaction, depending on your research question).
Rescale (i.e., center) predictor
variables as needed.
Assess the practical significance of your findings (i.e., calculate effect sizes
).
Report results regarding each step of the model building process including all coefficients, standard errors and variance components
.
54Slide55
ReferencesAguinis, H., Gottfredson, R. K., & Culpepper, S. A. (2013). Best-practice recommendations for estimating cross-level interaction effects using multilevel modeling. Journal of Management
, 0149206313478188.Bloom, H. S. (Ed.). (2005). Learning more from social experiments: Evolving analytic approaches. Russell Sage Foundation.Bloom, H. & Michalopoulos, M. (2013). When Is the Story in the Subgroups? MDRC Working Paper.
Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in cross-sectional multilevel models: a new look at an old issue.
Psychological methods
,
12
(2), 121.
Mathieu, J. E., Aguinis, H., Culpepper, S. A., & Chen, G. (2012). Understanding and estimating the power to detect cross-level interaction effects in multilevel modeling.
Journal of Applied Psychology
,
97(5), 951.55