For ease of presentation, assume the pilot study will involve n independent patients for which the probability of the adverse event of interest is π, where 0 <π < 1. A 100 × (1 - α)% confidence interval is to be generated for π and an estimate of the sample size, n, is desired. Denote X as the number of patients sampled who experience the adverse event of interest. Then, the probability of observing x events in n subjects follows the usual binomial distribution. Namely,
Denote π
u
as the upper limit of the exact one-sided 100 × (1 - α)% confidence interval for the unknown proportion, π [5]. Then π
u
is the value such that
A special case of the binomial distribution occurs when zero events of interest are observed. In pilot studies with relatively few patients, this is of practical concern and warrants particular attention. When zero events are realized (i.e., x = 0), equation (1) reduces to
(1 - π
u
)n= α.
Accordingly, the upper limit of a one-sided 100 × (1 - α)% confidence interval for π is
π
u
= 1 - α1/n. (2)
The resulting 100 × (1 - α)% one-sided confidence interval is (0, 1 - α1/n).
Graphically, one can represent this interval on a plot of π against n as illustrated in Figure 1 for α = 0.05, 0.10 and 0.25. As the figure illustrates, for relatively small sample sizes, there is a large amount of uncertainty in the true value of π. It is critical to convey this uncertainty in the findings and to guard against inferring a potential treatment is harmless when no adverse effects of interest are observed with limited data. Louis [6] also cautioned the clinical observation of zero false negatives in the context of diagnostic testing stating that zero false negatives may generate unreasonable optimism regarding the rate, particularly for smaller sample sizes.
Furthermore, one can consider using (2) in other clinically important manners. For instance, an investigator may be planning a pilot study and want to know how large it would need to be to infer with 100 × (1 - α)% confidence that the true rate did not exceed a pre-specified π, say π0, given that zero adverse events were observed. Using (2), it follows that:
To illustrate the utility of this solution, consider the following example. Ototoxicity is well documented with increasing doses of cisplatin, a platinum-containing antitumoral drug that is known to be effective against a variety of solid tumors. It is of clinical interest to identify augmentative therapies that can alleviate some of the cell death since up to 31% of patients receiving initial doses of 50 mg/m2 cisplatin are expected to have irreversible hearing loss [7, 8]. Therefore, it is desirable to rule out potential treatments not consistent with this rate of hearing loss before considering more conclusive testing. Using equation (3), we would conclude that the augmentative therapy has a hearing loss rate less than 0.31, at the 90% confidence level, if a total of 7 patients are recruited and all 7 do not experience ototoxicity. Therefore, an initial sample size of 7 patients would be sufficient to identify augmentative therapies, such as heat shock or antioxidant supplements, that demonstrate preliminary efficacy in humans. In the event one or more ototoxic events are observed, then the results in relationship to the historical rate (31% in this example) may not be statistically different. The results of several of these pilot studies could then be used to rank-order potential therapies thereby proving an empirically justified approach to therapy development.