Imagine you have data on how students performed in various courses - A, B, C, D, E

A, B, C, D, E
10, 10, 10, 1,10
10 9, 8, 8, 10
1, 2, 2, 1, 2
1, 1, 9, 1, 1

Here the first row got excellent results (10) in all but in course D, where they got only 1. This is an outlier. The last row got poor results (1) in all but got an excellent result (9) in course C. Is there some statistical analysis that can be used to find such outliers in a large such dataset with several rows and columns? Even better would be a way to also consider whether it is just this particular row or whether all rows tend to have higher/lower score in that particular course.

A, B, C, D, E
10, 9, 8, 8, 5
7, 3, 3, 2, 3,
6, 1, 1, 1, 1

In the last row, while the 6 in A is an outlier, it is generally the trend that everyone performs relatively better in the course A. So it would be also better if the analysis could identify such insights.

Note: I am not trying to ‘remove’ these outliers. I just need a way to find such interesting rows.


2 Answers 2


Use a robust additive model to fit the data and then explore its residuals.

An "additive model" approximates the value $y_{ij}$ at row $i,$ column $j$ as the sum of a row term $\rho_i$ and a column term $\gamma_j,$ with any difference known as the "residual" $r_{ij}.$ Thus,

$$r_{ij} = y_{ij} - (\rho_i + \gamma_j).$$

Relatively large residuals will crop up in any cell that is inconsistent with the typical values suggested by its row and column.

A "robust" method estimates the row and column terms in a way that is insensitive to values that would yield large residuals. There are many robust methods. For instance, John Tukey described a simple "median polish" in his EDA book in which you alternately estimate the terms as the medians of their rows and columns, make suitable adjustments, and repeat until the fit works well. I like this, but it's not readily available in software (because convergence is not guaranteed in all cases).

To fit the first example, I used the rlm ("robust linear model") function in the MASS library for R.

"Exploration" is usually most effective when done visually, by plotting suitable graphics.

Simple, standard graphical tools can help identify unusually large residuals. For instance, here is a boxplot of these 20 residuals that clearly shows the two extreme values--and just how extreme they are:

Figure 2

For two-way tables like your examples, map the residuals.

Figure 1

(For reference I have posted the original data values.) By making small-sized residuals nearly transparent, the largest ones really stand out in this map: the large negative residual (corresponding to the original value of 1 in row 1, column D) and the large positive residual (corresponding to the original value of 9 in row 4, column C).

To see how insightful this can be, apply it to your second example. You will be surprised at the entries with the most extreme residuals! (The value of 6 in the third row is not an outlier. That's because it is perfectly consistent with the large values that occur in column A.)

Here is the R code.

# X <- data.frame(y = c(
# 10, 9, 8, 8, 5,
# 7, 3, 3, 2, 3,
# 6, 1, 1, 1, 1),
# Row = factor(rep(1:3, each=5), levels=3:1),
# Column = rep(c("A","B","C","D","E"), 3))

X <- data.frame(y = c(
  10, 10, 10, 1, 10,
  10, 9, 8, 8, 10,
  1, 2, 2, 1, 2,
  1, 1, 9, 1, 1),
  Row = factor(rep(1:4, each=5), levels=4:1),
  Column = rep(c("A","B","C","D","E"), 4))

fit <- rlm(y ~ Column + Row - 1, X)
# plot(fit)

X$Residual <- residuals(fit)
boxplot(X$Residual, horizontal=TRUE)

ggplot(X) + 
  geom_raster(aes(Column, Row, fill=Residual, alpha=abs(Residual))) + 
  geom_text(aes(Column, Row, label=y), color="Gray") + 
  scale_fill_gradient2(low=muted("blue"), high=muted("red")) +
  ggtitle("First example")

One method of finding multivariate outliers involves the outlier detection algorithm suggested by Leland Wilkinson's paper Visualizing Outliers. I won't delve into the statistical details of it here since I've linked to the paper. Instead, I'll simply show you how you can implement the outlier detection algorithm using R's HDoutliers package.

The package performs multivariate outlier detection that "can handle a) data with a mixed categorical and continuous variables, b) many columns of data, c) many rows of data, d) outliers that mask other outliers, and e) both unidimensional and multidimensional datasets" so it should perform well on your data.

X <- data.frame(y = c(
  10, 10, 10, 1, 10,
  10, 9, 8, 8, 10,
  1, 2, 2, 1, 2,
  1, 1, 9, 1, 1),
  Row = factor(rep(1:4, each=5), levels=4:1),
  Column = rep(c("A","B","C","D","E"), 4))


out.W<-HDoutliers(X [,-1])
plotHDoutliers(X[,-1], out.W)

This produces the following plot where the outliers can be identified by star symbols, instead of blue dots as shown below:

enter image description here

  • $\begingroup$ Because the sense in which these starred points are any kind of outliers is obscure, please at least explain what you mean by "outlier" in this post. These stars don't seem to correspond to anything the OP would recognize as an outlying count. $\endgroup$
    – whuber
    Aug 24, 2020 at 14:10

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