How to Calculate Mahalanobis Distance in R

by Tutor Aspire February 17, 2016

The Mahalanobis distance is the distance between two points in a multivariate space.

It is often used to find outliers in statistical analyses that involve several variables.

This tutorial explains how to calculate the Mahalanobis distance in R.

Example: Mahalanobis Distance in R

Use the following steps to calculate the Mahalanobis distance for every observation in a dataset in R.

Step 1: Create the dataset.

First, we’ll create a dataset that displays the exam score of 20 students along with the number of hours they spent studying, the number of prep exams they took, and their current grade in the course:

#create data
df = data.frame(score = c(91, 93, 72, 87, 86, 73, 68, 87, 78, 99, 95, 76, 84, 96, 76, 80, 83, 84, 73, 74),
        hours = c(16, 6, 3, 1, 2, 3, 2, 5, 2, 5, 2, 3, 4, 3, 3, 3, 4, 3, 4, 4),
        prep = c(3, 4, 0, 3, 4, 0, 1, 2, 1, 2, 3, 3, 3, 2, 2, 2, 3, 3, 2, 2),
        grade = c(70, 88, 80, 83, 88, 84, 78, 94, 90, 93, 89, 82, 95, 94, 81, 93, 93, 90, 89, 89))

#view first six rows of data
head(df)

  score hours prep grade
1    91    16    3    70
2    93     6    4    88
3    72     3    0    80
4    87     1    3    83
5    86     2    4    88
6    73     3    0    84

Step 2: Calculate the Mahalanobis distance for each observation.

Next, we’ll use the built-in mahalanobis() function in R to calculate the Mahalanobis distance for each observation, which uses the following syntax:

mahalanobis(x, center, cov)

where:

x: matrix of data
center: mean vector of the distribution
cov: covariance matrix of the distribution

The following code shows how to implement this function for our dataset:

#calculate Mahalanobis distance for each observation
mahalanobis(df, colMeans(df), cov(df))

 [1] 16.5019630  2.6392864  4.8507973  5.2012612  3.8287341  4.0905633
 [7]  4.2836303  2.4198736  1.6519576  5.6578253  3.9658770  2.9350178
[13]  2.8102109  4.3682945  1.5610165  1.4595069  2.0245748  0.7502536
[19]  2.7351292  2.2642268

Step 3: Calculate the p-value for each Mahalanobis distance.

We can see that some of the Mahalanobis distances are much larger than others.

To determine if any of the distances are statistically significant, we need to calculate their p-values.

The p-value for each distance is calculated as the p-value that corresponds to the Chi-Square statistic of the Mahalanobis distance with k-1 degrees of freedom, where k = number of variables.

So, in this case we’ll use a degrees of freedom of 4-1 = 3.

#create new column in data frame to hold Mahalanobis distances
df$mahal #create new column in data frame to hold p-value for each Mahalanobis distance
df$p pchisq(df$mahal, df=3, lower.tail=FALSE)

#view data frame
df

   score hours prep grade      mahal            p
1     91    16    3    70 16.5019630 0.0008945642
2     93     6    4    88  2.6392864 0.4506437265
3     72     3    0    80  4.8507973 0.1830542407
4     87     1    3    83  5.2012612 0.1576392526
5     86     2    4    88  3.8287341 0.2805615121
6     73     3    0    84  4.0905633 0.2518495222
7     68     2    1    78  4.2836303 0.2324211504
8     87     5    2    94  2.4198736 0.4899458807
9     78     2    1    90  1.6519576 0.6476670033
10    99     5    2    93  5.6578253 0.1294978092
11    95     2    3    89  3.9658770 0.2651724541
12    76     3    3    82  2.9350178 0.4017530495
13    84     4    3    95  2.8102109 0.4218217836
14    96     3    2    94  4.3682945 0.2243432904
15    76     3    2    81  1.5610165 0.6682610031
16    80     3    2    93  1.4595069 0.6916471506
17    83     4    3    93  2.0245748 0.5673218169
18    84     3    3    90  0.7502536 0.8613248635
19    73     4    2    89  2.7351292 0.4342904353
20    74     4    2    89  2.2642268 0.5194087143

Typically a p-value that is less than .001 is considered to be an outlier.

We can see that the first observation is an outlier in the dataset because it has a p-value less than .001.

Depending on the context of the problem, you may decide to remove this observation from the dataset since it’s an outlier and could affect the results of the analysis.

How to Calculate Mahalanobis Distance in R

Example: Mahalanobis Distance in R

How to Find a P-Value from a Z-Score in Python

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