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Last Time:. Relationships between Variables Started Simple Least Squares Regression. Relationships among Variables: Interpretations. One variable is used to “explain” another variable. X Variable Independent Variable Explaining Variable Exogenous Variable Predictor Variable. Y Variable

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  1. Last Time: Relationships between Variables Started Simple Least Squares Regression

  2. Relationships among Variables: Interpretations One variable is used to “explain” another variable X Variable Independent Variable Explaining Variable Exogenous Variable Predictor Variable Y Variable Dependent Variable Response Variable Endogenous Variable Criterion Variable

  3. Scatter Plots Y X

  4. The 1970 Vietnam War Draft Lottery http://www.sss.gov/lotter1.htm http://lib.stat.cmu.edu/DASL/Stories/DraftLottery.html

  5. Scatter Plots Y Venus Mars X

  6. Example: Performance in Experiment PRACTICE: Performance Score in a Practice Session TRIAL: Performance Score in a Trial Session Suppose these scores are Interval Scale Case i = Respondent i Sample Size: 10 Respondents

  7. Simple Least-Squares Regression Y 1 X

  8. We will end up being reasonably confident that the true regression line is somewhere in the indicated region. Y X

  9. Estimated Regression Line Y errors/residuals X

  10. Estimated Regression Line Y X

  11. Estimated Regression Line Y Wrong Picture! Wrong Picture! X Error Terms have to be drawn vertically

  12. Estimated Regression Line Y X

  13. Y X

  14. How do we find a and b? In Least-Squares Regression:

  15. In Least-Squares Regression: Computational Formula

  16. Today Least Squares Regression (Simple Linear Regression) Correlation

  17. Last Time: Example: Performance in Experiment

  18. Can wedo this? Totals:

  19. Calculating the Least Squares Regression Line contd.

  20. Slope is 1.09 10.9 10 Intercept is -9

  21. TRIAL = 1.09 PRACTICE - 9 Slope is 1.09 10.9 10 Intercept is -9 You can’t see it in this graph

  22. A view from further away….

  23. Look at the residuals:

  24. Look at Residuals & Line Fit Residual Plot Problem: Relationship is not linear Line Fit Plot

  25. Look at Residuals & Line Fit Residual Plot Problem: Predictions are very precise for small predicted values, but very unprecise for large predicted values. (Not good)

  26. Look at Residuals Residual Plot 1 2 3 4 5 6 7 8 9 10 11 12 Problem: Lurking (third) variables (?) Here: Seasonal Trend?

  27. Outliers / Influential Data Points • An outlier is a data point with an exceptionally large residual. • An influential data point is a data point with the property that if you remove that point, then the least squares regression line changes a lot.

  28. Outliers? Influential Data Points?

  29. Interpretation of linear relationships? Number of babies Number of storks Problem: Lurking (third) variables (?)?

  30. Correlation How strong is the linear relationship between two variables X and Y? Y Slope? Depends on scale units of X and Y X

  31. Correlation Slope in regression of standardized variables How strong is the linear relationship between two variables X and Y? Does not depend on scale units of X and Y

  32. Correlation Slope in regression of standardized variables How strong is the linear relationship between two variables X and Y? This slope tells me How much a given change (in standardized units) of X translates into a change (in standardized units) of Y

  33. Let’s take a closer look…

  34. Let’s take a closer look… Formula for Regression of Y on X is replaced by the following formula for regression of standardized variables:

  35. Let’s take a closer look… Standardized Variables have Mean Zero Standardized Variables Have Variance One

  36. Let’s take a closer look… Correlation Coefficient

  37. Correlation How strong is the linear relationship between two variables X and Y? Correlation Coefficient Computational Formula:

  38. Properties of Correlation • Symmetric Measure (You can exchange X and Y and get the same value) • -1 ≤ r ≤ 1 • -1 is “perfect” negative correlation • 1 is “perfect” positive correlation • Not dependent on linear transformations of X and Y • Measures linear relationship only

  39. Let’s try it out on our X = PRACTICE, Y = TRIAL Data Set Check this calculation at home!

  40. Influence on Correlation

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