# Sample size calculations in studies with a baseline

## 2004-07-23

Many research studies evaluate all patients at baseline and then randomly assign the patients to groups, conduct the interventions, and then re-evaluate them at the end of the study. The sample size calculations for this type of study are a bit tricky.

One of the reasons that you measure the patients at baseline is that you are interested in the change or improvement that a specific intervention might produce. The change in a measure is almost always going to be less variable than the measurement itself. This gives you more precision and might allow you to use a smaller sample size.

You also might measure at baseline to use that value as a covariate. The covariate will also improve precision and might allow you to use a smaller sample size.

As a simple illustration of the increased precision, consider a study of acupuncture that appeared in the March 27, 2004 issue of the British Medical Journal.

• Acupuncture for chronic headache in primary care: large, pragmatic, randomised trial. Vickers AJ, Rees RW, Zollman CE, McCarney R, Smith CM, Ellis N, Fisher P, Van Haselen R. Bmj 2004: 328(7442); 744. [Medline] [Abstract] [Full text] [PDF]

In this study patients were randomly assigned to either normal standard of care or normal care plus additional visits to a accupuncturist. After three months, the control group had an average headache score of 23.7 (SD 16.8) and the acupuncture group had a score of 18.0 (14.8). A simple confidence interval for the difference in means is 2.0 to 9.4 which establishes that the accupuncture group did significantly better.

The authors considered a more complex model where the three month measurements were adjusted using the baseline measures as a covariate. The resulting confidence interval, 1.6 to 6.3, is shifted a bit towards zero because the control group had slightly higher headache scores at baseline. But notice also that the interval is much narrower. Using the baseline decreased the width of the confidence interval by about a third.

If you are planning a study with baseline measures, you should try to account for the greater precision you get by having a baseline. Otherwise, you would end up with an unnecessarily large sample size. This is not a trivial consideration. A reduction of one third in the width of the confidence interval, such as the one seen above, would cut your required sample by more than half.

To factor in the greater efficiency of a study with baseline measures, you need to specify

• the standard deviation of the change score,
• the within subject variation, or
• the intraclass correlation.

Typically, these numbers are hard to find. In a pinch, I have performed these calculations using a range of intraclass correlations between 0.7 and 0.9. The problem with this is that the sample size is highly sensitive to small changes in the intraclass correlation.

By the way, I like to use change scores in my analyses because they are easy to explain and to interpret. Change scores, however, are generally considered to be inferior to using the baseline as a covariate.