Linear Regressions

Tips before you start:

You can pull up documentation for a function by executing ?function (e.g. ?lm) in the Console.
Have the tidyverse package installed and the dplyr library loaded in RStudio.
```
install.packages('tidyverse')
library(dplyr)
```
dplyr cheat sheet

In this activity, we build linear models to predict the life expectancy with one or more independent/explanatory variables. Download the Life Expectancy dataset here and save it in your current directory or somewhere you can find later. Data source.

Import the dataset by typing the following:

life_expectancy <- read.csv("WHO Life Expectancy Data.csv") #change to the appropriate file path to the downloaded dataset on your computer
head(life_expectancy,2)

We are going to use only data from the year of 2015 as it is most recent. Do filtering by dplyr, which we cover in the workshop Introduction to RStudio/Data Manipulation:

life_expectancy_2015 <- life_expectancy %>% filter(Year == 2015)
head(life_expectancy_2015,2)

The lm(...) command creates linear regression models. It takes the following format:

lm([response variable] ~ [predictor variables], data = [data source])

Simple linear regression:
- A plot of Schooling vs Life expectancy shows that a linear relationship is reasonable. life_expectancy_2015$Schooling selects the Schooling column in the life_expectancy_2015 dataset; the sane applies for the Life expectancy column. Run:
```
plot(life_expectancy_2015$Schooling, life_expectancy_2015$`Life.expectancy`)
```
- We create a simple linear regression model where the response variable is Life expectancy and the predictor variable is Schooling. The model can be written as:
  
  life expectancy = slope * schooling + intercept In R:
```
lm_schooling <- lm(`Life.expectancy` ~ Schooling, data=life_expectancy_2015)
summary(lm_schooling)
```
  The small p-values (<0.001) indicate that the estimates for the intercept and slope estimates are statistically significant. The R-squared value of 0.6694 indicates that 66.94% of the variation in Life expectancy can be explained by Schooling. We can write the model mathematically as: Life expectancy = 2.2287 * Schooling + 42.9016
- Add this regression line to the plot with abline(lm_schooling)
- Get the 95% confidence interval for the coefficient estimates: confint(lm_schooling)
- Linear regression makes several assumptions about the data:
  
  Linearity of the data & constant variance: we want to check the Residuals vs Fitted plot for no pattern, the red line should be fairly flat, the points should be equally scattered.
  
  plot(lm_schooling, 1)
  
  Normality: points should be close to the line in the Normal Q-Q plot.
  
  plot(lm_schooling, 2)
  
  Overall the assumptions are met. However, there seem to be a few outliers seen in the Schooling vs. Life expectancy plot. We may want to examine these data points in further analysis.
Multiple linear regression:
- We want to expand our model to consider an additional predictor variable, BMI. Run the following code:
```
plot(select(life_expectancy_2015, one_of(c("Life expectancy", "BMI", "Schooling"))))
```
  From the plot, BMI doesn’t look as good as Schooling as a predictor of Life expectancy. But we will go ahead and fit a multiple regression model to have a concrete result.
- Create a multiple linear regression model where the response variable is Life expectancy and the independent variables are BMI and Schooling. The model can be written as: Life expectancy = slope_1 * BMI + slope_2 * Schooling + intercept
  
  Run the following code for that model:
```
lm_multiple <- lm(life_expectancy_2015$`Life expectancy` ~ life_expectancy_2015$Schooling + life_expectancy_2015$BMI, data=life_expectancy_2015)
summary(lm_multiple)
```
  As we thought, BMI is not a significant variable with a p-value of 0.234. The model is still significant however, with p-value of 2.2e^-16, because Schooling is included. We conclude that the simple regression model adequately fits the data. For the sake of completeness, the multiple regression model can be written as:
  
  Life expectancy = 2.16981 * Schooling + 0.02442 * BMI + 42.57196
- Similar to the simple model, this command produces graphs to check the model assumptions.
```
plot(lm_multiple)
```
Conduct a single or multiple regression analysis with other variables of your choosing:
- Make a scatter plot to explore their linear relationship
- Build a linear regression model
- Assess the results. Let the instructors know if you need help! This activity’s code in your Markdown file may look like this:

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