# Difference between revisions of "R Hackathon 1/Ancestral State Reconstruction"

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The primary ancestral state reconstruction algorithms in R are accessed through the function "ace" in package "ape". To work through the following example using the Geospiza dataset, make sure that you have installed and loaded ape into your R session and loaded the Geospiza phylogeny and tip data into memory. | The primary ancestral state reconstruction algorithms in R are accessed through the function "ace" in package "ape". To work through the following example using the Geospiza dataset, make sure that you have installed and loaded ape into your R session and loaded the Geospiza phylogeny and tip data into memory. | ||

Line 15: | Line 16: | ||

geodata <- geodata[geotree$tip.label, ] | geodata <- geodata[geotree$tip.label, ] | ||

− | The other, and better option is to extract the column of data of interest from the overall datatable, creating a new vector, and to transfer the row.names of the | + | The other, and better option is to extract the column of data of interest from the overall datatable, creating a new vector, and to transfer the row.names of the data frame to the NAMES of the vector. This is the option least likely to inadvertently disassociate the tip data from the tips. |

wingL <- geodata$wingL | wingL <- geodata$wingL | ||

Line 55: | Line 56: | ||

'''GENERALIZED LEAST SQUARES''' | '''GENERALIZED LEAST SQUARES''' | ||

− | The | + | WARNING: The current implementation of GLS in ace is unreliable, giving spurious results or error messages that seem to vary between computers and R platforms. Use these at your own risk. |

− | Reconstructions based upon generalized least squares require specification of a correlation structure for the generalized linear model. There are three basic options for the specification of the correlation structure: 1) corBrownian, which uses a simple Brownian motion model | + | Reconstructions based upon generalized least squares require specification of a correlation structure for the generalized linear model. There are three basic options for the specification of the correlation structure: 1) corBrownian, which uses a simple Brownian motion model; corMartins, which uses an Ornstein-Uhlenbeck (constrained random-walk) model; and corGrafen, which uses a modified Brownian motion model based upon an ultrametricization of tree as specified by Grafen (1989). |

This is the syntax for the simple Brownian model | This is the syntax for the simple Brownian model | ||

GLSreconstruction <- ace(wingL, geotree, type="continuous", method="GLS", corStruct = corBrownian(1, geotree)) | GLSreconstruction <- ace(wingL, geotree, type="continuous", method="GLS", corStruct = corBrownian(1, geotree)) | ||

+ | |||

+ | GLS ancestral state reconstruction using corBrownian currently gives spurious results near the root of the phylogeny, at least at present. Specifically, the root node is always reconstructed as possessing state zero, and internal nodes near the root may receive reconstructed values out of the range of values observed among the tips. | ||

The syntax for the Ornstein-Uhlenbeck model requires specifying an alpha parameter (here, 0.5). | The syntax for the Ornstein-Uhlenbeck model requires specifying an alpha parameter (here, 0.5). | ||

GLSreconstruction <- ace(wingL, geotree, type="continuous", method="GLS", corStruct = corMartins(0.5, geotree)) | GLSreconstruction <- ace(wingL, geotree, type="continuous", method="GLS", corStruct = corMartins(0.5, geotree)) | ||

+ | |||

+ | Note that Orstein-Uhlenbeck model in this application is currently not working correctly. If you execute the command above, you will receive a Not a Number warning, the reconstructed node values are unreasonably low, and NaN appears in the returned confidence interval. | ||

The syntax for the Grafen model requires specifying a rho parameter (here, 1) which exponentiates the recalculated branch lengths. | The syntax for the Grafen model requires specifying a rho parameter (here, 1) which exponentiates the recalculated branch lengths. | ||

Line 71: | Line 76: | ||

GLSreconstruction <- ace(wingL, geotree, type="continuous", method="GLS", corStruct = corGrafen(1, geotree)) | GLSreconstruction <- ace(wingL, geotree, type="continuous", method="GLS", corStruct = corGrafen(1, geotree)) | ||

+ | The use of corGrafen generates an error on some computers on which we have executed this code, but not all. We are currently investigating the cause. | ||

Line 98: | Line 104: | ||

Plot.phylo can scale symbols at tips and nodes of your tree to your character data. Try this to visualize the evolution of wing length (wingL) across the Geospiza phylogeny. | Plot.phylo can scale symbols at tips and nodes of your tree to your character data. Try this to visualize the evolution of wing length (wingL) across the Geospiza phylogeny. | ||

