# Sample Size for Paired Cohort Studies

This function gives you the minimum number of subject pairs that you require to detect a true relative risk RR with power POWER and two sided type I error probability ALPHA (Dupont, 1990; Breslow and Day, 1980).

Information required

• POWER: probability of detecting a real effect.
• ALPHA: probability of detecting a false effect (two sided: double this if you need one sided).
• r: correlation coefficient for failure between paired subjects.
• *: input either (P0 and RR) or (P0 and P1), where RR=P0/P1.
• P0: event rate in the control group.
• P1: event rate in experimental group.
• RR: risk of failure of experimental subjects relative to controls.

Practical issues

• Usual values for POWER are 80%, 85% and 90%; try several in order to explore/scope.
• 5% is the usual choice for ALPHA.
• r can be estimated from previous studies - note that r is the phi (correlation) coefficient that is given for a two by two table if you enter it into the StatsDirect r by c chi-square function. When r is not known from previous studies, some authors state that it is better to use a small arbitrary value for r, say 0.2, than it is to assume independence (a value of 0) (Dupont, 1988).
• P0 can be estimated as the population event rate. Note, however, that due to matching, the control sample is not a random sample from the population therefore population event rate can be a poor estimate of P0 (especially if confounders are strongly associated with the event).
• If possible, choose a range of relative risks that you want have the statistical power to detect.

Technical validation

The estimated sample size n is calculated as:

- where α = alpha, β = 1 - power and zp is the standard normal deviate for probability p. n is rounded up to the closest integer.