Handling Dependent Observations in Cluster Randomized Studies: Correlated Data - Prof. Mic, Study notes of Data Analysis & Statistical Methods

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Correlated Data Bios 662

Michael G. Hudgens, Ph.D. [email protected]

http://www.bios.unc.edu/∼mhudgens

2007-11-14 12:

Clustered Data

• Heretofore, all methods assume data iid
• What to do if dependencies exist between data?
• Typically correlated data occur in clusters/groups
• Examples:
  • Repeated measures on individuals over time
  • Natural groupings of individuals (e.g., litters, schools)
• Can occur in observational or randomized studies; ex- ample of later would be cluster randomized studies

Central Issue

• How do we do testing, estimation, sample size calcula- tions, etc allowing that responses within a cluster/group (e.g., school) may not be independent?
• Use methods allowing for dependency (correlation) within groups but assuming independence (no correlation) be- tween groups

Continuous response model

• Let Yijk = the response of the kth^ person in the jth cluster in the ith^ treatment level

i = 1, 2 ,... , t j = 1, 2 ,... , c k = 1, 2 ,... , m

• Let

Y¯ij =

∑m k=1 Yijk m

Continuous response model

• Then

V ar( Y¯ij) = E( Y¯ (^) ij^2 ) − μ^2 i

= m−^2 E

{∑m k=1 Yijk

− μ^2 i

= m−^2 E

m k=1 Y^ 2 ijk +^

k 6 =k′YijkYijk′

− μ^2 i

= m−^2

mσ^2 + m(m − 1)ρσ^2

= σ

2 m {1 + (m^ −^ 1)ρ}

Variance Inflation Factor (VIF)

• {1 + (m − 1)ρ} is the variance inflation factor (VIF)
• Measures the increase in the variance of the mean due to the within-subject correlation of measurements (ρ)
• VIF > 1 for ρ > 0 and m > 1

Continuous response model

• Suppose t = 2 and n 1 = n 2 = cm
• If we ignore the correlation within cluster

zignore =

Y¯ 1 − Y¯ 2

σ

1 /n 1 + 1/n 2

• Should instead use

ztrue = Y¯ 1 − Y¯ 2 σ

(1/n 1 +1/n 2 ){V IF }

= √zignoreV IF

Effect of correlation

• ztrue < zignore for ρ > 0 and m > 1
• Thus ignoring correlation will lead to inflated type I error
• Intuition: naive approach acts as if we have more infor- mation than we do

Sample Size when t = 2

• If ρ = 0, then V IF = 1 and

n = 2

(z 1 −α/ 2 +^ z^1 −β ∆

• If ρ = 1, then V IF = m

n = 2

(z 1 −α/ 2 +^ z^1 −β ∆

m

• Typically 0. 1 ≤ ρ ≤ 0. 4

Variance Inflation Factors

• Table 18.4 text:

ρ m 0.001 0.01 0.02 0.05 0. 2 1.001 1.01 1.02 1.05 1. 5 1.004 1.04 1.09 1.20 1. 10 1.009 1.09 1.18 1.45 1. 100 1.099 1.99 2.98 5.95 10. 1000 1.999 10.99 20.98 50.95 100.