Lecture 18 Simple Linear Regression (Chapters )

Interaction plots in ANOVA It is a good idea to always look at the interaction plots when doing a two-way ANOVA, regardless of whether or not the test for interactions is significant. Interaction plots display the basic results of study. If there really are no interactions, then the interaction plots will consist of parallel lines.

Regression Analysis The goal: Estimate E(Y|X) = conditional mean of Y given X based on a sample. Simple Linear Regression: Assumes E(Y|X) is a straight line in X.

Uses of Regression Analysis Descriptive. Describe the association between y and x in the population observed. Passive prediction. Predict y based on x where you do not plan to manipulate x, e.g., predict today’s stock price based on yesterday’s stock price. Control. Predict what y will be if you change x, e.g., predict what your earnings will be if you obtain different levels of education.

Lurking Variables A lurking variable is a variable that has an important effect on the relationship among the variables in a study but is not included among the variables studied. Examples: –Y=Salaries of Presbyterian Ministers over time, X=Price of rum in Havana over time, Lurking Variable = Inflation rate over time. –Y=Pellagra rate in village, X=Amount of flies in village, Lurking Variable = Amount of corn in diet.

Pitfalls in Regression Analysis (1) Descriptive: If using simple linear regression, need to make sure E(Y|X) is actually approximately a straight line. (2) Passive Prediction: Need to beware of pitfall for (1) plus extrapolation and lurking variables Control: Need to beware of pitfalls for (1) and (2) plus extra caution about lurking variables. Requires a cause-and-effect relationship. Best found through a controlled experiment.

Example of Pitfall A researcher measures the number of television sets per person X and the average life expectancy Y for the world’s nations. The regression line has a positive slope – nations with many TV sets have higher life expectancies. Could we lengthen the lives of people in Rwanda by shipping them TV sets?

Residual Plots Against Time Many lurking variables change systematically over time. Useful method for detecting lurking variables: Plot residuals against time order of observation is available. If a systematic pattern is found, an understanding of the background of the data might allow you to guess what the lurking variables are. Another useful residual plot: Plot residuals vs. location of observations.

Residual Plot vs. Time Example Goal: Predict elementary mathematics enrollment (X) at college based on number of freshman students (Y). Linear Fit Math enrollment = Freshman students

Residual Plot vs. Time Residual plot suggests that a change took place between 1994 and 1995 that caused higher proportion of students to take math courses. In fact, one of schools in university changed its program in 1995 to require entering students to take another math course. Conclusion: The math dept. shouldn’t use data from before 1995 for predicting future enrollment.