Design and Analysis of Cluster Randomization Trials in Health Research

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Design and Analysis of Cluster Randomization
Trials in Health Research

• Allan Donner, Ph.D., Professor and Chair
• Department of Epidemiology Biostatistics
• The University of Western Ontario, London,
  Ontario, Canada
• Neil Klar, Ph.D., Senior Biostatistician
• Division of Preventive Oncology
• Cancer Care Ontario, Toronto, Ontario, Canada

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• Dr. Allan Donner

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• Dr. Neil Klar

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Learning Objectives

• To distinguish experimental trials based on the
  unit of randomization (e.g. individual, family,
  community).
• To appreciate the consequences of cluster
  randomization on sample size estimation and data
  analysis.
• To identify key features of a cluster
  randomization trial which need to be included in
  reports.

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What Are Cluster Randomization Trials?

• Cluster randomization trials are experiments in
  which clusters of individuals rather than
  independent individuals are randomly allocated to
  intervention groups.

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Example 1 Study Purpose To
evaluate the
effectiveness of Vitamin A
supplements on childhood mortality.

• 450 villages in Indonesia were randomly assigned
  to either participate in a Vitamin A
  supplementation scheme, or serve as a control.
• One year mortality rates were compared in the two
  groups.
• Sommer et al.
  (Lancet, 1986)

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Example 2 Study Purpose To promote
smoking cessation using community resources.

• 11 paired communities were selected and one
  member of each pair was randomly assigned to the
  intervention group.
• 5-year smoking cessation rates were compared in
  the two groups.
• Communities were matched on demographic
  characteristics
• (e.g. size, population density) and
  geographical proximity.
• COMMIT Research Group (Am J Public Health,
  1995)

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Example 3 Study Purpose To evaluate the
effectiveness of treated
nasal tissues versus
standard tissues.

• 90 families were selected and randomized to one
  of the two intervention groups separately in
  each of three family size strata (2, 3, or 4
  members per family).
• 24-week incidence of respiratory illness were
  compared in the two groups.
• Farr et al. (Am. J. Epid., 1988)

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Reasons for Adopting Cluster Randomization

• Administrative convenience
• To obtain cooperation of investigators
• Ethical considerations
• To enhance subject compliance
• To avoid treatment group contamination
• Intervention is naturally applied at the cluster
  level

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Unit of Randomization vs. Unit of Analysis

• A key property of cluster randomization trials is
  that inferences are frequently intended to apply
  at the individual level while randomization is at
  the cluster or group level. Thus the unit of
  randomization may be different from the unit of
  analysis.
• In this case, the lack of independence among
  individuals in the same cluster, i.e.
  between-cluster variation, creates special
  methologic challenges in both design and analysis.

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Implications of Between-Cluster Variation

• Presence of between-cluster variation implies
• (i) Reduction in effective sample size.
• Extent depends on degree of within-cluster
  correlation and on average cluster size.
• (ii) Standard approaches for sample size
  estimation and statistical analysis do not apply.
• Application of standard sample size approaches
  leads to an underpowered study.
• Application of standard statistical methods
  generally tends to bias p-values downwards, i.e.
  could lead to spurious statistical significance.

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Possible Reasons for Between- Cluster
Variation

• 1. Subjects frequently select the clusters to
  which they belong
• e.g., Patient characteristics could be
  related to age or sex
• differences among physicians
• 2. Important covariates at the cluster level
  affect all individuals within the cluster in the
  same manner e.g. Differences in temperature
  between nurseries may be related to infection
  rates
• 3. Individuals within clusters frequently
  interact and, as a result, may respond similarly
  e.g. Education strategies or therapies provided
  in a group setting
• 4. Tendency of infectious diseases to spread
  more rapidly within than among families or
  communities.

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Quantifying the Effect of Clustering

• Consider a trial in which k clusters of size m
  are randomly assigned to each of an experimental
  and control group. Also assume the response
  variable Y is normally distributed with common
  variance ?2
• Aim is to test H0?1 ?2. Then appropriate
  estimates of ?1 and ?2 are given by Y1, Y2, the
  usual sample means.
• Also V(Yi) (?2/km) 1 (m - 1) ? , i 1,2
• where ? is the coefficient of intracluster
  correlation.
• Then IF 1 (m- 1) ? is the variance
  inflation factor or design effect associated
  with cluster randomization.

