# What Is Regression to the Mean?

Have you ever heard that “2 tall parents will have shorter children”?

This phenomenon, known as regression to the mean, has been used to explain everything from patterns in hereditary stature (as Galton first did in 1886) to why movie sequels or sophomore albums so often flop.

So just what is regression to the mean (RTM)?

RTM is a statistical phenomenon that occurs when unusually large or unusually small measurement values are followed by values that are closer to the population mean. This is due to random measurement error or, put another way, non-systematic fluctuations around the true mean.

The problem is that RTM can make the predictable change in repeated measures look like meaningful change due to a treatment.

RTM is a particular concern in two situations:

1. Pre-post intervention study designs that target “high risk” groups (e.g., individuals with high blood pressure)
2. Two-phase sampling designs where a subset of the first sample based on initial value is chosen for further study (e.g., when selecting the highest or lowest risk group for sub-analyses or follow-up).

Luckily, there are corrections you can make at the design phase and/or analysis phase to minimize the risk of RTM.

The best solution, say most authors, is an appropriately designated control group.

A classic example is Reader, et al.’s trial in mild hypertensive patients (1980). Subjects were classified into 3 groups based on their screening values and randomized to treatment or control.

The following two tables show the baseline and follow-up values for the treatment and control groups:

Treatment:

 Screening DBP group (mm Hg) Mean screening DBP (mm Hg) Mean DBP on treatment (mm Hg) Mean fall in DBP (mm Hg) 95-99 96.9 87.2 9.7 100-104 101.9 88.8 13.1 105-109 106.7 90.2 16.5

Control/Placebo:

 Screening DBP group (mm Hg) Mean screening DBP (mm Hg) Mean DBP on placebo (mm Hg) Mean fall in DBP (mm Hg) 95-99 97.0 92.1 5.0 100-104 101.9 94.5 7.4 105-109 106.7 97.5 9.2

In both groups, as you can see, the more extreme the initial diastolic blood pressure (DBP), the greater the decline. However, the treatment group experiences a much greater improvement.

Without the appropriate control group for comparison, it would be difficult to conclude the decline was not just due to regression to the mean. (It’s important to note, however, that using a control group may not be a complete solution when extreme values are chosen).

In the design phase, another approach to minimizing RTM would be to take 2 sets of baseline measurements at 2 different times — one baseline measurement to select groups, and the other baseline measure as a covariate. (Alternatively, using the mean of several baseline measurements for the pre-value will be closer to the true mean than a single measurement).

Once you get into the analysis phase, graphs are a helpful way of detecting RTM (for example, a scatter plot of post-pre values plotted against baseline). In this phase, some authors suggest using analysis of covariance as the best method rather than difference scores (the post-pre values).

For a sub-group analysis (i.e., subgroup based on extreme values), if the total population mean and the correlation for pre- and post- measurements are known, there are formulas for correcting for RTM. (There are other formulas for correcting for RTM using the within-subject and between-subject variance, but this only applies to normally distributed data.)

The goal of all this, of course, is to become aware of RTM so we know how it can affect results. I focused on health-related data here, but regression to the mean is not limited to biological data – it can occur in any setting. And even when we can detect RTM and understand the theory, seeing how it affects our own research may still be tricky.

References:

Regression to the mean: what it is and how to deal with it. Adrian G Barnett, Jolieke C van der Pols and Annette J Dobson. International Journal of Epidemiology 2005;34:215–220.

The need to control for regression to the mean in social psychology studies. Rongjun Yu and LiChen. Frontiers in Psychology. January 2015 Volume 5 1574.

Correcting for Regression to the Mean in Behavior and Ecology. Colleen Kelly and Trevor D. Price. The American Naturalist.  Vol. 166, no. 6 December 2005.

Regression towards the mean or Why was Terminator III such a disappointment? Martin Bland. Talk first presented to the Commission for Health Improvement. January 2004.

Assessing regression to the mean effects in health care initiatives. Ariel Linden. BMC Medical Research Methodology 2013, 13:119.

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