# Difference between revisions of "Phylogenetics: APE Lab"

Paul Lewis (Talk | contribs) (→Analyses involving tree shape) |
Paul Lewis (Talk | contribs) (→Thanks!) |
||

Line 313: | Line 313: | ||

I bet you wish you had known about this earlier! | I bet you wish you had known about this earlier! | ||

− | == | + | == Independent contrasts (PREVIOUS VERSION) == |

− | I | + | APE can compute Felsenstein's independent contrasts, as well as several other methods for assessing phylogenetically-corrected correlations between traits that I did not discuss in lecture (autocorrelation, generalized least squares, mixed models and variance partitioning, and the very interesting Ornstein-Uhlenbeck model, which can be used to assess the correlation between a continuous character and a discrete habitat variable). |

+ | |||

+ | Today, however, we will just play with independent contrasts. Since you did a homework on this topic earlier, let's try to get the same numbers using APE's <tt>pic</tt> command. Here are the numbers you should have computed for homework 9: | ||

+ | {| border="2" | ||

+ | |- | ||

+ | | | ||

+ | | X | ||

+ | | Y | ||

+ | | Var | ||

+ | | X* | ||

+ | | Y* | ||

+ | |- | ||

+ | | A-B | ||

+ | | 2.80 | ||

+ | | 8.30 | ||

+ | | 2.00 | ||

+ | | 1.98 | ||

+ | | 5.87 | ||

+ | |- | ||

+ | | C-G | ||

+ | | 1.10 | ||

+ | | -8.95 | ||

+ | | 3.50 | ||

+ | | 0.59 | ||

+ | | -4.78 | ||

+ | |- | ||

+ | | D-E | ||

+ | | 1.10 | ||

+ | | 7.70 | ||

+ | | 5.00 | ||

+ | | 0.49 | ||

+ | | 3.44 | ||

+ | |- | ||

+ | | F-H | ||

+ | | -4.13 | ||

+ | | -22.33 | ||

+ | | 26.06 | ||

+ | | -0.81 | ||

+ | | -4.38 | ||

+ | |} | ||

+ | In the table, X and Y denote the raw contrasts, while X* and Y* denote the rescaled contrasts (raw contrasts divided by the square root of the variance). You found that the correlation among the rescaled contrasts was 0.75. | ||

+ | |||

+ | ==== Enter the tree ==== | ||

+ | Start by entering the tree: | ||

+ | > t <- read.tree(text="(((A:1,B:1):1,C:2):10,(D:2,E:3):14);") | ||

+ | The attribute <tt>text</tt> is needed because we are entering the Newick tree description in the form of a string, not supplying a file name. | ||

+ | |||

+ | Plot the tree to make sure the tree definition worked: | ||

+ | > plot(t) | ||

+ | |||

+ | ==== Enter the data ==== | ||

+ | Now we must tell APE the X and Y values. Do this by supplying vectors of numbers. We will tell APE which tips these numbers are associated with in the next step: | ||

+ | > x <- c(11.3, 8.5, 8.8, 14.0, 12.9) | ||

+ | > y <- c(18.9,10.6,23.7,44.0,36.3) | ||

+ | |||

+ | Here's how we tell APE what taxa the numbers belong to: | ||

+ | > names(x) <- c("A","B","C","D","E") | ||

+ | > names(y) <- c("A","B","C","D","E") | ||

+ | If you want to avoid repitition, you can enter the names for both x and y simultaneously like this: | ||

+ | > names(x) <- names(y) <- c("A","B","C","D","E") | ||

+ | |||

+ | ==== Compute independent contrasts ==== | ||

+ | Now compute the contrasts with the APE function <tt>pic</tt>: | ||

+ | > cx <- pic(x,t) | ||

+ | > cy <- pic(y,t) | ||

+ | The variables cx and cy are arbitrary; you could use different names for these if you wanted. Let's see what values cx and cy hold: | ||

+ | > cx | ||

+ | 6 7 8 9 | ||

+ | -0.8093509 0.5879747 1.9798990 0.4919350 | ||

+ | > cy | ||

+ | 6 7 8 9 | ||

+ | -4.375308 -4.783976 5.868986 3.443545 | ||

+ | The top row in each case holds the node number in the tree, the bottom row holds the rescaled contrasts. | ||

