Rethinking phylogenetic comparative methods

Failure to consider alternative models in which the background diversification rate changes across the tree explains support for trait-dependent diversification.

“The work of Beaulieu and O'Meara (Beaulieu et al. 2013;Beaulieu andO'Meara 2014, 2016) has illuminated important underlying reasons behind Rabosky and Goldberg's findings: the failure to consider alternative models in which the”

• cites Beaulieu et al. 2013 sourceNot checked — no full text of the source
• cites Beaulieu and O'Meara 2016 sourceSupported

Accounting for background shifts in evolutionary regimes unrelated to the focal trait association is general and applies across comparative biology.

“We think that the solution proposed by Beaulieu and O'Meara (2016)-accounting for background shifts in evolutionary regimes unrelated to the focal trait association-is general and applies across comparative biology.”

• cites Beaulieu and O'Meara (2016) sourceSupported

The Brownian motion model is inconsistent with how we think of the process of adapting to an optimal state.

“While this model is useful (and has often been used) for testing for adaptation, it is inconsistent with how we think of the process of adapting to an optimal state (Lande 1976;Hansen 1997;Hansen and Orzack 2005;Hansen et al. 2008;Hansen and Bartoszek 2012).”

• cites Lande 1976 sourceNot checked — no full text of the source
• cites Hansen 1997 sourceSupported

A wide breadth of macroevolutionary data suggest that abrupt shifts and discontinuities have been a major feature of life on Earth.

“a wide breadth of macroevolutionary data suggest that abrupt shifts and discontinuities have been a major feature of life on Earth”

• cites Uyeda et al. 2011 sourcePartly supported

  The paper’s opening section reports that a model combining rare, substantial bursts of phenotypic change with shorter-term bounded fluctuations best fits its data. But those data chiefly concern body size in mammals, squamates, and birds; they do not establish that abrupt shifts and discontinuities have been a major feature of life on Earth generally. (LLM, conf=0.85)

  Experimental — accuracy not yet measured
• cites Landis and Schraiber 2017 sourcePartly supported

  The Abstract describes evidence of rapid evolutionary pulses in fossil and ecological data, and the Discussion reports that jump models best explained body-size evolution in 32% of the 66 vertebrate clades studied. These passages support abrupt shifts as a feature of evolution, but the claim extends the finding to life on Earth generally; the paper says the prevalence of pulsed evolution across the Tree of Life remains unknown. Page numbers were not available in our copy. (LLM, conf=0.86)

  Experimental — accuracy not yet measured
• cites Jablonski 2017 sourceSupported

Hansen's introduction of the Ornstein-Uhlenbeck process to comparative biology and the suite of methods built on his approach have been the only real attempts to capture the basic dynamics of adaptive trait evolution on phylogenies.

“Hansen's introduction of the Ornstein-Uhlenbeck (OU) process to comparative biology and the suite of methods built on his approach have been the only real attempts to actually try and capture the basic dynamics of adaptive trait evolution on phylogenies.”

• cites Hansen 1997 sourcePartly supported

  In “The Evolutionary Approach to a Primary Optimum” (page unavailable), Hansen identifies his model as an Ornstein–Uhlenbeck process and describes a novel biological interpretation of it. But he explicitly says the process had previously been suggested for comparative studies, so the paper does not support saying he introduced it to comparative biology. The Abstract describes his method, but does not establish that methods built on it were the only attempts to capture adaptive trait evolution on phylogenies. (LLM, conf=0.88)

  Experimental — accuracy not yet measured

AIC provides only relative support for a model given a set of alternatives.

“However, AIC provides only relative support for a model given a set of alternatives (see Pennell et al. 2015 for more on this point in the context of comparative methods).”

• cites Pennell et al. 2015 sourceSupported

Depending on the Bayesian network structure, the appropriate method of analysis can range from a nonphylogenetic regression, to commonly used comparative methods such as PGLS, to methods that require modeling both the evolutionary history of interaction of both trait X and trait Y.

