The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

66 CHAPTER 5 Estimating Means and Proportions

cost of the experiment. In a clinical trial, the sample size is usually deter-
mined by the number of patients that are recruited. Each patient must get
a regimen of drugs, have tests taken on each of a series of hospital visits,
and be examined by the investigating doctors and nurses. The patients
volunteer and do not incur much of the cost. Sometimes, the pharmaceu-
tical company will even pay the transportation cost. So the sample size
is one of the main cost drivers for the sponsor. Therefore, meeting objec-
tives with the smallest defensible sample size is important.
To illustrate the idea, let us consider a normal distribution with a
known variance, and we are simply interested in accurate estimation of
the population mean. Recall that for a sample size n , a two - sided 95%
confi dence interval for the mean is [/XnXnˆ−+ 196 .,σσˆ 196. /]. The
width of this interval is 2196 (. )σ/ n, and since the interval is sym-
metric, we can specify the requirement equally well by the half - width,
which is 196. σ/ n. We require the half - width to be no larger than d.
Then we have 196. σ/ nd≤. Since n is in the denominator of this
inequality, the minimum would occur when equality holds. But that
value need not always be an integer. To meet the requirement, we take
the next integer above the estimated value. So we solve the equation
196. σ/ nd= for n. Then ndn== 196 .(.)σσ/or 196 22 2/d.
Chernick and Friis ( 2003 , p. 177) also derive the required equal or
unequal sample sizes when considering confi dence intervals for the
difference of two normal means with a known common variance.
Without losing generality, we take n to be the smaller sample size, and
kn to be the larger sample size, with k ≥ 1 to be the ratio of the larger
to the smaller sample size. The resulting sample size n is the next integer
larger than (1.96)^2 ( k + 1 ) σ 2 /( kd 2 ). The total sample size is then ( k + 1 ) n.
This is minimized at k = 1 but for practical reasons, we may want a
larger number of patients in the treatment group in a clinical trial.

5.5 BOOTSTRAP PRINCIPLE AND BOOTSTRAP

CONFIDENCE INTERVALS

The bootstrap is a nonparametric method for making statistical infer-

ences without making parametric assumptions about the population

distribution. All that we infer about the population is the distribution

we obtain from the sample (the empirical distribution). The bootstrap

does it in a very different way than the parametric approach. It is also