1. Applied Statistics Week 4 Exercise 3 Tick bites and suspicion of Borrelia Mihaela Frincu 20.12.2011

2. Presentation of data set gender IgG F M neg 1660 1252 pos 57 50 IgG – presence of Borrelia infection neg/pos Gender – M=male, F=female Age [years]

3. 1. Perform a logistic regression with IgG as response and gender as explanatory variable fitlog<-glm(IgG~gender, family=binomial,data=borrelia) summary(fitlog) Deviance Residuals: Min 1Q Median 3Q Max -0.2798 -0.2798 -0.2599 -0.2599 2.6097 Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -3.3715 0.1347 -25.028 <2e-16 *** genderM 0.1510 0.1973 0.765 0.444 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Dispersion parameter for binomial family taken to be 1) Null deviance: 924.89 on 3018 degrees of freedom Residual deviance: 924.31 on 3017 degrees of freedom AIC: 928.31 Number of Fisher Scoring iterations: 6

4. 2. Assess the effect of gender by a likelihood ratio test First model: infection depends on gender fitlog<-glm(IgG~gender, family=binomial,data=borrelia) Second model: infection is independent of gender: fitno<-glm(IgG~1, family=binomial,data=borrelia) Comparison of the two models: anova(fitlog,fitno,test="Chisq") Analysis of Deviance Table Model 1: IgG ~ gender Model 2: IgG ~ 1 Resid. Df Resid. Dev Df Deviance Pr(>Chi) 1 3017 924.31 23018 924.89 -1 -0.58355 0.4449 P=0.4449 there is no significant difference between the two models -> infection does not depend on gender

5. 3. Calculate the relative change in odds of a positive IgG test (odds ratio) due to gender > summary(fitlog) Coefficients: Estimate Std. Error z value Pr(>|z|) (Intercept) -3.3715 0.1347 -25.028 <2e-16 *** genderM 0.1510 0.1973 0.765 0.444 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Relative change in odds ratio due to gender is: exp(0.1510)= 1.162997

6. 4. Provide the odds ratio with a 95% CI library(multcomp) gencomp<-glht(fitlog,IgG=mcp(gender)) exp(confint(gencomp,calpha=1.96)$confint) Estimate lwr upr (Intercept) 0.034337350.02636945 0.04471287 genderM 1.163051400.78997917 1.71230915

7. 5. What value of the odds ratio corresponds to no association between gender and IgG If infection does not depend on gender we expect the odds ratio to be 1 (equal odds). 6. Do we get the same conclusion from the 2 test, the likelihood ratio test, and the 95% CI for the odds ratio We got the same answer. 2 test: P=0.444 there is no significant difference between the two models -> infection does not depend on gender Likelihood ratio test: for the effect of gender p=0.4449 -> no significant gender effect 95% CI: genderM 1.163051400.78997917 1.71230915 -> 1 is in the confidence interval

8. Reporting the results The influence of gender on the incidence of Borrelia infections was investigated by a logistic regression with IgG as response and gender as explanatory variable. The influence of the gender on the incidence of infections was not found to be significant (p=0.444). The dataset was also ivestigated using a likelihood ratio test. The result was similar (p=0.4449), which indicates that gender does not have a significant influence on the incidence of borrelia infection. The odds ratio of infection male:female for the data set was found to be (95% CI) = 1.16 (0.79-1.71) The analysis was performed in R version 2.14.0 (www.R-project.org)