1. October 6, 2009 Session 6Slide 1 PSC 5940: Running Basic Multi- Level Models in R Session 6 Fall, 2009

2. October 6, 2009 Session 6Slide 2 Running Multilevel Models in R Using lmer: “linear mixed-effects in R” Identify a grouping variable: “state” levels(state) # will show the categories: > levels(state) [1] "AK" "AL" "AR" "AZ" "CA" "CO" "CT" "DC" "DE" "FL" "GA" [12] "HI" "IA" "ID" "IL" "IN" "KS" "KY" "LA" "MA" "MD" "ME" [23] "MI" "MN" "MO" "MS" "MT" "NC" "ND" "NE" "NH" "NJ" "NM" [34] "NV" "NY" "OH" "OK" "OR" "PA" "RI" "SC" "SD" "TN" "TX" [45] "UT" "VA" "VT" "WA" "WI" "WV" "WY” Texas is element #44; Oklahoma is element #37; etc.

3. October 6, 2009 Session 6Slide 3 Running Multilevel Models in R Re-name some variables for analysis income<-e130e_co educ<-e2b_edu Run a simple linear model for comparison: OLS1<-lm(income ~ educ) lm(formula = income ~ educ) Residuals: Min 1Q Median 3Q Max -9.2963 -2.5845 -0.5845 1.4600 16.5934 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 2.05071 0.27953 7.336 3.58e-13 *** educ 1.17794 0.07544 15.613 < 2e-16 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 3.704 on 1506 degrees of freedom (190 observations deleted due to missingness) Multiple R-squared: 0.1393,Adjusted R-squared: 0.1387 F-statistic: 243.8 on 1 and 1506 DF, p-value: < 2.2e-16

4. October 6, 2009 Session 6Slide 4 Running Multilevel Models in R For a simple-minded intercept-varying model (with no slope coefficients): ML1<-lmer(income ~ 1 + (1 | state)) Formula: income ~ 1 + (1 | state) AIC BIC logLik deviance REMLdev 8480 8496 -4237 8472 8474 Random effects: Groups Name Variance Std.Dev. state (Intercept) 0.19588 0.44258 Residual 15.68736 3.96073 Number of obs: 1513, groups: state, 51 Fixed effects: Estimate Std. Error t value (Intercept) 6.0937 0.1304 46.75

5. October 6, 2009 Session 6Slide 5 Running Multilevel Models in R To see the fixed effect: fixef(ML1) Returns the average intercept: 6.093686 ranef(ML1) Returns the variation for each state around the mean intercept: $state (Intercept) AK 0.03582853 AL -0.34874818 AR -0.35354326 AZ -0.09795315 CA 0.74016962 CO 0.22587276 (etc.)

6. October 6, 2009 Session 6Slide 6 Running Multilevel Models in R A somewhat more interesting ML model: ML2<-lmer(income ~ educ + (1 | state)) Returns a model with a fixed slope and varying intercepts. Summary gets you this: Formula: income ~ educ + (1 | state) AIC BIC logLik deviance REMLdev 8238 8259 -4115 8224 8230 Random effects: Groups Name Variance Std.Dev. state (Intercept) 0.13219 0.36357 Residual 13.59123 3.68663 Number of obs: 1508, groups: state, 51 Fixed effects: Estimate Std. Error t value (Intercept) 2.0361 0.2867 7.102 educ 1.1751 0.0757 15.524

7. October 6, 2009 Session 6Slide 7 Running Multilevel Models in R To observe the model estimates: fixef(ML2): (Intercept) educ 2.036075 1.175145 ranef(ML2): Calculation of the intercept for Texas (46 th state): coef(ML2)$state[46,1], returns: [1] 2.169662 $state (Intercept) AK 3.310271e-02 AL -3.366027e-01 AR -2.271760e-01 AZ -1.131920e-01 CA 4.937171e-01 CO 6.491345e-02 CT 2.490139e-01

8. October 6, 2009 Session 6Slide 8 Running Multilevel Models in R To calculate the 95% confidence interval for Texas: coef(ML2)$state[46,1]+c(-2,2)*se.ranef(ML2)$state[46][1] 1.527386 2.811937 The 95% confidence interval for the model slope is: fixef(ML2)["educ"]+c(-2,2)*se.fixef(ML2)["educ"] which returns: [1] 1.023752 1.326537

9. October 6, 2009 Session 6Slide 9 Running Multilevel Models in R A still more interesting ML model: ML2<-lmer(income ~ educ + (1 + educ | state)) Returns a model with both a varying slope and intercept for each state. Summary gets you this: Formula: income ~ educ + (1 + educ | state) AIC BIC logLik deviance REMLdev 8233 8265 -4111 8216 8221 Random effects: Groups Name Variance Std.Dev. Corr state (Intercept) 0.65751 0.81087 educ 0.13761 0.37096 -1.000 Residual 13.36960 3.65645 Number of obs: 1508, groups: state, 51 Fixed effects: Estimate Std. Error t value (Intercept) 2.1212 0.3172 6.687 educ 1.1431 0.1017 11.235

10. October 6, 2009 Session 6Slide 10 Running Multilevel Models in R To observe the model estimates: fixef(ML3): (Intercept) educ 2.121166 1.143087 ranef(ML3): Calculation of the intercept and slopes for Texas: coef(ML3)$state[46,1], returns: [1] 1.662176 coef(ML3)$state[46,2], returns: [2] 1.353068 $state (Intercept) educ AK -0.062791841 0.028726346 AL 1.064733054 -0.487099757 AR 0.716358907 -0.327723694 AZ -0.174025953 0.079614321 CA -0.970880883 0.444163765 CO -0.594929356 0.272171455 CT -0.951004214 0.435070481

11. October 6, 2009 Session 6Slide 11 Workshop 1: Build ML Model using Ideology to Predict GHG Risk Use the state variable as the group level How much is the model residual reduced by allowing states to vary? Present it to me in 20 min.

12. October 6, 2009 Session 6Slide 12 BREAK

13. October 6, 2009 Session 6Slide 13 Workshop 2: Data presentations Sources, characteristics Preliminary group-level models?

14. October 6, 2009 Session 6Slide 14 For Next Week Read Gelman & Hill Ch. 13 Build plots: Figure out how to replicate Figure 12.4 (p. 257) code is shown on p. 262. Present your initial group-level models