## Friday, March 3, 2017

### More on PGLS with error in y

Nearly two years ago I posted about fitting a PGLS model taking into account sampling variance (uncertainty) in y.

Inspired by a student dataset & problem here at los Andes, the following is a slightly differently implemented solution that should lead to the same result:

``````pgls.SEy<-function(model,data,corClass=corBrownian,tree=tree,
se=NULL,method=c("REML","ML"),interval=c(0,1000),...){
corfunc<-corClass
## preliminaries
data<-data[tree\$tip.label,]
if(is.null(se)) se<-setNames(rep(0,Ntip(tree),
tree\$tip.label))
## likelihood function
lk<-function(sig2e,data,tree,model,ve){
tree\$edge.length<-tree\$edge.length*sig2e
ii<-sapply(1:Ntip(tree),function(x,e) which(e==x),
e=tree\$edge[,2])
tree\$edge.length[ii]<-tree\$edge.length[ii]+
ve[tree\$tip.label]
v<-diag(vcv(tree))
vf<-varFixed(~v)
COR<-corfunc(1,tree,...)
fit<-gls(model,data=data,correlation=COR,weights=vf)
-logLik(fit)
}
## estimate sig2[e]
fit<-optimize(lk,interval=interval,
data=data,tree=tree,model=model,ve=se^2)
tree\$edge.length<-tree\$edge.length*fit\$minimum
ii<-sapply(1:Ntip(tree),function(x,e) which(e==x),
e=tree\$edge[,2])
tree\$edge.length[ii]<-tree\$edge.length[ii]+
se[tree\$tip.label]^2
v<-diag(vcv(tree))
vf<-varFixed(~v)
## fit & return model
gls(model,data,correlation=corfunc(1,tree),weights=vf,
method=method)
}
``````