− | plot | + | plot(geotree, show.tip.label=FALSE) |

− | tiplabels(pch = 21, cex=wingL) | + | tiplabels(pch = 21, cex=(wingL-3.9)*10) |

− | nodelabels(pch = 21, cex= | + | nodelabels(pch = 21, cex=(MLreconstruction$ace-3.9)*10) |

## (pch = 21 is just telling plot.phylo which symbol to use) | ## (pch = 21 is just telling plot.phylo which symbol to use) | ||

− | Note that the | + | Note that cex argument determines the size at which the symbols will be plotted. These have been rescaled so that the differences between species are visible on the screen. Subtracting 3.9 and multiplying by 10 rescales the wingL to range from a little less than 1 to about 5, which is a pretty good range for plotting. |

+ | |||

'''HOW DO I FIT A MODEL OF CHARACTER CHANGE TO CONTINUOUS DATA?''' | '''HOW DO I FIT A MODEL OF CHARACTER CHANGE TO CONTINUOUS DATA?''' | ||

− | While ace returns a rate of Brownian evolution (e.g. MLreconstruction$ | + | While ace returns a rate of Brownian evolution (e.g. MLreconstruction$sigma2), that rate and its associated likelihood value (e.g., MLreconstruction$loglik) is conditioned on the specific ancestral states reconstructed by ace. More generally applicable model fits using Brownian motion and the Ornstein-Uhlenbeck model are available in the packages ouch and geiger. [[R_Hackathon/ContinuousData|Please see here]] for more information. |

== Reconstructing Ancestral States for Discrete Variables == | == Reconstructing Ancestral States for Discrete Variables == | ||

Line 134: | Line 141: | ||

ARDreconstruction <- ace(char1, geotree, type="discrete", model="ARD") | ARDreconstruction <- ace(char1, geotree, type="discrete", model="ARD") | ||

− | + | These commands issue lots of output, of which you are likely to be most interested in the log likelihoods and the inferred transition rates. You can access the log likelihoods for each model as follows | |

+ | |||

+ | ERreconstruction$loglik | ||

+ | SYMreconstruction$loglik | ||

+ | ARDreconstruction$loglik | ||

+ | |||

+ | You should receive values of -9.00, -9.00 and -7.28, respectively. | ||

+ | |||

+ | The transition rates are accessed with | ||

+ | |||

+ | ERreconstruction$rates | ||

+ | SYMreconstruction$rates | ||

+ | ARDreconstruction$rates | ||

+ | |||

+ | ER and SYM return a single rate of 51.6, while ARD returns a forward and reverse rate (6.00 and 17.9, respectively). | ||

− | + | You may have noticed that the output for the ER model matches that of the SYM model. This is not surprising. For a binary character these models are identical, though they are distinct for characters with three or more states. For a three-state character, ER is a one parameter model, SYM a three parameter model, and ARD a six parameter model. | |

− | + | ''Which model should you prefer?'' | |

− | + | In this example, the ARD model gives the highest likelihood. However, it also includes more parameters that the ER and SYM models, and it is well known that adding parameters to a model generally increases its likelihood. To determine whether the use of the more heavily parameterized model is appropriate, you should execute a likelihood test. | |

− | + | Twice the difference in log likelihood between two models (known as the G statistic) is distributed as chi square, with degrees of freedom equal to the number of parameters that differ between the models. Thus a likelihood test of these models asks whether the difference in likelihoods is large enough to lie in the rightmost tail of the chisquare distribution, typically considered to be the largest 5% of values. | |

− | + | The command pchisq(value, df) gives the percentage of the cumulative distribution function for chisquare that lies to the left of the given value for a desired degree of freedom(df). Thus, the following command gives the percentage of the chisquare distribution that lies to the right of the observed likelihood difference for the ER and ARD models (which differ by one parameter), given the Geospiza data. | |

− | + | 1-pchisq(2*abs(ERreconstruction$loglik - ARDreconstruction$loglik), 1) | |

− | + | Evaluation of the equation above yields the p-value for the likelihood test, in this case 0.064. This is not a significant result at a 5% error rate (though it is very close to significance), so we do not accept the more heavily parameterized ARD model, and instead choose the single-parameter ER model. | |

+ | |||

+ | For a single degree of freedom, the significance cutoff for the likelihood values lies at about 1.93 log-likelihood units of difference. | ||

