Reaction Days

Transcription

1 Stat April 03 Week Fitting Individual Trajectories # Straight-line, constant rate of change fit > sdat = subset(sleepstudy, Subject == "37") > sdat Reaction Days Subject > lm.sdat = lm(reaction ~ Days) #OLS rate of 'decli > summary(lm.sdat) Call: lm(formula = Reaction ~ Days) Residuals: Min Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-0 ** Days e-05 ** Residual standard error:.8 on 8 d freedom > plot(days, Reaction) Multiple R-squared: 0.98, Adj R-squared: > abline(coef(lm.sdat)[], F-statistic: 8.7 on and 8 DF,p-val:.76e-05 coef(lm.sdat)[]) #see plot # Autocorrelation worries > # AR() with standard Durbin-Watson test from package lmtest > install.packages("lmtest") > dwtest(lm.sdat, alternative = "two.sided") Durbin-Watson test data: lm.sdat DW =.878, p-value = # approx DW = ( - r) alternative hypothesis: true autocorrelation is not 0 > acf(sdat) =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= # Polynomial (Quadratic, Cubic) Trajectories; Berkeley Growth Data > bgsdat = read.table(file="d:\\drr3\\stat\\week\\bgsdata", header = T) > attach(bgsdat) > plot(age, cog) #see plot #Data from the Berkeley Growth Study > lm.bgsq = lm(cog ~ age + I(age^)) #(Nancy Bailey). Data are for Child > lm.bgsc = lm(cog ~ age + I(age^) + I(age^3)) ##8 in the BGS study with age in mont > anova(lm.bgsq, lm.bgsc) #(ranging from to 60) and intellect Analysis of Variance Table #performance "cog". Model : cog ~ age + I(age^) cog age Model : cog ~ age + I(age^) + I(age^3) 4 Res.Df RSS Df Sum of Sq F Pr(>F) *** > summary(lm.bgsc) 85 9 Call: lm(formula = cog ~ age + I(age^) + I(age^3)) 88 0 Residuals: 95 Min Q Median 3Q Max Coefficients: 07 4 Estimate Std. Error t value Pr(> t ) 3 5 (Intercept) * 8 age e-3 *** 48 I(age^) e-07 *** 6 4 I(age^3) *** Residual standard error: on 7 degrees of freedom 05 4 Multiple R-squared: , Adjusted R-squared: F-statistic: 039 on 3 and 7 DF, p-value: <.e

2 Reaction Days

3 cog age

4 > dwtest(lm.bgsc, alternative = "two.sided") Durbin-Watson test data: lm.bgsc DW =., p-value = alternative hypothesis: true autocorrelation is not 0 =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= #######Count Data, Generalized Linear Models # slide for Poisson link function (log) > am = glm(cases ~ year,data=belg.aids, family=poisson(link=log)) > summary(am) Call: glm(formula = cases ~ year, family = poisson(link = log), data = belg.aids) Deviance Residuals: > belg.aids Min Q Median 3Q Max cases year Coefficients: 4 Estimate Std. Error z value Pr(> z ) (Intercept) <e-6 *** year <e-6 *** (Dispersion parameter for poisson family taken to be ) Null deviance: on degrees of freedom Residual deviance: on degrees of freedom AIC: Number of Fisher Scoring iterations: > plot(am) # gives you the set of diagnostic plots--resids vs fitted etc # Quadratic in year > am = glm(cases ~ year+i(year^),data=belg.aids, family=poisson(link=log)) > summary(am) Call: glm(formula = cases ~ year + I(year^), family = poisson(link = log), data = belg.aids) Deviance Residuals: Min Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) < e-6 *** year < e-6 *** I(year^) e-6 *** --- (Dispersion parameter for poisson family taken to be ) Null deviance: on degrees of freedom Residual deviance: 9.40 on 0 degrees of freedom AIC: Number of Fisher Scoring iterations: 4 > anova(am,am) # compare nested models Analysis of Deviance Table Model : cases ~ year Model : cases ~ year + I(year^) Resid. Df Resid. Dev Df Deviance

5 > anova(am,am, test = "Chisq") Analysis of Deviance Table Model : cases ~ year Model : cases ~ year + I(year^) Resid. Df Resid. Dev Df Deviance Pr(>Chi) <.e-6 *** > AIC(am,am) df AIC am am > # cubic doesn't help, see link > year = seq(,3,length=00) > fv = predict(am,newdata=data.frame(year=year),se=true) > plot(belg.aids$year+980,belg.aids$cases) # data > lines(year+980,exp(fv$fit),col=) # fit > lines(year+980,exp(fv$fit+*fv$se),col=3) # upper c.l. > lines(year+980,exp(fv$fit-*fv$se),col=3) # lower c.l. > # produces nice final plot, note the overlay of fit and CI bands (*se) =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= # Non-linear Models: Logistic Growth Trajectory ### ### Data on the growth of chicks on different diets. ### Hand and Crowder (996), Table A., p. 7 > Time = c(0,, 4, 6, 8, 0,, 4, 6, 8, 0, ) > weight = c(4, 5, 59, 64, 76, 93, 06, 5, 49, 7, 99, 05) > plot(time, weight) > Chick. = as.data.frame(cbind(time, weight)) > Asym = 668; xmid = 9; scal = 6 #fit not sensitive to choices of initial vals > fm3 <- nls(weight ~ SSlogis(Time, Asym, xmid, scal), data = Chick.) > summary(fm3) Formula: weight ~ SSlogis(Time, Asym, xmid, scal) > Chick. Parameters: Time weight Estimate Std. Error t value Pr(> t ) 0 4 Asym xmid ** scal e-07 *** Residual standard error:.99 on 9 d f Number of iterations to convergence: Achieved convergence tolerance: 6.6e > predict(fm3, Time) [] > #at 0 weight = 3, four parameter logistic SSfpl Self-Starting Nls Four-Parameter Log

6 AIDS model example belg.aids <- data.frame(cases=c(,4,33,50,67,74,3, 4,65,04,53,46,40),year=:3) am <- glm(cases ~ year,data=belg.aids, family=poisson(link=log)) plot(am) Residuals vs Fitted Normal Q Q Scale Location Residuals vs Leverage Residuals Cook s distance Predicted values Theoretical Quantiles Predicted values Leverage...clear trend in the residual mean + some overly influential points.

7 AIDS model example II Try a quadratic time dependence? am <- glm(cases ~ year+i(year^),data=belg.aids, family=poisson(link=log)) plot(am) Residuals vs Fitted Normal Q Q Scale Location Residuals vs Leverage Residuals Cook s distance Predicted values Theoretical Quantiles Predicted values Leverage...much better.

8 Fitted AIDS model cases year

9 weight Time