Today I was asked the following question:

*“I am interested in testing the correlation between 2 traits taking
into account for the phylogenetic structure in my dataset. PGLS is a good
way to test it, but I was wondering if a simple correlation test (using a
Pearson r) would be possible. Your phyl.vcv computes a
phylogenetic trait variance-covariance matrix between two variables. Can I
use this matrix to compute a phylogenetic Pearson r value? If so, can this
r-value be used to compute the statistic to test the significance of the
correlation (with n-2 df in a t distribution)?”*

The question was also posted here.

The answer is basically, yes & yes - but it is not strictly necessary as PGLS will give us the same p-value. Let's see how:

```
library(phytools)
## our data
tree
```

```
##
## Phylogenetic tree with 26 tips and 25 internal nodes.
##
## Tip labels:
## A, B, C, D, E, F, ...
##
## Rooted; includes branch lengths.
```

```
X
```

```
## x y z
## A -0.723942909 0.96553388 0.50467165
## B -0.034600412 1.48143249 0.66533528
## C -0.001282309 0.62094719 0.60352347
## D -0.100971061 1.23102030 0.01866007
## E -0.363619319 0.06831715 0.91782821
## F -0.400834652 -0.07312087 0.77584466
## G -0.037785409 0.18406320 1.18509956
## H -1.284112703 0.26824899 0.15203366
## I -0.573447373 1.56312743 0.58464867
## J -0.275070942 0.05165640 -0.24120290
## K -1.050699705 0.31715393 -0.63071969
## L -0.689968116 0.22713074 -0.04087866
## M -0.572979102 0.16002629 -0.67389194
## N -0.353661739 0.31157683 -0.60997825
## O -1.720295244 0.57079006 -2.24647340
## P -2.830254149 0.78574612 -2.39094415
## Q -0.104129552 0.93488329 1.66983162
## R -0.324883255 1.00993141 -1.87523279
## S -1.330114250 0.53192888 -2.47716184
## T -0.394635253 0.84544343 -2.05151552
## U -2.788040191 0.75579827 -1.90934711
## V 0.076717481 -1.13500947 -0.01313445
## W 1.388265725 -1.15038331 0.50393587
## X 1.423667629 -1.41685533 1.29662644
## Y 1.459867791 -0.53755989 1.83210421
## Z 2.235835060 -1.28003540 0.84061852
```

First, let's compute the evolutionary (phylogenetic) correlation between
`x`

& `y`

:

```
## our covariance matrix
obj<-phyl.vcv(X,vcv(tree),1)
obj$R
```

```
## x y z
## x 0.6810667 -0.0592844 0.3212055
## y -0.0592844 0.4238996 0.0413417
## z 0.3212055 0.0413417 0.6319725
```

```
## correlation between x & y
r.xy<-cov2cor(obj$R)["x","y"]
## t-statistic & P-value
t.xy<-r.xy*sqrt((Ntip(tree)-2)/(1-r.xy^2))
P.xy<-2*pt(abs(t.xy),df=Ntip(tree)-2,lower.tail=F)
P.xy
```

```
## [1] 0.5915613
```

As noted before, this is the *same* P-value as we would obtain fitting
a linear model using PGLS of `y~x`

*or* `x~y`

.

```
fit.yx<-gls(y~x,data=as.data.frame(X),correlation=corBrownian(1,tree))
anova(fit.yx)
```

```
## Denom. DF: 24
## numDF F-value p-value
## (Intercept) 1 0.2645170 0.6117
## x 1 0.2957732 0.5916
```

```
fit.xy<-gls(x~y,data=as.data.frame(X),correlation=corBrownian(1,tree))
anova(fit.xy)
```

```
## Denom. DF: 24
## numDF F-value p-value
## (Intercept) 1 0.1812473 0.6741
## y 1 0.2957732 0.5916
```

Now let's see an example in which a weak correlation exists between the two characters:

```
## correlation between x & z
r.xz<-cov2cor(obj$R)["x","z"]
## t-statistic & P-value
t.xz<-r.xz*sqrt((Ntip(tree)-2)/(1-r.xz^2))
P.xz<-2*pt(abs(t.xz),df=Ntip(tree)-2,lower.tail=F)
P.xz
```

```
## [1] 0.01112811
```

```
fit.zx<-gls(z~x,data=as.data.frame(X),correlation=corBrownian(1,tree))
anova(fit.zx)
```

```
## Denom. DF: 24
## numDF F-value p-value
## (Intercept) 1 0.003181 0.9555
## x 1 7.566719 0.0111
```

```
## visualize it
phylomorphospace(tree,X[,c("x","z")],node.size=c(0,0))
points(X[,c("x","z")],pch=21,cex=1.5,bg="grey")
text(-3.2,2,paste("evolutionary correlation\nr = ",round(r.xz,4),
", P = ",round(P.xz,4),sep="" ),pos=4)
```

Neat.

The data for this example were simulated as follows:

```
tree<-pbtree(n=26,tip.label=LETTERS)
X<-sim.corrs(tree,
vcv=matrix(c(1.0,0.0,0.3,
0.0,1.0,0.0,
0.3,0.0,1.0),3,3))
colnames(X)<-c("x","y","z")
```

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