Читать книгу Introduction to Linear Regression Analysis - Douglas C. Montgomery - Страница 21

This chapter considers the simple linear regression model, that is, a model with a single regressor x that has a relationship with a response y that is a straight line. This simple linear regression model is

(2.1)

where the intercept β₀ and the slope β₁ are unknown constants and ε is a random error component. The errors are assumed to have mean zero and unknown variance σ². Additionally we usually assume that the errors are uncorrelated. This means that the value of one error does not depend on the value of any other error.

It is convenient to view the regressor x as controlled by the data analyst and measured with negligible error, while the response y is a random variable. That is, there is a probability distribution for y at each possible value for x. The mean of this distribution is

(2.2a)

and the variance is

(2.2b)

Thus, the mean of y is a linear function of x although the variance of y does not depend on the value of x. Furthermore, because the errors are uncorrelated, the responses are also uncorrelated.

The parameters β₀ and β₁ are usually called regression coefficients. These coefficients have a simple and often useful interpretation. The slope β₁ is the change in the mean of the distribution of y produced by a unit change in x. If the range of data on x includes x = 0, then the intercept β₀ is the mean of the distribution of the response y when x = 0. If the range of x does not include zero, then β₀ has no practical interpretation.