Yesterday I received the following question:

*I have three character states (1,2,3) and Iâ€™d like to create stochastic
maps for a situation where character change is constrained to go through
state 2, so that the 1->3 and 3->1 transitions are not allowed. In my
analyses so far Iâ€™ve just been using and ARD model, but I wondered if there
is a way to implement these constrained pathways so that the Q matrix only
allows certain specified transitions? I was also wondering if there is an
easy way to constrain the root state for the analysis?*

Both of these things are pretty easy to do. Here, I'll demonstrate with simulated data & tree:

```
library(phytools)
tree
```

```
##
## Phylogenetic tree with 100 tips and 99 internal nodes.
##
## Tip labels:
## t14, t15, t51, t67, t68, t55, ...
##
## Rooted; includes branch lengths.
```

```
x
```

```
## t14 t15 t51 t67 t68 t55 t46 t26 t11 t1 t2 t3 t4 t97 t98
## "1" "1" "1" "1" "1" "1" "1" "1" "2" "1" "1" "1" "1" "2" "2"
## t78 t93 t94 t41 t79 t80 t47 t48 t24 t25 t29 t30 t31 t58 t59
## "2" "2" "2" "2" "2" "2" "2" "2" "2" "2" "2" "2" "2" "3" "3"
## t6 t53 t54 t87 t88 t32 t81 t82 t95 t96 t83 t84 t21 t22 t18
## "3" "2" "2" "3" "3" "3" "2" "2" "3" "3" "2" "3" "2" "2" "2"
## t19 t99 t100 t44 t45 t42 t43 t64 t69 t70 t7 t73 t74 t75 t17
## "2" "2" "2" "2" "2" "2" "3" "2" "2" "2" "2" "2" "2" "2" "2"
## t85 t86 t12 t10 t5 t39 t91 t92 t20 t40 t62 t63 t89 t90 t38
## "2" "2" "2" "3" "2" "2" "2" "2" "1" "2" "3" "2" "3" "3" "3"
## t71 t72 t8 t9 t36 t37 t33 t34 t35 t60 t61 t65 t66 t52 t13
## "3" "3" "2" "2" "3" "2" "2" "2" "2" "2" "2" "2" "2" "2" "2"
## t49 t50 t16 t23 t76 t77 t27 t28 t56 t57
## "2" "2" "2" "3" "3" "3" "2" "2" "2" "2"
```

First, let's make the model we want to fit. Here, I assume that there is two rates - one of forward transition, and a second of backward transition- but I could have also fit a model with a single rate, or one in which each type of permitted change had a different rate. (This just depends on my preference.)

```
model<-matrix(c(0,1,0,2,0,1,0,2,0),3,3)
rownames(model)<-colnames(model)<-1:3
model
```

```
## 1 2 3
## 1 0 2 0
## 2 1 0 2
## 3 0 1 0
```

```
prior<-setNames(c(1,0,0),1:3)
prior
```

```
## 1 2 3
## 1 0 0
```

OK, now we are ready to do our stochastic mapping:

```
trees<-make.simmap(tree,x,model=model,pi=prior,nsim=100)
```

```
## make.simmap is sampling character histories conditioned on the transition matrix
##
## Q =
## 1 2 3
## 1 -1.0920856 1.0920856 0.0000000
## 2 0.1074346 -1.1995202 1.0920856
## 3 0.0000000 0.1074346 -0.1074346
## (estimated using likelihood);
## and (mean) root node prior probabilities
## pi =
## 1 2 3
## 1 0 0
```

```
## Done.
```

First, here is our result:

```
par(mfrow=c(10,10))
colors<-setNames(c("blue","purple","red"),1:3)
plot(trees,lwd=1,ftype="off",colors=colors)
```

We can also do any specific tree:

```
plot(trees[[1]],colors=colors,fsize=0.5)
markChanges(trees[[1]],colors=colors)
```

Or summarize our results across trees:

```
obj<-summary(trees)
obj
```

```
## 100 trees with a mapped discrete character with states:
## 1, 2, 3
##
## trees have 18.27 changes between states on average
##
## changes are of the following types:
## 1,2 1,3 2,1 2,3 3,1 3,2
## x->y 5.63 0 1.19 11.25 0 0.2
##
## mean total time spent in each state is:
## 1 2 3 total
## raw 4.3721937 11.2234788 1.7854335 17.38111
## prop 0.2515486 0.6457287 0.1027227 1.00000
```

```
plot(obj,colors=colors,ftype="off")
add.simmap.legend(colors=colors,prompt=FALSE,x=0,y=Ntip(tree))
```

Finally, it is also possible to use Bayesian MCMC to sample the parameters of the transition process from their posterior distribution rather than setting them to their ML values (as is the default). This otherwise is done in a very similar way to the above. For instance:

```
trees.mcmc<-make.simmap(tree,x,model=model,pi=prior,Q="mcmc",nsim=100)
```

```
## Running MCMC burn-in. Please wait....
## Running 10000 generations of MCMC, sampling every 100 generations.
## Please wait....
##
## make.simmap is simulating with a sample of Q from
## the posterior distribution
##
## Mean Q from the posterior is
## Q =
## 1 2 3
## 1 -1.0707285 1.0707285 0.0000000
## 2 0.2334955 -1.3042240 1.0707285
## 3 0.0000000 0.2334955 -0.2334955
## and (mean) root node prior probabilities
## pi =
## 1 2 3
## 1 0 0
```

```
## Done.
```

```
obj<-summary(trees.mcmc)
plot(obj,colors=colors,ftype="off")
add.simmap.legend(colors=colors,prompt=FALSE,x=0,y=Ntip(tree))
```

(Note that for this to work is written you will need a bug fix for
`make.simmap`

that can be obtained by installing from
GitHub.)

BTW, these were done on simulated data. Simulation code was as follows:

```
library(phytools)
Q<-matrix(c(-1,1,0,0.2,-1.2,1,0,0.2,-0.2),3,3)
rownames(Q)<-colnames(Q)<-1:3
tree<-pbtree(n=100,scale=1)
x<-sim.history(tree,t(Q),anc="1")$states
```

That's all for now.

Hello! Thank you for your useful post!

ReplyDeleteI have a similar situation, the character states are 1, 2 and the change is allow only for 1->2 and not vice versa. I build the model like this

> model<-matrix(c(0,0,2,0),2,2)

> rownames(model)<-colnames(model)<-1:2

> model

## 1 2

#1 0 2

#2 0 0

Is it correct?

But then for the prior? Which is the meaning of your matrix?

Thank you again!

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