An Introduction to TITAN2

Note on Loading Your Own Data

As noted in the previous section, titan() requires two datasets: a site by taxa matrix and a environmental gradient dataset. The site by taxa matrix should ideally be of class data.frame, the standard data structure in R. These can be read into R in several ways, for example the read.table() or read.csv() functions. The readr package has other similar functions to help read data into R. The environmental gradient dataset can either be a data.frame object with one column or simply an atomic vector of class numeric. In addition to the the data-reading functions listed above, scan() can be helpful for reading data of this type into R. In addition to these, the file.choose() function allows users to select files interactively, which can help avoid pathing issues.

plot_sumz() and plot_sumz_density()

The community-level output may be viewed using the plot_sumz() or plot_sumz_density() functions. The plot_sumz_density() function is new to this version of TITAN2. We recommend its use because we think it will be both easier to understand than previous output and encourage appropriate interpretation of results.

There are many options for changing the look of the graphs; see their full documentations with ?plot_sumz or ?plot_sumz_density. An abbreviated version is presented below. To begin, here’s how you use plot_sumz_density():

plot_sumz_density(glades.titan)

plot_sumz_density() plots community level change identified with TITAN2. There are three component panels in the default output and we will consider them from the bottom to the top. The first plot is a slight modification of the original sum(z) plot first presented by Baker and King (2010). It shows the magnitude of change among taxa declining along the gradient (z-) and those increasing along the gradient. Peaks in the values indicate points along the environmental gradient that produce large amounts of change in community composition and/or structure. These are the nominal community change points. Plateaus denote regions of similar change. The sums are converted to areas using the logical argument ribbon, a line plot can be obtained by setting ribbon to FALSE, and the original point plot can be obtained by setting the logical argument points to TRUE.

plot_sumz_density(glades.titan, ribbon = FALSE, points = TRUE)

The second change from earlier versions of TITAN2 is that the logical argument filter defaults to TRUE. This argument ensures that the only taxa contributing to patterns in the sum(z) are pure and reliable taxa identified by the bootstrap results rather than all taxa. Setting filter to FALSE allows impure or unreliable taxa to contribute to apparent patterns of commuity change. In general, the filtered scores should show a similar pattern but be somewhat lower in magnitude depending on contributions from impure or unreliable taxa.

The middle panel shows the estimated probability density of the sum(z) across all bootstrap replicates. Previously these distributions were represented by cumulative frequency distributions of the sum(z) maxima. A narrow spread indicates a high degree of precison regarding the location of the community change point.

The top panel shows the observed sum(z-) and sum(z+) maxima as circles with the 95th percentile of their distributions as horzontal lines. The primary reason for this plot is to illustrate how much information is hidden by this technique relative to the probability density functions.

There are several additional plotting options in plot_sumz_density(). First, any of the three panels can be dropped by setting logical agruments sumz, density, or change_points to FALSE. Second, it is possible to display only the z- or z+ taxa through the use of the logical arguments sumz1 and sumz2. Finally, the x axis label can be edited using the xlabel aregument.

plot_sumz_density(glades.titan, 
  ribbon = TRUE, points = FALSE, sumz1 = FALSE, change_points = FALSE, 
  xlabel = expression(paste("Surface Water Total Phosphorus ("*mu*"g/l)"))
)

Besides the mandatory titan object, arguments include:

• filter - controls whether filtered (default) or unfiltered sum(z) scores are plotted

• sumz - logical whether sum(z) values are plotted

• points - logical whether point values of the sum(z) should be plotted

• ribbon - logical whether values of sum(z) should be plotted as a ribbon

• density - logical whether estimated probability densities should be plotted

• change_points - logical whether community change points should be plotted

• sumz1 - logical whether sum(z) for maxgrp = 1 should be plotted

• sumz2 - logical whether sum(z) for maxgrp = 2 should be plotted

• xmin - controls the x-axis minimum

• xmax - controls the x-axis maximum

• xlabel - text string for x-axis label

• y1label - text string for sum(z) y-axis, default labels are provided

• y2label - text string for density plot

• alpha1 - controls the transparancy of sum(z-)

• alpha2 - controls the transparancy of sum(z+)

• col1 - color for `maxgrp = 1

• col2 - color for `maxgrp = 2

• axis_lab_size - controls size of axis labels

• legend.position - controls location of legend

For plot_sumz, the older plotting function in TITAN2, filled and hollow symbols denote the magnitude of summed z scores of increasing (z+) or decreasing (z-) taxa. Peaks in the values indicate points along the environmental gradient that produce large amounts of change in community composition and/or structure. Solid and dashed lines are cumulative frequency distributions of sum(z-) and sum(z+) maxima (respectively) across bootstrap replicates. Vertical CFDs indicate narrow uncertainty about where the maximum chnage occurs, sloping or stair-step CFDs suggest broad uncertainty regarding the location of maximum change.

