MS Module 17: Regression analysis confidence intervals and hypothesis testing (overview)

(The attached PDF file has better formatting.)

Reading: §12.3 Inferences about the regression coefficient β1

β1 is a parameter of the regression equation (its slope). The estimate of β1 (shown an 1 in the textbook) is a sample statistic with a sampling distribution. Inferences from regression analysis depend on the attributes of this sample statistic, such as its standard deviation, and the attributes of the estimate of σ2.

Know the degrees of freedom for each statistical test. Most problems in previous modules subtract 1 from the number of observations or the number of classes. A regression equation has two parameters, so the degrees of freedom is the number of observations minus 2. Multiple regression has more parameters, and the degrees of freedom decrease accordingly.

β1 can be expressed as a linear combination the observed Y values, where the coefficients in this combination are functions of the X values. Know the formula for these coefficients, which appears in confidence intervals, prediction intervals, and standardized residuals as well.

Final exam problems test three types of confidence intervals for regression analysis:

●    The confidence interval for β1, which is a range with upper and lower bounds.
●    The confidence interval for ŷ (the fitted value), which depends on the x value.
●    The prediction interval, part of which depends on the x value and part of which is constant.

This module computes the confidence interval for β1, t values to test hypotheses about β1, and the p values to estimate the probability of a Type I error. The next module derives the other two intervals. All three intervals are tested on the final exam.

Distinguish between the standard deviation of the error term (σ) and the standard deviation of the estimate of β1. The standard deviation of the estimate of β1 = s / √Sxx. A common error is to forget to divide by √Sxx when testing hypotheses about β1.

Know the relation of the t value for hypothesis testing and the F test for the regression analysis (t2 = f).

Know how a logistic regression is fit. The fitting procedure cannot be done by pencil and paper, so the final exam tests the odds ratios.