In statistics, an observation is considered an **outlier** if it has a value for the response variable that is much larger than the rest of the observations in the dataset.

Similarly, an observation is considered to have high **leverage** if it has a value (or values) for the predictor variables that are much more extreme compared to the rest of the observations in the dataset.

One of the first steps in any type of analysis is to take a closer look at the observations that have high leverage since they could have a large impact on the results of a given model.

This tutorial shows a step-by-step example of how to calculate and visualize the leverage for each observation in a model in R.

**Step 1: Build a Regression Model**

First, we’ll build a multiple linear regression model using the built-in **mtcars** dataset in R:

#load the dataset data(mtcars) #fit a regression model model <- lm(mpg~disp+hp, data=mtcars) #view model summary summary(model) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 30.735904 1.331566 23.083 < 2e-16 *** disp -0.030346 0.007405 -4.098 0.000306 *** hp -0.024840 0.013385 -1.856 0.073679 . --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 3.127 on 29 degrees of freedom Multiple R-squared: 0.7482, Adjusted R-squared: 0.7309 F-statistic: 43.09 on 2 and 29 DF, p-value: 2.062e-09

**Step 2: Calculate the Leverage for each Observation**

Next, we’ll use the **hatvalues()** function to calculate the leverage for each observation in the model:

#calculate leverage for each observation in the model hats <- as.data.frame(hatvalues(model)) #display leverage stats for each observation hats hatvalues(model) Mazda RX4 0.04235795 Mazda RX4 Wag 0.04235795 Datsun 710 0.06287776 Hornet 4 Drive 0.07614472 Hornet Sportabout 0.08097817 Valiant 0.05945972 Duster 360 0.09828955 Merc 240D 0.08816960 Merc 230 0.05102253 Merc 280 0.03990060 Merc 280C 0.03990060 Merc 450SE 0.03890159 Merc 450SL 0.03890159 Merc 450SLC 0.03890159 Cadillac Fleetwood 0.19443875 Lincoln Continental 0.16042361 Chrysler Imperial 0.12447530 Fiat 128 0.08346304 Honda Civic 0.09493784 Toyota Corolla 0.08732818 Toyota Corona 0.05697867 Dodge Challenger 0.06954069 AMC Javelin 0.05767659 Camaro Z28 0.10011654 Pontiac Firebird 0.12979822 Fiat X1-9 0.08334018 Porsche 914-2 0.05785170 Lotus Europa 0.08193899 Ford Pantera L 0.13831817 Ferrari Dino 0.12608583 Maserati Bora 0.49663919 Volvo 142E 0.05848459

Typically we take a closer look at observations that have a leverage value greater than 2.

An easy way to do this is to sort the observations based on their leverage value, descending:

#sort observations by leverage, descending hats[order(-hats['hatvalues(model)']), ] [1] 0.49663919 0.19443875 0.16042361 0.13831817 0.12979822 0.12608583 [7] 0.12447530 0.10011654 0.09828955 0.09493784 0.08816960 0.08732818 [13] 0.08346304 0.08334018 0.08193899 0.08097817 0.07614472 0.06954069 [19] 0.06287776 0.05945972 0.05848459 0.05785170 0.05767659 0.05697867 [25] 0.05102253 0.04235795 0.04235795 0.03990060 0.03990060 0.03890159 [31] 0.03890159 0.03890159

We can see that the largest leverage value is **0.4966**. Since this isn’t greater than 2, we know that none of the observations in our dataset have high leverage.

**Step 3: Visualize the Leverage for each Observation**

Lastly, we can create a quick plot to visualize the leverage for each observation:

#plot leverage values for each observation plot(hatvalues(model), type = 'h')

The x-axis displays the index of each observation in the dataset and the y-value displays the corresponding leverage statistic for each observation.

**Additional Resources**

How to Perform Simple Linear Regression in R

How to Perform Multiple Linear Regression in R

How to Create a Residual Plot in R