Solved – Comparing unbalanced groups with ANOVA/Kruskal-Wallis when one group has only 1 observation

anovagroup-differenceshypothesis testingkruskal-wallis test”small-sample

I would like to compare a continuous variable across 5 health status groups, but one of the groups has only 1 observation. Would an ANOVA/Kruskal-Wallis be valid? What can I do about the group with only 1 observation?

The boxplots for the continuous variable across the groups look as follows:

The group sizes are:

A: 12
B: 8
C: 9
D: 7
E: 1

Background info: I work in biostatistics and my datasets are fairly small, and some outcomes are rare in biology. Collecting samples is also dependent on patients and is an expensive procedure, so getting a larger sample is not a feasible solution.

Edit: What I currently do is to omit the group with only one observation in my analysis, however I am not sure about the validity of that.

Best Answer

You can perform inference in one-way-ANOVA type designs where there's a group with only one observation if you make the equal-variance assumption. If you don't assume equal variance (or some other informative variance structure), then you won't have information about variance in the singleton group.

Not all packages will deal with it in their implementation of ANOVA, it depends on how they're set up, but that doesn't mean it can't be done. [I gave an example of performing a t-test with a singleton in another answer on site.]

Here's an example in R with a singleton group both included and omitted for both one-way ANOVA and Kruskal-Wallis:

x=rnorm(100)
g=as.factor(rep(1:5,c(40,30,20,9,1)))
anova(lm(x~g))
Analysis of Variance Table

Response: x
          Df Sum Sq Mean Sq F value Pr(>F)
g          4  6.554 1.63839  1.6588  0.166
Residuals 95 93.830 0.98769               

anova(lm(x[-100]~g[-100]))
Analysis of Variance Table

Response: x[-100]
          Df Sum Sq Mean Sq F value Pr(>F)
g[-100]    3  3.498 1.16608  1.1806 0.3213
Residuals 95 93.830 0.98769               

kruskal.test(x~g)

        Kruskal-Wallis rank sum test

data:  x by g
Kruskal-Wallis chi-squared = 5.9232, df = 4, p-value = 0.205

kruskal.test(x[-100]~g[-100])

        Kruskal-Wallis rank sum test

data:  x[-100] by g[-100]
Kruskal-Wallis chi-squared = 3.1894, df = 3, p-value = 0.3633

Related Solutions

Solved – ANOVA / Kruskal-Wallis: one of four groups has different distribution

As ttnphns commented, neither Kruskal-Wallis nor rank sum tests have any assumptions about distributional similarity between groups. There is a point of confusion that somtimes arises in these tests because, while in the most general sense they are tests for stochastic dominance (e.g., H$_{0} \text{: P}(X_{A} > X_{B}) = \frac{1}{2})$, with two additional assumptions—(1) that the distributions are the same shape, and (2) that any differences between the distributions of the groups are differences of central location—the tests can be interpreted as tests for median difference (e.g., H$_{0} \text{: } \tilde{x}_{A} = \tilde{x}_{b}$).

Therefore, significance is not an issue, and there is nothing to "mitigate." However, substantive interpretation (i.e. stochastic dominance versus median, mean, etc. difference) will entail.

Solved – How to plot the results of a Kruskal-Wallis or a Welch’s ANOVA

As you already mentioned the Kruskal-Wallis test is a test of significance based on the ranks. In my opinion however, plotting the ranks isn't really that helpful for the reader in order to understand the underlying response variable. Instead, what I would do is to plot the individual data points (including the median for descriptive purposes) plus the ranks as differently colored points. To make it clear, you could also place the letters indicating significant difference next to those points indicating the ranks. You can also obviously report everything you don't want to plot (e.g. the ranks as separate points, etc.) in a separate table (see example below).

I am not sure which software package you are using but below is an example using R to illustrate what I mentioned above (note: this approach may not look nice if the numerical values of the data points and the ranks are largely different. In that case, I would plot the data points and the significant differences via letters, and report the ranks in a separate table.

### required packages
require(tidyverse)
#> Loading required package: tidyverse
require(agricolae)
#> Loading required package: agricolae
### set seed for reproducibility
set.seed(564)

### subset the PlantGrowth dataset (available in R) to replicate your n=5 scenario
PlantGrowth %>% 
  group_by(group) %>% 
  slice(sample(1:5)) -> d_sub

### run Kruskal test from the agricolae package
k <- kruskal(d_sub$weight, d_sub$group, console = TRUE)
#> 
#> Study: d_sub$weight ~ d_sub$group
#> Kruskal-Wallis test's
#> Ties or no Ties
#> 
#> Critical Value: 3.290877
#> Degrees of freedom: 2
#> Pvalue Chisq  : 0.192928 
#> 
#> d_sub$group,  means of the ranks
#> 
#>      d_sub.weight r
#> ctrl          8.3 5
#> trt1          5.3 5
#> trt2         10.4 5
#> 
#> Post Hoc Analysis
#> 
#> t-Student: 2.178813
#> Alpha    : 0.05
#> Minimum Significant Difference: 5.816519 
#> 
#> Treatments with the same letter are not significantly different.
#> 
#>      d_sub$weight groups
#> trt2         10.4      a
#> ctrl          8.3      a
#> trt1          5.3      a

### create summary table incl. mean rank sums and significant differences letters
(t_comp <- k$means %>% 
    rownames_to_column(var = "group") %>%
    rename(weight = d_sub.weight) %>%
    as_tibble() %>% 
    left_join(as_tibble(k$groups), by = c("rank" = "d_sub$weight")))
#> # A tibble: 3 x 11
#>   group weight  rank   std     r   Min   Max   Q25   Q50   Q75 groups
#>   <chr>  <dbl> <dbl> <dbl> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <fct> 
#> 1 ctrl    5.11   8.3 0.788     5  4.17  6.11  4.5   5.18  5.58 a     
#> 2 trt1    4.57   5.3 0.851     5  3.59  5.87  4.17  4.41  4.81 a     
#> 3 trt2    5.57  10.4 0.446     5  5.12  6.31  5.37  5.5   5.54 a

### create plot with ranks as blue dots and align the letters next to them
d_sub %>% 
  ggplot(aes(x = group, y = weight)) +
  geom_point(color = "grey50", size = 2) +
  # add ranks as separate points
  geom_point(data = t_comp, aes(x = group, y = rank), col = "blue", size = 3) + 
  # add median as horizontal line
  stat_summary(fun.y = median, geom = "errorbar", aes(ymax = ..y.., ymin = ..y..),
               width = .75, col = "red") +
  # add letters
  geom_text(data = t_comp, aes(x = group, y = rank, label = groups), size = 6, nudge_x = -0.1)

^{Created on 2020-01-30 by the reprex package (v0.3.0)}

Best Answer

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Solved – ANOVA / Kruskal-Wallis: one of four groups has different distribution

Solved – How to plot the results of a Kruskal-Wallis or a Welch’s ANOVA

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