1. Fit the linear regression model

\[MORT=\beta_0+\beta_1*Lat+\beta_2*Long+\beta_{12}Lat*Long+\varepsilon_i\]

cancer.dat <- read.csv("Data/skincancer.csv",header=T)
cancer.lm <- lm(Mort ~ Lat*Long,data=cancer.dat)
summary(cancer.lm)

## 
## Call:
## lm(formula = Mort ~ Lat * Long, data = cancer.dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -34.649 -13.091   0.052  12.465  41.004 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)  
## (Intercept) 489.37255  185.66083   2.636   0.0115 *
## Lat          -8.08381    4.49542  -1.798   0.0789 .
## Long         -1.10396    1.98941  -0.555   0.5817  
## Lat:Long      0.02318    0.04794   0.483   0.6312  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 19.36 on 45 degrees of freedom
## Multiple R-squared:  0.6857, Adjusted R-squared:  0.6647 
## F-statistic: 32.72 on 3 and 45 DF,  p-value: 2.233e-11

2. Obtain the confidence intervals for the regression parameters (the \(\beta\)’s).

confint(cancer.lm)

##                    2.5 %      97.5 %
## (Intercept) 115.43244935 863.3126482
## Lat         -17.13804469   0.9704284
## Long         -5.11084292   2.9029310
## Lat:Long     -0.07339017   0.1197413

3. Create a plot of the residuals vs the fitted values.

cancer.resid <- residuals(cancer.lm)
cancer.fitted <- fitted(cancer.lm)
plot(cancer.fitted,cancer.resid,xlab="Predicted Values",ylab="Residuals",pch=16)
abline(h=0)

4. Create a histogram and QQ plot of the residuals.

hist(cancer.resid,col="grey",main="Histogram of Residuals",freq=FALSE)
dens <- dnorm(seq(-40,45,by=0.2),mean=mean(cancer.resid),sd=sd(cancer.resid))
lines(seq(-40,45,by=.2),dens,col="red")

qqnorm(cancer.resid)
qqline(cancer.resid)

5. Calculate the confidence interval for the mean mortality rate and the prediction interval when Lat = 33 and Long = 86.

predict(cancer.lm,data.frame(Lat=33,Long=86),interval="confidence")

##       fit      lwr      upr
## 1 193.439 182.6452 204.2327

predict(cancer.lm,data.frame(Lat=33,Long=86),interval="prediction")

##       fit      lwr      upr
## 1 193.439 152.9872 233.8907

6. Fit the reduced regression model without the interaction term.

cancer.lm2 <- lm(Mort ~ Lat+Long,data=cancer.dat)
summary(cancer.lm2)

## 
## Call:
## lm(formula = Mort ~ Lat + Long, data = cancer.dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -36.551 -13.525  -0.757  14.055  41.700 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 400.6755    28.0512  14.284  < 2e-16 ***
## Lat          -5.9308     0.6038  -9.822 7.18e-13 ***
## Long         -0.1467     0.1873  -0.783    0.438    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 19.19 on 46 degrees of freedom
## Multiple R-squared:  0.684,  Adjusted R-squared:  0.6703 
## F-statistic: 49.79 on 2 and 46 DF,  p-value: 3.101e-12