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Do different values of the smoothing parameter lambda for estimating the divergence times (in situ Sanderson 2002) will affect trait conservatism (e.g., Blomberg's K)?

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I don't know if lambda for estimating divergence times works in the same way as it does for scaling internal tree nodes, but I'm going to assume it does given the context of your question.

My gut reaction was: of course it must because K is the ratio of observed to expected variance and tip variance is moderated in part by branch lengths. So if you change the node positions, the expected tip variance should also change.

A quick simulation shows that this appears to be true.

library("ape")
library("picante")
library("geiger")

set.seed(5)

tree <- rtree(30)
y <- rTraitCont(tree)

lambdas <- seq(1.0, 0.05, by = -0.05)
M <- data.frame(lambda = lambdas, k = numeric(length(lambdas)))

for (lambda in lambdas){
  tree_trans <- rescale(x = tree,
                        model = "lambda", lambda = lambda)
  M$k[M$lambda == lambda] <- phylosignal(y, tree_trans)$K
}

plot(k ~ lambda, data = M)

Generate a tree with 30 tips and generate random data along the branches. Then rescale the tree for a series of lambda values and calculate k for each lambda scaling.

K vs lambda

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