“Depending on the Bayesian network structure, the appropriate method of analysis can range from a nonphylogenetic regression (Fig. 7A), to commonly used comparative methods such as PGLS (Fig. 7B), to methods that require modeling both the evolutionary history of interaction of both trait X and trait Y (Fig. 7C) (Hansen 1997;Butler and King 2004;Hansen et al. 2008;Revell 2010;Hansen and Bartoszek 20”

• cites Hansen 1997 sourcePartly supported

  The Phylogenetic Covariance section describes generalized least squares for phylogenetically correlated trait data, and the Discussion argues that comparative analyses should consider the history of selective regimes. The paper does not set out a Bayesian-network-based choice among nonphylogenetic regression, PGLS, and joint modeling of the evolutionary histories of two traits. Page numbers were not available in our copy. (LLM, conf=0.84)

  Experimental — accuracy not yet measured
• cites Butler and King 2004 sourceNot checked — no full text of the source
• cites Hansen et al. 2008 sourceNot checked — no full text of the source

Modeling the errors using fat-tailed distributions may mitigate the impact of singular evolutionary events on the estimation of the slope in PGLS and other linear modeling approaches.

“For PGLS and other linear modeling approaches, modeling the errors using fat-tailed distributions (Blomberg et al. 2012;Landis et al. 2012;Elliot and Mooers 2014;Duchen et al. 2017) may mitigate the impact of singular evolutionary events on the estimation of the slope (also see Slater and Pennell 2013, for an alternative approach using robust regression).”

• cites Blomberg et al. 2012 sourceNot checked — no full text of the source
• cites Landis et al. 2012 sourcePartly supported

  The paper’s introduction (page not available in our copy) says that Lévy-process jumps give trait-change distributions fat tails and account for abrupt evolutionary shifts. Its Discussion addresses detecting such jumps, but the paper does not show that fat-tailed errors mitigate their effect on slope estimates in PGLS or other linear models. (LLM, conf=0.82)

  Experimental — accuracy not yet measured
• cites Elliot and Mooers 2014 sourcePartly supported

  The Background toy example shows that a heavy-tailed stable model is less affected than a Brownian model by an unusually large change on one branch, but its outcome is ancestral-state reconstruction, not a regression slope. The Discussion proposes heavy-tailed methods for future studies of correlated evolution and notes that stable distributions lack the covariance matrix used in phylogenetic regression. It does not show that fat-tailed errors mitigate the effect of singular events on slope estimates in PGLS or other linear models; page numbers were not provided in our copy. (LLM, conf=0.85)

  Experimental — accuracy not yet measured

The probability of the dependent model P(M_dep|A) under Darwin's scenario is (t/T).

“The probability of the dependent model P(M dep |A) under Darwin's scenario is P(M dep |A) = (t/T) (7 )”

When data were simulated under the PE hypothesis, the estimated weight given to the PE hypothesis was high.

“When we simulated data under the PE hypothesis, the estimated weight given to the PE hypothesis was likewise high (Fig. 3B).”

Repeating the exercise with more phylogenetically structured data where the two clades are fully bifurcating following a Yule process yields identical results.

“We can repeat the same exercise with more phylogenetically structured data (where the two clades of interest are fully bifurcating following a Yule process) and obtain identical results (Fig. 2, see Supplementary Material available on Dryad at http://dx.doi.org/10.5061/dryad.p8066hd).”

Under the assumptions above, the probability of the independent model P(M_ind|A) is (t/T)^2.

“Therefore, the probability of the independent model denoted as P(M ind |A) given the assumptions above is P(M ind |A) = P(N x (t)|N x (T) ≥ 1)P(N y (t)|N y (T) ≥ 1) = (t/T) 2 , (6 )”

• AssumptionThe authors state this as an assumption: The traits were selected for study without first looking at their phylogenetic distribution.
• AssumptionThe authors state this as an assumption: The likelihood of evolving these traits at all is quite small.

