Author: Martin R. Smith

# Comparing sets of trees from different analyses

#### Martin R. Smith

A common application of tree space analysis is to compare the outputs of different analyses – for instance, trees obtained from different gene sequences, or results obtained using different models or methods (e.g. Bayesian, maximum likelihood, or parsimony).

## Shiny app

This can be accomplished quickly using the `MapTrees()` graphical user interface:

- Load trees from file: Select first tree file

- Select an appropriate sample size

- Select Replace existing

- Load each additional set of trees from file using Add batch to existing

Load tree batches

- On the Display tab, select Point symbols: One per batch, or Colour points by: Batch

Tree batch styles

## Scripting at the R command line

More control over the mapping can be obtained at the command line:

    # Load trees
    library("TreeTools", quietly = TRUE)
    batch1 <- as.phylo(1:60, 8) # Generate 60 similar trees
    batch2 <- as.phylo(seq(200, 800, length.out = 30), 8) # A separate batch of 30 trees
    styles <- c(1, 2) # Select plotting colours / symbols
    treeStyle <- rep(styles, c(length(batch1), length(batch2)))

    # Calculate distances
    library("TreeDist")
    distances <- ClusteringInfoDistance(c(batch1, batch2))

    # Construct over-simple 2D PCoA mapping
    mapping <- cmdscale(distances, k = 2)

    # Plot mapping
    par(mar = rep(0, 4))
    plot(mapping,
         asp = 1, # Preserve aspect ratio - do not distort distances
         ann = FALSE, axes = FALSE, # Don't label axes: dimensions are meaningless
         col = treeStyle, # Colour
         pch = treeStyle # Plotting symbol
         )
    legend("left", c("Batch 1", "Batch 2"), col = styles, pch = styles)

For more robust analyses than the (potentially misleading!) 2D plot above, consult the companion vignette. Note also that mapped areas and their regions of overlap may not correspond to reality; see the warnings and recommendations in Smith (2022a).

## Comparing trees’ dispersal / hypervolume

Interpreting and comparing the areas of tree space from a projection can be misleading – the expanded apparent area of Greenland under the Mercator projection being a familiar example.

Mercator map projection, with true size of each country overlaid. Credit: Neil Kaye

As such, it is always best to work with original distances when interpreting whether sets of trees occupy larger or smaller regions of tree space.

### Distances from median

One approach is to plot distances from a median tree:

    # Calculate median trees
    median1 <- median(batch1)
    median2 <- median(batch2)

    # Compute distance from each tree to the median of its batch
    dist1 <- ClusteringInfoDist(batch1, median1)
    dist2 <- ClusteringInfoDist(batch2, median2)

    # Set resolution of histogram
    nBreaks <- 10
    breaks <- seq(0, max(dist1, dist2), length.out = nBreaks)

    # Plot first distance set
    hist(dist1, col = "#00000022", breaks = breaks,
         main = "Distance from median of batch",
         xlab = "Clustering information distance",
         ylim = c(0, 25) # Omit this line to infer Y axis limit from first batch.
         )

    # Add second distance set
    hist(dist2, col = "#ff000022", breaks = breaks, add = TRUE)

    # Add legend
    legend("topleft", c("Batch 1", "Batch 2"),
           fill = c("#00000022",  "#ff000022"))

In the plotted example, distances to the median tree are greater for batch 2 than batch 1, indicating a more dispersed set of trees that occupies a greater hypervolume. Note that the increased frequency at higher distances is expected: the outer shell of a sphere contains more volume than a layer of equivalent thickness closer to the centre, and this phenomenon becomes more pronounced as the dimensionality of tree space increases.

### Consensus resolution

A complementary approach is to identify the resolution of the consensus of each batch of trees. This approach shares many of the problems with the Robinson–Foulds distance: in particular, resolution can be decimated by a single “rogue” taxon whose position is poorly defined (Smith, 2022b). Detecting and removing rogue taxa can provide a more meaningful point of comparison.

    # Create tree set with a rogue taxon
    batch3 <- AddTipEverywhere(as.phylo(7, 7), "t8")

    # Set up plotting area
    par(mfrow = c(2, 2), mar = rep(0.4, 4))

    # Plot naive strict consensus
    plot(consensus(batch1, p = 1))
    plot(consensus(batch3, p = 1))

    if (requireNamespace("Rogue", quietly = TRUE)) {
      cons1 <- ConsensusWithout(batch1, p = 1,
                                Rogue::QuickRogue(batch1, p = 1)[-1, "taxon"])
      cons3 <- ConsensusWithout(batch3, p = 1,
                                Rogue::QuickRogue(batch3, p = 1)[-1, "taxon"])

      # The information content of each tree gives a measure of its resolution,
      # accounting for omitted rogue leaves
      SplitwiseInfo(cons1) # 8.5 bits
      SplitwiseInfo(cons3) # 15.1 bits: higher resolution indicates that these
      # trees are more similar, notwithstanding rogue taxa.

      # Plot the trees
      plot(cons1)
      plot(cons3)
    } else {
      message("The package 'Rogue' is required to run this example.")
    }

Whereas a direct interpretation of this analysis is not straightforward, it can provide a complementary way of understanding the distribution of trees across tree space.

## References

Smith, M. R. (2022a). Robust analysis of phylogenetic tree space. Systematic Biology, 71(5), 1255–1270. https://doi.org/10.1093/sysbio/syab100
Smith, M. R. (2022b). Using information theory to detect rogue taxa and improve consensus trees. Systematic Biology, 71(5), 1088–1094. https://doi.org/10.1093/sysbio/syab099