Introduction to linear regression


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Linear regression is the most widely used of all statistical techniques: it is the study of linear (i.e., straight-line) relationships between variables, usually under an assumption of normally distributed errors.

The first thing you ought to know about linear regression is how the strange term regression came to be applied to the subject of linear statistical models. This type of predictive model was first studied in depth by a 19th-Century scientist, Sir Francis Galton. Galton was a self-taught naturalist, anthropologist, astronomer, and statistician--and a real-life Indiana Jones character. He was famous for his explorations, and he wrote a best-selling book on how to survive in the wilderness entitled "Shifts and Contrivances Available in Wild Places." (The book is still in print and still considered a useful resource--you can find a copy in Perkins Library. Among other handy hints for staying alive--such as how to treat spear-wounds or extract your horse from quicksand--it introduced the concept of the sleeping bag to the Western World.)

Galton was a pioneer in the application of statistical methods to human measurements, and in studying data on relative heights of fathers and their sons, he observed the following phenomenon: a taller-than-average father tends to produce a taller-than-average son, but the son is likely to be less tall than the father in terms of his relative position within his own population. Thus, for example, if the father's height is x standard deviations from the mean within his own population, then you should predict that the son's height will be rx (r times x) standard deviations from the mean within his own population, where r is a number less than 1 in magnitude. (r is what will be defined below as the correlation between the height of the father and the height of the son.) The same is true of virtually any physical measurement than can be performed on parents and their offspring. This seems at first glance like evidence of some genetic or sociocultural mechanism for damping out extreme physical traits, and Galton therefore termed it a "regression toward mediocrity," which in modern terms is a "regression to the mean." But the phenomenon discovered by Galton is a mathematical inevitability: unless every son is exactly as tall as his father in a relative sense (i.e., unless the correlation is exactly equal to 1), the predictions must regress to the mean regardless of the underlying mechanisms of inheritance or culture.


Regression to the mean is an inescapable fact of life. Your children can be expected to be less exceptional (for better or worse) than you are. Your score on a final exam in a course can be expected to be less good (or bad) than your score on the midterm exam. A baseball player's batting average in the second half of the season can be expected to be closer to the mean (for all players) than his batting average in the first half of the season. And so on. The key word here is "expected." This does not mean it's certain that regression to the mean will occur, but that's the way to bet! (More precisely, that's the way to bet if you wish to minimize squared error.)

We have already seen a suggestion of regression-to-the-mean in some of the time series forecasting models we have studied already: plots of forecasts tend to be smoother--i.e., they exhibit less variability--than the plots of the original data. (This is not true of random walk models, but it is generally true of smoothing models and other models in which coefficients other than a constant term are estimated.) The intuitive explanation for the regression effect is simple: the thing we are trying to predict usually consists of a predictable component ("signal") and a statistically independent unpredictable component ("noise"). The best we can hope to do is to predict the value of the signal, and then let the noise fall where it may. Hence our forecasts will tend to exhibit less variability than the actual values, which implies a regression to the mean.

Another way to think of the regression effect is in terms of sampling bias. Suppose we select a sample of baseball players whose batting averages were much higher than the mean (or students whose grades were much higher than the mean) in the first half of the year. Presumably their averages were unusually high in part because they were "good" and in part because they were "lucky." The fact that they did so well in the first half of the year makes it probable that both their ability and their luck were better than average during that period. In the second half of the year they may be just as good, but they probably will not be as lucky. So we should predict that in the second half their performance will be closer to the mean.


Now, why do we often assume that relationships beween variables are linear?

  1. Because linear relationships are the simplest non-trivial relationships that can be imagined (hence the easiest to work with), and.....
  2. Because the "true" relationships between our variables are often at least approximately linear over the range of values that are of interest to us, and...
  3. Even if they're not, we can often transform the variables in such a way as to linearize the relationships.

And why do we often assume the errors of linear models are normally distributed?

