Tutorial 6

Thursday February 21MBP 1010

Kevin Brown

Linear Regression

Requires you to define?

• Y – independent variable• X – dependent variable(s)

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.

Assumes

•Linearity•Constant residual variance (homoscedasticity) / residuals normal

•Errors are independent (i.e. not clustered)

Homogeneity of variance

Outputs “estimates”

• intercept•slope•standard errors•t values•p-values•residual standard error (SSE – what is this?)•R2

Linear regression example: height vs. weightExtract 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 freedomMultiple R-squared: 0.2925, Adjusted R-squared: 0.2777 F-statistic: 19.84 on 1 and 38 DF, p-value: 5.022e-05

Linear regression example: height vs. weightExtract 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 freedomMultiple R-squared: 0.2925, Adjusted R-squared: 0.2777 F-statistic: 19.84 on 1 and 38 DF, p-value: 5.022e-05

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

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

Do these treatments interact?

• Standard approach: ANOVA

Treatment #1

Trea

tmen

t #2 Interaction