APPENDIX 3
SIMULATION EXPERIMENT DEMONSTRATING THE EQUIVALENCE BETWEEN THE CENSORED
APPROACH (O’MEARA ET AL. 2006) AND THE RATE METHOD USING A DESIGN MATRIX FOR
GROUP MEANS (THOMAS ET AL. 2006, 2009)
Maximum-likelihood methods for comparing evolutionary rates across groups defined by
a discrete, binary trait across a phylogeny were simultaneously developed by O’Meara et al.
(2006) and Thomas et al. (2006). Both studies proposed slightly different models for rate
comparison. O’Meara et al. (2006) proposed two different approaches. In the “non-censored”
approach, each branch on the tree is assigned a rate parameter based on a single phylogenetic
mean for the whole tree (e.g. the expected value E(X), corresponding to the phenotypic value at
the root). In contrast, the “censored” approach works by eliminating the branches not included in
each of the defined groups and calculating a rate estimate based on this reduced tree, also
calculating a separate phylogenetic mean for each group (O’Meara et al. 2006). This approach is
more flexible in terms of the assumptions made regarding the exact point where a rate shift
occurred, but is restricted to monophyletic subtrees. Simultaneously, Thomas et al. (2006)
proposed a similar method, where rate comparison between groups is performed using a design
matrix to calculate the expected values (phylogenetic means) for each group (Thomas et al. 2006,
2009). While the model of Thomas et al. 2006 allows for any type of distribution of groups
across the tree (e.g. there is no restriction to monophyly), it coincides with the censored approach
in using different phylogenetic means for each of the examined groups. As such, both models are
mathematically expected to have equal likelihoods and provide the same relative rate estimates
in cases when they are comparable, e.g. when at least one of the examined groups is
monophyletic. However, the identity of these two methods has never been formally tested.
1
We used a simulation experiment to confirm whether this was in fact the case.
Simulations were conducted on a random phylogeny with 64 species, divided in two groups
originating in a random node relatively deep in the tree, where at least one of the groups was
monophyletic. We then used the transformPhylo.sim function of R-package motmot (Thomas
and Freckleton 2012) to simulate a continuous phenotypic trait evolving under a BM with
varying degrees of mean and rate differentiation between groups. The mean for the first group
was always set to â1 = 0 and the rate of the first group was always set to σ21 = 1. The mean for the
second group was set to either 0.5, 1, 2, 3, 4, 5, or 6 standard deviations larger than â1, whereas
the rate of the second group (σ22 = θσ21) was set to: 1.1, 1.5, 2, 3, 4, 5, or 6 times larger than σ21.
For each of the above combinations, 1000 phenoytpic datasets were simulated. We then used the
censored approach, as implemented in R-package RBrownie (Stack et al. 2011), and the function
ML.RatePhylo of motmot to fit the two evolutionary models for rate comparison mentioned
above. Finally, we compared the obtained relative rate estimates (θcen and θmotmot), as well as the
corresponding model log-likelihoods (loglikcen and loglikmotmot, by examining the correlation
across 1000 datasets and the mean ratio of the values obtained from each implementation.
The results obtained clearly demonstrate a full correspondence between the two methods
when comparing evolutionary rates for continuous phenotypic traits between monophyletic
groups on a phylogeny. Across 1000 simulated datasets, and across varying conditions of
differentiation in phylogenetic means and rates between groups, the correlation and mean ratio of
both relative rate estimates and log-likelihoods obtained by the two methods approached unit
value (Table A3, Fig. A3), showing that the two implementations are fully coincident.
2
Relative rate (θ)
Relative rate (θ)
Relative rate (θ)
Relative rate (θ)
Table A3: Correlations and mean ratios of relative rate estimates and likelihoods obtained from fitting a two-rate model using
the censored approach (O’Meara et al. 2006) or the rate test using a group design matrix for phylogenetic means (Thomas et al.
2006) across different simulation conditions for 1000 datasets simulated under a two-means, two-rates BM process.
Relative rate estimates
Difference in means (in St.Devs)
Difference in means (in St.Devs)
Correlation
Ratio
0.5
1
2
3
4
5
6
0.5
1
2
3
4
5
6
1.1
1.1 1.020 1.020 1.020 1.020 1.020 1.020 1.021
1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.5
1.5 1.024 1.024 1.024 1.024 1.024 1.024 1.024
1.000 1.000 1.000 1.000 1.000 1.000 1.000
2
2 1.027 1.028 1.027 1.028 1.027 1.027 1.028
1.000 1.000 1.000 1.000 1.000 1.000 1.000
3
3 1.032 1.032 1.032 1.032 1.032 1.032 1.032
1.000 1.000 1.000 0.999 1.000 0.999 1.000
4
4 1.035 1.035 1.035 1.035 1.035 1.035 1.035
0.999 1.000 1.000 1.000 1.000 1.000 1.000
5
5 1.037 1.037 1.037 1.037 1.037 1.037 1.037
1.000 0.999 1.000 1.000 1.000 1.000 1.000
6
6 1.039 1.038 1.038 1.039 1.038 1.038 1.038
1.000 1.000 1.000 1.000 1.000 1.000 0.999
Log-likelihoods
Difference in means (in St.Devs)
Difference in means (in St.Devs)
Correlation
Ratio
0.5
1
2
3
4
5
6
0.5
1
2
3
4
5
6
1.1
1.1 1.110 1.141 1.144 1.121 1.184 1.115 1.123
0.997 0.997 0.997 0.997 0.997 0.997 0.997
1.5
1.5 1.089 1.033 1.060 1.206 1.146 1.080 1.113
0.998 0.998 0.998 0.998 0.998 0.998 0.998
2
2 1.066 1.063 1.164 1.066 1.065 1.065 1.066
0.998 0.998 0.998 0.998 0.998 0.998 0.998
3
3 1.048 1.059 1.060 1.021 1.058 1.057 1.055
0.999 0.999 0.999 0.999 0.999 0.999 0.999
4
4 1.050 1.055 1.046 1.059 1.046 1.043 1.053
0.999 0.999 0.999 0.999 0.999 0.999 0.999
5
5 1.053 1.048 1.003 1.034 1.049 1.046 1.043
1.000 1.000 1.000 1.000 1.000 1.000 1.000
6
6 1.048 0.994 1.043 1.038 1.044 1.010 1.060
1.000 1.000 1.000 1.000 1.000 1.000 1.000
3
Figure A3: Comparison of relative rate estimates and log-likelihoods obtained from the censored
test (θcen and loglikcen) and from ML.RatePhylo (θmotmot and loglikmotmot). Results are shown for
the case of â2 / â1 = 4*stdev and θ=4, but were similar for all the simulation conditions examined
(Table A3).
References
Stack, C.J, Harmon, L.J. & O’Meara, B. 2011. RBrownie: an R package for testing hypotheses
about rates of evolutionary change. Methods in Ecology and Evolution 2: 660-662.
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