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 za and zb 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 zb value for a type II error of 10% is –1.28. As the acceptable level of error decreases, za and zb increase.

Note that in equation (1), the larger the za and zb—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 5 years from 20% to 10% using intensive rather than standard insulin therapy?” This can be restated as follows: If the true difference in rate of retinopathy at 5 years is 10% versus 20%, then the probability that the researchers will find no difference between the proportion of subjects developing retinopathy during the first 5 years of therapy with an equally divided sample size of 610 would be only 10%.

Because the diabetes trial had over 305 subjects in each group, the likelihood of not finding a true difference of this magnitude was actually less than 10%.

If average glucose levels 5 years after beginning therapy had been the measure chosen to compare the two treatment groups, the required sample size could have been determined using the following equation:

where n is the number of subjects for each treatment group, μ1 - μ2 is the detectable difference between the means of the two groups, σ is the common standard deviation of each group, and za and zb have the same meaning as in equation (1).

Again, without memorizing this formula, we can intuitively understand how its various components contribute to sample size. The greater the absolute values of za, zb, and σ, and the smaller the difference in the means, μ1 - μ2, the larger the n, or sample size, required (see Table 7–3). This makes sense, as smaller differences in means between groups would be harder to detect, and greater variability within the groups would tend to blur intergroup differences. As in all sample size calculations, the larger the values of za and zb—that is, the smaller the acceptable type I and type II errors—the larger the sample size required.

Suppose the investigators estimated that the mean glucose levels at 5 years after the start of therapy would be 200 mg/dL for patients on standard insulin therapy and 175 mg/dL for those on intensive therapy, and the pooled standard deviation would be 45 mg/dL. The sample size for each group would be calculated using equation (3):

and n = 68. This end point would have required far fewer patients to be enrolled in the trial. At the conclusion of the trial, treating physicians may not have considered a reduction in glucose levels to be a sufficiently important outcome to warrant a change to the more intensive therapy; however, if the trial found that intensive therapy reduced the onset of retinopathy, intensive therapy would be judged to be a superior treatment regimen.