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Cook’s Distance

Purpose

Cook’s distance is useful for identifying outliers in the X values (observations for predictor variables). It also shows the influence of each observation on the fitted response values. An observation with Cook’s distance larger than three times the mean Cook’s distance might be an outlier.

Definition

Cook’s distance is the scaled change in fitted values. Each element in `CooksDistance` is the normalized change in the vector of coefficients due to the deletion of an observation. The Cook’s distance, Di, of observation i is

`${D}_{i}=\frac{\sum _{j=1}^{n}{\left({\stackrel{^}{y}}_{j}-{\stackrel{^}{y}}_{j\left(i\right)}\right)}^{2}}{p\text{\hspace{0.17em}}MSE},$`

where

• ${\stackrel{^}{y}}_{j}$ is the jth fitted response value.

• ${\stackrel{^}{y}}_{j\left(i\right)}$ is the jth fitted response value, where the fit does not include observation i.

• MSE is the mean squared error.

• p is the number of coefficients in the regression model.

Cook’s distance is algebraically equivalent to the following expression:

`${D}_{i}=\frac{{r}_{i}^{2}}{p\text{\hspace{0.17em}}MSE}\left(\frac{{h}_{ii}}{{\left(1-{h}_{ii}\right)}^{2}}\right),$`

where ri is the ith residual, and hii is the ith leverage value.

`CooksDistance` is an n-by-1 column vector in the `Diagnostics` table of the `LinearModel` object.

How To

After obtaining a fitted model, say, `mdl`, using `fitlm` or `stepwiselm`, you can:

• Display the Cook’s distance values by indexing into the property using dot notation,

`mdl.Diagnostics.CooksDistance`

• Plot the Cook’s distance values using

`plotDiagnostics(mdl,'cookd')`
For details, see the `plotDiagnostics` method of the `LinearModel` class.

Determine Outliers Using Cook's Distance

This example shows how to use Cook's Distance to determine the outliers in the data.

Load the sample data and define the independent and response variables.

```load hospital X = double(hospital(:,2:5)); y = hospital.BloodPressure(:,1); ```

Fit the linear regression model.

```mdl = fitlm(X,y); ```

Plot the Cook's distance values.

```plotDiagnostics(mdl,'cookd') ```

The dashed line in the figure corresponds to the recommended threshold value, `3*mean(mdl.Diagnostics.CooksDistance)`. The plot has some observations with Cook's distance values greater than the threshold value, which for this example is 3*(0.0108) = 0.0324. In particular, there are two Cook's distance values that are relatively higher than the others, which exceed the threshold value. You might want to find and omit these from your data and rebuild your model.

Find the observations with Cook's distance values that exceed the threshold value.

```find((mdl.Diagnostics.CooksDistance)>3*mean(mdl.Diagnostics.CooksDistance)) ```
```ans = 2 13 28 44 58 70 71 84 93 95 ```

Find the observations with Cook's distance values that are relatively larger than the other observations with Cook's distances exceeding the threshold value.

```find((mdl.Diagnostics.CooksDistance)>5*mean(mdl.Diagnostics.CooksDistance)) ```
```ans = 2 84 ```

References

[1] Neter, J., M. H. Kutner, C. J. Nachtsheim, and W. Wasserman. Applied Linear Statistical Models. 4th ed. Chicago: Irwin, 1996.