+ | |||

+ | ''How do I define and use a custom transition model?'' | ||

+ | |||

+ | What if you want to use a transition model that isn't hardwired into ace? You can do so by defining a custom transition matrix. For example, let's create a custom four-parameter matrix for a three state character, in which the the probability of transitioning from state 1 to 2 equals the probability of state 2 to 1 (parameter 1), the probability of transitioning from 1 to 3 equals the probability of transitioning from state 3 to 1 (parameter 2), but the probability of transitioning from state 2 to 3 (parameter 3) does not necessarily equal the probability of transitioning from state 3 to 2 (parameter 4). | ||

+ | |||

+ | This syntax will set up such a transition matrix. | ||

transitions <- matrix (c(0, 1, 2, 1, 0, 3, 2, 4, 0), nrow=3) | transitions <- matrix (c(0, 1, 2, 1, 0, 3, 2, 4, 0), nrow=3) | ||

+ | transitions | ||

− | + | You should see a matrix that looks like the following output to your screen. | |

[,1] [,2] [,3] | [,1] [,2] [,3] | ||

Line 158: | Line 188: | ||

[2,] 1 0 4 | [2,] 1 0 4 | ||

[3,] 2 3 0 | [3,] 2 3 0 | ||

+ | |||

+ | Let's now define a three-state discrete character for the Geospiza and apply this custom transition model to the ancestral state reconstruction. | ||

+ | |||

+ | char2<-c(3,3,1,2,1,1,3,3,3,3,2,2,2,3) | ||

+ | names(char2)<- geotree$tip.label | ||

+ | CUSreconstruction <- ace(char2, geotree, type="discrete", model=transitions) | ||

+ | |||

+ | You should obtain a log likelihood of -11.49 | ||

+ | |||

+ | ''Are there any caveats about the reconstruction of discrete characters that I should be aware of?'' | ||

+ | |||

+ | Yes. Ace deals with this analysis by parameterizing each node within the phylogeny and reconstructing its ancestral state separately. This is a computationally intensive problem, and ace sometime returns spurious results when attempting a solution. If the returned log likelihood is tiny or enormous, or some of the reconstructed ancestral states have greater than 100% probability (or negative probability) at certain nodes, it is likely that ape is not reconstructing an accurate solution for the character in question. You are most likely to run into this issue when using complicated transition models or trees that imply many character state changes. | ||

+ | |||

+ | You can see the probability of each of the possible ancestral states at each of the nodes by typing the following: | ||

+ | |||

+ | CUSreconstruction$lik.anc | ||

+ | |||

+ | In other cases, the likelihood surface for the rates will be essentially flat, and you can place little confidence in the specific rates being reconstructed for each node. If the rate values are important in your application, check their standard errors. | ||

+ | |||

+ | CUSreconstruction$se | ||

+ | |||

+ | If the standard errors are very large, or represented by NaN (not a number), then you are dealing with a flat or nearly flat likelihood surface, and should not place any confidence in the reconstructed rates. This was true for the equal rates (ER) reconstruction of char1 above. Type | ||

+ | |||

+ | ERreconstruction$se | ||

+ | |||

+ | to see the standard error for that reconstruction. R should returned NaN for this value. | ||

+ | |||

+ | |||

+ | == Citations == | ||

+ | |||

+ | Felsenstein, J. 1985. Phylogenies and the comparative method. ''American Naturalist'' 125, 1-15. | ||

+ | |||

+ | Grafen, A. 1989. The phylogenetic regression. ''Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences'', Vol. 326, No. 1233. (Dec. 21, 1989), pp. 119-157. | ||

+ | |||

+ | Paradis, E. 2006. ''Analysis of Phylogenetics and Evolution using R''. New York, Springer . 211 pp. | ||

+ | Much of the text above is paraphrased from this source. | ||

+ | |||

+ | Schluter D., Price T. Mooers A. O. and Ludwig D. 1997. Likelihood of ancestral states in adaptive radiation. ''Evolution'' 51: 1699-1711. | ||

+ | |||

+ | [[Category:R Hackathon 1]] | ||

+ | [[Category:Comparative Methods Help]] |

## Latest revision as of 12:22, 12 February 2008

## Contents

The primary ancestral state reconstruction algorithms in R are accessed through the function "ace" in package "ape". To work through the following example using the Geospiza dataset, make sure that you have installed and loaded ape into your R session and loaded the Geospiza phylogeny and tip data into memory.

