There are multiple functions in R to fit the so-called M*k* model for discrete character
evolution. For instance, there is `ace`

in the ape package, `fitDiscrete`

in geiger, and now `fitMk`

in my package, phytools - which has a lot of similarity
to `ace`

, but some important differences. This does not even consider the functions
of phangorn and diversitree, which can also be used to fit this model.

A casual user may discovery, however, that although these methods all purport to be fitting the
same model, parameter estimates and likelihoods often differ ever so *slightly* (but more
than one would expect based on the limits of numerical precision) between methods.

What accounts for this, it turns out, is different implicit or explicit assumptions about the
prior probability distribution at the global root of the tree. This is difficult to track down
in documentation for any of these functions, although in `fitMk`

the assumed prior
(`pi`

) is reported explicitly (and can be modified), and (as I show below) it is
also possible to use `fitDiscrete`

to fit the M*k* model with any prior.

First, load packages

```
## load packages
library(phytools)
library(geiger)
```

Next, let's simulate some data under the M*k* model:

```
## simulate some data
tree<-pbtree(n=100,scale=1)
Q<-matrix(c(-2,1,1,1,-2,1,1,1,-2),3,3)
rownames(Q)<-colnames(Q)<-letters[1:3]
Q
```

```
## a b c
## a -2 1 1
## b 1 -2 1
## c 1 1 -2
```

```
x<-sim.history(tree,Q,anc="a")$states
```

```
## Done simulation(s).
```

```
x
```

```
## t53 t54 t89 t90 t9 t10 t6 t11 t24 t25 t12 t44 t62 t67 t74
## "a" "a" "c" "c" "a" "a" "b" "b" "a" "a" "a" "a" "a" "b" "a"
## t75 t65 t66 t45 t81 t82 t69 t35 t36 t70 t71 t1 t5 t17 t23
## "a" "c" "c" "a" "a" "c" "a" "a" "a" "b" "c" "c" "a" "a" "a"
## t91 t92 t55 t56 t46 t47 t50 t51 t22 t78 t85 t86 t61 t63 t64
## "c" "b" "c" "c" "c" "c" "c" "c" "c" "c" "c" "c" "c" "c" "c"
## t18 t95 t96 t13 t42 t43 t93 t94 t99 t100 t15 t4 t40 t41 t2
## "c" "b" "b" "c" "c" "c" "c" "c" "a" "a" "b" "c" "b" "b" "b"
## t3 t76 t77 t39 t38 t30 t27 t31 t83 t84 t57 t58 t37 t14 t19
## "b" "b" "a" "a" "a" "b" "a" "a" "a" "a" "a" "a" "c" "a" "a"
## t20 t7 t79 t80 t21 t8 t68 t87 t88 t52 t72 t73 t33 t34 t16
## "a" "c" "b" "b" "b" "a" "a" "a" "a" "a" "b" "b" "b" "b" "b"
## t26 t28 t29 t48 t49 t97 t98 t32 t59 t60
## "b" "c" "a" "a" "a" "b" "b" "b" "a" "b"
```

Now, we can fit each model under the generating model (`model="ER"`

):

```
## fit models
fit.phytools<-fitMk(tree,x,model="ER",pi="equal") ## flat root
fit.phytools
```

```
## Object of class "fitMk".
##
## Fitted (or set) value of Q:
## a b c
## a -1.773072 0.886536 0.886536
## b 0.886536 -1.773072 0.886536
## c 0.886536 0.886536 -1.773072
##
## Fitted (or set) value of pi:
## a b c
## 0.3333333 0.3333333 0.3333333
##
## Log-likelihood: -86.449618
```

```
fit.ape<-ace(x,tree,type="discrete",model="ER")
fit.ape
```

```
##
## Ancestral Character Estimation
##
## Call: ace(x = x, phy = tree, type = "discrete", model = "ER")
##
## Log-likelihood: -85.35101
##
## Rate index matrix:
## a b c
## a . 1 1
## b 1 . 1
## c 1 1 .
##
## Parameter estimates:
## rate index estimate std-err
## 1 0.8865 0.1524
##
## Scaled likelihoods at the root (type '...$lik.anc' to get them for all nodes):
## a b c
## 0.4114971 0.3071304 0.2813725
```

```
fit.geiger<-fitDiscrete(tree,x,model="ER")
fit.geiger
```

```
## GEIGER-fitted comparative model of discrete data
## fitted Q matrix:
## a b c
## a -1.7668479 0.8834239 0.8834239
## b 0.8834239 -1.7668479 0.8834239
## c 0.8834239 0.8834239 -1.7668479
##
## model summary:
## log-likelihood = -86.421424
## AIC = 174.842847
## AICc = 174.883663
## free parameters = 1
##
## Convergence diagnostics:
## optimization iterations = 100
## failed iterations = 0
## frequency of best fit = 1.00
##
## object summary:
## 'lik' -- likelihood function
## 'bnd' -- bounds for likelihood search
## 'res' -- optimization iteration summary
## 'opt' -- maximum likelihood parameter estimates
```

We can see that each is slightly different - either in the fitted model, or the likelihood, or both!

To show that this is due to different priors on the root node, we can change the prior using
the object created by `fitDiscrete`

. This is not super-straightforward, but kind
of neat in the end. (There may be some way to do this within a call of `fitDiscrete`

,
but I was not able to do that.)

```
## the likelihood function
lik<-fit.geiger$lik
## optimize it using a flat/"equal" prior
fit.flat<-optimize(lik,c(1e-8,1000),root=ROOT.FLAT,maximum=TRUE)
fit.flat
```

```
## $maximum
## [1] 0.8865379
##
## $objective
## [1] -86.44962
```

If we compare this rate parameter to `fitMk`

or `ace`

, or the likelihood
to `fitMk`

, we should see that it is equal. Cool.

That's all I have to say about this for now.

Great post Liam,

ReplyDeleteAnother option out their is the way that diversitree handles this by default. Where the likelihoods at the root are weighted in proportion to how well they explain the observed tip data. Rich does a nice job of explaining the approach in the first appendix of Fitzjohn, Maddison and Otto 2009. Also in diversitree you can choose just to return the multiple probabilites calculated at the root rather than summing over them. This can be nice at least for exploration so you can see how or to what extent the choice of root prior is going to effect your result

cheers

This comment has been removed by the author.

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ReplyDeleteWhy is the likelihood returned by ace (-85.35101) not the same as the likelihood returned by the geiger likelihood function (-86.44962), when evaluated at the same rate value (0.8865379)? The examples in the help page for fitDiscrete give a clue: one can get the same likelihood as fitMk (-86.44962) like this:

ReplyDeletefit(0.8865379, root="given", root.p=c(1/3,3))

or one can get the same likelihood as ace (-85.35101) like this, using un-normalized root probabilities (1,1,1 instead of .33,.33,.33):

fit(0.8865379, root="given", root.p=c(1,3))

So it looks like the likelihood output by ace has an extra term log(3) = 1.098612 = -85.35101 - (- 86.44962). That's log(3) for 3 states. Weird. It's the same constant term for all choices of rate parameters, so the rates estimated by ace are correct. Any likelihood ratio would still be correct, too. But still, I find this strange.

Thanks Liam for your post, and thanks Luke for the help in the manual page of fitDiscrete!