In statistics, we often want to know how influential different observations are in regression models.

One way to calculate the influence of observations is by using a metric known as **DFFITS**, which stands for “difference in fits.”

This metric tells us how much the predictions made by a regression model change when we leave out an individual observation.

This tutorial shows a step-by-step example of how to calculate and visualize DFFITS 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 DFFITS for each Observation**

Next, we’ll use the built-in **dffits()** function to calculate the DFFITS value for each observation in the model:

#calculate DFFITS for each observation in the model dffits <- as.data.frame(dffits(model)) #display DFFITS for each observation dffits dffits(model) Mazda RX4 -0.14633456 Mazda RX4 Wag -0.14633456 Datsun 710 -0.19956440 Hornet 4 Drive 0.11540062 Hornet Sportabout 0.32140303 Valiant -0.26586716 Duster 360 0.06282342 Merc 240D -0.03521572 Merc 230 -0.09780612 Merc 280 -0.22680622 Merc 280C -0.32763355 Merc 450SE -0.09682952 Merc 450SL -0.03841129 Merc 450SLC -0.17618948 Cadillac Fleetwood -0.15860270 Lincoln Continental -0.15567627 Chrysler Imperial 0.39098449 Fiat 128 0.60265798 Honda Civic 0.35544919 Toyota Corolla 0.78230167 Toyota Corona -0.25804885 Dodge Challenger -0.16674639 AMC Javelin -0.20965432 Camaro Z28 -0.08062828 Pontiac Firebird 0.67858692 Fiat X1-9 0.05951528 Porsche 914-2 0.09453310 Lotus Europa 0.55650363 Ford Pantera L 0.31169050 Ferrari Dino -0.29539098 Maserati Bora 0.76464932 Volvo 142E -0.24266054

Typically we take a closer look at observations that have DFFITS values greater than a threshold of 2√p/n where:

**p:**Number of predictor variables used in the model**n:**Number of observations used in the model

In this example, the threshold would be **0.5**:

#find number of predictors in model p <- length(model$coefficients)-1 #find number of observations n <- nrow(mtcars) #calculate DFFITS threshold value thresh <- 2*sqrt(p/n) thresh [1] 0.5

We can sort the observations based on their DFFITS values to see if any of them exceed the threshold:

#sort observations by DFFITS, descending dffits[order(-dffits['dffits(model)']), ] [1] 0.78230167 0.76464932 0.67858692 0.60265798 0.55650363 0.39098449 [7] 0.35544919 0.32140303 0.31169050 0.11540062 0.09453310 0.06282342 [13] 0.05951528 -0.03521572 -0.03841129 -0.08062828 -0.09682952 -0.09780612 [19] -0.14633456 -0.14633456 -0.15567627 -0.15860270 -0.16674639 -0.17618948 [25] -0.19956440 -0.20965432 -0.22680622 -0.24266054 -0.25804885 -0.26586716 [31] -0.29539098 -0.32763355

We can see that the first five observations have a DFFITS value greater than 0.5, which means we may want to investigate these observations closer to determine if they’re highly influential in the model.

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

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

#plot DFFITS values for each observation plot(dffits(model), type = 'h') #add horizontal lines at absolute values for threshold abline(h = thresh, lty = 2) abline(h = -thresh, lty = 2)

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

**Additional Resources**

How to Perform Simple Linear Regression in R

How to Perform Multiple Linear Regression in R

How to Calculate Leverage Statistics in R

How to Create a Residual Plot in R