  1. This assumption is often justified by appeal to the central limit theorem of statistics, which states that the sum of a sufficiently large number of independent random variables--whatever their individual distributions--approaches a normal distribution. Much data in business and economics is obtained by aggregation--e.g., adding up the effects of all transactions occurring in a month, quarter, year, or whatever. Insofar as individual transactions may occur randomly and somewhat independently, we might expect the variations in the aggregate data to be approximately normal in distribution.
  2. It is (again) mathematically convenient: it implies that the optimal coefficient estimates for a linear model are those that minimize the mean squared error (which are easily calculated), and it justifies the use of a host of statistical tests based on the normal family of distributions. (This family includes the t distribution, the F distribution, and the Chi-square distribution.)
  3. Even if the "true" error process is not normal in terms of the original units of the data, it may be possible to transform the data so that your model's prediction errors are approximately normal.

A variable is, by definition, a quantity that varies. (If it didn't vary, it would be a constant, not a variable.) In fitting statistical models in which some variables are used to predict others, what we hope to find is that the different variables do not vary independently (in a statistical sense), but that they tend to vary together.

In particular, when fitting linear models, we hope to find that one variable (say, Y) is varying as a straight-line function of another variable (say, X). In other words, if all other possibly-relevant variables could be held fixed, we would hope to find the graph of Y versus X to be a straight line (apart from the inevitable random errors or "noise").

A measure of the absolute amount of "variability" in a variable is (naturally) its variance, which is defined as its average squared deviation from its own mean. Equivalently, we can measure variability in terms of the standard deviation, which is defined as the square root of the variance. The standard deviation has the advantage that it is measured in the same units as the original variable, rather than squared units.

Our task in predicting Y might be described as that of "explaining" some or all of its variance--i.e., why, or under what conditions, does it deviate from its mean? Why is it not constant? That is, we would like to be able to improve on the "naive" predictive model: Ý(t) = CONSTANT, in which the best value for the constant is presumably the historical mean of Y. More precisely, we hope to find a model whose prediction errors are smaller, in a mean square sense, than the deviations of the original variable from its mean.


In using linear models for prediction, it turns out very conveniently that the only statistics of interest (at least for purposes of estimating coefficients) are the mean and variance of each variable and the correlation coefficient between each pair of variables. The coefficient of correlation between X and Y is commonly denoted by the letter r.

The correlation coefficient between two variables is a statistic that measures the strength of the linear relationship between them, on a relative (i.e., unitless) scale of -1 to +1. That is, it measures the extent to which a linear model can be used to predict the deviation of one variable from its mean given knowledge of the other's deviation from its mean at the same point in time.

The correlation coefficient is most easily computed if we first standardize each of the variables--i.e., express it in units of standard deviations from its own mean. The standardized value of X is commonly denoted by X*, and the value of X* at period t is defined as:

X*(t) = (X(t) - AVERAGE(X))/STDEV(X)

where AVERAGE(X) and STDEV(X) are the sample mean (average) and standard deviation of X, in Excel notation. For example, suppose that AVERAGE(X) = 20 and STDEV(X) = 5. If X(t) = 25, then X*(t) = 1, if X(t) = 10, then X*(t) = -2, and so on.

Now, the correlation coefficient is equal to the average product of the standardized values of the two variables. That is, if we let X* and Y* denote the standardized values of X and Y, we have:

r = AVERAGE(X*Y*) = (X*(1)Y*(1) + X*(2)Y*(2) + ... + X*(n)Y*(n))/n

... where n is the sample size. Thus, for example, if X and Y are stored in columns on a spreadsheet, you can use the AVERAGE and STDEV functions to compute their averages and standard deviations, then you can create two new columns in which the values of X* and Y* in each period are computed according to the formula above. Then create a third new column in which X* is multiplied by Y* in every period. The average of the values in the last column is the correlation between X and Y. (Of course, in Excel, you can just use the =CORREL(x,y) function to calculate a correlation coefficient!)


If the two variables tend to vary on the same sides of their respective means at the same time, then the average product of their deviations (and hence the correlation between them) will be positive, since the product of two numbers with the same sign is positive. Conversely, if they tend to vary on opposite sides of their respective means at the same time, their correlation will be negative. If they vary independently with respect to their means--that is, if one is equally likely to be above or below its mean regardless of what the other is doing--then the correlation will be zero.

