The *phytools* package has a little-used function called `mcmcMk`

for doing a
Bayesian MCMC analysis of the (extended) M*k* model of Lewis (2001). A user contacted me
about some weird results he had obtained from the function. It turns out this was due to a
strange choice of default prior that I can't really explain but turned out to be much more
informative than intended.

In fixing this issue I discovered that in other ways the function is quite neat, and so I have developed a bunch of different new S3 methods that I think could make it much more useful.

Here I'll demonstrate this a little bit, first using some simulated data - and then subsequently moving on to an empirical dataset for feeding mode in eels.

Load the *phytools* package:

```
library(phytools)
packageVersion("phytools")
```

```
## [1] '0.6.75'
```

(Obviously, since this is new functionality if you are reading this shortly after it was written
you may have to update *phytools* from GitHub
to get this package version.)

Now let's inspect our data which consists of a simulated 200 taxon tree & two character vectors,
`x`

& `y`

:

```
tree
```

```
##
## Phylogenetic tree with 200 tips and 199 internal nodes.
##
## Tip labels:
## t6, t103, t186, t189, t190, t56, ...
##
## Rooted; includes branch lengths.
```

```
x
```

```
## t6 t103 t186 t189 t190 t56 t57 t16 t127 t128 t43 t29 t14 t25 t76
## b a b b b b b b a a a b b a a
## t77 t82 t83 t137 t138 t101 t27 t122 t123 t4 t10 t17 t94 t95 t75
## b a a b b b a b b a a b a a a
## t131 t132 t104 t105 t69 t65 t66 t26 t187 t188 t34 t81 t109 t110 t180
## a a a a a a a a a a a a a a a
## t181 t5 t8 t44 t45 t149 t150 t89 t169 t170 t111 t112 t61 t193 t194
## a a a a a a a a a a a a a a a
## t18 t2 t72 t113 t114 t151 t152 t32 t41 t42 t167 t168 t7 t30 t31
## a a a a a a a a b b a a a b a
## t40 t106 t107 t38 t39 t19 t67 t68 t53 t195 t196 t100 t161 t162 t159
## b a a a a a a a a a a a a a a
## t160 t173 t174 t148 t87 t88 t64 t46 t165 t166 t47 t48 t139 t157 t158
## a a a a a a a a a b a b b b b
## t184 t185 t175 t125 t28 t51 t52 t9 t36 t176 t177 t92 t93 t71 t197
## a a a a b b b b b b b b b b b
## t198 t37 t86 t191 t192 t3 t163 t164 t20 t21 t178 t179 t35 t79 t80
## b b b b b a a a b a b b a a a
## t59 t58 t70 t129 t130 t78 t33 t54 t55 t115 t116 t12 t182 t183 t96
## a a a a a a a a a a a a a a a
## t97 t171 t172 t90 t91 t98 t99 t140 t141 t11 t13 t117 t153 t154 t126
## a a a b a a a a a a a a a b a
## t22 t146 t147 t23 t24 t124 t144 t145 t135 t136 t108 t50 t142 t143 t73
## a a a a a a a a a a a a a a a
## t74 t84 t85 t120 t121 t133 t134 t60 t155 t156 t199 t200 t102 t62 t63
## a a a a a a a a a a b b b a a
## t15 t118 t119 t49 t1
## a a a a a
## Levels: a b
```

```
y
```

```
## t6 t103 t186 t189 t190 t56 t57 t16 t127 t128 t43 t29 t14 t25 t76
## B A A A A A A A B B C A A B A
## t77 t82 t83 t137 t138 t101 t27 t122 t123 t4 t10 t17 t94 t95 t75
## A A A A A A A C C A A B B B B
## t131 t132 t104 t105 t69 t65 t66 t26 t187 t188 t34 t81 t109 t110 t180
## B B A A A A A A A A A A A A A
## t181 t5 t8 t44 t45 t149 t150 t89 t169 t170 t111 t112 t61 t193 t194
## A A A B B B B B B B B B B A A
## t18 t2 t72 t113 t114 t151 t152 t32 t41 t42 t167 t168 t7 t30 t31
## B A A A A A A A A A A A A A A
## t40 t106 t107 t38 t39 t19 t67 t68 t53 t195 t196 t100 t161 t162 t159
## B A A A A B A A B A A B A A C
## t160 t173 t174 t148 t87 t88 t64 t46 t165 t166 t47 t48 t139 t157 t158
## C C C C B A B B B B C B C C C
## t184 t185 t175 t125 t28 t51 t52 t9 t36 t176 t177 t92 t93 t71 t197
## A A A B B B C B B B B B C A B
## t198 t37 t86 t191 t192 t3 t163 t164 t20 t21 t178 t179 t35 t79 t80
## B B B B B B A A A C A A A A A
## t59 t58 t70 t129 t130 t78 t33 t54 t55 t115 t116 t12 t182 t183 t96
## A A A A A A A A A A A A A A A
## t97 t171 t172 t90 t91 t98 t99 t140 t141 t11 t13 t117 t153 t154 t126
## A A A A A A A A A A A A A A A
## t22 t146 t147 t23 t24 t124 t144 t145 t135 t136 t108 t50 t142 t143 t73
## A A A A A A A A A A A A A A A
## t74 t84 t85 t120 t121 t133 t134 t60 t155 t156 t199 t200 t102 t62 t63
## A A A A A A A B A A B A A A A
## t15 t118 t119 t49 t1
## A A A A C
## Levels: A B C
```