+ | |||

+ | ==== Label interior nodes with the contrasts ==== | ||

+ | APE makes it fairly easy to label the tree with the contrasts: | ||

+ | > plot(t) | ||

+ | > nodelabels(round(cx,2), adj=c(0,-1), frame="n") | ||

+ | > nodelabels(round(cy,2), adj=c(0,1), frame="n") | ||

+ | In the nodelabels command, we supply the numbers with which to label the nodes. The vectors cx and cy contain information about the nodes to label, so APE knows from this which numbers to place at which nodes in the tree. The round command simply rounds the contrasts to 2 decimal places. The <tt>adj</tt> setting adjusts the spacing so that the contrasts for X are not placed directly on top of the contrasts for Y. The command <tt>adj=c(0,-1)</tt> causes the labels to be horizontally displaced 0 lines and vertically displaced one line up (the -1 means go up 1 line) from where they would normally be plotted. The contrasts for Y are displaced vertically one line down from where they would normally appear. Finally, the <tt>frame="n"</tt> just says to not place a box or circle around the labels. | ||

+ | |||

+ | You should find that the contrasts are the same as those you calculated on homework 9. Computing the correlation coefficient is as easy as: | ||

+ | > cor(cx, cy) | ||

+ | [1] 0.7522686 | ||

+ | I bet you wish you had known about this earlier! | ||

== Literature Cited == | == Literature Cited == |

## Revision as of 18:50, 4 April 2016

EEB 349: Phylogenetics | |

This lab is an introduction to some of the capabilities of APE, a phylogenetic analysis package written for the R language. You may want to review the R Primer lab if you've already forgotten everything you learned about R. |

## Contents

## Installing APE and apTreeshape

**APE** is a package largely written and maintained by Emmanuel Paradis, who has written a very nice book^{[1]} explaining in detail how to use APE. APE is designed to be used inside the R programming language, to which you were introduced earlier in the semester (see Phylogenetics: R Primer). APE can do an impressive array of analyses. For example, it is possible to estimate trees using neighbor-joining or maximum likelihood, estimate ancestral states (for either discrete or continuous data), perform Sanderson's penalized likelihood relaxed clock method to estimate divergence times, evaluate Felsenstein's independent contrasts, estimate birth/death rates, perform bootstrapping, and even automatically pull sequences from GenBank given a vector of accession numbers! APE also has impressive tree plotting capabilities, of which we will only scratch the surface today (flip through Chapter 4 of the Paradis book to see what more APE can do).

**apTreeshape** is a different R package (written by Nicolas Bortolussi et al.) that we will also make use of today.

To install APE and apTreeshape, start R and type the following at the R command prompt:

> install.packages("ape") > install.packages("apTreeshape")

Assuming you are connected to the internet, R should locate these packages and install them for you. After they are installed, you will need to load them into R in order to use them (note that no quotes are used this time):

> library(ape) > library(apTreeshape)

You should never again need to issue the `install.packages` command for APE and apTreeshape, but you will need to use the `library` command to load them whenever you want to use them.

## Reading in trees from a file

Download this tree file and save it as a file named `yule.tre` in a new folder somewhere on your computer. Tell R where this folder is using the `setwd` (set working directory) command. For example, I created a folder named `apelab` on my desktop, so I typed this to make that folder my working directory:

> setwd("/Users/plewis/Desktop/apelab")

Now you should be able to read in the tree using this ape command (the `t` is an arbitrary name I chose for the variable used to hold the tree; you could use `tree` if you want):

> t <- read.nexus("yule.tre")

APE stores trees as an object of type "phylo".

#### Getting a tree summary

Some basic information about the tree can be obtained by simply typing the name of the variable you used to store the tree:

> t Phylogenetic tree with 20 tips and 19 internal nodes. Tip labels: B, C, D, E, F, G, ... Rooted; includes branch lengths.

#### Obtaining vectors of tip and internal node labels

The variable `t` has several attributes that can be queried by following the variable name with a dollar sign and then the name of the attribute. For example, the vector of tip labels can be obtained as follows:

> t$tip.label [1] "B" "C" "D" "E" "F" "G" "H" "I" "J" "K" "L" "M" "N" "O" "P" "Q" "R" "S" "T" "U"

The internal node labels, if they exist, can be obtained this way:

> t$node.label NULL

The result above means that labels for the internal nodes were not stored with this tree.

