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.

Mapping can introduce distortion

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