Statistics Chapter 2
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1 Statistics 9055 Chapter 2
2 Example: Children and Malaria A random sample of 100 children aged 3 15 years was taken from a village in Ghana. The children were followed for a period of eight months. At the beginning of the study, values of a particular antibody were assessed. Based on observations during the study period, the children were categorized into two groups: individuals with and without symptoms of malaria
3 Variables in the Dataset subject subject code age ab mal age in years antibody level 1 if the subject has malaria, 0 if not Note: the response variable mal is Bernoulli
4 Reading the Data into R > library(iswr) > data(malaria) > attach(malaria) > head(malaria) subject age ab mal
5 Treat age as a Factor > malglm_full<glm(mal~factor(age)+ab,family=binomial) > summary(malglm_full)
6 Output Call: glm(formula = mal ~ factor(age) + ab, family = binomial) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) e e factor(age) e e factor(age) e e factor(age) e e factor(age) e e factor(age) e e factor(age) e e factor(age) e e factor(age) e e factor(age) e e factor(age) e e factor(age) e e factor(age) e e ab e e Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 99 degrees of freedom Residual deviance: on 86 degrees of freedom AIC: Number of Fisher Scoring iterations: 17
7 Run the Analysis without age > malglm_ab<-glm(mal~ab,family=binomial) > summary(malglm_ab) Call: glm(formula = mal ~ ab, family = binomial) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) ab * --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 99 degrees of freedom Residual deviance: on 98 degrees of freedom AIC: Number of Fisher Scoring iterations: 6
8 Likelihood Ratio Test for age Recall Residual deviance: on 86 degrees of freedom Residual deviance: on 98 degrees of freedom Likelihood ratio test calculation > as.numeric(-2*loglik(malglm_full)) [1] > as.numeric(-2*loglik(malglm_ab)) [1] > lrt<-as.numeric(-2*(loglik(malglm_ab)-loglik(malglm_full))) > lrt [1] > pchisq(lrt,12,lower=false) [1]
9 Example: Animal Testing
10 Data File dead alive dose spleen
11 Initial Manipulations > animals<read.table("animaltesting.tx t",header=t) > attach(animals) > head(animals) dead alive dose spleen > y<-cbind(dead,alive) > p<-dead/(dead+alive) > stripchart(p~dose) > stripchart(p~spleen) p p
12 Analysis I > glm_animal1<-glm(y~dose+spleen,family=binomial(link=logit)) > summary(glm_animal1) Call: glm(formula = y ~ dose + spleen, family = binomial(link = logit)) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) e-09 *** dose e-09 *** spleen e-07 *** --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 29 degrees of freedom Residual deviance: on 27 degrees of freedom AIC: Number of Fisher Scoring iterations: 6
13 Analysis II > glm_animal2<-glm(y~factor(dose)+factor(spleen),family=binomial(link=logit)) > summary(glm_animal2) Call: glm(formula = y ~ factor(dose) + factor(spleen), family = binomial(link = logit)) Deviance Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error z value Pr(> z ) (Intercept) e-07 *** factor(dose) factor(dose) *** factor(dose) e-06 *** factor(dose) e-07 *** factor(dose) e-07 *** factor(spleen) * factor(spleen) *** factor(spleen) e-05 *** factor(spleen) e-07 *** --- Signif. codes: 0 *** ** 0.01 * (Dispersion parameter for binomial family taken to be 1) Null deviance: on 29 degrees of freedom Residual deviance: on 20 degrees of freedom AIC: Number of Fisher Scoring iterations: 6
14 Significance Tests: Analysis II > glm_animal2_spleen<-glm(y~factor(spleen),family=binomial) > glm_animal2_dose<-glm(y~factor(dose),family=binomial) > glm_animal2$deviance [1] > glm_animal2_dose$deviance [1] > glm_animal2_spleen$deviance [1] > devfull<-glm_animal2$deviance > devdose<-glm_animal2_dose$deviance > devspleen<-glm_animal2_spleen$deviance Testing for the significance of the dosages > pchisq(devspleen-devfull,5,lower=false) [1] e-17 Testing for the significance of the amount of spleen that is removed > pchisq(devdose-devfull,4,lower=false) [1] e-11
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