Читать книгу Introduction to Linear Regression Analysis - Douglas C. Montgomery - Страница 55

Now suppose that y and x are jointly distributed according to the bivariate normal distribution. That is,

(2.60)

where μ₁ and the mean and variance of y, μ₂ and the mean and variance of x, and

is the correlation coefficient between y and x. The term σ₁₂ is the covariance of y and x.

The conditional distribution of y for a given value of x is

(2.61)

where

(2.62a)

(2.62b)

and

(2.62c)

That is, the conditional distribution of y given x is normal with conditional mean

(2.63)

and conditional variance . Note that the mean of the conditional distribution of y given x is a straight-line regression model. Furthermore, there is a relationship between the correlation coefficient ρ and the slope β₁. From Eq. (2.62b) we see that if ρ = 0, then β₁ = 0, which implies that there is no linear regression of y on x. That is, knowledge of x does not assist us in predicting y.

The method of maximum likelihood may be used to estimate the parameters β₀ and β₁. It may be shown that the maximum-likelihood estimators of these parameters are

(2.64a)

and

(2.64b)

The estimators of the intercept and slope in Eq. (2.64) are identical to those given by the method of least squares in the case where x was assumed to be a controllable variable. In general, the regression model with y and x jointly normally distributed may be analyzed by the methods presented previously for the model where x is a controllable variable. This follows because the random variable y given x is independently and normally distributed with mean β₀ + β₁x and constant variance . As noted in Section 2.12.1, these results will also hold for any joint distribution of y and x such that the conditional distribution of y given x is normal.

It is possible to draw inferences about the correlation coefficient ρ in this model. The estimator of ρ is the sample correlation coefficient

(2.65)

Note that

(2.66)

so that the slope is just the sample correlation coefficient r multiplied by a scale factor that is the square root of the spread of the y’s divided by the spread of the x’s. Thus, and r are closely related, although they provide somewhat different information. The sample correlation coefficient r is a measure of the linear association between y and x, while measures the change in the mean of y for a unit change in x. In the case of a controllable variable x, r has no meaning because the magnitude of r depends on the choice of spacing for x. We may also write, from Eq. (2.66),

which we recognize from Eq. (2.47) as the coefficient of determination. That is, the coefficient of determination R² is just the square of the correlation coefficient between y and x.

While regression and correlation are closely related, regression is a more powerful tool in many situations. Correlation is only a measure of association and is of little use in prediction. However, regression methods are useful in developing quantitative relationships between variables, which can be used in prediction.

It is often useful to test the hypothesis that the correlation coefficient equals zero, that is,

(2.67)

The appropriate test statistic for this hypothesis is

(2.68)

which follows the t distribution with n − 2 degrees of freedom if H₀: ρ = 0 is true. Therefore, we would reject the null hypothesis if |t₀| > t_{α/2, n−2}. This test is equivalent to the t test for H₀: β₁ = 0 given in Section 2.3. This equivalence follows directly from Eq. (2.66).

The test procedure for the hypotheses

(2.69)

where ρ₀ ≠ 0 is somewhat more complicated. For moderately large samples (e.g., n ≥ 25) the statistic

(2.70)

is approximately normally distributed with mean

and variance

Therefore, to test the hypothesis H₀: ρ = ρ₀, we may compute the statistic

(2.71)

and reject H₀: ρ = ρ₀ if |Z₀| > Z_α/2.

It is also possible to construct a 100(1 − α) percent CI for ρ using the transformation (2.70). The 100(1 − α) percent CI is

(2.72)

where tanh u = (eu − e^−u)/(eu + e^−u).