1. Tutorial 5 Thursday February 14 MBP 1010 Kevin Brown

2. Linear Regression

3. Requires you to define? Y – independent variable X – dependent variable(s)

4. Allows you to answer what questions? Is there an association (same question as the Pearson correlation coefficient) What is the association? Measured as the slope.

5. Assumes Linearity Constant residual variance (homoscedasticity) / residuals normal Errors are independent (i.e. not clustered)

6. Homogeneity of variance

7. Outputs “estimates” intercept slope standard errors t values p-values residual standard error (SSE – what is this?) R 2

8. Linear regression example: height vs. weight Extract information: > summary(lm(HW[,2] ~ HW[,1])) Call: lm(formula = HW[, 2] ~ HW[, 1]) Residuals: Min 1Q Median 3Q Max -36.490 -10.297 3.426 9.156 37.385 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -2.860 18.304 -0.156 0.876 HW[, 1] 42.090 9.449 4.454 5.02e-05 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 16.12 on 48 degrees of freedom Multiple R-squared: 0.2925,Adjusted R-squared: 0.2777 F-statistic: 19.84 on 1 and 38 DF, p-value: 5.022e-05

9. Linear regression example: height vs. weight Extract information: > summary(lm(HW[,2] ~ HW[,1])) Call: lm(formula = HW[, 2] ~ HW[, 1]) Residuals: Min 1Q Median 3Q Max -36.490 -10.297 3.426 9.156 37.385 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -2.860 18.304 -0.156 0.876 HW[, 1] 42.090 9.449 4.454 5.02e-05 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 16.12 on 48 degrees of freedom Multiple R-squared: 0.2925,Adjusted R-squared: 0.2777 F-statistic: 19.84 on 1 and 38 DF, p-value: 5.022e-05

10. Example Televisions, Physicians and Life Expectancy (World Almanac Factbook 1993) example – Residuals & Outliers – High leverage points & influential observations – Dummy variable coding – Transformations Take home messages – Regression is a very flexible tool – correlation ≠ causation

11. Dummy coding Creates an alternate variable that’s used for analysis For 2 categories you set values of … – reference level to 0 – level of interest to 1

12. Do these treatments interact? Standard approach: ANOVA Treatment #1 Treatment #2 Interaction