In the simulation, a single contrast results in a high-leverage statistical outlier that drives significance as the size of the shift increases.

“That single contrast results in a very highleverage statistical outlier that drives significance as the size of the shift increases (Fig. 2).”

The test statistic D for Pagel's discrete correlation test comparing M_ind and M_dep is D = 2(ln(T)-ln(t)).

“Therefore, the test statistic D used in the likelihood ratio test for Pagel's discrete correlation test comparing models M ind and M dep is simply proportional to the ratio of the length of the branch where the shift occurred to the total length of the tree: D = 2(lnL(M dep )-lnL(M ind )) = 2(ln(t)-ln(T))-4(ln(t)-ln(T)) = 2(ln(T)-ln(t)).(8)”

Data simulated under the singular event model can result in highly significant regressions for OLS, PGLS, and IC, regardless of whether errors are simulated as independent or phylogenetically correlated.

“even for the simple case of two continuous traits, we have shown how easily data simulated under the singular event model can result in highly significant regressions for OLS, PGLS, and IC regressions, regardless if the errors are simulated as independent or phylogenetically correlated with respect to the model and phylogeny.”

• AssumptionThe authors state this as an assumption: Trait correlations can result from a single random shift drawn from a different distribution than the one used to model trait evolution across the rest of the branches.

The analysis recovers high weights for the Phrynosoma-only hypothesis regardless of whether the model is fit to the Scales data set or to data simulated under either the Phrynosoma-only hypothesis or the PE hypothesis.

“In this case, we recover high weights for the Phrynosomaonly hypothesis regardless if the model is fit to the Scales data set, or to data simulated under either the Phrynosoma-only hypothesis or the PE hypothesis.”

The likelihood value at the maximum likelihood estimate of the dependent model is L(M_dep) = t/T.

“The likelihood value at the maximum likelihood estimate of this model turns out to be just the probability from Eq. ( 7), that is L(M dep ) = t/T.”

The authors' Bayesian mixture model places intermediate weight on the PE hypothesis for the original Scales data set.

“We find that our approach places intermediate weight on the PE hypothesis for the original Scales data set.”

The RJ portion of the model fit to the Scales data set recovers only a single highly supported shift on the stem branch of the Phrynosoma lizards.

“Furthermore, the RJ portion of the model fit to the Scales data set recovers only a single highly supported shift on the stem branch of the Phrynosoma lizards (Fig. 3C,D).”

When data were simulated under the Phrynosoma-only hypothesis, the weight given to the PE hypothesis was low.

“When data were simulated under the Phrynosoma-only hypothesis, the weight given to the PE hypothesis was low, as predicted (Fig. 3B).”

The PE hypothesis is the simplest model that allows a shift on the branch leading to Phrynosoma.

“Looking at the reconstructions, it is also apparent that the PE hypothesis is the simplest model that allows a shift on the branch leading to Phrynosoma, a group that any herpetologist would identify as”

In no analysis did the reversible-jump portion recover support for any additional shifts.

“In no analysis did the reversible-jump portion recover support for any additional shifts.”

Plotting the empirically estimated differences in the MLEs against the predictions making the simplifying assumptions above reveals a strong modal correlation between them.

“Plotting the empirically estimated differences in the MLEs against the predictions making the simplifying assumptions above reveals a strong modal correlation between them (Fig. 6).”

The empirical data set has intermediate weights.

“Notice that the empirical data set has intermediate weights.”

Points falling off the line represent slight violations of the assumptions used to derive the prediction.

“Points falling off the line represent slight violations of the assumptions we used to derive our prediction.”

• AssumptionThe authors state this as an assumption: The rates of gain of the traits are so low that only one shift is ever observed (in the Figure 6 analysis).

