Week 3 Review Question Constrained models R version 3.4.4 (2018-03-15) -- "Someone to Lean On" # Class example, for reference > summary(sleeplmer) Linear mixed model fit by REML ['lmerMod'] Formula: Reaction ~ Days + (1 + Days | Subject) Data: sleepstudy REML criterion at convergence: 1743.6 Scaled residuals: Min 1Q Median 3Q Max -3.9536 -0.4634 0.0231 0.4634 5.1793 Random effects: Groups Name Variance Std.Dev. Corr Subject (Intercept) 612.09 24.740 Days 35.07 5.922 0.07 Residual 654.94 25.592 Number of obs: 180, groups: Subject, 18 Fixed effects: Estimate Std. Error t value (Intercept) 251.405 6.825 36.838 Days 10.467 1.546 6.771 Correlation of Fixed Effects: (Intr) Days -0.138 ## For sleepstudy the constrained model (refer to the sleeplmer in class example) that constrains ## no individual difference in time gradient (increase in reaction time per day) > sleeplmerC = lmer(Reaction ~ 1 + Days + (1 |Subject), sleepstudy) ## In terms of our model formulation guide sleeplmerC appears in the random effects to set each individual with a separate flat line # and then adds in a constant gradient in the fixed effects #comparison with class examples shows same fixed effects, #larger residual variance and variability of intercept (initial level) # and as indicated no variability for gradient (slope) over subjects > summary(sleeplmerC) Linear mixed model fit by REML ['lmerMod'] Formula: Reaction ~ 1 + Days + (1 | Subject) Data: sleepstudy REML criterion at convergence: 1786.5 Scaled residuals: Min 1Q Median 3Q Max -3.2257 -0.5529 0.0109 0.5188 4.2506 Random effects: Groups Name Variance Std.Dev. Subject (Intercept) 1378.2 37.12 Residual 960.5 30.99 Number of obs: 180, groups: Subject, 18 Fixed effects: Estimate Std. Error t value (Intercept) 251.4051 9.7467 25.79 Days 10.4673 0.8042 13.02 Correlation of Fixed Effects: (Intr) Days -0.371 # compare results by looking at confidence intervals > confint(sleeplmer, oldNames = FALSE) # to get good labels Computing profile confidence intervals ... 2.5 % 97.5 % sd_(Intercept)|Subject 14.3815822 37.715996 cor_Days.(Intercept)|Subject -0.4815007 0.684986 sd_Days|Subject 3.8011641 8.753383 sigma 22.8982669 28.857997 (Intercept) 237.6806955 265.129515 Days 7.3586533 13.575919 > confint(sleeplmerC, oldNames = FALSE) # to get good labels Computing profile confidence intervals ... 2.5 % 97.5 % sd_(Intercept)|Subject 26.007120 52.93598 sigma 27.813847 34.59105 (Intercept) 231.992326 270.81788 Days 8.886551 12.04802 Warning message: In optwrap(optimizer, par = start, fn = function(x) dd(mkpar(npar1, : convergence code 3 from bobyqa: bobyqa -- a trust region step failed to reduce q # which model better fits these data? clearly the class example, as our # plots showed considerable indiviual difference in change > anova(sleeplmerC, sleeplmer) refitting model(s) with ML (instead of REML) Data: sleepstudy Models: sleeplmerC: Reaction ~ 1 + Days + (1 | Subject) sleeplmer: Reaction ~ Days + (1 + Days | Subject) Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq) sleeplmerC 4 1802.1 1814.8 -897.04 1794.1 sleeplmer 6 1763.9 1783.1 -875.97 1751.9 42.139 2 7.072e-10 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #Further constraint would be to require all individuals to have same growth curve # which can be done via lm or better geee (which tries to account for correlation) # average growth curve point estimates identical to fixed effects > sleeplm = lm(Reaction ~ 1 + Days, sleepstudy) > summary(sleeplm) Call: lm(formula = Reaction ~ 1 + Days, data = sleepstudy) Residuals: Min 1Q Median 3Q Max -110.848 -27.483 1.546 26.142 139.953 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 251.405 6.610 38.033 < 2e-16 *** Days 10.467 1.238 8.454 9.89e-15 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 47.71 on 178 degrees of freedom Multiple R-squared: 0.2865, Adjusted R-squared: 0.2825 F-statistic: 71.46 on 1 and 178 DF, p-value: 9.894e-15