Introductory Biostatistics

survival rates. The comparision of the five-year survival rate from an experi-
mental treatment versus the five-year survival rate from a standard regimen,
say in the analysis of results from a trial of cancer treatments, fits the frame-
work of a comparison of two population proportions as seen in Section 12.4.2.
The problem is that for survival data, studies have staggered entry and subjects
are followed for varying lengths of time; they do not have the same probability
for the event to occur. Therefore, similar to the process of data analysis, the
process of sample size determination should be treated di¤erently for trials
where the endpoint is binary from trials when the endpoint is survival time.
As seen in Chapter 11, the log-rank test has become commonly used in the
analysis of clinical trials where event or outcome become manifest only after a
prolonged time interval. The method for sample size determination, where the
di¤erence in survival experience of the two groups in a clinical trial is tested
using the log-rank test, proceeds as followed. Suppose that the two treatments
give rise to survival ratesP 1 (for the experimental treatment) andP 2 (for the
standard regimen), respectively, at some conventional time point: say, five
years. If the ratio of the two hazards (or risk functions) in the two groups are
assumed not changing with time in a proportional hazards model (PHM) and is
y:1, the quantitiesP 1 ,P 2 , andyare related to each other by

y¼

lnP 1

lnP 2

1. The total number of eventsdfrom both treatment arms needed to be
   observed in the trial is given by

d¼ðz 1 aþz 1 bÞ^2

1 þy

1 y

 2

In this formula the quantitiesz 1 aandz 1 b are defined as in Section

12.9.2, andyis the hazards ratio.

2. Once the total number of eventsdfrom both treatment arms has been
   estimated, the total number of patientsNrequired in the trial can be cal-
   culated from

N¼

2 d

2 P 1 P 2

assuming equal numbers of patients randomized into the two treatment

arms, withN=2 in each group. In this formula the quantitiesP 1 andP 2

are five-year (or two- or three-year) survival rates.

Example 12.12 Consider the planning of a clinical trial of superficial bladder
cancer. With the current method of treatment (resection of tumor at cysto-
scopy), the recurrence-free rate [i.e., the survival rate when the event under
investigation is recurrency (the tumor comes back)] is 50% at two years. Inves-

SAMPLE SIZE DETERMINATION FOR PHASE III TRIALS 467