#### Obtaining the nodes attached to each edge

The nodes at the ends of all the edges in the tree can be had by asking for the edge attribute:

> t$edge [,1] [,2] [1,] 21 22 [2,] 22 23 [3,] 23 1 . . . . . . . . . [38,] 38 12

#### Obtaining a vector of edge lengths

The edge lengths can be printed thusly:

> t$edge.length [1] 0.07193600 0.01755700 0.17661500 0.02632500 0.01009100 0.06893900 0.07126000 0.03970200 0.01912900 [10] 0.01243000 0.01243000 0.03155800 0.05901300 0.08118600 0.08118600 0.00476400 0.14552600 0.07604800 [19] 0.00070400 0.06877400 0.06877400 0.02423800 0.02848800 0.01675100 0.01675100 0.04524000 0.19417200 [28] 0.07015000 0.12596600 0.06999200 0.06797400 0.00201900 0.00201900 0.12462600 0.07128300 0.00004969 [37] 0.00004969 0.07133200

#### About this tree

This tree in the file `yule.tre` was obtained using PAUP from 10,000 nucleotide sites simulated from a Yule tree. The model used to generate the simulated data (HKY model, kappa = 4, base frequencies = 0.3 A, 0.2 C, 0.2 G, and 0.3 T, no rate heterogeneity) was also used in the analysis by PAUP (the final ML tree was made ultrametric by enforcing the clock constraint).

## Fun with plotting trees in APE

You can plot the tree using all defaults with this ape command:

> plot(t)

Let's try changing a few defaults and plot the tree in a variety of ways. All of the following change just one default option, but feel free to combine these to create the plot you want.

#### Left-facing, up-facing, or down-facing trees

> plot(t, direction="l") > plot(t, direction="u") > plot(t, direction="d")

The default is to plot the tree right-facing (`direction="r"`).

#### Hide the taxon names

> plot(t, show.tip.label=FALSE)

The default behavior is to show the taxon names.

#### Make the edges thicker

> plot(t, edge.width=4)

An edge width of 1 is the default. If you specify several edge widths, APE will alternate them as it draws the tree:

> plot(t, edge.width=c(1,2,3,4))

#### Color the edges

> plot(t, edge.color="red")

Black edges are the default. If you specify several edge colors, APE will alternate them as it draws the tree:

> plot(t, edge.color=c("black","red","green","blue"))

#### Make taxon labels smaller or larger

> plot(t, cex=0.5)

The cex parameter governs relative scaling of the taxon labels, with 1.0 being the default. Thus, the command above makes the taxon labels half the default size. To double the size, use

> plot(t, cex=2.0)

#### Plot tree as an unrooted or radial tree

> plot(t, type="u")

The default type is "p" (phylogram), but "c" (cladogram), "u" (unrooted), "r" (radial) are other options. Some of these options (e.g. "r") create very funky looking trees, leading me to think there is something about the tree description in the file `yule.tre` that APE is not expecting.

#### Labeling internal nodes

> plot(t) > nodelabels()

This is primarily useful if you want to annotate one of the nodes:

> plot(t) > nodelabels("Clade A", 22) > nodelabels("Clade B", 35)

To put the labels inside a circle rather than a rectangle, use `frame="c"` rather than the default (`frame="r"`). To use a background color of white rather than the default "lightblue", use `bg="white"`:

> plot(t) > nodelabels("Clade A", 22, frame="c", bg="white") > nodelabels("Clade B", 35, frame="c", bg="yellow")

#### Adding a scale bar

> plot(t) > add.scale.bar(length=0.05)

The above commands add a scale bar to the bottom left of the plot. To add a scale going all the way across the bottom of the plot, try this:

> plot(t) > axisPhylo()

## Diversification analyses

APE can perform some lineage-through-time type analyses. The tree read in from the file `yule.tre` that you already have in memory is perfect for testing APE's diversification analyses because we know (since it is based on simulated data) that this tree was generated under a pure-birth (Yule) model.

#### Lineage through time plots

This is a rather small tree, so a lineage through time (LTT) plot will be rather crude, but let's go through the motions anyway.

> ltt.plot(t)

LTT plots usually have a log scale for the number of lineages (y-axis), and this can be easily accomplished:

> ltt.plot(t, log = "y")

Now add a line extending from the point (t = -0.265, N = 2) to the point (t = 0, N = 20) using the command `segments` (note that the "segments" command is a core R command, not something added by the APE package):

> segments(-0.265, 2, 0, 20, lty="dotted")

The slope of this line should (ideally) be equal to the birth rate of the yule process used to generate the tree, which was .

- Calculate the slope of this line. Is it close to the birth rate 10? answer

If you get something like 68 for the slope, then you forgot to take the natural log of 2 and 20. The plot uses a log scale for the y-axis, so the two endpoints of the dotted line are really (-0.265, log(2)) and (0, log(20)).

