Understanding and Interpreting Compact Letter Displays

library(cld)

What is a Compact Letter Display?

A Compact Letter Display (CLD) visualizes multiple pairwise comparison results in a compact format by assigning letters to groups:

Groups sharing at least one letter are NOT significantly different
Groups with no shared letters are significantly different

This convention is widely used in agricultural research, biology, and statistics to present post-hoc test results.

Basic Example

# Example with shared letters
result <- pairwise.wilcox.test(chickwts$weight, chickwts$feed, exact = FALSE)
cld_result <- make_cld(result)
cld_result
#> Compact Letter Display (CLD)
#> Signif. level (alpha):  0.05 
#> Method:  Wilcoxon rank sum test with continuity correction 
#> 
#>      group cld spaced_cld
#>     casein   a        a__
#>  horsebean   b        _b_
#>    linseed  bc        _bc
#>   meatmeal  ac        a_c
#>    soybean   c        __c
#>  sunflower   a        a__

Interpretation:

Groups with shared letters → not significantly different, e.g.:
- casein (a) and sunflower (a) share letter “a”
- linseed (bc) and meatmeal (ac) share letter “c”
- horsebean (b) and linseed (bc) share letter “b”
Groups with no shared letters → significantly different, e.g.:
- horsebean (b) and soybean (c) share no letters
- horsebean (b) and sunflower (a) share no letters
- casein (a) and horsebean (b) share no letters

Common Patterns

Pattern 1: All different (a, b, c) → all groups significantly different

set.seed(123)
data1 <- data.frame(
  value = c(rnorm(10, 10, 1), rnorm(10, 20, 1), rnorm(10, 30, 1)),
  group = rep(c("Low", "Medium", "High"), each = 10)
)
make_cld(pairwise.t.test(data1$value, data1$group))
#> Compact Letter Display (CLD)
#> Signif. level (alpha):  0.05 
#> Method:  t tests with pooled SD 
#> 
#>   group cld spaced_cld
#>    High   a        a__
#>     Low   b        _b_
#>  Medium   c        __c

Pattern 2: One different (a, a, b) → first two groups not different from each other

set.seed(455)
data2 <- data.frame(
  value = c(rnorm(10, 10, 1.5), rnorm(10, 10.1, 1.5), rnorm(10, 20, 1.5)),
  group = rep(c("A", "B", "C"), each = 10)
)
make_cld(pairwise.t.test(data2$value, data2$group))
#> Compact Letter Display (CLD)
#> Signif. level (alpha):  0.05 
#> Method:  t tests with pooled SD 
#> 
#>  group cld spaced_cld
#>      A   a         a_
#>      B   a         a_
#>      C   b         _b

Pattern 3: Overlapping (a, ab, bc, c) → creates chain of non-significant differences

set.seed(889)
data3 <- data.frame(
  value = c(rnorm(10, 10, 3), rnorm(10, 13, 3), rnorm(10, 16, 3), rnorm(10, 19, 3)),
  group = rep(c("G1", "G2", "G3", "G4"), each = 10)
)
make_cld(pairwise.t.test(data3$value, data3$group))
#> Compact Letter Display (CLD)
#> Signif. level (alpha):  0.05 
#> Method:  t tests with pooled SD 
#> 
#>  group cld spaced_cld
#>     G1   a        a__
#>     G2  ab        ab_
#>     G3  bc        _bc
#>     G4   c        __c

Spaced CLD Format

The spaced_cld column aligns letters vertically using underscores (_) for spaces, making patterns easier to visualize in monospaced fonts.

Statistical Considerations

Significance Level (Alpha)

Lower alpha values are more conservative, resulting in fewer significant differences:

result <- pairwise.wilcox.test(chickwts$weight, chickwts$feed, exact = FALSE)

make_cld(result, alpha = 0.05)   # Standard
#> Compact Letter Display (CLD)
#> Signif. level (alpha):  0.05 
#> Method:  Wilcoxon rank sum test with continuity correction 
#> 
#>      group cld spaced_cld
#>     casein   a        a__
#>  horsebean   b        _b_
#>    linseed  bc        _bc
#>   meatmeal  ac        a_c
#>    soybean   c        __c
#>  sunflower   a        a__

make_cld(result, alpha = 0.01)   # Stringent
#> Compact Letter Display (CLD)
#> Signif. level (alpha):  0.01 
#> Method:  Wilcoxon rank sum test with continuity correction 
#> 
#>      group cld spaced_cld
#>     casein  ab        ab_
#>  horsebean   c        __c
#>    linseed  ac        a_c
#>   meatmeal  ab        ab_
#>    soybean  ab        ab_
#>  sunflower   b        _b_

make_cld(result, alpha = 0.001)  # Very stringent
#> Compact Letter Display (CLD)
#> Signif. level (alpha):  0.001 
#> Method:  Wilcoxon rank sum test with continuity correction 
#> 
#>      group cld spaced_cld
#>     casein   a          a
#>  horsebean   a          a
#>    linseed   a          a
#>   meatmeal   a          a
#>    soybean   a          a
#>  sunflower   a          a

Multiple Comparison Adjustment

P-value adjustment methods (Bonferroni, Holm, FDR, etc.) control Type I error inflation and can significantly impact results. Use the p.adjust.method argument in your pairwise test function.

Common Misinterpretations

❌ Wrong: “Group ‘a’ has higher values than group ‘b’”
✅ Correct: Letters indicate grouping, not magnitude or rank. They only show which groups differ statistically.

💡 Tip: Sort groups by their original means/medians to understand which groups have higher or lower values.

Accessing Metadata

result <- pairwise.wilcox.test(chickwts$weight, chickwts$feed, exact = FALSE)
cld_result <- make_cld(result)

# Access stored information
attr(cld_result, "alpha")           # Significance level
#> [1] 0.05
attr(cld_result, "method")          # Statistical method
#> [1] "Wilcoxon rank sum test with continuity correction"
attr(cld_result, "n_comparisons")   # Total comparisons
#> [1] 15
attr(cld_result, "n_significant")   # Significant comparisons
#> [1] 8

Quick Reference

Comparison	Shared Letters?	Interpretation
“a” vs “a”	Yes	Not significantly different
“a” vs “b”	No	Significantly different
“ab” vs “bc”	Yes (letter “b”)	Not significantly different
“ab” vs “cd”	No	Significantly different