Modest support for two additional shifts is recovered for the empirical data only.

“Modest support for two additional shifts are recovered for the empirical data only (red circles).”

All analyses recover high weights.

“All analysis recover high weights.”

Most comparative research questions have a poorly defined scope of inference: it is unclear to what population a model or inference should generalize.

“For example, we find that most comparative research questions have a poorly defined scope of inference: it is unclear to what population a model or inference should generalize to.”

The two causal structures in the illustrative example are observationally equivalent: any distribution simulated under one can be replicated under the alternative causal structure.

“These two causal structures are observationally equivalent-meaning that any distribution simulated under one can be replicated under the alternative causal structure.”

Singular and unreplicated evolutionary events are incompatible with models typically used to describe change over time, such as Brownian motion or the Mk model.

“Their singular and unreplicated nature seems incompatible with models that we typically use to describe change over time, such as Brownian motion (BM; Felsenstein 1973) or the Mk model (Pagel 1994;Lewis 2001).”

• cites Felsenstein 1973Not checked — no full text of the source
• cites Pagel 1994 sourceNot checked — no full text of the source
• cites Lewis 2001 sourceNot checked — no full text of the source

Rare evolutionary events may deceive comparative methods and distort interpretations.

“We will show in each Case Study that rare evolutionary events may deceive our methods and distort our interpretations.”

The phenomenon that made Felsenstein's argument intuitive is not the violation of i.i.d. errors but the realization that unreplicated differences colocalized on a single branch provide only weak evidence of a causal relationship between traits.

“Nevertheless, our primary point here is to suggest that the phenomenon that made Felsenstein's argument so intuitive is not the violation of i.i.d.errors but rather the biologically intuitive realization that unreplicated differences colocalized on a single branch provide only weak evidence of a causal relationship between traits.”

Common statistical tests for the evolutionary correlation of discrete characters are prone to reporting a significant association even when the pattern is driven by a single or very few independent transitions.

“Maddison and FitzJohn (2015) recently demonstrated that common statistical tests (e.g., Maddison 1990;Pagel 1994) for the evolutionary correlation of discrete characters are prone to reporting a significant association even when the pattern is driven by a single (or, very few) independent transition(s) from one character state to another.”

• cites Maddison and FitzJohn (2015) sourceNot checked — no full text of the source

This is a clear case of phylogenetic pseudoreplication.

“This seems to be a clear case of phylogenetic pseudoreplication (Read and Nee 1995;Maddison and FitzJohn 2015).”

• cites Read and Nee 1995 sourceNot checked — no full text of the source
• cites Maddison and FitzJohn 2015 sourceNot checked — no full text of the source

The unresolved challenge of singular events permeates nearly every method of finding associations in comparative methods.

“We will argue that this unresolved challenge permeates not just tests for discrete character correlations, but nearly every method of finding associations in comparative methods (Fig. 1).”

Guessing correctly which of the 42 branches on the phylogeny has a single shift with the hypothesis is rewarded by the analysis.

“Guessing correctly which of the 42 branches on the phylogeny has a single shift with our hypothesis is rewarded by the analysis (we will return to this issue in Case Study III).”

The PE hypothesis fails to fully explain the shift leading to the Phrynosoma lizards, which are more extreme than expected given other taxa in their regime.

“In particular, the PE hypothesis fails to fully explain the shift leading to the Phrynosoma lizards (Fig. 3C,D), which are more extreme than they should be considering the other taxa in their regime (there is only one, Holbrookia maculata, which does not show such an extreme shift).”

Phylogenetic comparative methods are susceptible to unreplicated evolutionary events.

“Not only has this sowed widespread confusion in the literature about what PCMs are doing but has led us to develop methods that are susceptible to the very thing we sought to build defenses against-unreplicated evolutionary events.”

Maddison and FitzJohn (2015) simulated comparative data and reported a preponderance of significant results using Pagel's correlation test (1994) and Maddison's (1990) concentrated changes test.

