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Simple Linear Regression

STAT 101 Dr. Kari Lock Morgan 11/6/12. Simple Linear Regression. SECTION 2.6 Interpreting coefficients Prediction Cautions Least Squares regression. Crickets and Temperature. Can you estimate the temperature on a summer evening, just by listening to crickets chirp?

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Simple Linear Regression

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  1. STAT 101 Dr. Kari Lock Morgan 11/6/12 Simple Linear Regression • SECTION 2.6 • Interpreting coefficients • Prediction • Cautions • Least Squares regression

  2. Crickets and Temperature • Can you estimate the temperature on a summer evening, just by listening to crickets chirp? • We will fit a model to predict temperature based on cricket chirp rate

  3. Crickets and Temperature Response Variable, y Explanatory Variable, x

  4. Linear Model A linear modelpredicts a response variable, y, using a linear function of explanatory variables Simple linear regressionpredicts on response variable, y, as a linear function of one explanatory variable, x

  5. Regression Line • Goal: Find a straight line that best fits the data in a scatterplot

  6. Equation of the Line The estimated regression line is Intercept Slope where x is the explanatory variable, and is the predicted response variable. • Slope: increase in predicted y for every unit increase in x • Intercept: predicted y value when x = 0

  7. Regression in StatKey

  8. Regression in RStudio

  9. Regression Model Which is a correct interpretation? • The average temperature is 37.68 • For every extra 0.23 chirps per minute, the predicted temperature increases by 1 degree • Predicted temperature increases by 0.23 degrees for each extra chirp per minute • For every extra 0.23 chirps per minute, the predicted temperature increases by 37.68

  10. Units • It is helpful to think about units when interpreting a regression equation y units y units x units degrees/ chirps per min chirps per minute degrees degrees

  11. Prediction • The regression equation can be used to predict y for a given value of x • If you listen and hear crickets chirping about 140 times per minute, your best guess at the outside temperature is

  12. Prediction

  13. Prediction • If the crickets are chirping about 180 times per minute, your best guess at the temperature is • 60 • 70 • 80

  14. Prediction • The intercept tells us that the predicted temperature when the crickets are not chirping at all is 37.68. Do you think this is a good prediction? • (a) Yes • (b) No None of the observed chirp rates are anywhere close to 0, so we have no way of knowing if the linear trend continues all the way down to 0.

  15. Regression Caution 1 • Do not use the regression equation or line to predict outside the range of x values available in your data (do not extrapolate!) • If none of the x values are anywhere near 0, then the intercept is meaningless!

  16. Duke Rank and Duke Shirts • Are the rank of Duke among schools applied to and the number of Duke shirts owned • positively associated • negatively associated • not associated • other

  17. Duke Rank and Duke Shirts • Are the rank of Duke among schools applied to and the number of Duke shirts owned • positively associated • negatively associated • not associated • other

  18. Regression Caution 2 • Computers will calculate a regression line for any two quantitative variables, even if they are not associated or if the association is not linear • ALWAYS PLOT YOUR DATA! • The regression line/equation should only be used if the association is approximately linear

  19. Regression Caution 3 • Outliers (especially outliers in both variables) can be very influential on the regression line • ALWAYS PLOT YOUR DATA! • http://illuminations.nctm.org/LessonDetail.aspx?ID=L455

  20. Life Expectancy and Birth Rate Which of the following interpretations is correct? A decrease of 0.89 in the birth rate corresponds to a 1 year increase in predicted life expectancy Increasing life expectancy by 1 year will cause birth rate to decrease by 0.89 Both Neither Coefficients: (Intercept) LifeExpectancy 83.4090 -0.8895 The model is predicting birth rate based on life expectancy, not the other way around Can only make conclusions about causality from a randomized experiment.

  21. Regression Caution 4 • Higher values of x may lead to higher (or lower) predicted values of y, but this does NOT mean that changing x will cause y to increase or decrease • Causation can only be determined if the values of the explanatory variable were determined randomly (which is rarely the case for a continuous explanatory variable)

  22. Explanatory and Response • Unlike correlation, for linear regression it does matter which is the explanatory variable and which is the response

  23. Regression Line • How do we find the best fitting line???

  24. Predicted and Actual Values • The actual response value, y, is the response value observed for a particular data point • The predicted response value, , is the response value that would be predicted for a given x value, based on a model • In linear regression, the predicted values fall on the regression line directly above each x value • The best fitting line is that which makes the predicted values closest to the actual values

  25. Predicted and Actual Values

  26. Residual The residualfor each data point is actual – predicted = • The residual is also the vertical distance from each point to the line

  27. Residual • Want to make all the residuals as small as possible. • How would you measure this?

  28. Least Squares Regression Least squares regressionchooses the regression line that minimizes the sum of squared residuals

  29. Least Squares Regression

  30. Presidential Elections • Can we use the current approval rating (50%, according to a Rasmussen Poll yesterday) to predict whether he will win the election tonight? • We can build a model using data from past elections to predict an incumbent’s margin of victory based on approval rating

  31. Presidential Elections What is Obama’s predicted margin of victory, based on his approval rating?

  32. Presidential Elections • For this number to be useful, we need an interval estimatefor Obama’s margin of victory • We’ll learn how to be compute these next class • For now: if Obama’s approval rating really is 50%, then we are 95% confident that Obama’s margin of victory will be between -8.8 and 19.4

  33. Project 2 • Group project on regression (modeling) to be done in your lab groups • Need a data set with a quantitative response variable and multiple explanatory variables; must have at least one categorical and at least one quantitative explanatory variable

  34. To Do • Read Section 2.6, 9.1 • Complete the anonymous midterm evaluation TODAY by 5pm • Do Homework 7 (due Tuesday, 11/13) • NO LATE HOMEWORK ACCEPTED – SOLUTIONS WILL BE POSTED IMMEDIATELY AFTER CLASS • Find data for Project 2 • Presentations last lab (Monday, 12/3) • Report due last day of class (Thursday, 12/6)

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