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Variance Component Interpretation of ?

• The overall response variance ?2 may be
  expressed as the sum of two components, i.e.,
• ?2 ?2A ?2W ,
• where
• ?2A between-cluster component of variance
• ?2W within-cluster component of variance
• then
• ? ?2A / (?2A ?2W)

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Sample Size Requirements for Completely
Randomized Designs

• Comparison of Means
• Suppose k clusters of size m are to be
  assigned to each of two intervention groups.
• Then the number of subjects required per
  intervention group to test H0?1 ?2 is given by
• n (Z?/2 Z?)2 (2?2) 1 (m 1) ? / (?1 -
  ?2)2
• where ?2 ?2A ?2W
• Equivalently, the number of required clusters is
  given by k n/m.

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Example

• Hsieh (1988) reported on the results of a pilot
  study for a planned 5-year trial examining
  cardiovascular risk factors, obtaining
  cholesterol levels from 754 individuals in 4
  worksites.
• Estimated variance components were
• S2W 2209, S2A 93.
• ? value of ? assessed as ? 93 / (93 2209)
  0.04
• Assuming 70 subjects/worksite,
• IF 1 (70 - 1) 0.04 3.76

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? To obtain 80 power at ? .05 (2 sided) for
detecting a mean difference of 20 mg/dl between
intervention groups, the number of required
worksites per group is given byk (1.96
0.84)2 2(2302) (3.76) / 70 (20) 2 4.8 ? 5To
adjust for the use of normal distribution
critical values, and possible loss of follow-up,
might enroll 7 clusters per group.
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Impact on Power of Increasing the Number of
Clusters vs. Increasing Cluster Size

• Let d mean difference between intervention
  groups then,
• Var (d) (2?2 / km) 1 ( m 1) ?
• As the number of clusters k ? ? , Var(d) ? 0
  but,
• as the cluster size size m ? ? , Var (d) ? (2?2
  ?) / k 2?2A/k
• Trial randomizing between 30 and 50 individuals
  will tend to have almost the same statistical
  power as trials randomizing the same number of
  much larger units. But clusters of larger size
  are often recruited for very practical reasons
  (to reduce contamination, to avoid logistic or
  ethical problems, etc.).

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Factors Influencing Loss of Precision

• 1. Interventions often applied on a group basis
  with little or no attention given to individual
  study participants.
• 2. Some studies permit the immigration of new
  subjects after baseline.
• 3. Entire clusters, rather than just
  individuals, may be lost to follow-up.
• 4. Over-optimistic expectations regarding
  effect size.

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Strategies for Improving Precision in Cluster
Randomization Trials

• 1. Establish cluster-level eligibility criteria
  so as to reduce between-cluster variability
  e.g., geographical restrictions.
• 2. Consider increasing the number of clusters
  randomized, even if only in the control group.
• 3. Consider matching or stratifying in the design
  by baseline variable having prognostic
  importance.
• 4. Obtain baseline measurements on other
  potentially important prognostic variables.
• 5. Take repeated assessments over time from the
  same clusters or from different clusters of
  subjects.
• 6. Develop a detailed protocol for ensuring
  compliance and minimizing loss to follow-up.

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The Importance of Cluster Level Replication

• Some investigators have designed community
  intervention trials in which exactly one cluster
  has been assigned to each intervention group.
• Such trials invariably result in interpretational
  difficulties caused by the total confounding of
  two sources of variation
• the variation in response due to the effect of
  intervention, and
• the natural variation that exists between the two
  communities (clusters) even in the absence of an
  intervention effect.
• Analysis is only possible under the untenable
  assumption that there is no clustering of
  individuals responses within communities.
• More attention to the effects of clustering when
  determining sample size might help to eliminate
  designs which lack replication.

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Analysis of Binary Outcomes

• Objective
• To assess the effects of interventions on
  individuals when clusters are the sampling unit.
• Statistical Issue
• Responses on individuals within the same
  cluster tend to be positively correlated,
  violating the assumption of independence required
  for the application of standard statistical
  methods.

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• Example
• Data obtained from a study evaluating the effect
  of school-based interventions in reducing
  adolescent tobacco use.
• 12 school units were randomly assigned to each of
  four conditions, including three intervention
  conditions and a control condition (existing
  curriculum).
• We compare here the effect of the SFG (Smoke Free
  Generation) intervention to the existing
  curriculum (EC) with respect to reducing the
  proportion of children who report using smokeless
  tobacco after four years of follow-up.