#### Birth/death analysis

APE's `birthdeath` command estimates the birth and death rates using the node ages in a tree:

> birthdeath(t) Estimation of Speciation and Extinction Rates with Birth-Death Models Phylogenetic tree: t Number of tips: 20 Deviance: -120.4538 Log-likelihood: 60.22689 Parameter estimates: d / b = 0 StdErr = 0 b - d = 8.674513 StdErr = 1.445897 (b: speciation rate, d: extinction rate) Profile likelihood 95% confidence intervals: d / b: [0, 0.5286254] b - d: [5.25955, 13.32028]

The output indicates that it was correctly able to detect that the death rate was 0, and the estimated birth rate should be very close to the value you calculated for the slope of the dotted line in the LTT plot.

- What is the death rate estimated by APE? answer
- Is the true birth rate within one standard deviation of the estimated birth rate? answer
- Are the true birth and death rates within the profile likelihood 95% confidence intervals? answer

A "profile" likelihood is obtained by varying one parameter in the model and re-estimating all the other parameters conditional on the current value of the focal parameter. This is, technically, not the correct way of getting a confidence interval, but is easier to compute and may be more stable for small samples than getting confidence intervals the correct way.

- What is the correct way to interpret the 95% confidence interval for b - d: [5.25955, 13.32028]? Is it that there is 95% chance that the true value of b - d is in that interval? answer
- Or, does it mean that our estimate (8.674513) is within the middle 95% of values that would be produced if the true b - d value was in that interval? answer

#### Analyses involving tree shape

The apTreeshape package (as the name applies) lets you perform analyses of tree shape (which measure how balanced or imbalanced a tree is). apTreeshape stores trees differently than APE, so you can't use a tree object that you created with APE in functions associated with apTreeshape. You can, however, convert a "phylo" object from APE to a "treeshape" object used by apTreeshape:

> ts <- as.treeshape(t)

Here, I'm assuming that `t` still refers to the tree you read in from the file `yule.tre` using the APE command `read.nexus`. We can now obtain a measure of **tree imbalance** known as Colless's index:

> c <- colless(ts) > c [1] 44

The formula for Colless's index is easy to understand. Each internal node branches into a left and right lineage. The absolute value of the difference between the number of left-hand descendants and right-hand descendants provides a measure of how imbalanced the tree is with respect to that particular node. Adding these imbalance measures up over all internal nodes yields Colless's overall tree imbalance index:

apTreeshape can do an analysis to assess whether the tree has the amount of imbalance one would expect from a Yule tree:

> colless.test(ts, model = "yule", alternative="greater", n.mc = 1000)

This generates 1000 trees from a Yule process and compares the Colless index from our tree (44) to the distribution of such indices obtained from the simulated trees. The p-value is the proportion of the 1000 trees generated from the null distribution that have indices greater than 44 (i.e. the proportion of Yule trees that are more *im*balanced than our tree). If the p-value was 0.5, for example, then our tree would be right in the middle of the distribution expected for Yule trees. If the p-value was 0.01, however, it would mean that very few Yule trees are as imbalanced as our tree, which would make it hard to believe that our tree is a Yule tree.

- Does the p-value indicate that we should reject the hypothesis that a Yule process generated our tree? answer

You can also test one other model with the `colless` function: the "proportional to distinguishable" (or PDA) model. This null model produces random trees by starting with three taxa joined to a single internal node, then building upon that by adding new taxa to randomly-chosen (discrete uniform distribution) edges that already exist in the (unrooted) tree. The edge to which a new taxon is added can be an internal edge as well as a terminal edge, which causes this process to produce trees with a different distribution of shapes than the Yule process, which only adds new taxa to the tips of a growing rooted tree.

> colless.test(ts, model = "pda", alternative="greater", n.mc = 1000)

- Does the p-value indicate that we should reject the hypothesis that our tree is a PDA tree? answer

You might also wish to test whether our tree is more *balanced* than would be expected under the Yule or PDA models. apTreeshape let's you look at the other end of the distribution too:

> colless.test(ts, model = "yule", alternative="less", n.mc = 1000) > colless.test(ts, model = "pda", alternative="less", n.mc = 1000)

- Does the p-value for the first test above indicate that our tree is more balanced than would be expected were it a Yule tree? answer
- Does the p-value for the second test above indicate that our tree is more balanced than would be expected were it a PDA tree? answer

