Sample Size and Power

Often the main aim of a medical study is to estimate or test an unknown parameter of interest, such as the incidence of a certain disease, the effect of a certain treatment, or the relative risk associated with a certain exposure. An important issue in the design of a study is the choice of the number of subjects to include. The larger the study sample, the more precise a parameter estimate will be, and the choice of the sample size will depend on how much precision is required. If the aim of the study is to demonstrate that a certain treatment is effective, the power of the study is very important. The power is the chance that the study, through a statistically significant treatment effect, will prove that the treatment is effective, if the treatment really is effective. The sample size should be chosen large enough to have sufficient power in case the real treatment effect is clinically relevant. Small studies can be inadequate because of too small power or too low precision. On the other hand, very large studies can have more precision than really needed or have high power even against treatment effects that are too small to be clinically relevant, leading to a waste of money. Therefore, the choice of the sample size should be well balanced. Sample size and power are strongly related; the larger the sample size, the higher the power. This relationship is quantified by what are called sample size formulae. Below, we give the ones that are most important in practice. These formulae can be very helpful in determining the

In sample size formulae, there are always one or two z scores involved. One of them regards the significance level of the statistical test or the confidence level of the confidence interval, and the other one regards the power of the test. These z scores are associated with the standard normal distribution, which is probably the most important distribution in statistics. It has mean 0 and standard deviation 1 and is graphically represented by the well-known bell-shaped curve. The total area under the curve is equal to 1. A zβ score z is defined as the point in the distribution such that on the right of zβ, the area under the curve is equal to (3, and therefore the area on the left-hand side of zβ is 1 − β. Consequently, zα/2 is the value such that the area on the right-hand side is equal to α/2. A table giving the value of zβ for different values of β or vice versa can be found in any introductory statistics book.