Now we can proceed to run our MCMC for the first character vector `x`

. We can assume any
of the standard extended M*k* models, but here I will set `model="ARD"`

:

```
mcmc.x<-mcmcMk(tree,x,model="ARD",prop.var=0.005,
ngen=10000,print=500)
```

```
## Running MCMC....
## gen [b,a] [a,b] logLik
## 1 0.0492 0.0492 -107.356
## 501 0.8713 1.1106 -78.4027
## 1001 1.5521 0.9809 -75.4808
## 1501 1.3616 0.9887 -75.8263
## 2001 1.6947 0.9337 -74.9609
## 2501 1.6446 0.7872 -74.0923
## 3001 1.205 0.9015 -75.3249
## 3501 3.1329 0.9437 -75.7518
## 4001 1.9646 0.4388 -74.3684
## 4501 3.7514 1.0544 -77.2086
## 5001 4.9081 1.248 -80.1114
## 5501 3.8791 0.9513 -77.2952
## 6001 3.6613 0.9009 -76.7296
## 6501 2.9479 0.3835 -76.8355
## 7001 1.3162 0.8561 -74.8016
## 7501 1.6166 0.6805 -73.7287
## 8001 0.7741 0.6074 -74.6178
## 8501 0.9214 0.3527 -75.8891
## 9001 1.6804 0.386 -74.73
## 9501 1.6623 0.8166 -74.2324
## Done.
```

```
print(mcmc.x)
```

```
##
## Posterior sample from mcmcMk consisting of a posterior sample obtained using
## Bayesian MCMC in the form of a matrix.
##
## 1. plot('object_name') will create a likelihood profile plot.
## 2. summary('object_name') will compute a summary of the MCMC.
## 3. density('object_name') will calculate a posterior density from the sample.
## 4. Finally, plot(density('object_name')) will plot the posterior density and
## and high probability density intervals.
##
## To work best, we recommend users install the package 'coda'.
```

The S3 `plot`

method gives us a likelihood profile for our MCMC. Let's try it:

```
plot(mcmc.x)
```

It's a bit hard to tell if this is well-mixed or not, but let's just assume that it is!

