# Expected Events (Two-arm Study)

### Program Code

The program is written in R.

### Source

George SL and Desu MM. Planning the size and duration of a clinical trial studying the time to some critical event. J Chronic Dis. 1971; 27(1): 15-24.

### Description

This program calculates an analysis time for a given proportion of total events, or a proportion of total events for a given analysis time. The calculations are provided both under the assumption of the null hypothesis (equal hazards in the two arms) and under the alternative hypothesis (hazard rate in the experimental arm is different from the standard arm). A table of expected events by month for the entire course of accrual and follow-up can be generated. For a time t with associated accrual period T, the expected events can be calculated by:

$$E[D(t)] = {\gamma t* \over 2} (p_1(t) + p_2(t))$$

where

• γ is the accrual rate,
• t* = min(T, t), and
• $$p_i(t) = 1-(\lambda _it^*)^{-1}exp(-\lambda _it)[exp(\lambda _it^*)-1]$$

### Input Items

Note: The specific inputs required will depend upon which calculation is requested.

• Time Unit: Select the time unit for accrual, follow-up, analysis time, and survival probabilities. Must be consistent throughout.
• Sample Size: Enter n, the planned sample size.
• Accrual Time: Enter the number of years or months of patient accrual.
• Follow-up Time: Enter the number of years or months between the last patient accrual to the time of the final analysis.
• Analysis Time: Enter the number of years or months between the start of accrual and a desired interim analysis time.
• Proportion of Total Events: Enter the proportion (percent) of total expected events for which to calculate an interim analysis time.
• Hazard Rate: Enter λ, the exponential event rate.
• Survival Probability and Time: Enter a time, t (in years or months consistent with above), and the probability of patients alive at that time, S(t). The program will calculate λ as follows.
• $$\lambda = -1n(S(t))/t$$

### Output Items

There are three options for calculations. The default option is a table of results by month, under the null and under the alternative hypothesis. The second and third options provide expected events for a given time, or time at which the expected proportion of events have occurred.