library(ape) geotree <- read.nexus("geospiza.nex") geodata <- read.table("geospiza.txt")

Tip data is not available for the outgroup "olivacea", so drop that taxon from the analysis.

geotree <- drop.tip(geotree, "olivacea")

IMPORTANT. The row.names of your dataframe must match the tip.labels of your phylogeny. In the example above, the row.names for geodata will match the tip.labels for geotree after "olivacea" has been culled. However, the individual columns of geodata (e.g. geodata$wingL) do not automatically have the row.names of the whole data table associated with them! If you call a column of the data.table with ace without first dealing with this issue, the analysis will run, but the tip data will be disassociated from the proper tips! There are two workarounds.

One option is to sort the datatable so that the taxa appear in the same order in the table as in the phylogeny. For example:

geodata <- geodata[geotree$tip.label, ]

The other, and better option is to extract the column of data of interest from the overall datatable, creating a new vector, and to transfer the row.names of the data frame to the NAMES of the vector. This is the option least likely to inadvertently disassociate the tip data from the tips.

wingL <- geodata$wingL names(wingL) <- row.names(geodata)

The worked examples below assume that you have used the second solution (vector extraction and name assignment).

## Reconstructing Ancestral States for Continuous Variables

There are three general options for continuous variables: reconstructions based on maximum likelihood (ML), reconstructions based upon phylogenetic independent contrasts (pic) and reconstructions based on generalized least squares (GLS). Be aware that the generalized least squares algorithms seem to give spurious results near the root of the phylogeny at present.

**MAXIMUM LIKELIHOOD**

This syntax will reconstruct the ancestral states for the variable "wingL" (extracted from geodata) using the Brownian motion-based maximum likelihood (ML) estimator of Schluter et. al. (1997). This is the default method.

MLreconstruction <- ace(wingL, geotree, type="continuous", method="ML")

*Why am I getting all these warning messages?*

ML reconstruction using ace tends to generate a large number of not-a-number error messages. These result when the program calculates the likelihood of particularly poor fits. You can safely ignore these messages.

**PHYLOGENETIC INDEPENDENT CONSTRASTS**

This syntax will reconstruct the ancestral states for the variable "wingL" (extracted from geodata) using Felsenstein's (1985) phylogenetic independent contrasts (pic). This is also a Brownian-motion based estimator, but it only takes descendants of each node into account in reconstructing the state at that node. More basal nodes are ignored.

picreconstruction <- ace(wingL, geotree, type="continuous", method="pic")

**GENERALIZED LEAST SQUARES**

WARNING: The current implementation of GLS in ace is unreliable, giving spurious results or error messages that seem to vary between computers and R platforms. Use these at your own risk.

Reconstructions based upon generalized least squares require specification of a correlation structure for the generalized linear model. There are three basic options for the specification of the correlation structure: 1) corBrownian, which uses a simple Brownian motion model; corMartins, which uses an Ornstein-Uhlenbeck (constrained random-walk) model; and corGrafen, which uses a modified Brownian motion model based upon an ultrametricization of tree as specified by Grafen (1989).

This is the syntax for the simple Brownian model

GLSreconstruction <- ace(wingL, geotree, type="continuous", method="GLS", corStruct = corBrownian(1, geotree))

GLS ancestral state reconstruction using corBrownian currently gives spurious results near the root of the phylogeny, at least at present. Specifically, the root node is always reconstructed as possessing state zero, and internal nodes near the root may receive reconstructed values out of the range of values observed among the tips.

The syntax for the Ornstein-Uhlenbeck model requires specifying an alpha parameter (here, 0.5).

GLSreconstruction <- ace(wingL, geotree, type="continuous", method="GLS", corStruct = corMartins(0.5, geotree))

Note that Orstein-Uhlenbeck model in this application is currently not working correctly. If you execute the command above, you will receive a Not a Number warning, the reconstructed node values are unreasonably low, and NaN appears in the returned confidence interval.

The syntax for the Grafen model requires specifying a rho parameter (here, 1) which exponentiates the recalculated branch lengths.

GLSreconstruction <- ace(wingL, geotree, type="continuous", method="GLS", corStruct = corGrafen(1, geotree))

The use of corGrafen generates an error on some computers on which we have executed this code, but not all. We are currently investigating the cause.

