# Appendix A: Estimation of Sample Size Requirements for Randomized Controlled Clinical Trials

The formulas used to estimate sample size requirements are provided in this appendix. Also provided are illustrative calculations relative to the Diabetes Control and Complications Trial described in Chapter 7: Clinical Trials.

Prior to undertaking this study, the investigators specified an alpha level (0.05, or 5%), statistical power (90%, and thus a beta level of 10%), and the outcome difference that should be detected by the trial (a reduction in the proportion of patients diagnosed with diabetic retinopathy from 20% to 10%). The baseline proportion of subjects who would develop retinopathy is derived from previous literature. The amount of reduction in retinopathy is based on clinical judgment; the following question was posed: “What would be a clinically important difference in the proportion of patients who would suffer this complication?”

The equation for sample size for a comparison of two proportions is as follows:

where *n* is the number of subjects
for each treatment group, πc and πt are the proportion
of patients that develops retinopathy within 5 years in the control
group (standard therapy) and treatment group (intensive therapy),
respectively, and *z*a and *z*b are the values that include
alpha in the two tails and beta in the lower tail of the standard
normal distribution. These values can be determined from tables
available in most statistical texts (see Dawson and Trapp, 2004;
complete publication data can be found at the end of Chapter 7: Clinical Trials).
The value for a type I error of 5% is 1.96, and the *z*b value for a type II error
of 10% is –1.28. As the acceptable level of error
decreases, *z*a and *z*b increase.

Note that in equation (1), the larger the *z*a and *z*b—that is, the smaller
the acceptable type I and type II errors—the larger the
sample size required; also the smaller the difference in πc and
πt, the larger the sample size required. What may not be
so intuitively obvious is the relation of sample size to the distance
of πc from 0.5. The part of the equation πc(1 – πc)
is maximized, and therefore the numerator is greater when πc = 0.5.
Movement of πc away from 0.5 reduces the required sample
size.

If we expected the proportion of patients on standard insulin therapy for diabetes that would develop retinopathy by year 5 to be 0.20, and we wanted this trial to be able to detect a reduction in retinopathy at 5 years from 0.20 to 0.10, then the sample size would be calculated as follows:

and *n* = 305. Therefore,
a total of 610 subjects equally divided between groups would be
required to answer the following question: “Is there a
reduction in the rate of retinopathy at ...

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