Cluster Randomized StudiesBios 662Michael G. Hudgens, [email protected]://www.bios.unc.edu/∼mhudgens2006-11-20 12:13BIOS 662 1 Cluster Randomized StudiesBackground: RCT• Chapter 19 text• RCT - randomized clinical/controlled trial• Cf Friedman et al. “Fundamentals of Clinical Trials”• Randomized: treatment assignments given by some ran-dom process• Provide strongest evidence re efficacy of treatmentBIOS 662 2 Cluster Randomized StudiesBackground: Simple Randomization1. Generate n Uniform(0,1) random deviates: u1, . . . , un2. If ui< 0.5, assign intervention A to unit i; otherwise,assign B to i3. Note nAis a random variable with E(nA) = 0.5n• Disadvantage: possible imbalance• Advantage: independence of Rx assignmentsBIOS 662 3 Cluster Randomized StudiesBackground: Restricted Randomization• Aka Permutation/Block randomization1. Generate n Uniform(0,1) random deviates: u1, . . . , un2. Sort deviates from smallest to largest3. Assign intervention A to the units with n/2 smallestui’s; assign intervention B to the remaining units• Advantage: balance• Disadvantage: dependence of Rx assignmentsBIOS 662 4 Cluster Randomized StudiesCluster Randomized Studies• Aka Group Allocation designs• Section 18.4 text• Suppose we want to compare two methods of smokingprevention in teenagers• We randomly assign intervention to schools, but mea-sure smoking in childrenBIOS 662 5 Cluster Randomized StudiesCentral Issue• How do we do testing, estimation, sample size calcula-tions, etc allowing that responses within a group (e.g.,school) may not be independent?BIOS 662 6 Cluster Randomized StudiesContinuous response model• Let Yijk= the response of the kthperson in the jthcluster in the ithtreatment groupi = 1, 2, . . . , tj = 1, 2, . . . , ck = 1, 2, . . . , m• Let¯Yij=Pmk=1YijkmBIOS 662 7 Cluster Randomized StudiesContinuous response model• Assume:E(Yijk) = µ; V ar( Yijk) = σ2Cov(Yijk, Yijk0) = ρσ2; Cov(Yijk, Yij0k0) = 0BIOS 662 8 Cluster Randomized StudiesContinuous response model• Assume µ = 0• ThenV ar(¯Yij) = E(¯Y2ij) =1m2EPmk=1Yijk2= m−2EnPmk=1Y2ijk+PPk6=k0YijkYijk0o= m−2mσ2+ m(m − 1)ρσ2=σ2m{1 + (m − 1)ρ}BIOS 662 9 Cluster Randomized StudiesVariance Inflation Factor (VIF)• {1 + (m −1)ρ} is the variance inflation factor (VIF)• Measures the increase in the variance of the mean dueto the within-subject correlation of measurements (ρ)• VIF > 1 for ρ > 0 and m > 1BIOS 662 10 Cluster Randomized StudiesContinuous response model continued• Let¯Yi=Pj¯Yijm=Pj,kYijkcm• ThenV (¯Yi) =σ2cmV IFBIOS 662 11 Cluster Randomized StudiesContinuous response model• Suppose t = 2 and n1= n2= cm• If we ignore the c orrelation within clusterzignore=¯Y1−¯Y2σp1/n1+ 1/n2,• Should instead useztrue=¯Y1−¯Y2σ√(1/n1+1/n2){V IF }=zignore√V IFBIOS 662 12 Cluster Randomized StudiesEffect of correlation• ztrue< zignorefor ρ > 0 and m > 1• Thus ignoring correlation will lead to inflated type Ierror• Intuition: naive approach acts as if we have more infor-mation than we doBIOS 662 13 Cluster Randomized StudiesSample Size• Sample size per armn = 2z1−α/2+ z1−β∆2V IFwhere∆ =|µ1− µ2|σBIOS 662 14 Cluster Randomized StudiesSample Size• If ρ = 0, then V IF = 1 andn = 2z1−α/2+ z1−β∆2• If ρ = 1, then V IF = mn = 2z1−α/2+ z1−β∆2m• Typically 0.1 ≤ ρ ≤ 0.4BIOS 662 15 Cluster Randomized StudiesVariance Inflation Factors• Table 18.4 text:ρm 0.001 0.01 0.02 0.05 0.12 1.001 1.01 1.02 1.05 1.105 1.004 1.04 1.09 1.20 1.4010 1.009 1.09 1.18 1.45 1.90100 1.099 1.99 2.98 5.95 10.901000 1.999 10.99 20.98 50.95 100.90BIOS 662 16 Cluster Randomized StudiesConcluding Re marks• What if cluster/group sizes vary? i.e., k = 1, . . . , mj• Use expected cluster size; Cf Manatunga, Hudgens, Chen(Biometrical Journal 2001)• Methods for group randomization include mixed modelsand generalized estimating equations (BIOS 762/3/7)BIOS 662 17 Cluster Randomized
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