The S3 `summary`

method gives (unsurprisingly) a summary of our posterior sample -
in particularly an average value of the transition matrix ** Q** and HPD intervals
around each of the transition rates between states:

```
summary(mcmc.x)
```

```
## Assuming 20% burn-in as no burn-in was specified....
##
## Mean value of Q from the post burn-in posterior sample:
## a b
## a -0.8434544 0.8434544
## b 2.4166152 -2.4166152
##
## 95% HPD interval computed either from the post burn-in
## samples or using 'coda':
## lower upper
## [b,a] 0.6779169 4.591337
## [a,b] 0.2849471 1.553939
```

Note (as indicated) that the method discards the first 20% of our generations of MCMC as burn-in unless we tell it otherwise, which we can do, of course:

```
summary(mcmc.x,burnin=5000)
```

```
## Mean value of Q from the post burn-in posterior sample:
## a b
## a -0.8293531 0.8293531
## b 2.2536256 -2.2536256
##
## 95% HPD interval computed either from the post burn-in
## samples or using 'coda':
## lower upper
## [b,a] 0.6173297 4.345430
## [a,b] 0.2849471 1.631915
```

I also added a `density`

and associated `plot`

function as follows:

```
pd.x<-density(mcmc.x,bw=0.1)
```

```
## Assuming 20% burn-in as no burn-in was specified....
```

```
print(pd.x)
```

```
##
## Density [b,a]
##
## x y
## Min. :0.1327 Min. :0.0000277
## 1st Qu.:1.4867 1st Qu.:0.0855801
## Median :2.8407 Median :0.1548434
## Mean :2.8407 Mean :0.1844559
## 3rd Qu.:4.1947 3rd Qu.:0.2933771
## Max. :5.5487 Max. :0.4307307
##
## Density [b,a]
##
## x y
## Min. :0.1327 Min. :0.0000277
## 1st Qu.:1.4867 1st Qu.:0.0855801
## Median :2.8407 Median :0.1548434
## Mean :2.8407 Mean :0.1844559
## 3rd Qu.:4.1947 3rd Qu.:0.2933771
## Max. :5.5487 Max. :0.4307307
##
## To plot enter plot('object_name') at the command line interface.
```

But the `plot`

method for this custom density object class is definitely cooler. Let's check
it out:

```
plot(pd.x)
```

Neat. Now, let's do something similar for `y`

which contains a three-state character.
In this case, I happen to know that the data were simulated under an ordered process in which transitions
are only permitted between adjacent states.