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Data
Overall rates of tobacco use Control 91/1479
.062 Experimental 58/1341 .043
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Procedures Reviewed

• 1. Standard chi-squared test
• 2. Two sample t-test on school-specific
• event rates
• 3. Direct adjustment of standard chi-
• square test
• 4. Generalized estimating equations
• approach

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Hypothesis to be tested
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Two-sample T-test Comparing Average Values of the
Event Rates
Pooled variance (.0352 .0262) / 2
.00095 t22 (.060 - .039) / ?.00095(1/12
1/12) 1.66 (p .11)
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Adjusted Chi-square Test

• An adjustment which depends on computing
  clustering correction factors for each group, is
  given by
• C1 ?m1j 1 (m1j - 1) ?/ ?m1j ,
• C2 ?m2j 1 (m2j - 1) ?/ ?m2j ,
• where m1j, m2j are the cluster sizes within
  groups 1 and 2, respectively, and ? is an
  estimate of within-cluster correlation.

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For Data in the Example

• ? .011, C1 2.599, C2 2.536
• Then the adjusted one degree of freedom
  chi-square statistic is given by
• ?2A M1 (P1 P)2 / C1 P (1 P) M2
  (P2 - P)2 / C2 P (1 - P)
• 1479 (.062 - .053)2 / 2.599
  (.053) (1 - .053) 1341 (.043 - .053) 2 /
  2.536 (.053) (1 - .053) 1.83 (p .18)
• Comments
• 1. Reduces to standard Pearson chi-square test
  when ? 0
• 2. Imposes no distributional assumptions on the
  responses within a cluster.
• 3. Assumes the Ci , i 1,2 are not
  significantly different, i.e. estimate the same
  population design effect.

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Generalized Estimating Equations Approach

• The generalized estimating equations (GEE)
  approach, developed by Liang and Zeger (1986),
  can be used to construct an extension of standard
  logistic regression which adjusts for the effect
  of clustering, and does not require parametric
  assumptions.
• Consider the logistic-regression model given by
• log P / (1 - P) ?0 ?1 ?
• where ? 1 if experimental
• and ? 0 if control
• The odds ratio for the effect of intervention is
  given
• by exp (?1)

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Result

• The resulting robust one degree of freedom
  chi-square statistic, constructed using a working
  exchangeable correlation matrix is given by
• X2LZ 2.62 (? 0.11).
• Comments
• 1) The GEE extensions of logistic regression
  were fit using the statistical package SAS. These
  procedures are also available in several other
  statistical packages (e.g., STATA, SUDAAN).
• 2) Robust statistical inference are valid so
  long as there are large numbers of clusters even
  if the working correlation is not correctly
  specified.
• 3) Can be extended to model dependence on
  individual level covariates.

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Summary of Results
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Reporting Cluster Randomization Trials

• Reporting of study design
• Justify the use of cluster randomization.
• Provide a clear definition of the unit of
  randomization.
• Explain how the chosen sample size or statistical
  power calculations accounts for between-cluster
  variation.

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Reporting of study results

• Provide the number of clusters randomized, the
  average cluster size and the number of subjects
  selected for study from each cluster.
• Provide the values of the intracluster
  correlation coefficient as calculated for the
  primary outcome variables.
• Explain how the reported statistical analyses
  account for between-cluster variations.

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References Links

• References
• 1. Donner A, Klar N. Design and Analysis of
  Cluster Randomization Trials in Health Research.
  Arnold, London 2000
• 2. Statistics notes Sample size in cluster
  randomisation Sally M Kerry and J Martin Bland
  BMJ 1998 316 549.

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• 3. Cluster randomised trials time for
  improvement Marion K Campbell and Jeremy M
  Grimshaw BMJ 1998 317 1171- 1172
• 4. Ethical issues in the design and conduct of
  cluster randomised controlled trials Sarah J L
  Edwards, David A Braunholtz, Richard J Lilford,
  and Andrew J Stevens BMJ 1999 318 1407-1409.

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• Links
• The last three articles are available at
  http//www.bmj.com/
• Arnold Publishers
• http//www.arnoldpublishers.com/