Note that users can adjust colors (e.g. col1 and fil1) and labels for the left hand y-axis using the argument y1label or the x-axis with xlabel. Setting filter = TRUE will display the filtered sum(z) scores. In general, the filtered scores should show a similar pattern but be somewhat lower in magnitude depending on contributions from impure or unreliable taxa:

plot_sumz(glades.titan, filter = TRUE)

plot_taxa_ridges()

Pure (purity>=0.95 or higher) and reliable (reliability>=0.95 or higher) indicator taxa that contribute to the overall sum(z) scores can be plotted using the function plot_taxa_ridges().

Taxa that meet purity (pur.cut) and reliability (rel.cut) criteria are extracted from the full taxa output object titan.out$sppmax, separated into two groups: z- (negative responders, or decliners) and z+ (positive responders, or increasers).

These two groups of negative and positive responders are arranged into separate, stacked panels for plotting. Within each panel, taxa change points (values of x resulting in the largest indicator z value) are visualized as a probability density function of the nBoot bootstrapped replicates (n=500 or however many nBoot replicates specified by the user).

Taxa names are arranged discretely along the y-axis based on the rank order of the central tendency of the probability density function for z- (negative, decliners) and z+ (positive, increasers), respectively.

There are many options for changing the look of the graph, so see the full documentation with ?plot_taxa_ridges, but you can get started by simply calling plot_taxa_ridges() on the output of titan():

plot_taxa_ridges(glades.titan, axis.text.y = 8)
# There are 90 indicator taxa of 90 possible for plotting
# Number of Decreasers = 41
# Number of Increasers = 49

Optional arguments include:

• z1 - logical whether maxgrp = 1 taxa (z-, negative, decliners) should be plotted

• z2 - logical whether maxgrp = 2 taxa (z-, negative, decliners) should be plotted

• z1_fill_low - low value of the color fill gradient for z- (negative) taxa

• z1_fill_high - high value of the color fill gradient for z- (negative) taxa

• z2_fill_low - low value of the color fill gradient for z+ (positive) taxa

• z2_fill_high - high value of the color fill gradient for z+ (positive) taxa

• pur.cut - the purity value cutoff for selecting taxa for plotting (defaults to value selected in titan.out). We do not recommend selecting values less than 0.95 for purity; higher values up to 1.0 may be used to refine your selection to the purest responders in terms of direction.

• rel.cut - the reliability value cutoff for selecting taxa for plotting (defaults to value selected in titan.out). We do not recommend selecting values less than 0.95 for purity; higher values up to 1.0 may be used to refine your selection to the purest responders in terms of direction.

• xlabel - the x axis label; the default is “Environmental Gradient”

• n_ytaxa - the maximum number of taxa that can be plotted on the y-axes (combined between the two panels if both z1 and z2 = TRUE, or just for one panel if z1 or z2 = FALSE). If the total number of pure and reliable indicator taxa exceeds the value of n_ytaxa, then the function will select only the most pure and reliable taxa to match the value of n_ytaxa based on rank order of purity, reliability, and finally z.med (sequential sorting, respectively). The selection of n_ytaxa is independent of group membership, so that only the best indicators are displayed, regardless of z- or z+ grouping.

• ytxt.sz - size of the taxa labels on the y-axis. Defaults to ytxt.sz=10. Depending upon the height and width of the plot, labels may need to set to a smaller size to avoid overlap (e.g., see the plot above; here, ytxt.sz = 8).

• printspp - a logical determining whether the sppmax output for the plotted taxa is printed to the console; default = FALSE.

• grid - a logical determining whether gridlines are plotted; default=TRUE

• xlim - the upper and lower limits of the x-axis, e.g., xlim=c(0,180). The default is computed internally based on the lower and upper limits of the x variable.

• bw - short for “bandwidth”, it controls the degree of smoothing of the probability density plots. This is computed internally but can be adjusted manually using this argument. We strongly advise against changing this parameter from the default.

Below is the same function with a user-specified x-label and n_ytaxa set to 50

plot_taxa_ridges(glades.titan, 
  xlabel = expression(paste("Surface water total phosphorus ("*mu*"g/l)")), 
  n_ytaxa = 50
)
# There are 90 indicator taxa of 50 possible for plotting
# Number of Decreasers = 41
# Number of Increasers = 49

There are a total of 90 pure and reliable indicator taxa in the glades.titan output, thus 40 taxa were excluded from this plot because we set n_ytaxa = 50. The 50 taxa plotted are the top 50 out of 90 taxa based on descending rank order of purity, reliability, and z.median, respectively.This argument can be useful when there are too many pure and reliable taxa to effectively visualize in one plot.