**WHAT ABOUT SQUARED CHANGE PARSIMONY?**

Squared change parsimony is mathematically equivalent to a special case of Schluter et. al.'s maximum likelihood method in which branch length information is ignored. Thus, to perform a squared change parsimony reconstruction, set all branch lengths equal to 1 and calculate a maximum likelihood solution.

geotreeones <- compute.brlen(geotree, 1) SQPreconstruction <- ace(wingL, geotreeones, type="continuous", method="ML")

**HOW DO I USE THE OUTPUT FROM ACE?**

Each of the objects (for example: MLreconstruction) that we reconstructed above is a list of several elements. These are loglik (the log-likelihood of the most likely reconstruction), ace (the vector of reconstructed node values), sigma2 (a two element vector of the rate estimate and its standard error) and CI95, the 95% confidence intervals around the vector of reconstructed node values. Thus

MLreconstruction$ace

returns the vector of node values for wing length (wingL) that were reconstructed using maximum likelihood, and

wingLfinal <- c(wingL, MLreconstruction$ace)

concatenates the original tip data with the reconstructed node data into a single vector.

**HOW DO I PLOT THE OUTPUT FROM ACE? (drawn from a course handout by Gene Hunt)**

Plot.phylo can scale symbols at tips and nodes of your tree to your character data. Try this to visualize the evolution of wing length (wingL) across the Geospiza phylogeny.

plot(geotree, show.tip.label=FALSE) tiplabels(pch = 21, cex=(wingL-3.9)*10) nodelabels(pch = 21, cex=(MLreconstruction$ace-3.9)*10) ## (pch = 21 is just telling plot.phylo which symbol to use)

Note that cex argument determines the size at which the symbols will be plotted. These have been rescaled so that the differences between species are visible on the screen. Subtracting 3.9 and multiplying by 10 rescales the wingL to range from a little less than 1 to about 5, which is a pretty good range for plotting.

**HOW DO I FIT A MODEL OF CHARACTER CHANGE TO CONTINUOUS DATA?**

While ace returns a rate of Brownian evolution (e.g. MLreconstruction$sigma2), that rate and its associated likelihood value (e.g., MLreconstruction$loglik) is conditioned on the specific ancestral states reconstructed by ace. More generally applicable model fits using Brownian motion and the Ornstein-Uhlenbeck model are available in the packages ouch and geiger. Please see here for more information.

## Reconstructing Ancestral States for Discrete Variables

The ace command in ape can reconstruct ancestral states for discrete characters using maximum likelihood.

First, ensure that you have ape loaded.

library(ape)

Load or reload the Geospiza phylogeny, which will restore the taxon "olivacea" that you may have removed in the continuous example above.

geotree <- read.nexus("geospiza.nex")

These commands will set up a new discrete character vector "char1" and assign the names of the terminal taxa in the Geospiza phylogeny to the elements in the new vector. This is same vector used in the example on modeling discrete character evolution in R.

char1<-c(1,1,1,1,1,0,1,0,0,1,0,1,1,1) names(char1)<-geotree$tip.label

You will need to choose among several different options for the matrix that specifies the transition probabilitites between the states of your discrete character. Ace offers shortcuts for a one-parameter equal rates model (ER), a symmetric model (SYM) in which forwards and reverse transitions between states are constrained to be equal, and an all rates different matrix (ARD) where all possible transitions between states receive distinct parameters. You can also specify your own matrix.

Let's reconstruct ancestral states for the Geospiza data using the three standard methods (ER, SYM and ARD).

ERreconstruction <- ace(char1, geotree, type="discrete", model="ER") SYMreconstruction <- ace(char1, geotree, type="discrete", model="SYM") ARDreconstruction <- ace(char1, geotree, type="discrete", model="ARD")

These commands issue lots of output, of which you are likely to be most interested in the log likelihoods and the inferred transition rates. You can access the log likelihoods for each model as follows

ERreconstruction$loglik SYMreconstruction$loglik ARDreconstruction$loglik

You should receive values of -9.00, -9.00 and -7.28, respectively.

The transition rates are accessed with

ERreconstruction$rates SYMreconstruction$rates ARDreconstruction$rates

ER and SYM return a single rate of 51.6, while ARD returns a forward and reverse rate (6.00 and 17.9, respectively).

You may have noticed that the output for the ER model matches that of the SYM model. This is not surprising. For a binary character these models are identical, though they are distinct for characters with three or more states. For a three-state character, ER is a one parameter model, SYM a three parameter model, and ARD a six parameter model.

