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 loglike <- function(lambda, sumx=100, n=15)
 {
 sumx* log(lambda) + -n*lambda
 }

This function is used to find the loglikelihood when you have lamda

lambdavals <- seq(.1, 10, by = .1)
llvals <- loglike(lambdavals)

This is a way to create a vector with different lamda values to test

plot(lambdavals, llvals, type = "l", xlab = expression(lambda),
 ylab = "log likelihood")

Here is a plot of a bunch of different lamda values being plotted on the loglikelihood function

Question 3

attach(swiss)
names(swiss)
## [1] "Fertility"        "Agriculture"      "Examination"     
## [4] "Education"        "Catholic"         "Infant.Mortality"
mod1<-lm(Infant.Mortality ~ Fertility + Agriculture + Examination + Education + Catholic)
mod2<-lm(Infant.Mortality ~ Fertility + Examination + Education + Catholic)

I used names(swiss) to get all the variables to put into mod1. mod2 is the same as mod1 except without agriculture

A <- logLik(mod2)
B<- logLik(mod1)
teststat <- -2 * (as.numeric(A)-as.numeric(B))
p.val <- pchisq(teststat, df = 1, lower.tail = FALSE)
p.val
## [1] 0.6549756

The P value of the LRT is .6549 which is very high indicating Agriculture should not be removed

anova(mod2,mod1)
## Analysis of Variance Table
## 
## Model 1: Infant.Mortality ~ Fertility + Examination + Education + Catholic
## Model 2: Infant.Mortality ~ Fertility + Agriculture + Examination + Education + 
##     Catholic
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1     42 296.32                           
## 2     41 295.07  1    1.2563 0.1746 0.6783

The P value of the ANOVA test is .6783 which is very high indicating Agriculture should not be removed

AIC(mod1)
## [1] 233.7217
AIC(mod2)
## [1] 231.9214
BIC(mod1)
## [1] 246.6727
BIC(mod2)
## [1] 243.0222

There is not a significant decrease in the AIC or BIC which suggests that it is best to keep Agriculutre in the model.