Now let's try to apply it:

``````library(phytools)
library(nlme)
## here's our data
tree
``````
``````##
## Phylogenetic tree with 100 tips and 99 internal nodes.
##
## Tip labels:
##  t65, t66, t29, t30, t13, t78, ...
##
## Rooted; includes branch lengths.
``````
``````X
``````
``````##               ye           x1          x2
## t65  -0.77900192 -0.933177258 -0.50289849
## t66   0.84646891 -0.360779073 -0.67343194
## t29   1.95846040 -0.004324650 -0.86464732
## t30   0.24373967 -0.283674892 -0.05950573
## t13  -5.54444003 -0.260222231  1.69823520
## t78   1.88453970 -0.030577008 -0.12238217
## t79  -0.59317590 -0.242023433  0.32537283
## t5    1.17182511 -0.351473922 -0.86081240
## t26   3.36099814 -0.417929491 -0.87860040
## t67  -0.21357307 -0.494481915 -1.04212209
## t68   0.73039688 -0.588127345 -0.71373910
## t86   2.81142943 -0.045597241 -1.28701218
## t87   4.89158144 -0.284624727 -1.19177925
## t25   2.69692032 -0.099326903 -1.26492190
## t16   1.27290355 -0.329391279 -0.69189543
## t47  -0.57529243 -0.998675854 -0.60213264
## t48  -4.91449264 -1.135659972 -0.90384083
## t10   2.88459685 -0.170786099 -0.84404497
## t20   1.11432889 -0.116176393 -1.43439903
## t63   0.67056966 -0.271788410 -0.48803237
## t64   2.25646332 -0.122474607 -0.87832126
## t53   2.52828710 -0.188851979 -0.48504631
## t21   1.78891253 -0.666761834 -1.27216994
## t23   1.45124819 -0.263860837 -1.16042313
## t49   1.03438725 -0.143231501 -1.19002853
## t50   0.61071781  0.258295208 -0.60080140
## t80   1.00845570 -0.326915114  0.10754248
## t81  -1.01437643 -0.507894799  0.26522058
## t19  -1.71820534 -0.811681993 -0.26318414
## t84   0.71358338 -0.737986864 -1.02763382
## t85   1.56004661 -0.712068874 -0.77728970
## t40  -0.38524799 -0.933452671 -0.83696879
## t41   0.49296392 -0.229120475 -1.49848945
## t18   2.76362994 -0.204012732 -1.64088635
## t76  -0.38900308 -0.117917709 -0.77909630
## t77   0.16193894  0.287088622 -0.72342580
## t27   2.64224382 -0.358620911 -1.71171019
## t92  -0.08353593 -0.337589308 -1.45006236
## t93   0.31220124 -0.312388755 -1.60509872
## t8   -1.04234066 -0.676186538 -0.41457179
## t2   -0.24958117 -1.382336906 -0.60476399
## t95   0.30844124  0.071824226 -0.04136211
## t96   0.17344643 -0.000369423  0.04677993
## t6   -4.30672086 -1.228130839  1.24440860
## t44  -0.39925053 -0.778840190 -0.10432506
## t51  -0.06845018 -1.529691824 -0.63523463
## t52  -0.33920397 -0.671563524  0.25110920
## t72  -0.22233595 -0.208175485  0.21888791
## t73  -0.74246626 -0.280504466 -0.05425123
## t35   0.58660363 -0.432133697 -0.60337432
## t38  -0.02995503 -1.599070451 -1.12646600
## t39  -0.99818963 -1.019397388 -0.14879313
## t99   2.02180211  0.819938089  0.10457809
## t100  3.03396613  0.797765277  0.09518138
## t24  -2.28364797  0.132350015  1.25951182
## t11  -1.03843807 -0.338010923  0.63895729
## t36   3.08690868 -0.109222940 -0.84517257
## t37   1.66001718 -0.783798640  0.25056370
## t28  -0.42294326 -0.624356825 -0.18617947
## t7    1.63649995 -0.995914763 -0.82088749
## t12   0.30240267 -0.430696448 -0.37673980
## t33  -0.90310597 -0.459191936  0.14792529
## t34  -0.46084029 -1.369205857  0.15734648
## t15  -0.95017011 -0.668597155 -0.19397849
## t42  -0.40629518 -1.006046065 -0.59102696
## t43  -1.45997398 -0.657881494 -0.76035426
## t22  -0.06754584 -0.549687736 -0.49300322
## t1   -0.52189221 -0.735847227 -0.10188643
## t3    1.42350778 -1.255254006 -1.36243767
## t4   -1.66965072  0.036571658  0.12260330
## t74   3.03252085  1.178218512 -0.76133067
## t75   2.48876980  1.064090251 -0.95706527
## t31  -0.97972153 -0.435236571  0.54331428
## t59  -3.89650669 -0.567740108  0.91285849
## t60  -1.92222453 -0.516435916  0.64042685
## t9    3.53788280  1.375046920 -1.24425619
## t90   6.09565095  0.878034624 -2.01772447
## t91   6.88955565  0.935005458 -2.09172115
## t56  -0.35774725 -0.517446554  0.30596671
## t57   1.23478325 -0.116159889  0.05991413
## t14  -1.27430128 -0.906819465  0.46607917
## t61  -4.01746299 -1.312009985  1.32878564
## t62  -2.40971394 -1.612880369  1.08221968
## t69  -2.04466742 -0.242584724  0.76102366
## t70  -4.84522907 -0.396631201  1.07417172
## t88  -0.17336430  0.237257899  0.41246936
## t89  -2.05031514  0.434875538  0.30737619
## t54  -0.80195450 -0.569749931  1.07419352
## t55  -1.48926859 -0.152194258  1.08008287
## t32  -0.87454629  0.602680886  0.75156851
## t45  -3.25099532 -1.438964529  2.01395563
## t46  -2.70322247 -1.404376125  1.86329858
## t17   0.38924791 -0.217652443  0.32900077
## t58  -3.40005156 -1.498640558  1.17233451
## t82  -1.89445105 -1.348265012  1.08151593
## t83  -2.05781804 -1.381394718  1.22167672
## t71  -1.44719125 -1.170708866  1.02083131
## t94  -2.16041746 -1.200085947  1.03456095
## t97  -1.44175507 -1.213089703  1.18276260
## t98  -3.88229471 -1.402582088  1.24727944
``````