*Which model should you prefer?*

In this example, the ARD model gives the highest likelihood. However, it also includes more parameters that the ER and SYM models, and it is well known that adding parameters to a model generally increases its likelihood. To determine whether the use of the more heavily parameterized model is appropriate, you should execute a likelihood test.

Twice the difference in log likelihood between two models (known as the G statistic) is distributed as chi square, with degrees of freedom equal to the number of parameters that differ between the models. Thus a likelihood test of these models asks whether the difference in likelihoods is large enough to lie in the rightmost tail of the chisquare distribution, typically considered to be the largest 5% of values.

The command pchisq(value, df) gives the percentage of the cumulative distribution function for chisquare that lies to the left of the given value for a desired degree of freedom(df). Thus, the following command gives the percentage of the chisquare distribution that lies to the right of the observed likelihood difference for the ER and ARD models (which differ by one parameter), given the Geospiza data.

1-pchisq(2*abs(ERreconstruction$loglik - ARDreconstruction$loglik), 1)

Evaluation of the equation above yields the p-value for the likelihood test, in this case 0.064. This is not a significant result at a 5% error rate (though it is very close to significance), so we do not accept the more heavily parameterized ARD model, and instead choose the single-parameter ER model.

For a single degree of freedom, the significance cutoff for the likelihood values lies at about 1.93 log-likelihood units of difference.

*How do I define and use a custom transition model?*

What if you want to use a transition model that isn't hardwired into ace? You can do so by defining a custom transition matrix. For example, let's create a custom four-parameter matrix for a three state character, in which the the probability of transitioning from state 1 to 2 equals the probability of state 2 to 1 (parameter 1), the probability of transitioning from 1 to 3 equals the probability of transitioning from state 3 to 1 (parameter 2), but the probability of transitioning from state 2 to 3 (parameter 3) does not necessarily equal the probability of transitioning from state 3 to 2 (parameter 4).

This syntax will set up such a transition matrix.

transitions <- matrix (c(0, 1, 2, 1, 0, 3, 2, 4, 0), nrow=3) transitions

You should see a matrix that looks like the following output to your screen.

[,1] [,2] [,3] [1,] 0 1 2 [2,] 1 0 4 [3,] 2 3 0

Let's now define a three-state discrete character for the Geospiza and apply this custom transition model to the ancestral state reconstruction.

char2<-c(3,3,1,2,1,1,3,3,3,3,2,2,2,3) names(char2)<- geotree$tip.label CUSreconstruction <- ace(char2, geotree, type="discrete", model=transitions)

You should obtain a log likelihood of -11.49

*Are there any caveats about the reconstruction of discrete characters that I should be aware of?*

Yes. Ace deals with this analysis by parameterizing each node within the phylogeny and reconstructing its ancestral state separately. This is a computationally intensive problem, and ace sometime returns spurious results when attempting a solution. If the returned log likelihood is tiny or enormous, or some of the reconstructed ancestral states have greater than 100% probability (or negative probability) at certain nodes, it is likely that ape is not reconstructing an accurate solution for the character in question. You are most likely to run into this issue when using complicated transition models or trees that imply many character state changes.

You can see the probability of each of the possible ancestral states at each of the nodes by typing the following:

CUSreconstruction$lik.anc

In other cases, the likelihood surface for the rates will be essentially flat, and you can place little confidence in the specific rates being reconstructed for each node. If the rate values are important in your application, check their standard errors.

CUSreconstruction$se

If the standard errors are very large, or represented by NaN (not a number), then you are dealing with a flat or nearly flat likelihood surface, and should not place any confidence in the reconstructed rates. This was true for the equal rates (ER) reconstruction of char1 above. Type

ERreconstruction$se

to see the standard error for that reconstruction. R should returned NaN for this value.

## Citations

Felsenstein, J. 1985. Phylogenies and the comparative method. *American Naturalist* 125, 1-15.

Grafen, A. 1989. The phylogenetic regression. *Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences*, Vol. 326, No. 1233. (Dec. 21, 1989), pp. 119-157.

Paradis, E. 2006. *Analysis of Phylogenetics and Evolution using R*. New York, Springer . 211 pp.
Much of the text above is paraphrased from this source.

Schluter D., Price T. Mooers A. O. and Ludwig D. 1997. Likelihood of ancestral states in adaptive radiation. *Evolution* 51: 1699-1711.