1. Efficiency of an Unbalanced Design in Collecting Time to Event Data with Interval Censoring
Present by: Peiyao Cheng
University of South Florida
Jaeb Center for Health Research

2. Longitudinal Study Design
Balanced design
Every subject has the same frequency of examinations over the course of study, which can be evenly spaced or unevenly spaced
Unbalanced design
The visit schedule for a subject depend on the treatment method (in RCT) or certain risk factor measured at baseline or during the study

3. Interval-Censored Time to Event Data
Interval-censored data arises when a failure time T can not be directly observed, but can only be determined to lie in an interval obtained from a sequence of examination times
Previous study show that the precision of event rate estimation can be increased when the total number of examinations in fixed study duration increases and this increase is greater when event rate per study duration is higher

4. Unbalanced design with only increasing the examination frequency in the high risk group based on a baseline risk factor is more efficient than balanced design when estimating the covariate effect from interval-censored time to event data

5. Sampling Variance of Covariate Effect --- Theoretical Formula
Assumptions
A longitudinal study with evenly spaced visit schedules
The primary study outcome is time to an event, which follows exponential distribution
Study duration is fixed time T, no skipping of visit or early dropout
A baseline risk factor separates the study cohort into two strata, and a common binary covariate of interest Z in both strata and has common effect on event rate as 𝑒 𝛽
stratum 1 – low risk group with baseline event rate 𝜆 1 , sample size is 2 𝑛 1 (𝑁= 𝑛 1 for each category of covariate Z)
stratum 2 – high risk group with baseline event rate 𝜆 2 , sample size is 2 𝑛 2 (𝑁= 𝑛 2 for each category of covariate Z)
In balanced design, assume number of visits is m for every subject
In unbalanced design, assume number of visits is m in stratum 1 and c*m in stratum 2, where c is a constant and c > 1

6. Fisher information matrix can be obtained as below:
where

8. 𝐼 23 = 𝐼 32 =0
Therefore
𝑉 𝛽 = 𝑰 11 −1 =f( 𝜆 1 , 𝜆 2 ,𝛽,𝑚,𝑐, 𝑛 1 , 𝑛 2 )
When 𝑐=1, this formula provides the sampling variance of 𝛽 for regular balanced design

9. Asymptotic Sampling Variance of 𝜷 Unbalanced Design vs. Balanced Design
The vertical axis is sampling variance of 𝛽 from unbalanced design/balanced design. The horizontal axis is the number of evenly spaced visits in the low risk group throughout the study.
In all 4 plots, assume 𝜆 1 =1, 𝑛 1 = 𝑛 2 =50, 𝑇=1, and the 4 lines in each plot represent different baseline event rates in the high risk stratum ( 𝜆 2 ).
Plot A) 𝛽=1, 𝑐=2;
Plot B) 𝛽=1, 𝑐=3;
Plot C) 𝛽=−1, 𝑐=2;
Plot D) 𝛽=−1, 𝑐=3.

10. Empirical Variances of 𝜷 Unbalanced Design vs. Balanced Design
In all 4 plots, assume
𝑛 1 = 𝑛 2 =50, 𝑇=1,
𝜆 1 =1, 𝑚 𝐻 =2 𝑚 𝐿

11. Asymptotic Variances vs. Empirical Variances
Assuming 𝜆 1 =1, 𝜆 2 =4, 𝛽=1,𝑇=1, 𝑚 𝐿 =3, 𝑚 𝐻 =2 𝑚 𝐿 , 𝑛 1 = 𝑛 2

12. Limitations Conclusion
Losses to follow-up and accrual times are not considered
Only considered evenly spaced visits throughout the study
The follow-up schedules are only based on a baseline risk factor
Conclusion
The unbalanced design should be considered when designing a longitudinal study with interval-censored time to event data, since it can help to improve efficiency with cost controlled
This mainly points to a new direction when designing medical studies with interval- censored time to event outcome. In complex situations, extensive simulations are required in order to estimate the sampling error and power under unbalanced design.

13. Thank You!