Let's impose that model on our Bayesian inference. I'm also going to adjust the variance of the normal
proposal distribution. By default this would be 0.01/*total tree length* (in this case, also 0.01),
but here I'm going to set it to 0.05:

```
model<-matrix(c(
0,1,0,
2,0,3,
0,4,0),3,3,byrow=TRUE,
dimnames=list(levels(y),levels(y)))
model
```

```
## A B C
## A 0 1 0
## B 2 0 3
## C 0 4 0
```

```
mcmc.y<-mcmcMk(tree,y,model=model,prop.var=0.05,
ngen=20000,print=500)
```

```
## Running MCMC....
## gen [A,B] [B,A] [B,C] [C,B] logLik
## 1 0.0738 0.0738 0.0738 0.0738 -139.9116
## 501 0.7929 2.3011 1.2086 3.0696 -100.0168
## 1001 0.2424 1.9118 1.968 1.3895 -106.8495
## 1501 0.8968 0.8617 1.2571 1.4952 -100.8796
## 2001 0.9987 3.5494 0.4307 2.5344 -103.6955
## 2501 0.3571 0.9037 1.4929 3.1541 -101.0945
## 3001 0.527 1.8505 0.8831 1.6118 -99.9354
## 3501 0.9003 2.9273 3.3041 2.5723 -107.8218
## 4001 0.303 3.7478 0.8807 6.2265 -105.8216
## 4501 0.4373 2.6688 2.6178 5.7192 -102.596
## 5001 0.6424 1.5677 0.5674 7.6649 -104.0805
## 5501 0.9437 4.3249 1.5454 9.1119 -108.0044
## 6001 0.3885 5.0376 2.8744 10.2261 -112.6765
## 6501 2.1845 4.8637 3.6409 9.5295 -116.2657
## 7001 1.4617 4.7083 4.1376 8.6136 -111.5319
## 7501 0.4512 1.653 3.5337 12.5136 -103.1212
## 8001 1.1108 2.6291 5.2321 11.1377 -107.3971
## 8501 0.4506 0.6254 5.3749 11.2217 -105.4739
## 9001 0.669 0.9857 1.4964 9.7234 -102.6163
## 9501 0.5219 1.6389 2.2034 8.0083 -100.9076
## 10001 0.591 1.6407 1.3879 8.3047 -101.3362
## 10501 1.3114 2.2078 1.5595 7.6294 -105.3352
## 11001 1.3235 2.3554 1.6809 8.5757 -105.9546
## 11501 0.5321 0.6261 2.9792 11.9312 -103.3113
## 12001 0.5009 2.491 1.2613 8.5838 -102.7391
## 12501 0.8139 5.936 0.6449 1.6821 -111.3256
## 13001 0.565 0.2615 1.5711 0.6184 -102.4311
## 13501 0.7858 2.3639 0.3216 2.8565 -101.902
## 14001 0.9089 1.315 1.5638 1.41 -101.2064
## 14501 0.6014 1.5487 2.2075 3.0507 -100.8305
## 15001 0.751 0.3838 1.5209 2.6174 -101.5443
## 15501 0.2071 2.9983 0.6016 3.4198 -102.2142
## 16001 0.3475 1.6659 1.9359 5.5227 -100.872
## 16501 0.6605 1.7695 1.6263 9.6205 -102.0407
## 17001 0.6212 0.8155 1.1363 6.6569 -101.3515
## 17501 0.471 2.7336 3.3019 5.6302 -104.1388
## 18001 0.4604 2.6996 4.9081 5.4362 -108.5106
## 18501 0.6036 1.4582 4.7579 4.6932 -106.5785
## 19001 0.193 3.2469 4.4128 6.0448 -110.3741
## 19501 1.0267 1.7573 4.7954 5.2047 -108.3454
## Done.
```

```
mcmc.y
```

```
##
## Posterior sample from mcmcMk consisting of a posterior sample obtained using
## Bayesian MCMC in the form of a matrix.
##
## 1. plot('object_name') will create a likelihood profile plot.
## 2. summary('object_name') will compute a summary of the MCMC.
## 3. density('object_name') will calculate a posterior density from the sample.
## 4. Finally, plot(density('object_name')) will plot the posterior density and
## and high probability density intervals.
##
## To work best, we recommend users install the package 'coda'.
```

```
plot(mcmc.y)
```

```
summary(mcmc.y,burnin=5000)
```

```
## Mean value of Q from the post burn-in posterior sample:
## A B C
## A -0.8078591 0.8078591 0.000000
## B 2.3106788 -4.8984632 2.587784
## C 0.0000000 6.6205242 -6.620524
##
## 95% HPD interval computed either from the post burn-in
## samples or using 'coda':
## lower upper
## [A,B] 0.1839739 1.680220
## [B,A] 0.2294231 5.016908
## [B,C] 0.2186470 5.469995
## [C,B] 1.0242727 12.421114
```

```
pd.y<-density(mcmc.y,burnin=5000)
plot(pd.y)
```

Probably convergence looks kind of bad here. If so, we might try to adjust our proposal distributions (which we can do independently for each variable in the model) to get rejection rates around 75%. We can easily compute rejection rates, but when doing so we need to remember that only one variable is updated for each generation so we should first thin our posterior sample accordingly:

```
mcmc.thinned<-mcmc.y[seq(5001,20000,by=4),]
library(coda)
rejectionRate(as.mcmc(mcmc.thinned))
```

```
## gen [A,B] [B,A] [B,C] [C,B] logLik
## 0.00000000 0.25313417 0.29394505 0.09975994 0.10962923 0.09975994
```

Pretty cool. Finally, as promised we can try this out with some empirical data for elopomorph eels and feeding mode. Luke Harmon & I have used these data in a bunch of different workshops, but they were originally kindly provided to me by Dave Collar.

```
eel.tree<-read.tree(file="http://www.phytools.org/TS2018/data/elopomorph.tre")
eel.data<-read.csv(file="http://www.phytools.org/TS2018/data/elopomorph.csv",
row.names=1)
plotTree(eel.tree,ftype="i",fsize=0.7)
```

```
head(eel.data)
```

```
## feed_mode Max_TL_cm
## Albula_vulpes suction 104
## Anguilla_anguilla suction 50
## Anguilla_bicolor suction 120
## Anguilla_japonica suction 150
## Anguilla_rostrata suction 152
## Ariosoma_anago suction 60
```

```
fmode<-setNames(eel.data$feed_mode,rownames(eel.data))
```

