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 Y ik = min T ik, C ik and δ ik, where T ik is the survival time, C ik the censoring time, and δ ik the failure indicator of the ith individual in treatment arm k, a 1 indicating T ik C ik 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 = z 2 α/2+z β, where π π 1 π 2 log HR 2 i 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 (π 1 S 1 T + π 2 S 2 T ) S i (T) is the survivorship function (1-CDF), often presumed to be exponential Often historical or pilot data aids in estimating the parameter(s) of S i (T), or just assume some
HOW DOES CRT CHANGE THINGS? Now have clusters, naïve method involves averaging their sizes Observe Y ijk = min T ijk, C ijk and δ ijk, where T ijk is the survival time, C ijk the censoring time, and δ ijk the failure indicator of the jth individual in cluster i in treatment arm k, a 1 indicating T ijk C ijk and a 0 otherwise. Generalize to the Clayton-Oakes model S t 1, t 2 = S(t 1 ) 1 θ + S(t 2 ) 1 θ 1 1/(θ 1) θ is the measure of association between T 1 and T 2. 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 Where Λ k = U k(λ k ) Γ k λ k 2, U k λ k = E n k λ k λ k N(0, Λ k ) 1 λ k i j δ ijk i j y ijk 2, Γk λk = 1 λ k 2 E i j δ ijk. Much algebra later, in the framework of a hypothesis test, we arrive at N log λ 1 λ 2 = z α γ 2 U 2 λ 2 + γ 1 U 1 (λ 1 ) γ 1 λ 1 Γ 1 λ 1 + γ 2 λ 2 Γ 2 λ 2 2 1 + 1 1 + z γ 1 γ β 2 2 λ Λ 2 + 1 2 2 λ Λ 1 1 Solve for N or z β as desired. Where N = n 1 + n 2, γ 1 = n 1 /N, γ 2 = n 2 /N.
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>.