You might want to see a histogram of the Colless index like that used to determine the p-values for the tests above. apTreeshape lets you generate 10 trees each with 20 tips under a Yule model as follows:

> rtreeshape(10,20,model="yule")

That spits out a summary of the 10 trees created, but what we really wanted was to know the Colless index for each of the trees generated. To do this, use the R command `sapply` to call the apTreeshape command `colless` for each tree generated by the `rtreeshape` command:

> sapply(rtreeshape(10,20,model="yule"),FUN=colless) [1] 38 92 85 91 73 71 94 75 72 93

That's more like it! Now, generate 1000 Yule trees instead of just 10, and create a histogram using the standard R command `hist`:

> yulecolless <- sapply(rtreeshape(1000,20,model="yule"),FUN=colless) > hist(yulecolless)

Now create a histogram for PDA trees:

> pdacolless <- sapply(rtreeshape(1000,20,model="pda"),FUN=colless) > hist(pdacolless)

Use the following to compare the mean Colless index for the PDA trees to the Yule trees:

> summary(yulecolless) > summary(pdacolless)

- Which generates the most balanced trees, on average: Yule or PDA? answer

apTreeshape provides one more function (`likelihood.test`) that performs a likelihood ratio test of the PDA model against the Yule model null hypothesis. This test says that we cannot reject the null hypothesis of a Yule model in favor of the PDA model:

> likelihood.test(ts)

- Does the p-value indicate that we should reject the hypothesis that our tree is a Yule tree? answer

## Independent contrasts

APE can compute Felsenstein's independent contrasts, as well as several other methods for assessing phylogenetically-corrected correlations between traits that I did not discuss in lecture (autocorrelation, generalized least squares, mixed models and variance partitioning, and the very interesting Ornstein-Uhlenbeck model, which can be used to assess the correlation between a continuous character and a discrete habitat variable).

Today, however, we will just play with independent contrasts. Since you did a homework on this topic earlier, let's try to get the same numbers using APE's `pic` command. Here are the numbers you should have computed for homework 9:

X | Y | Var | X* | Y* | |

A-B | 2.80 | 8.30 | 2.00 | 1.98 | 5.87 |

C-G | 1.10 | -8.95 | 3.50 | 0.59 | -4.78 |

D-E | 1.10 | 7.70 | 5.00 | 0.49 | 3.44 |

F-H | -4.13 | -22.33 | 26.06 | -0.81 | -4.38 |

In the table, X and Y denote the raw contrasts, while X* and Y* denote the rescaled contrasts (raw contrasts divided by the square root of the variance). You found that the correlation among the rescaled contrasts was 0.75.

#### Enter the tree

Start by entering the tree:

> t <- read.tree(text="(((A:1,B:1):1,C:2):10,(D:2,E:3):14);")

The attribute `text` is needed because we are entering the Newick tree description in the form of a string, not supplying a file name.

Plot the tree to make sure the tree definition worked:

> plot(t)

#### Enter the data

Now we must tell APE the X and Y values. Do this by supplying vectors of numbers. We will tell APE which tips these numbers are associated with in the next step:

> x <- c(11.3, 8.5, 8.8, 14.0, 12.9) > y <- c(18.9,10.6,23.7,44.0,36.3)

Here's how we tell APE what taxa the numbers belong to:

> names(x) <- c("A","B","C","D","E") > names(y) <- c("A","B","C","D","E")

If you want to avoid repitition, you can enter the names for both x and y simultaneously like this:

> names(x) <- names(y) <- c("A","B","C","D","E")

#### Compute independent contrasts

Now compute the contrasts with the APE function `pic`:

> cx <- pic(x,t) > cy <- pic(y,t)

The variables cx and cy are arbitrary; you could use different names for these if you wanted. Let's see what values cx and cy hold:

> cx 6 7 8 9 -0.8093509 0.5879747 1.9798990 0.4919350 > cy 6 7 8 9 -4.375308 -4.783976 5.868986 3.443545

The top row in each case holds the node number in the tree, the bottom row holds the rescaled contrasts.

#### Label interior nodes with the contrasts

APE makes it fairly easy to label the tree with the contrasts:

> plot(t) > nodelabels(round(cx,2), adj=c(0,-1), frame="n") > nodelabels(round(cy,2), adj=c(0,1), frame="n")

In the nodelabels command, we supply the numbers with which to label the nodes. The vectors cx and cy contain information about the nodes to label, so APE knows from this which numbers to place at which nodes in the tree. The round command simply rounds the contrasts to 2 decimal places. The `adj` setting adjusts the spacing so that the contrasts for X are not placed directly on top of the contrasts for Y. The command `adj=c(0,-1)` causes the labels to be horizontally displaced 0 lines and vertically displaced one line up (the -1 means go up 1 line) from where they would normally be plotted. The contrasts for Y are displaced vertically one line down from where they would normally appear. Finally, the `frame="n"` just says to not place a box or circle around the labels.