Alternatively, set z1 or z2 = FALSE.

plot_taxa_ridges(glades.titan, 
  xlabel = expression(paste("Surface water total phosphorus ("*mu*"g/l)")), 
  z2 = FALSE
)
# There are 90 indicator taxa of 90 possible for plotting
# Number of Decreasers = 41
# Number of Increasers = 49

Gridlines are set to TRUE as the default, but can be removed if desired:

plot_taxa_ridges(glades.titan, 
  xlabel = expression(paste("Surface water total phosphorus ("*mu*"g/l)")), 
  z2 = FALSE, grid = FALSE
)
# There are 90 indicator taxa of 90 possible for plotting
# Number of Decreasers = 41
# Number of Increasers = 49

plot_taxa()

Pure and reliable indicator taxa change points can be plotted using an older plotting function that was integral to the original TITAN2 release, plot_taxa(). We have replaced this plot with the distributions in plot_taxa_ridges() because it tended to empahsize the observed change points over the bootrap replicates. Again, there are many options for changing the look of the graph, so see the full documentation with ?plot_taxa, but you can get started by simply calling plot_taxa() on the output of titan():

plot_taxa(glades.titan, xlabel = "Surface Water TP (ug/l)")

Each plot includes robust declining taxa on the left axis, increasers on the right. Each observed change point (or median of bootstrap replicates if z.med = TRUE) is indicated by the circular symbol; the horizontal lines suggest 5-95% quantiles from the bootstrapped change point distribution.

Optional arguments include:

• z1 - logical whether maxgrp = 1 taxa change points should be plotted

• z2 - logical whether maxgrp = 2 taxa change points should be plotted

• interval - logical whether 5%-95% quantile intervals should be plotted

• prob95 - controls whether change point symbols are plotted at observed value or at quantile extremes, the extremes are useful for thinking about change from a risk perspective

• z.med - controls whether change point symbols are plotted at median of bootstrap replicates and using median \(z\) score instead of observed value

• cex.taxa - controls size of taxon names in graph.

log, at, xmin, xmax, xlabel, tck, bty, ntick, prtty, dig, leg.x, leg.y, cex, cex.axis, cex.leg, cex.lab, legend, col1, fil1, col2, and fil2 are the same as in plot_sumz() above.

Here is the same function with z.med = TRUE:

plot_taxa(glades.titan, xlabel = "Surface Water TP (ug/l)", z.med = TRUE)

Alternatively, with prob95 = TRUE:

plot_taxa(glades.titan, xlabel = "Surface Water TP (ug/l)", z.med = FALSE, prob95 = TRUE)

Importantly, some of the control over the plotting that was optional in the original version has been internalized and warnings are issued when users plot results without using the bootstrap results. Tables of significant z- and z+ taxa used to make graph are automatically returned but no longer automatically written as text files (these can be easily extracted from the glades.titan$sppmax table).

plot_cps()

In TITAN2 2.0 and later, there are several plotting options; all contained within a single function centered on interpreting taxon-specific change points in the context of the results from the bootstrap resampling procedure.

• taxa.dist - logical controls if taxon-specific change point distributions are plotted

• z.weights - logical controls if the median z-sore magnitude is used to weight the PDF from the distribution of resampled change points

• taxaID - if a number or name, creates a single taxon-specific abundance plot

• cp.med - logical controls if change point locations should be plotted using the median value across all bootstrap replicates instead of the observed value

• cp.trace - logical controls if IndVal and zscores are plotted for all candidate change points

• cp.hist - logical controls if histogram of change point PDF is plotted

• stacked - logical controls if change point PDFs for increasing and decreasing taxa are stacked or plotted separately

The remaining arguments: xlabel, xmin, xmax, tck, bty, ntick, cex, cex.axis, cex.leg, cex.lab, leg.x, leg.y and legend are the same as in the previous plotting functions.

