Correction to: Regression to the mean: what it is and how to deal with it

RTM is a statistical phenomenon that occurs when repeated measurements are made on the same subject or unit of observation. It happens because values are observed with random error. By random error we mean a non-systematic variation in the observed values around a true mean (e.g. random measurement error, or random fluctuations in a subject). Systematic error, where the observed values are consistently biased, is not the cause of RTM. It is rare to observe data without random error, which makes RTM a common phenomenon. Figure 1 illustrates a simple example of RTM using an artificial but realistic1 distribution of high density cholesterol (HDL) cholesterol in a single subject. The first panel shows a Normal distribution of observations for the same subject. The true mean for this subject (shown here as 50 mg/dl) is unknown in practice and we assume it remains constant over time. We assume that the variation is only due to random error (e.g. fluctuations in the HDL cholesterol measurements, or the subject’s diet). In the second panel we show an observed HDL cholesterol value (from this Normal distribution) of 30 mg/dl, a relatively low reading for this subject. If we were to observe another value in the same subject it would more likely be 30 mg/dl than 30 mg/dl (third panel). That is, the next observed value would probably be closer to the mean of 50 mg/dl (third panel). In general, when observing repeated measurements in the same subject, relatively high (or relatively low) observations are likely to be followed by less extreme ones nearer the subject’s true mean. This phenomenon was first discussed by Sir Francis Galton in 1877 (see Stigler2 for an historical account of RTM), and it was Galton who coined the phrase ‘regression to the mean’. The practical problem caused by RTM is the need to distinguish a real change from this expected change due to the natural variation. For example, in the third panel of Figure 1 we might think that the subject’s HDL cholesterol has increased when in fact the first measurement was just unusually low and the subject’s true mean HDL cholesterol has remained constant.