You should find that the contrasts are the same as those you calculated on homework 9. Computing the correlation coefficient is as easy as:

> cor(cx, cy) [1] 0.7522686

I bet you wish you had known about this earlier!

## Independent contrasts (PREVIOUS VERSION)

APE can compute Felsenstein's independent contrasts, as well as several other methods for assessing phylogenetically-corrected correlations between traits that I did not discuss in lecture (autocorrelation, generalized least squares, mixed models and variance partitioning, and the very interesting Ornstein-Uhlenbeck model, which can be used to assess the correlation between a continuous character and a discrete habitat variable).

Today, however, we will just play with independent contrasts. Since you did a homework on this topic earlier, let's try to get the same numbers using APE's `pic` command. Here are the numbers you should have computed for homework 9:

X | Y | Var | X* | Y* | |

A-B | 2.80 | 8.30 | 2.00 | 1.98 | 5.87 |

C-G | 1.10 | -8.95 | 3.50 | 0.59 | -4.78 |

D-E | 1.10 | 7.70 | 5.00 | 0.49 | 3.44 |

F-H | -4.13 | -22.33 | 26.06 | -0.81 | -4.38 |

In the table, X and Y denote the raw contrasts, while X* and Y* denote the rescaled contrasts (raw contrasts divided by the square root of the variance). You found that the correlation among the rescaled contrasts was 0.75.

#### Enter the tree

Start by entering the tree:

> t <- read.tree(text="(((A:1,B:1):1,C:2):10,(D:2,E:3):14);")

The attribute `text` is needed because we are entering the Newick tree description in the form of a string, not supplying a file name.

Plot the tree to make sure the tree definition worked:

> plot(t)

#### Enter the data

Now we must tell APE the X and Y values. Do this by supplying vectors of numbers. We will tell APE which tips these numbers are associated with in the next step:

> x <- c(11.3, 8.5, 8.8, 14.0, 12.9) > y <- c(18.9,10.6,23.7,44.0,36.3)

Here's how we tell APE what taxa the numbers belong to:

> names(x) <- c("A","B","C","D","E") > names(y) <- c("A","B","C","D","E")

If you want to avoid repitition, you can enter the names for both x and y simultaneously like this:

> names(x) <- names(y) <- c("A","B","C","D","E")

#### Compute independent contrasts

Now compute the contrasts with the APE function `pic`:

> cx <- pic(x,t) > cy <- pic(y,t)

The variables cx and cy are arbitrary; you could use different names for these if you wanted. Let's see what values cx and cy hold:

> cx 6 7 8 9 -0.8093509 0.5879747 1.9798990 0.4919350 > cy 6 7 8 9 -4.375308 -4.783976 5.868986 3.443545

The top row in each case holds the node number in the tree, the bottom row holds the rescaled contrasts.

#### Label interior nodes with the contrasts

APE makes it fairly easy to label the tree with the contrasts:

> plot(t) > nodelabels(round(cx,2), adj=c(0,-1), frame="n") > nodelabels(round(cy,2), adj=c(0,1), frame="n")

In the nodelabels command, we supply the numbers with which to label the nodes. The vectors cx and cy contain information about the nodes to label, so APE knows from this which numbers to place at which nodes in the tree. The round command simply rounds the contrasts to 2 decimal places. The `adj` setting adjusts the spacing so that the contrasts for X are not placed directly on top of the contrasts for Y. The command `adj=c(0,-1)` causes the labels to be horizontally displaced 0 lines and vertically displaced one line up (the -1 means go up 1 line) from where they would normally be plotted. The contrasts for Y are displaced vertically one line down from where they would normally appear. Finally, the `frame="n"` just says to not place a box or circle around the labels.

You should find that the contrasts are the same as those you calculated on homework 9. Computing the correlation coefficient is as easy as:

> cor(cx, cy) [1] 0.7522686

I bet you wish you had known about this earlier!

## Literature Cited

- ↑ Paradis, E. 2006. Analysis of phylogenetics and evolution with R. Springer. ISBN: 0-387-32914-5