The default graphic from plot_cps() is a \(z\)-score weighted probability density function derived from the empirical distribution of change points across all bootstrap replicates plotted as a histogram for each pure and reliable taxon (blue for decreasers and red for increasers). NOTE: When you try to plot the matrix of plots consecutively you can generate an error about “Figure margins too large”. This is easily addressed by closing the plot window, or turning the graphics device off.

plot_cps(glades.titan)

Typically, these histograms are visually scanned to distinguish those taxa whose bootstrapped change point distributions are clearly unimodal from those that are multimodal or more uniform. For any taxon of interest, a closer inspection can be achieved by setting the argument “taxaID” to a taxon ID number or label name. For example, in the plot above, if the declining (i.e., maxgrp = 1) taxon "ENALCIVI" is of interest, the user can set taxaID = 57 or taxaID = "ENALCIVI" and either will produce the plot below. Again, you may need to close the plotting window or turn graphics off because we are switching plot margins.

plot_cps(glades.titan, taxaID = "ENALCIVI", xlabel = "Surface Water TP (ug/l)")

The base plot is simply taxon-specific abundance (black circles) along the environmental gradient. Overlaid near the top of the plot in red is the observed change point, or if cp.med = TRUE, the median change point across all bootstrap replicates. Overlaid on the abundance plot is the \(z\)-weighted (if z.weights = TRUE) probability density function of change point locations across all bootstrap replicates (blue histogram). Setting cp.hist = FALSE will remove the histogram. On the other hand, setting cp.trace = TRUE will plot the observed IndVal \(z\) scores (in red; what TITAN2 uses by default if imax = FALSE) and the observed IndVal scores (in grey) for every candidate change point along the environmental gradient (see Baker and King 2013 for examples):

plot_cps(glades.titan, taxaID = "ENALCIVI", cp.trace = TRUE, xlabel = "Surface Water TP (ug/l)")

Below is an increasing taxon from the same data set with a conspicuously bimodal distribution of replicate change points:

plot_cps(glades.titan, taxaID = "OSTRASP5", cp.trace = TRUE, xlabel = "Surface Water TP (ug/l)")

Here it appears as if two potential change points were predominantly identified: one that kept nearly all of the abundance to the right, and another (the observed) than kept nearly all the absences to the left.

Finally, if taxa.dist = FALSE, instead of a taxon-specific plot, plot_cps() will produce a plot of the summed \(z\)-weighted (if z.weights = TRUE) probability densities across increasing (maxgrp = 1; blue) decreasing (maxgrp = 2; red) and all pure and reliable taxa (filter>0; grey):

plot_cps(glades.titan, taxa.dist = FALSE, xlabel = "Surface Water TP (ug/l)")
# [1] "Summary of Summed Density Functions"
#          uw.env uw.max zw.env zw.max
# Group 1   14.55   1.00  14.55  10.22
# Group 2   28.80   1.34  28.80  11.13
# P&R Taxa  28.80   2.01  28.80  17.14

The accompanying table summarizes the location of both the \(z\)-weighted and unweighted maxima along the environmental gradient, as well as the magnitude of those maxima for decreasing taxa (maxgrp = 1), increasing taxa (maxgrp = 2), and all pure and reliable taxa combined. In this case there appears to be good correspondence between the sum(z) maxima and the summed \(z\)-weighted PDF maxima. In other words, the points at which community change appears to be greatest corresponds to where the sum of individual taxa changes tend to occur, even when different sets of observations are used. In practice, this is a desirable result we would expect to occur most of the time, yet exceptions to this pattern can also be informative. For example, in the plot above, there appears to be another candidate change point closer to ~10 ug/l where declining taxa tend to change frequently, and at both ~25 ug/l and ~40 ug/l there are large numbers of frequently occurring change points across both decliners and increasers even though they are not particularly large for either group by itself.

Alternatively, these different sums can be represented as a stacked bar chart by setting stacked = TRUE:

plot_cps(glades.titan, taxa.dist = FALSE, xlabel = "Surface Water TP (ug/l)", stacked = TRUE)
# [1] "Summary of Summed Density Functions"
#          uw.env uw.max zw.env zw.max
# Group 1   14.55   1.00  14.55  10.22
# Group 2   28.80   1.34  28.80  11.13
# P&R Taxa  28.80   2.01  28.80  17.14

Here the blue and red circles correspond to the location of the respective maxima for decreasing and increasing taxa change points. In either case, the summed PDFs are meant to provide complementarity to the plot_sumz() graphic about where there are key values along the environmental gradient where many coincident change points occur regardless of the sample.

Citations:

Baker, M.E. and R. S. King. 2010. A new method for detecting and interpreting biodiversity and ecological community thresholds. Methods in Ecology and Evolution 1: 25-43.Z

Baker, M.E. and R. S. King. 2013. Of TITAN and strawmen: an appeal for greater understanding of community data. Freshwater Science 32(2):489-506.

King, R.S. and C. J. Richardson. 2003. Integrating bioassessment and ecological risk assessment: an approach to developing numerical water-quality criteria. Environmental Management 31(6):795-809.

Dufrene, M. and P. Legendre. 1997. Species assemblages and indicator species: the need for a flexible asymmetrical approach. Ecological Monographs. 67:345-366.