1. Marathon Miles per Hour at NAPA Valley 2015 Marathon by Age and Gender
Gamma Regression
Marathon Miles per Hour at NAPA Valley 2015 Marathon by Age and Gender

2. Data Description
Y = Race speed (Miles per Hour) at Napa Valley 2015 Marathon (3/1/2015) for 1882 Runners (977 M, 905 F)
Predictor Variables:
Age in Years (18-76)
Gender (1=Male, 0=Female)
Age x Gender Interaction
Distribution: Gamma (Strictly Positive values, skewed right)
Potential Link Functions
Inverse Link (Conjugate)
Log Link

5. Gamma Distribution – Likelihood Function

6. Gamma Distribution – Inverse Link

7. Gamma Distribution – Inverse Link

8. Estimating f

9. Gamma Distribution – Log Link

10. Results – Inverse Link
The main effects of Age and gender are significant, the interaction is not
Deviance, Goodness-of-Fit and Likelihood Ratio Tests are Described Below

11. Results – Log Link
The main effects of Age and Gender are significant, the interaction is not.
Deviance, Goodness-of-Fit and Likelihood Ratio Tests are Described Below

12. Deviance
Deviance measures the discrepancy between the observed and fitted values for a model
Both (link) models provide a good fit (both p > 0.50)

14. Additive Model Results – Log Link

16. R Program
napaf2015 <- read.csv(" header=T)
attach(napaf2015); names(napaf2015)
gender <- factor(Gender)
napa.mod1 <- glm(mph~1,family=Gamma); summary(napa.mod1); deviance(napa.mod1)
napa.mod2 <- glm(mph~Age,family=Gamma); summary(napa.mod2)
napa.mod3 <- glm(mph ~ Age, family=Gamma(link="log")); summary(napa.mod3)
napa.mod4 <- glm(mph~gender,family=Gamma); summary(napa.mod4)
napa.mod5 <- glm(mph~Age + gender,family=Gamma); summary(napa.mod5)
napa.mod6 <- glm(mph ~ Age + gender, family=Gamma(link="log")); summary(napa.mod6)
napa.mod7 <- glm(mph~Age*gender,family=Gamma); summary(napa.mod7)
napa.mod8 <- glm(mph ~ Age*gender, family=Gamma(link="log")); summary(napa.mod8)
age1 <- min(Age):max(Age)
yhat.F <- exp( *age1)
yhat.M <- exp(( ) - ( )*age1)
plot(Age,mph,col=gender)
lines(age1,yhat.F,col=1)
lines(age1,yhat.M,col=2)
anova(napa.mod5,napa.mod7,test="Chisq")
anova(napa.mod6,napa.mod8,test="Chisq")
par(mfrow=c(2,2))
plot(Age[Gender=="F"],log(mph[Gender=="F"]))
plot(Age[Gender=="M"],log(mph[Gender=="M"]))
plot(Age[Gender=="F"],1/mph[Gender=="F"])
plot(Age[Gender=="M"],1/mph[Gender=="M"])