Sample solutions. Stat 8051 Homework 8
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1 Sample solutions Stat 8051 Homework 8 Problem 1: Faraway Exercise 3.1 A plot of the time series reveals kind of a fluctuating pattern: Trying to fit poisson regression models yields a quadratic model if we only consider significant polynomial effects. > n = as.numeric(discoveries) > discoveries1 = data.frame(n) > discoveries1$year = 1860:1959 > > m1.pois = glm(n~poly(year,2), family=poisson, data=discoveries1) > summary(m1.pois) glm(formula = n ~ poly(year, 2), family = poisson, data = discoveries1) (Intercept) < 2e-16 *** 1
2 Sample solutions 8 2 poly(year, 2) ** poly(year, 2) e-06 *** Null deviance: on 99 degrees of freedom Residual deviance: on 97 degrees of freedom AIC: Number of Fisher Scoring iterations: 5 There is not much overdispersion as we can see from the value of the dispersion parameter: > (dp <- sum(residuals(m1.pois,type="pearson")^2)/m1.pois$df.res) [1] The significant year effects indicate that the discovery rates vary significantly from constant with respect to year. Note Doing the half-normal plot and testing for outliers using studentized residuals reveals point 26 as a potential outlier. Removing this point actually improves the quadratic fit (check). Problem 2: Faraway Exercise 3.2 A poission model with linear effect of dose does very badly in terms of deviance: > m2.pois = glm(colonies~dose, family=poisson, data=salmonella) > summary(m2.pois) glm(formula = colonies ~ dose, family = poisson, data = salmonella) (Intercept) <2e-16 *** dose
3 Sample solutions 8 3 Null deviance: on 17 degrees of freedom Residual deviance: on 16 degrees of freedom AIC: Number of Fisher Scoring iterations: 4 The value of the dispersion parameter is very high, and we can actually check for its significance using the function dispersiontest from package AER. > require(aer) > dispersiontest(m2.pois, alternative="greater") Overdispersion test data: m2.pois z = 1.913, p-value = alternative hypothesis: true dispersion is greater than 1 sample estimates: dispersion Fitting a negative binomial model doesn t improve the fit. Instead we can try fitting polynomial link functions. Doing so reveals that a cubic fit gives all polynomial effects significant and decreases the deviance as well: > m21.pois = glm(colonies~poly(dose,3), family=poisson, data=salmonella) > summary(m21.pois) glm(formula = colonies ~ poly(dose, 3), family = poisson, data = salmonella) (Intercept) < 2e-16 *** poly(dose, 3) * poly(dose, 3) e-07 *** poly(dose, 3) e-06 ***
4 Sample solutions 8 4 Null deviance: on 17 degrees of freedom Residual deviance: on 14 degrees of freedom AIC: Number of Fisher Scoring iterations: 4 > dispersiontest(m21.pois, alternative="greater") Overdispersion test data: m21.pois z = 1.808, p-value = alternative hypothesis: true dispersion is greater than 1 sample estimates: dispersion The dispersion parameter is still high, so let us now fit a cubic negative-binomial model: > m21.nb = glm.nb(colonies~poly(dose,3), data=salmonella) > summary(m21.nb) glm.nb(formula = colonies ~ poly(dose, 3), data = salmonella, init.theta = , link = log) (Intercept) < 2e-16 *** poly(dose, 3) poly(dose, 3) *** poly(dose, 3) ** (Dispersion parameter for Negative Binomial( ) family taken to be 1) Null deviance: on 17 degrees of freedom
5 Sample solutions 8 5 Residual deviance: on 14 degrees of freedom AIC: Number of Fisher Scoring iterations: 1 Theta: 28.8 Std. Err.: x log-likelihood: Here also the higher powers of dose turn out to be significant. Problem 3: Faraway Exercise 3.7 Here the number of complaints linearly depends on the number of visits, so we fit a rate model (log(visits) as offset). The summary is as given below: glm(formula = complaints ~ residency + gender + revenue + hours + offset(log(visits)), family = poisson, data = esdcomp) (Intercept) <2e-16 *** residencyy genderm revenue hours * Null deviance: on 43 degrees of freedom Residual deviance: on 39 degrees of freedom AIC: Number of Fisher Scoring iterations: 5 Number of hours turns out to be the only significant variables. Also there is no significant overdispersion, so the poisson model is sufficient, though the fit is not good.
6 Sample solutions 8 6 > dispersiontest(m32.pois) Overdispersion test data: m32.pois z = , p-value = alternative hypothesis: true dispersion is greater than 1 sample estimates: dispersion
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