SLR: Application in R

Hands-on practice with pre-loaded R data

Paulo Fagandini

Lisbon Accounting and Business School

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Simple Linear Regression

Learning goals

Fit a simple linear regression model in R.
Interpret the slope and intercept in context.
Use residual plots to check whether the model is reasonable.
Make predictions with predict().
Practice reading summary(lm()).

Dataset

Pre-loaded R data

We will use a pre-loaded dataset in R.

For this class, we will use mtcars, because it is already available in base R and has numeric variables that work well for regression.

data(mtcars)

Inspecting: Observe

head(mtcars)

                   mpg cyl disp  hp drat    wt  qsec vs am gear carb
Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

Summary Statistics

summary(mtcars)

      mpg             cyl             disp             hp       
 Min.   :10.40   Min.   :4.000   Min.   : 71.1   Min.   : 52.0  
 1st Qu.:15.43   1st Qu.:4.000   1st Qu.:120.8   1st Qu.: 96.5  
 Median :19.20   Median :6.000   Median :196.3   Median :123.0  
 Mean   :20.09   Mean   :6.188   Mean   :230.7   Mean   :146.7  
 3rd Qu.:22.80   3rd Qu.:8.000   3rd Qu.:326.0   3rd Qu.:180.0  
 Max.   :33.90   Max.   :8.000   Max.   :472.0   Max.   :335.0  
      drat             wt             qsec             vs        
 Min.   :2.760   Min.   :1.513   Min.   :14.50   Min.   :0.0000  
 1st Qu.:3.080   1st Qu.:2.581   1st Qu.:16.89   1st Qu.:0.0000  
 Median :3.695   Median :3.325   Median :17.71   Median :0.0000  
 Mean   :3.597   Mean   :3.217   Mean   :17.85   Mean   :0.4375  
 3rd Qu.:3.920   3rd Qu.:3.610   3rd Qu.:18.90   3rd Qu.:1.0000  
 Max.   :4.930   Max.   :5.424   Max.   :22.90   Max.   :1.0000  
       am              gear            carb      
 Min.   :0.0000   Min.   :3.000   Min.   :1.000  
 1st Qu.:0.0000   1st Qu.:3.000   1st Qu.:2.000  
 Median :0.0000   Median :4.000   Median :2.000  
 Mean   :0.4062   Mean   :3.688   Mean   :2.812  
 3rd Qu.:1.0000   3rd Qu.:4.000   3rd Qu.:4.000  
 Max.   :1.0000   Max.   :5.000   Max.   :8.000

Variables to use

Question: how does car weight relate to fuel efficiency?

We will study:

mpg as the response variable (Y).
wt as the explanatory variable (X).

Plot

Code

ggplot(mtcars, aes(x = wt, y = mpg)) +
  geom_point(color = iscal_burgundy, size = 2.5) +
  theme_minimal(base_size = 14) +
  theme(panel.grid.minor = element_blank()) +
  labs(x = "Weight (1000 lbs)", y = "Miles per gallon")

Exercise 1

First look at the data

Create a scatter plot of mpg against wt.
Describe the direction of the relationship.
Is it roughly linear?
Are there any unusual observations?

Hint

plot(mtcars$wt, mtcars$mpg)

Exercise 2

Fit the model

Fit a simple linear regression model using lm().

Task

Estimate the model: [ mpg_i = _0 + _1 wt_i + _i ]
Display the regression summary.

Hint

model <- lm(mpg ~ wt, data = mtcars)
summary(model)

Exercise 3

Interpret the coefficients

Answer the following:

What does the slope mean in this context?
What does the intercept mean?
Is the intercept meaningful here?

Hint

coef(model)

Exercise 4

Compare fitted and observed values

Extract the fitted values.
Extract the residuals.
Check whether residuals sum to approximately zero.

Hint

fitted(model)
resid(model)
sum(resid(model))

Exercise 5

Diagnostic plots

Plot residuals against fitted values.
Create a normal Q-Q plot of residuals.
Decide whether the homoscedasticity and normality assumptions look reasonable.

Hint

par(mfrow = c(2, 2))
plot(model)
par(mfrow = c(1, 1))

Exercise 6

Prediction

Predict fuel efficiency for a car weighing 3.0 and 4.0 units.

Task

Create a new data frame.
Use predict() to obtain predicted values.
Try both confidence intervals and prediction intervals.

Hint

newcars <- data.frame(wt = c(3.0, 4.0))
predict(model, newcars, interval = "confidence")
predict(model, newcars, interval = "prediction")

Plot the data vs prediction

Make a plot where you can observe the data, and also the predicte values as the estimated regression line.

Hint

plot(mtcars$wt, mtcars$mpg)
abline(model, col = 'darkred', lwd = 2)

Cheat sheet

Core commands

# load data
data(mtcars)

# plot
plot(mtcars$wt, mtcars$mpg)

# fit model
model <- lm(mpg ~ wt, data = mtcars)

# regression output
summary(model)

# coefficients
coef(model)

# fitted values
fitted(model)

# residuals
resid(model)

# predictions
newcars <- data.frame(wt = c(3.0, 4.0))
predict(model, newcars)
predict(model, newcars, interval = "confidence")
predict(model, newcars, interval = "prediction")

Useful checks

# residual plot
plot(fitted(model), resid(model))
abline(h = 0, lty = 2)

# normality check
qqnorm(resid(model))
qqline(resid(model))

# diagnostics
par(mfrow = c(2, 2))
plot(model)
par(mfrow = c(1, 1))

Homework (not graded)

Explore the dataset, and find a suitable alternative regression to predict mpg. Compare both results, and we will discuss next lecture. You do not need to prepare slides or anything like that, it can be printed, or you can send it by email.