20.4 Multilevel Power - Simplified
20.5 Simulation of Power
Continuous response
Set number of groups and number within groups.
s
σy2
σ2
SE(y ) =
+ α
n
J
Recall split plot design and corCompSymm:
2
α
ρ = σ2σ+σ
2 so let ni = m and
y
α
q
2
σtotal
SE(y ) =
Jm [1 + (m − 1)ρ].
NYC example they pick ρ = 0.15. Reasoning?
Stat 506
Gelman & Hill, Chapter 20
Quick & Dirty
yjt ∼ N(αj + βj t, σy2 ) is the sqrt CD4% of children w/out zinc
supplement. Using lmer we get
σ
by = 0.7, σ
bα = 1.3, σ
bβ = 0.7, ρb = 0.1, with means for intercepts
and slopes of 4.8 and -0.5, respectively.
Step 1: Does the fitted model generate data which looks like the
original data? Fig 20.5.
Step 2: Find sample size to detect a shift in mean for slope of
+0.5 with 80% power
atα = .05. nj = 7 for each kid.
αj
with correlation ρ and shift γ1β under
Assume MVN for
βj
treatment.
Stat 506
Gelman & Hill, Chapter 20
R function to simulate fake data
βj ’s are data points – one per kid – with SD = .7. To detect a
shift of ∆, we need
σy2
2 · 2.8σ 2
2 · 2.8 2 2
J≥
=
(σβ +
)=
∆
∆
SSX
2 · 2.8
∆
2
Build function taking J, K = 7 as inputs returns data matrix of
fake data.
Generate 1000 datasets, test each to see if H0 gets rejected. Keep
track of rejection rate = power. power we need 130 kids.
(0.72 + .7 × 1.13)2 = 150
Stat 506
Gelman & Hill, Chapter 20
Stat 506
Gelman & Hill, Chapter 20