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Linear Regression and Correlation: Exploring the Relationship Between Math Scores and LSD Concentration

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  1. Chapter 11 Linear Regression and Correlation

  2. Linear Regression and Correlation • Explanatory and Response Variables are Numeric • Relationship between the mean of the response variable and the level of the explanatory variable assumed to be approximately linear (straight line) • Model: • b1 > 0  Positive Association • b1 < 0  Negative Association • b1 = 0  No Association

  3. Least Squares Estimation of b0, b1 • b0 Mean response when x=0 (y-intercept) • b1 Change in mean response when x increases by 1 unit (slope) • b0,b1 are unknown parameters (like m) • b0+b1x Mean response when explanatory variable takes on the value x • Goal: Choose values (estimates) that minimize the sum of squared errors (SSE) of observed values to the straight-line:

  4. Example - Pharmacodynamics of LSD • Response (y) - Math score (mean among 5 volunteers) • Predictor (x) - LSD tissue concentration (mean of 5 volunteers) • Raw Data and scatterplot of Score vs LSD concentration: Source: Wagner, et al (1968)

  5. Least Squares Computations Parameter Estimates Summary Calculations

  6. Example - Pharmacodynamics of LSD (Column totals given in bottom row of table)

  7. SPSS Output and Plot of Equation

  8. Inference Concerning the Slope (b1) • Parameter: Slope in the population model(b1) • Estimator: Least squares estimate: • Estimated standard error: • Methods of making inference regarding population: • Hypothesis tests (2-sided or 1-sided) • Confidence Intervals

  9. 2-Sided Test H0: b1 = 0 HA: b1 0 1-sided Test H0: b1 = 0 HA+: b1> 0 or HA-: b1< 0 Hypothesis Test for b1

  10. (1-a)100% Confidence Interval for b1 • Conclude positive association if entire interval above 0 • Conclude negative association if entire interval below 0 • Cannot conclude an association if interval contains 0 • Conclusion based on interval is same as 2-sided hypothesis test

  11. Example - Pharmacodynamics of LSD • Testing H0: b1 = 0 vs HA: b1 0 • 95% Confidence Interval for b1 :

  12. Confidence Interval for Mean When x=x* • Mean Response at a specific level x* is • Estimated Mean response and standard error (replacing unknown b0 and b1 with estimates): • Confidence Interval for Mean Response:

  13. Prediction Interval of Future Response @ x=x* • Response at a specific level x* is • Estimated response and standard error (replacing unknown b0 and b1 with estimates): • Prediction Interval for Future Response:

  14. Correlation Coefficient • Measures the strength of the linear association between two variables • Takes on the same sign as the slope estimate from the linear regression • Not effected by linear transformations of y or x • Does not distinguish between dependent and independent variable (e.g. height and weight) • Population Parameter: ryx • Pearson’s Correlation Coefficient:

  15. Correlation Coefficient • Values close to 1 in absolute value  strong linear association, positive or negative from sign • Values close to 0 imply little or no association • If data contain outliers (are non-normal), Spearman’s coefficient of correlation can be computed based on the ranks of the x and y values • Test of H0:ryx = 0 is equivalent to test of H0:b1=0 • Coefficient of Determination (ryx2) - Proportion of variation in y “explained” by the regression on x:

  16. Example - Pharmacodynamics of LSD Syy SSE

  17. Example - SPSS OutputPearson’s and Spearman’s Measures

  18. 2-Sided Test H0: ryx = 0 HA: ryx 0 1-sided Test H0: ryx = 0 HA+: ryx> 0 or HA-: ryx< 0 Hypothesis Test for ryx

  19. Analysis of Variance in Regression • Goal: Partition the total variation in y into variation “explained” by x and random variation • These three sums of squares and degrees of freedom are: • Total (TSS) DFT = n-1 • Error (SSE) DFE = n-2 • Model (SSR) DFR = 1

  20. Analysis of Variance for Regression • Analysis of Variance - F-test • H0: b1 = 0 HA: b1 0

  21. Example - Pharmacodynamics of LSD • Total Sum of squares: • Error Sum of squares: • Model Sum of Squares:

  22. Example - Pharmacodynamics of LSD • Analysis of Variance - F-test • H0: b1 = 0 HA: b1 0

  23. Example - SPSS Output

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