APPENDIX 4
SIMULATION EXPERIMENT TO INVESTIGATE WHETHER R-PACKAGES OUWIE AND MOTMOT
PROVIDE COMPARABLE PARAMETER ESTIMATES AND LIKELIHOOD SCORES
In order to fit the different evolutionary models of interest, we had to use two different Rpackages, namely OUwie (Beaulieu et al. 2012) and motmot (Thomas and Freckleton 2012).
This is because, although the models fit by the two packages highly overlap, OU models with
varying σ and β parameters are only implemented in OUwie, while the general (e.g. allowing for
non-monophyletic groups) model with group phylogenetic means (BMSG) is only implemented
in motmot. This poses a practical problem, because the way for calculating model parameters
and, above all, the optimizing algorithms used by each implementation to obtain ML estimators,
may not coincide. In such a case, the likelihood scores (and corresponding parameter estimates)
would not be comparable and thus one would not be able to evaluate the relative fit of models
obtained using different software packages.
To address this potential obstacle, we performed a series of simulations in order to
compare the parameter estimates and likelihood scores provided by the two packages for a model
implemented by both, e.g. a BM with different rates for each group, calculated considering a
single phylogenetic mean for the entire phylogeny (BMS). Simulations were conducted on a pure
birth, random phylogeny with 64 species, divided in two groups originating in a random node
relatively deep in the tree. 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 a
single phylogenetic mean, but varying rate differentiation between groups. The rate of the first
group was always set to σ21 = 1, whereas the rate of the second group (σ22 = θσ21) was set to: 1.5,
1
2, 3, 4, 5, or 6 times larger than σ21. For each of the above rate-difference conditions, 1000
phenoytpic datasets were simulated. We then fit the two aforementioned models (BM1 and
BMS) in both packages, and compared parameter estimates and likelihood scores using the same
approach as above.
Across 1000 simulated datasets, and across varying conditions of differentiation in
evolutionary rates between groups, the correlation and mean ratio of both relative rate estimates
and log-likelihoods obtained by the two packages approached unit value (Table A4, Fig.
A4).These results show that, except for a very few cases where problems of convergence occur
during the optimization process (e.g. outliers in Fig. A4), OUwie and motmot provide practically
the same likelihood scores and parameter estimates. As such, models fitted in the two packages
can be directly compared using likelihood criteria. This is very relevant for evolutionary
inference both with regards to empirical and simulation studies, as it broadens the range of
models of phenotypic evolution on phylogenies available for comparison.
Table A4: Correlations and mean ratios of relative rate estimates and
likelihoods obtained from fitting a two-rate model using OUwie (Beaulieu
et al. 2021) or motmot (Thomas and Freckleton 2012) R-packages across
different simulation conditions for 1000 datasets simulated under a singlemean, two-rate BM process.
Simulated relative rate (θ)
1.5
2
3
4
5
6
θ estimate
Correlation 0.983 0.999 0.996 0.999 0.992 0.996
Ratio
0.994 0.991 0.986 0.983 0.980 0.981
Log-likelihood Correlation 0.999 0.999 0.999 0.997 0.909 0.969
Ratio
1.000 0.999 0.999 0.998 0.998 0.998
2
Figure A4: Comparison of relative rate estimates and log-likelihoods obtained from OUwie
(Beaulieu et al. 2021) and motmot (Thomas and Freckleton 2012) for a two-rate BM
evolutionary model. Results are shown for the case of θ=2, but were similar for all the simulation
conditions examined (Table A4).
3