---
title: "likelihood_ratio"
author: "Prof. Eric A. Suess"
format:
  html:
    self-contained: true
---

## Examples of using the generalized likelihood ratio test.

Using the `mtcars` dataset, we will compare the full model to the reduced model.  The full model includes four of the variables in the dataset.  The reduced model includes only the variables `disp` and `carb`.  The null model includes only the intercept.  We will use the `lrtest()` function from the `lmtest` package to perform the likelihood ratio test.

# Likelihood Ratio Test for Linear Regression

```{r}
#| messages = FALSE, warning = FALSE
library(lmtest)

#fit full model
model_full <- lm(mpg ~ disp + carb + hp + cyl, data = mtcars)
model_full 

#fit reduced model
model_reduced <- lm(mpg ~ disp + carb, data = mtcars)
model_reduced 

#fit nullmodel
model_null <- lm(mpg ~ 1, data = mtcars)
model_null 

#perform likelihood ratio test for differences in models
lrtest(model_full, model_reduced) 

lrtest(model_full, model_null) 

lrtest(model_reduced, model_null) 

```

Check the calculations.

```{r}

-2*log( exp(-78.60301) / exp(-77.55790) )

-2*( - 78.60301 - (- 77.55790))

2*(  78.60301 + (- 77.55790))

pchisq(2.04522, df = 2, lower.tail = FALSE)

```



# Likelihood ratio test for logistic regression

Using the `infert` dataset, we will compare the full model to the reduced model.  The full model includes two of the variables in the dataset.  The reduced model includes only the variable `induced`.  The null model includes only the intercept.  We will use the `lrtest()` function from the `lmtest` package to perform the likelihood ratio test.

```{r}
infert

model_full <- glm(case ~ induced + spontaneous, family=binomial, data=infert)
model_full

model_reduced <- glm(case ~ induced, family=binomial, data=infert)
model_reduced

model_null <- glm(case ~ 1, family=binomial, data=infert)
model_null

lrtest (model_full, model_reduced)

lrtest (model_full, model_null)

lrtest (model_reduced, model_null)

```