It turns out the `"ER"`

model fits these data pretty well - so let's just assume that from
the get-go:

```
mcmc.fmode<-mcmcMk(eel.tree,fmode,model="ER",
ngen=10000,print=500)
```

```
## Running MCMC....
## gen [suction,bite] logLik
## 1 0.001 -59.2835
## 501 0.038 -38.8745
## 1001 0.0109 -37.4891
## 1501 0.101 -41.3336
## 2001 0.056 -40.1027
## 2501 0.0158 -37.0331
## 3001 0.0184 -37.0997
## 3501 0.0268 -37.7966
## 4001 0.0118 -37.3078
## 4501 0.0167 -37.0422
## 5001 0.0212 -37.2865
## 5501 0.0286 -37.9776
## 6001 0.0196 -37.1705
## 6501 0.0153 -37.0366
## 7001 0.0155 -37.0348
## 7501 0.0117 -37.3271
## 8001 0.0241 -37.5334
## 8501 0.0215 -37.311
## 9001 0.0169 -37.0475
## 9501 0.0166 -37.0406
## Done.
```

```
plot(mcmc.fmode)
```

```
summary(mcmc.fmode)
```

```
## Assuming 20% burn-in as no burn-in was specified....
##
## Mean value of Q from the post burn-in posterior sample:
## bite suction
## bite -0.03000165 0.03000165
## suction 0.03000165 -0.03000165
##
## 95% HPD interval computed either from the post burn-in
## samples or using 'coda':
## lower upper
## [suction,bite] 0.005749017 0.08181328
```

```
pd.fmode<-density(mcmc.fmode,bw=0.002)
```

```
## Assuming 20% burn-in as no burn-in was specified....
```

```
plot(pd.fmode)
```

One thing you might notice is that the posterior distribution for *q* is quite asymmetric. This
(predictably) means that our average from the posterior sample may not be the best way to
summarize the posterior distribution. It will also tend to cause the mean from the posterior sample
to be higher than our MLEs. For instance:

```
fitMk(eel.tree,fmode,model="ER")
```

```
## Object of class "fitMk".
##
## Fitted (or set) value of Q:
## bite suction
## bite -0.015828 0.015828
## suction 0.015828 -0.015828
##
## Fitted (or set) value of pi:
## bite suction
## 0.5 0.5
##
## Log-likelihood: -37.033072
##
## Optimization method used was "nlminb"
```

This is *not* an error. In fact, if we compare out likelihood surface for *q* to the posterior
distribution we should see that they match quite closely:

```
q<-seq(0.001,0.2,by=0.001)
lik<-sapply(q,function(q,t,x) logLik(fitMk(t,x,
fixedQ=matrix(c(-q,q,q,-q),2,2))),
t=eel.tree,x=fmode)
par(mfrow=c(2,1),mar=c(4.1,4.1,3.1,1.1))
plot(q,exp(lik),type="l",ylab="likelihood",xlab="q",
bty="l",col="blue",lwd=2,main="likelihood surface",
font.main=1,xlim=c(0,0.2))
plot(pd.fmode,xlim=c(0,0.2))
```

which is what we'd expect if our prior probability density is fairly uninformative. (Though it's not an error, it might suggest that summarizing the posterior sample with its mean could be misleading.)

For the reader's reference, my *simulated* data were generated as follows:

```
set.seed(1001)
tree<-pbtree(n=200,scale=1)
Q<-matrix(c(
-0.5,0.5,
2,-2),2,2,byrow=TRUE,
dimnames=list(letters[1:2],letters[1:2]))
x<-sim.Mk(tree,Q)
Q<-matrix(c(
-0.5,0.5,0,
2,-2.5,0.5,
0,2,-2),3,3,byrow=TRUE,
dimnames=list(LETTERS[1:3],
LETTERS[1:3]))
y<-sim.Mk(tree,Q)
```

That's all for now!