Unit 14: Regression Revisited

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Connections

Models
  • The tests for slopes and intercepts use the t-distribution, and have a very similar structure to tests for means.  Note that, with some algebra, one can show that the estimates for the slope and estimates for the intercept are linear combinations of the y-observations, and therefore the CLT applies. This is why the t-distribution provides a good approximation for the sampling distribution.

  • This might be the most explicit example we have in the course of a statistical model.  The regression model provides both a deterministic component  (y = a + bx) and a random component (errors are N(0,sigma)).  The two added together explain how an observation is "generated".

  • Students who continue with regression will learn about multiple regression, in which there are more than one predictors of the same response variable.  By including additional predictors, scientists can provide what is called a "statistical control" for variables that might affect the response variable, and potentially isolate the effect of a single predictor while holding all other predictors fixed.
Inference
  • Clearly, this builds on regression in Units 2 & 3.  In those units, regression is treated purely as a descriptive means.  In this unit, we acknowledge that the observations have variations, and therefore slope and intercept estimates have variation, and so we use hypothesis tests to test whether slopes and intercepts are non-zero.

  • The hypothesis that the slope is 0 is used to test whether two numerical variables are independent (the null hypothesis) or associated.