**Cook’s distance**, often denoted D_{i}, is used in regression analysis to identify influential data points that may negatively affect your regression model.

The formula for Cook’s distance is:

**D _{i}** = (r

_{i}

^{2}/ p*MSE) * (h

_{ii}/ (1-h

_{ii})

^{2})

where:

**r**_{i }is the i^{th}residual**p**is the number of coefficients in the regression model**MSE**is the mean squared error**h**_{ii}is the i^{th}leverage value

Although the formula looks a bit complicated, the good news is that most statistical softwares can easily compute this for you.

Essentially, Cook’s Distance does one thing: **it measures how much all of the fitted values in the model change when the i ^{th} data point is deleted.**

A data point that has a large value for Cook’s Distance indicates that it strongly influences the fitted values. A general rule of thumb is that any point with a Cook’s Distance over 4/n (*where n is the total number of data points*) is considered to be an outlier.

It’s important to note that Cook’s Distance is often used as a way to *identify *influential data points. Just because a data point is influential doesn’t mean it should necessarily be deleted – first you should check to see if the data point has simply been incorrectly recorded or if there is something strange about the data point that may point to an interesting finding.

**How to Calculate Cook’s Distance in R**

The following example illustrates how to calculate Cook’s Distance in R.

First, we’ll load two libraries that we’ll need for this example:

library(ggplot2) library(gridExtra)

Next, we’ll define two data frames: one with two outliers and one with no outliers.

#create data frame with no outliers no_outliers <- data.frame(x = c(1, 2, 2, 3, 4, 5, 7, 3, 2, 12, 11, 15, 14, 17, 22), y = c(22, 23, 24, 23, 19, 34, 35, 36, 36, 34, 32, 38, 41, 42, 44)) #create data frame with two outliers outliers <- data.frame(x = c(1, 2, 2, 3, 4, 5, 7, 3, 2, 12, 11, 15, 14, 17, 22), y = c(190, 23, 24, 23, 19, 34, 35, 36, 36, 34, 32, 38, 41, 42, 180))

Next, we’ll create a scatterplot to display the two data frames side by side:

#create scatterplot for data frame with no outliers no_outliers_plot <- ggplot(data = no_outliers, aes(x = x, y = y)) + geom_point() + geom_smooth(method = lm) + ylim(0, 200) + ggtitle("No Outliers") #create scatterplot for data frame with outliers outliers_plot <- ggplot(data = outliers, aes(x = x, y = y)) + geom_point() + geom_smooth(method = lm) + ylim(0, 200) + ggtitle("With Outliers") #plot the two scatterplots side by side gridExtra::grid.arrange(no_outliers_plot, outliers_plot, ncol=2)

We can see how outliers negatively influence the fit of the regression line in the second plot.

To identify influential points in the second dataset, we can can calculate **Cook’s Distance** for each observation in the dataset and then plot these distances to see which observations are larger than the traditional threshold of 4/n:

#fit the linear regression model to the dataset with outliers model <- lm(y ~ x, data = outliers) #find Cook's distance for each observation in the dataset cooksD <- cooks.distance(model) # Plot Cook's Distance with a horizontal line at 4/n to see which observations #exceed this thresdhold n <- nrow(outliers) plot(cooksD, main = "Cooks Distance for Influential Obs") abline(h = 4/n, lty = 2, col = "steelblue") # add cutoff line

We can clearly see that the first and last observation in the dataset exceed the 4/n threshold. Thus, we would identify these two observations as influential data points that have a negative impact on the regression model.

If we would like to remove any observations that exceed the 4/n threshold, we can do so using the following code:

#identify influential points influential_obs <- as.numeric(names(cooksD)[(cooksD > (4/n))]) #define new data frame with influential points removed outliers_removed <- outliers[-influential_obs, ]

Next, we can compare two scatterplots: one shows the regression line with the influential points present and the other shows the regression line with the influential points removed:

#create scatterplot with outliers present outliers_present <- ggplot(data = outliers, aes(x = x, y = y)) + geom_point() + geom_smooth(method = lm) + ylim(0, 200) + ggtitle("Outliers Present") #create scatterplot with outliers removed outliers_removed <- ggplot(data = outliers_removed, aes(x = x, y = y)) + geom_point() + geom_smooth(method = lm) + ylim(0, 200) + ggtitle("Outliers Removed") #plot both scatterplots side by side gridExtra::grid.arrange(outliers_present, outliers_removed, ncol = 2)

We can clearly see how much better the regression line fits the data with the two influential data points removed.

**Technical Notes**

- Most statistical softwares have the ability to easily compute Cook’s Distance for each observation in a dataset.
- Keep in mind that Cook’s Distance is simply a way to
*identify*influential points. - There are many ways to
*deal with*influential points including: removing these points, replacing these points with some value like the mean or median, or simply keeping the points in the model but making a careful note about this when reporting the regression results.