In statistics, multidimensional scaling is a way to visualize the similarity of observations in a dataset in an abstract cartesian space (usually a 2-D space).

The easiest way to perform multidimensional scaling in R is by using the built-in cmdscale() function, which uses the following basic syntax:

cmdscale(d, eig = FALSE, k = 2, …)

where:

• d: A distance matrix usually calculated by the dist() function.
• eig: Whether or not to return eigenvalues.
• k: The number of dimensions to visualize the data in. Default is 2.

The following example shows how to use this function in practice.

Example: Multidimensional Scaling in R

Suppose we have the following data frame in R that contains information about various basketball players:

#create data frame
df <- data.frame(points=c(4, 4, 6, 7, 8, 14, 16, 19, 25, 25, 28),
                 assists=c(3, 2, 2, 5, 4, 8, 7, 6, 8, 10, 11),
                 blocks=c(7, 3, 6, 7, 5, 8, 8, 4, 2, 2, 1),
                 rebounds=c(4, 5, 5, 6, 5, 8, 10, 4, 3, 2, 2))

#add row names
row.names(df) <- LETTERS[1:11]

#view data frame
df

  points assists blocks rebounds
A      4       3      7        4
B      4       2      3        5
C      6       2      6        5
D      7       5      7        6
E      8       4      5        5
F     14       8      8        8
G     16       7      8       10
H     19       6      4        4
I     25       8      2        3
J     25      10      2        2
K     28      11      1        2

We can use the following code to perform multidimensional scaling with the cmdscale() function and visualize the results in a 2-D space:

#calculate distance matrix
d <- dist(df)

#perform multidimensional scaling
fit <- cmdscale(d, eig=TRUE, k=2)

#extract (x, y) coordinates of multidimensional scaleing
x <- fit$points[,1]
y <- fit$points[,2]

#create scatter plot
plot(x, y, xlab="Coordinate 1", ylab="Coordinate 2",
     main="Multidimensional Scaling Results", type="n")

#add row names of data frame as labels
text(x, y, labels=row.names(df))

Players from the original data frame who have similar values across the original four columns (points, assists, blocks, and rebounds) are located close to each other in the plot.

For example, players A and C are located close to each other. Here are their values from the original data frame:

#view data frame values for players A and C
df[rownames(df) %in% c('A', 'C'), ]

  points assists blocks rebounds
A      4       3      7        4
C      6       2      6        5

Their values for points, assists, blocks, and rebounds are all quite similar, which explains why they’re located so close together in the 2-D plot.

By contrast, consider players B and K who are located far apart in the plot.

#view data frame values for players B and K
df[rownames(df) %in% c('B', 'K'), ]

  points assists blocks rebounds
B      4       2      3        5
K     28      11      1        2

Thus, the 2-D plot is a nice way to visualize how similar each players are across all of the variables in the data frame.

Players who have similar stats are grouped close together while players who have very different stats are located far apart from each other in the plot.

Note that you can also extract the exact (x, y) coordinates for each player in the plot by typing fit, which is the name of the variable that we stored the results of cmdscale() function in:

#view (x, y) coordinates of points in the plot
fit

         [,1]       [,2]
A -10.6617577 -1.2511291
B -10.3858237 -3.3450473
C  -9.0330408 -1.1968116
D  -7.4905743  1.0578445
E  -6.4021114 -1.0743669
F  -0.4618426  4.7392534
G   0.8850934  6.1460850
H   4.7352436 -0.6004609
I  11.3793381 -1.3563398
J  12.0844168 -1.5494108
K  15.3510585 -1.5696166