SAMPLE SIZE ESTIMATION FOR SURVIVAL OUTCOMES IN
CLUSTER-RANDOMIZED STUDIES WITH SMALL CLUSTER SIZES
BIOMETRICS (JUNE 2000)
AMITA K. MANATUNGA – THE ROLLINS SCHOOL OF PUBLIC HEALTH OF EMORY UNIVERSITY
SHANDE CHEN – RUSH-PRESBYTEIAN-ST. LUKE’S MEDICAL CENTER
PRESENTATION BY EVALYN VAERA BREIKŠS
DUKE UNIVERSITY – 24 MARCH 2017 – CURRENT PROBLEMS IN BIOSTATISTICS (BIOS 900)
OUTLINE
 Review of Survival Data
 Power Calculation Differences
 Non-CRT Sample Size Estimation Methods
 How CRT Changes Things
 Discussion of Simulation Studies
WHAT IS SURVIVAL DATA?
 Interested in Time-to-Event outcomes rather than typical measurements
 We observe 𝑌𝑖𝑘 = min 𝑇𝑖𝑘 , 𝐶𝑖𝑘 and 𝛿𝑖𝑘 , where 𝑇𝑖𝑘 is the survival time, 𝐶𝑖𝑘 the censoring
time, and 𝛿𝑖𝑘 the failure indicator of the ith individual in treatment arm k, a 1
indicating 𝑇𝑖𝑘 ≤ 𝐶𝑖𝑘 and a 0 otherwise.
 Censoring is when an individual’s survival time is not observed due to leaving the
study for other reasons, including the conclusion of the study.
 Generally fit Cox-Proportional Hazards models using the Kaplan-Meier Product-Limit
estimator to estimate the survivorship curve, that is, modeling the proportion of
individuals that have not yet experienced a failure event as a function of time and
other covariates.
HOW POWER CALCULATIONS IN SURVIVORSHIP STUDIES DIFFER
 Power is a function of number of events rather than of number of sampled individuals.
 Our treatment effect of interest is the difference in rates of survival, that is, the hazard
ratios, rather than some directly measureable difference between groups using
means.
 Necessary sample sizes are estimated by making assumptions of the event rates.
CURRENT METHODS FOR SAMPLE SIZE ESTIMATION
 Required number of events: 𝐸𝑣𝑒𝑛𝑡𝑠 =
𝑧𝛼/2 +𝑧𝛽
2
𝜋1 𝜋2 log 𝐻𝑅 2
, where 𝜋𝑖 is the proportion of events
in treatment arm 𝑖, HR is the assumed Hazard Ratio between the two groups, and the
z’s are from the standard normal distribution.
 Need an event rate, P(event)
 Work with bivariate marginal distributions
𝑃 𝑒𝑣𝑒𝑛𝑡 = 1 − (𝜋1 𝑆1 𝑇 + 𝜋2 𝑆2 𝑇 )
 𝑆𝑖 (𝑇) is the survivorship function (1-CDF), often presumed to be exponential
 Often historical or pilot data aids in estimating the parameter(s) of 𝑆𝑖 (𝑇), or just
assume some
HOW DOES CRT CHANGE THINGS?
 Now have clusters, naïve method involves averaging their sizes
 Observe 𝑌𝑖𝑗𝑘 = min 𝑇𝑖𝑗𝑘 , 𝐶𝑖𝑗𝑘 and 𝛿𝑖𝑗𝑘 , where 𝑇𝑖𝑗𝑘 is the survival time, 𝐶𝑖𝑗𝑘 the censoring
time, and 𝛿𝑖𝑗𝑘 the failure indicator of the 𝑗th individual in cluster 𝑖 in treatment arm 𝑘, a
1 indicating 𝑇𝑖𝑗𝑘 ≤ 𝐶𝑖𝑗𝑘 and a 0 otherwise.
 Generalize to the Clayton-Oakes model
𝑆 𝑡1 , 𝑡2 =
𝑆(𝑡1 )
1−𝜃
+ 𝑆(𝑡2 )
1−𝜃
−1
−1/(𝜃−1)
 𝜃 is the measure of association between 𝑇1 and 𝑇2 .

𝜃 = 1 means 𝑇’s independent, 𝜃 → ∞ means they approach maximal (+) dependence
HOW DOES CRT CHANGE THINGS? (CON’T)
 Asymptotic normality assumption, 𝜆’s are the hazard ratios for each treatment arm
𝑛𝑘 𝜆𝑘 − 𝜆𝑘 → 𝑁(0, Λ𝑘 )
Where Λ𝑘 =
𝑈𝑘 (𝜆𝑘 )
,
Γ𝑘 𝜆𝑘 2
𝑈𝑘 𝜆𝑘 = 𝐸
1
𝜆𝑘
𝑖
𝑗 𝛿𝑖𝑗𝑘 −
𝑖
𝑗 𝑦𝑖𝑗𝑘
2
,
Γ𝑘
𝜆𝑘
=
1
𝐸
𝜆2𝑘
 Much algebra later, in the framework of a hypothesis test, we arrive at
𝜆1
𝑁 𝑙𝑜𝑔
𝜆2
= 𝑧𝛼
𝛾2 𝑈2 𝜆2 + 𝛾1 𝑈1 (𝜆1 )
𝛾1 𝜆1 Γ1 𝜆1 + 𝛾2 𝜆2 Γ2 𝜆2
Where 𝑁 = 𝑛1 + 𝑛2 ,
 Solve for 𝑁 or 𝑧𝛽 as desired.
2
1
1
1
1
+
+ 𝑧𝛽 2 Λ2 + 2 Λ1
𝛾1 𝛾2
𝜆2
𝜆1
𝛾1 = 𝑛1 /𝑁,
𝛾2 = 𝑛2 /𝑁.
𝑖
𝑗 𝛿𝑖𝑗𝑘
.
SIMULATION STUDY DISCUSSION
 Compare their expected power of 90% to empirical power in simulation
 Numerically solved for N, then used a uniform distribution to simulate censorship and
compute empirical power to detect imposed clinical difference in hazard ratios
 Authors used a Weibull distribution for S(T), varying shape parameter to obtain
different event rates, more events = more power
 Overall their method is more conservative, especially with smaller cluster sizes and
lower risk ratios (that is, smaller clinical differences)

They chalked it up to non-normality conformity in small samples
REFERENCES
 Manatunga, Amita K., and Shande Chen. "Sample Size Estimation for Survival Outcomes in
Cluster‐Randomized Studies with Small Cluster Sizes." Biometrics 56.2 (2000): 616-621.
 Weaver, Mark A., PhD. "Sample Size Calculations for Survival Analysis." Family Health
International. India, Goa. Web. 24 Mar. 2017.
<http://www.icssc.org/Documents/AdvBiosGoa/Tab%2026.00_SurvSS.pdf>.