``````SE
``````
``````##       t65       t66       t29       t30       t13       t78       t79
## 1.5378419 0.6315825 0.8879493 0.5928441 1.0271921 1.0799337 0.2015944
##        t5       t26       t67       t68       t86       t87       t25
## 1.4215427 1.6140998 1.1763704 0.3116548 0.3712812 2.1127712 1.7325671
##       t16       t47       t48       t10       t20       t63       t64
## 0.6036392 0.2305401 2.1632606 0.5891664 1.6319491 0.1739720 0.2086460
##       t53       t21       t23       t49       t50       t80       t81
## 1.4821123 0.4555341 0.3351531 1.2322628 0.3969949 0.8365659 0.2839710
##       t19       t84       t85       t40       t41       t18       t76
## 0.3487848 1.1665896 1.5049262 0.5291481 0.5503178 0.7437542 0.9244183
##       t77       t27       t92       t93        t8        t2       t95
## 0.4384969 0.6420107 0.6988273 0.7246152 0.8660270 1.5637886 1.8151181
##       t96        t6       t44       t51       t52       t72       t73
## 1.3655987 0.9416576 0.7521436 0.4052548 0.7403313 0.3570909 1.2056779
##       t35       t38       t39       t99      t100       t24       t11
## 0.8792995 0.8277867 1.3258279 0.1751977 0.8815140 0.4549706 0.1941065
##       t36       t37       t28        t7       t12       t33       t34
## 1.4607861 1.1021203 0.6038450 0.9676804 0.2473128 0.1840373 1.6307541
##       t15       t42       t43       t22        t1        t3        t4
## 0.7664964 1.1059112 1.1145843 0.9188596 1.5650796 0.4641816 0.6615129
##       t74       t75       t31       t59       t60        t9       t90
## 0.9735615 1.0394457 1.5125566 1.8766065 0.5621988 0.9335404 0.2087170
##       t91       t56       t57       t14       t61       t62       t69
## 0.7721703 1.8066518 0.4828995 0.8829816 0.7459172 2.1854232 0.7337223
##       t70       t88       t89       t54       t55       t32       t45
## 1.4301866 0.2787410 1.0083834 0.5330130 1.1466571 1.7584170 0.3040079
##       t46       t17       t58       t82       t83       t71       t94
## 0.8741375 0.8283463 0.5284757 1.1277393 0.3540923 0.3703903 1.0910862
##       t97       t98
## 0.9403271 1.3459290
``````
``````fit<-pgls.SEy(ye~x1+x2,data=X,se=SE,tree=tree,method="ML")
fit
``````
``````## Generalized least squares fit by maximum likelihood
##   Model: model
##   Data: data
##   Log-likelihood: -166.1253
##
## Coefficients:
## (Intercept)          x1          x2
##   0.6837662   1.5101552  -1.6872891
##
## Correlation Structure: corBrownian
##  Formula: ~1
##  Parameter estimate(s):
## numeric(0)
## Variance function:
##  Structure: fixed weights
##  Formula: ~v
## Degrees of freedom: 100 total; 97 residual
## Residual standard error: 1.774718
``````
``````summary(fit)
``````
``````## Generalized least squares fit by maximum likelihood
##   Model: model
##   Data: data
##        AIC      BIC    logLik
##   340.2505 350.6712 -166.1253
##
## Correlation Structure: corBrownian
##  Formula: ~1
##  Parameter estimate(s):
## numeric(0)
## Variance function:
##  Structure: fixed weights
##  Formula: ~v
##
## Coefficients:
##                  Value Std.Error    t-value p-value
## (Intercept)  0.6837662 0.1650265   4.143373   1e-04
## x1           1.5101552 0.1365145  11.062231   0e+00
## x2          -1.6872891 0.1153723 -14.624738   0e+00
##
##  Correlation:
##    (Intr) x1
## x1 0.122
## x2 0.078  0.376
##
## Standardized residuals:
##        Min         Q1        Med         Q3        Max
## -2.7961712 -0.8297315 -0.2312799  0.4027741  3.0330635
##
## Residual standard error: 1.774718
## Degrees of freedom: 100 total; 97 residual
``````

In a separate post, I'm going to look at power & type I error when SEs are ignored vs. taken into account.

Note that also taking uncertainty in the xs is account is considerably more complicated. For that we would need to employ an approach similar to that of Ives et al. (2007).

The data & tree for this exercise were simulated as follows:

``````## simulate tree & data
set.seed(999)
tree<-pbtree(n=100,scale=1)
X<-fastBM(tree,nsim=2)
colnames(X)<-c("x1","x2")
beta<-c(1,1,-2)
y<-cbind(rep(1,Ntip(tree)),X)%*%beta+fastBM(tree)
v<-setNames(rexp(n=Ntip(tree)),tree\$tip.label)
ye<-setNames(sampleFrom(xbar=y,xvar=v,n=rep(1,length(y))),
rownames(y))
X<-as.data.frame(cbind(ye,X))
SE<-sqrt(v)
``````

#### 1 comment:

1. This function is now on GitHub. I recommend installing it from there & not using the above version. - Liam

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