The correlation coefficient is not only the average product of the standardized values, but also: the correlation coefficient is the "best" coefficient for multiplying the standardized value of one variable in order to predict the standardized value of the other. That is, the "best" linear model (in a minimum-squared-error sense) for predicting Y* from X* turns out to be:

Ý*(t) = rX* (t)

Thus, if X is observed to be 1 standard deviation above its own mean, then we should predict that Y will be r standard deviations above its own mean; if X is 2 standard deviations below its own mean, then we should be predict that Y will be 2r standard deviations below its own mean, and so on.

In graphical terms, this means that, on a scatter plot of Y* versus X*, the best-fit line for predicting Y* from X* is the line that passes through the origin and has slope r. (Note: this fact is not supposed to be obvious, but it is easily proved by elementary differential calculus of several variables.)


If we want to obtain a formula for predicting Y from X in unstandardized terms, we just need to substitute the formulas for the standardized values in the prededing equation, which then becomes:

(Ý(t) - AVERAGE(Y))/STDEV(Y) = r (X(t) - AVERAGE(X))/STDEV(X)

If we now rearrange this equation and collect constant terms, we obtain:

Ý(t) = a + bX(t)

where:

b = r(STDEV(Y)/STDEV(X))

a = AVERAGE(Y) - b(AVERAGE(X))

Notice that, as we claimed earlier, the coefficients in the linear equation for predicting Y from X depend only on the means and standard deviations of X and Y and on their coefficient of correlation. (By the way, these formulas for a and b are mathematically equivalent to the formulas for "b0" and "b", respectively, given on page 206 of your text. Personally, I find it much easier to remember the formulas in terms of the means and standard deviations and the correlation coefficient, and I find it easier to write the formula for the correlation coefficient in terms of the standardized values X* and Y* than in terms of the original values X and Y, as given on page 204 of your text.)


Perfect positive correlation (r = +1) or perfect negative correlation correlation (r = -1) is only obtained if one variable is an exact linear function of the other, without error. In such a case, one variable is merely a linear transformation of the other--they aren't really "different" variables at all!

In general, therefore, we find less-than-perfect correlation, which is to say, we find that r is less than 1 in absolute value. Therefore our prediction for Y* will typically be smaller in absolute value than our observed value for X*. That is, we will always predict Y to be closer to its own mean, in units of its own standard deviation, than X was observed to be, which is Galton's phenomenon of regression to the mean.

So, the technical explanation of the regression-to-the-mean effect hinges on two mathematical facts: (i) the correlation coefficient, calculated in the manner described above, happens to be the coefficient that minimizes the squared error in predicting Y* from X*, and (ii) the correlation coefficient is never larger than 1 in absolute value, and it is only equal to 1 when Y* is an exact (noiseless) linear function of X*.

The term "regression" has stuck and has even mutated from an intransitive verb into a transitive one since Galton's time. We don't merely say that the predictions for Y "regress to the mean"--we now say that we are "regressing Y on X" when we estimate a linear equation for predicting Y from X, and we refer to X as a "regressor" in this case.


When we have fitted a linear model, we can compute its mean squared prediction error and compare this to the variance of the original variable. As noted above, we hope to find that the MSE is less than the original variance. The relative amount by which the mean squared error is less than the variance of the original variable is referred to as the fraction of the variance that was explained by the model. For example, if the MSE is 20% less than the original variance, we say we have "explained 20% of the variance."

It turns out that in a simple regression model (a linear model with only one "X" variable), the fraction of variance explained is precisely the square of the correlation coefficient--i.e., the square of r. Hence, the fraction-of-variance-explained has come to be known as "r-squared."

In a multiple regression model (a linear model with two or more "regressors"), there are many correlation coefficients that must be computed, in addition to all the means and variances. For example, we must consider the correlation between each X variable and the Y variable, and also the correlation between each pair of X variables. In this case, it still turns out that the model coefficients and the fraction-of-variance-explained statistic can be computed entirely from knowledge of the means, standard deviations, and correlation coefficients among the variables--but the computations are no longer easy. We will leave the details to the computer...