HSEM 3010 notes 005 February 24, 2007 Regression Towards the Mean --------------------------- One day you begin to have a toothache. At first it's just a dull ache that maybe hurts only when you drink some cold water. But over the course of a few days it gets worse. The pain is more or less continuous and about the same intensity as a bad headache. It gets so bad you cannot sleep. You take some aspirin or tylenol, and it helps a little bit. The pain comes and goes, but mostly it keeps getting worse. Finally you decide you have to go see the dentist. So you call for an appointment. The receptionist says you can get in to see Dr. Payne next Wednesday. It's Friday now. You groan, but what else can you do? Dr. Payne doesn't have any other openings till Wednesday. So finally it's Wednesday. You spent a pretty miserable weekend. You are waiting in the dental chair. Dr. Payne comes in looking very cheerful and friendly. Dr. Payne says, "Well, Ms. Luckless, how are you today?" You say, "I had this toothache last Friday. It's still there, but it seems some better than it was when I called." Dr. Payne says, "That often happens. Let's take a look." And so your visit to the dentist begins. It does often happen that toothache pain diminishes a little bit by the time you get in to see the dentist. You may even feel a little guilty about going in to see the dentist when really, by the time you get there, it's fairly mild. It doesn't always happen - sometimes things just keep getting worse - but if the pain is somewhat variable to start with, it is likely that it will be less when you actually see the dentist than it was when it was at its worst. This is even more true for other conditions, like a sore throat. Sometimes a sore throat will heal over the course of a few days - it resolves without any treatment - so by the time you see the doctor, it isn't sore at all. ----------------------------------------------------------------------- Here's another situation. Some investigators are screening for people who have high blood pressure. They screen people at a health fair. Those people who have a diastolic blood pressure of 100 mm Hg or more are asked to come back in a week for a re-check of their blood pressure. The investigators find that, among the people who are asked to come back for a second measurement, their first screening measurement averages 104 mm Hg. But when they come back a week later, the average DBP is 97 mm Hg. This seems odd. Why did their blood pressure decrease over a 1-week period? Does asking people to come back for a second measurement somehow cure them of having high blood pressure? ------------------------------------------------------------------------ Another situation. Francis Galton, a 19th century scientist and relative of Charles Darwin, measured heights of men and their adult sons. The really tall men tended to have tall sons. It seemed likely that to some extent a person inherits his/her height from his/her parents. Tall men (and women) tend to have tall sons (and daughters). There is a lot of variability in height, but there is a definite tendency for tall parents to have tall children. But Galton noticed that although the tallest fathers had tall sons, the sons were not quite as tall as their fathers. A father who is 6'4" might have a son who is 6'1" (as happened in my own case). Once in a while a really tall father will have a son who is even taller, but on average, the height of sons of very tall fathers is a bit less than that of their fathers. The same is said to happen with IQ. The average IQ (by design) is 100. Geniuses supposedly have IQs of 140 or 160 or bigger. Suppose your IQ is 130. You are well above the population mean. What would you predict your mother's IQ would be? Some people say that a good estimate is, halfway between 100 and 130 - that is, 115, or the average of your IQ and the population mean IQ. So your mother, like you, is smarter than the typical person, but she's not smarter than you are. Don't tell your mother I said this. Actually, this rule of thumb (taking the average of your IQ and the population mean IQ) does not have much of a scientific basis. But it is at least in the right direction. ------------------------------------------------------------------------ All these situations - tooth pain, diastolic blood pressure, heights of parents and children, the IQ relationship - these are all examples of something called regression to the mean. It is a natural predictable phenomenon. Here is an explanation, in terms of blood pressure, for why it happens: Each person has a 'true' blood pressure. You can estimate what your 'true' blood pressure is by taking many repeated measurements of your blood pressure, every day over a period of, say, 60 days. Of course your blood pressure varies from one day to the next. One day it might be 84, the next day 88, the next day 80. The variability in the measurement of your blood pressure from one day to the next is called within-person variability. This is expressed in terms of a standard deviation. The within-person standard deviation of the measurement of blood pressure might be, 6 mm Hg. If a person's 'true' blood pressure is 80 mm Hg, you would expect his MEASURED blood pressure to be between 80 - 6 = 74 and 80 + 6 = 86, approximately 68% of the time. There is an