Inference about the slope parameter and correlation

Introduction The slope is the true average change in the dependent variable y associated with a 1-unit increase in x. The slope of the least squares line gives an estimate of the true slope This estimate depends on the values of Y, which are random. If we can determine the sampling distribution of the estimate, we can perform inference for the true slope.

Estimator of slope is a linear combination of normal variables The estimator of the slope is where . This is a linear combination of normal random variables , and thus it has a normal distribution.

Mean and variance of estimator of slope

Estimated variance of estimator Recall that in simple linear regression we estimate using (which the book calls ). Then the estimated standard deviation of the estimator is .

The t statistic The assumptions of the simple linear regression model then imply that the standardized variable has a t distribution with n-2 d.f.

Confidence interval for slope Confidence intervals and hypothesis tests for are then carried out in the usual manner. A confidence interval for is

Hypothesis test procedures Null hypothesis: Test statistic: Alternative hypothesis Rejection region The test versus tests the usefulness of the model.

Correlation The sample correlation coefficient gives a measure of the linear relationship among X and Y. Whereas for linear regression the X variable is fixed, here it doesn’t matter which variable is called X, and which is called Y. The statistic is related to the coefficient of determination in simple linear regression, and forms an estimate of the population correlation coefficient .

The sample correlation coefficient The sample correlation coefficient for the n pairs is Recall that , so that the estimated slope and have the same sign.

Properties of r The value of r is independent of the units in which x and y are measured r lies in the interval r = 1 if and only if all pairs lie on a straight line with positive slope, and r = -1 if and only if all pairs lie on a straight line with negative slope. The square of the correlation coefficient gives the value of the coefficient of determination from fitting the simple linear regression model. r measures the degree of the linear relationship

When is the correlation strong? Weak Moderate Strong The rationale for calling correlations weak even when they are as large in absolute value as .5 is that even in that case , so that if the linear model explains at most 25% of the observed variation, which is not very impressive.

Inferences about the population correlation coefficient We can think of the pairs as having been drawn from a bivariate population of pairs, with some joint pmf or pdf, and correlation . When the joint pdf is bivariate normal, one can carry out inference for . Let (X,Y) be bivariate normal with respective means , variances , and correlation coefficient .

Inferences about the population correlation coefficient (continued) If X = x, it can be shown that the (conditional) distribution of Y is normal with mean and variance This fits the simple linear regression model with , , and .

Inferences about the population correlation coefficient (continued) The implication is that if the observed pairs are actually drawn from a bivariate normal distribution, then the simple linear regression model is an appropriate way of studying the behavior of Y given X=x. If , then , independent of x.

Testing for the absence of correlation When is true, the test statistic has a t distribution with n-2 d.f. Alternative hypothesis Rejection region

Testing for absence of correlation (continued) The null hypothesis states that there is no linear relationship between X and Y in the population. In the context of regression analysis, we used to test for the absence of a linear relationship ( ). Since the tests are equivalent.