overall mean value of the 'true' blood pressures. The 'true' blood pressures vary around this overall mean. The variability again is expressed in terms of a standard deviation, called the *between person* standard deviation. For diastolic blood pressure, this standard deviation is appromately 8 mm Hg. Measured blood pressures on a population sample thus have two 'sources' of variation: between person and within person. As it turns out, these sources can be combined. The standard deviation of the measured diastolic blood pressure for people drawn at random from the population will be: s = sqrt(6^2 + 8^2) = sqrt(36 + 64 ) = sqrt(100) = 10 mm Hg. Now, here is how regression towards the mean occurs. You sample, say, 200 people. You measure their diastolic blood pressure. You select out the top 10% - that is, you select the 20 people with the highest diastolic blood pressure in the sample. You measure their blood pressure again a week later. There are essentially two ways to get into the top 10%. One is to have a 'true' blood pressure which is high, and to have it measured accurately. The other is to have a normal 'true' blood pressure, but also to be having a bad day - that is, a day when the blood pressure is unusually high, perhaps because of stress or having drunk too much liquid or having just done some vigorous exercise. Certainly some of the people in the top 10% were selected simply because, on the day of measurement, their blood pressure was well above their own 'true' blood pressure. When you measure those people's blood pressure again a week, later, chances are it will be lower, closer to their true mean value, and in fact closer to the overall mean of the population. Of course their blood pressure is not exactly regressing toward the population mean. It is regressing toward their own true mean. Now if you measure the blood pressure of the 20 who were selected for being in the top 10%, you will find that in most cases their measured blood pressures are reduced from what they were the week before. That is regression toward the mean. It is a consequence of within-person variability. If there were no within-person variability, there would be no regression toward the mean. The table below shows the results of a computer simulation of an experiment with blood pressure. A total of 1000 people were screened. The 'true' blood pressure mean was 80 mm Hg, with a between-person standard deviation of 8 mm Hg. The within-person standard deviation was 6 mm Hg. The 100 people with the highest blood pressure were selected to have a second measurement made (with the same 'true' mean that they had the first time). The simulated distributions were normal, N(mean, sdev^2). The variables in the table are defined as follows: mu = mean of 'true' DBPs truedbp = simulated values of 'true' DBPs for each person measdbp1 = simulated first measurement of DBP measdbp2 = simulated second measurement of DBP diffmu1 = measdbp1 - truedbp diffmu2 = measdbp2 - truedbp The thing to notice here is the mean value of diff12. It is about 6.40. This means that, among the people with in the highest 10% of blood pressure on the first measurement, the second measurement on the average is 6.40 mm Hg lower. It almost seems like a miracle cure for hypertension! Notice, though, that the mean of the 'true' DBPs of this subset is 90.6 mm Hg - that is, over 10 mm Hg higher than the background population mean. This implies that the people in the highest 10% did tend to have higher-than-average 'true' DBPs. Notice too, though, that the mean of the first DBP measurment, measdbp1, is higher still: 97.2 mm Hg. The second measurement is 90.8, again reflecting the regression toward the mean phenomenon. ======================================================================== Illustration of regression toward the mean. 1 17:51 Saturday, February 24, 2007 The MEANS Procedure Variable N Mean Std Dev Minimum Maximum ------------------------------------------------------------------------------- mu 100 80.0000000 0 80.0000000 80.0000000 truedbp 100 90.5804279 5.9167301 74.7357387 105.2704359 measdbp1 100 97.1799623 3.9558439 92.7421211 113.5536197 measdbp2 100 90.7761687 8.6975372 70.3339536 111.9141746 diffmu1 100 17.1799623 3.9558439 12.7421211 33.5536197 diffmu2 100 10.7761687 8.6975372 -9.6660464 31.9141746 diff12 100 6.4037936 8.1586498 -14.6665606 28.9820860 ------------------------------------------------------------------------------- Regression toward the mean plays a signficant role in many clinical trials. People are often screened for a clinical trial on the basis of having high values of a risk factor (like diastolic blood pressure). The people with the highest values - which need treatment of some kind - are entered into the clinical trial. In some cases there will be only one measurement, at a single screening visit. In other cases, there are 2 or 3 screening visits which are intended to confirm that the person really does have high values of the risk factor of interest. If people are randomized strictly on the basis of first screen values, their true risk status will not be as high as that first measurement would indicate. This can have an effect on the power of the clinical trial. An example of this kind, from the MRFTT study, will be discussed in class. Last date revised: February 24, 2007