“In their article, Maddison and FitzJohn (2015) simulated comparative data and reported a preponderance of significant results using Pagel's correlation test (1994) and Maddison's (1990) concentrated changes test.”

• cites Maddison and FitzJohn (2015) sourceNot checked — no full text of the source
• cites Pagel (1994) sourceNot checked — no full text of the source
• cites Maddison (1990) sourceNot checked — no full text of the source

The authors' framework does not automatically reward more complex, freely estimated models.

“Thus, our framework certainly does not automatically reward more complex, freely estimated models.”

The signal for an association between muscle fiber composition and predation escape behavior is generated in part, but not completely, by variation specific to the genus Phrynosoma.

“In summary, the signal for an association between muscle fiber composition and predation escape behavior is generated in part, but not completely, by variation that is specific to the genus Phrynosoma.”

The PE hypothesis has statistically supported explanatory power, as its estimated weight is well bounded away from 0.

“This suggests that the PE hypothesis has statistically supported explanatory power as its estimated weight is well bounded away from 0.”

The authors show the value in combining a hypothesis-testing framework with a natural history approach to identifying patterns of evolution.

“Nevertheless, we show the value in combining a hypothesis-testing framework with a natural history approach to identifying patterns of evolution.”

Maddison and FitzJohn (2015) demonstrated that commonly used methods return significant correlations all the time, in scenarios that defy statistical intuition.

“As we mention above, Maddison and FitzJohn (2015) recently demonstrated that commonly used methods return significant correlations all the time—and in scenarios that seem to defy our statistical intuition.”

• cites Maddison and FitzJohn (2015) sourceNot checked — no full text of the source

Current comparative methods are not designed to deal with abrupt shifts and discontinuities.

“But as recent controversies in phylogenetic comparative biology have highlighted, our current methods (reviewed in O'Meara 2012;Pennell and Harmon 2013;Garamszegi 2014) are not designed to deal with such dynamics.”

• cites O'Meara 2012 sourceNot checked — no full text of the source
• cites Pennell and Harmon 2013 sourceNot checked — no full text of the source
• cites Garamszegi 2014 sourceNot checked — no full text of the source

The problems and potential solutions identified apply just as well to situations where background evolutionary dynamics change more frequently.

“But the problems (and potential solutions) we identify apply just as well to situations where the background evolutionary dynamics change more frequently.”

Scales et al. found strong support for the predator escape hypothesis, which is 13.0 AICc units better than the next closest model, FMPE.

“After fitting the multioptimum OU models to the muscle fiber data, they find strong support for the PE hypothesis, which is 13.0 Akaike's Information Criterion (AICc) units better than the next closest model (FMPE).”

• cites Scales et al. 2009 sourceNot checked — no full text of the source

Hypothesis-driven and phylogenetic natural history approaches are complementary.

“For these reasons, we argue that hypothesis-driven and phylogenetic natural history approaches are complementary: we must pit our particular causal hypotheses against a descriptive”

Accounting for the Phrynosoma shift is the primary feature of all three data sets.

“This is because accounting for the Phrynosoma shift is the primary feature of all three data sets (though weights are somewhat higher for data simulated under the Phrynosoma-only hypothesis than others).”

The analysis favors the Phrynosoma-only hypothesis simply because of the vague priors placed on the number and location of shifts in the reversible-jump analysis.

“However, the analysis favors the Phrynosoma-only hypothesis simply because of the vague priors placed on the number and location of shifts in the reversible-jump analysis.”

Major shifts are likely to drown out any biological signal in a data set if they are unaccounted for by hypothesis-driven models.

“The reasons for this are covered in Case Study I: these major shifts are likely to drown out any biological signal in a data set if they are unaccounted for by our hypothesisdriven models.”

Pagel's (1994) correlation test would find the phylogenetic codistribution of milk production and middle ear bones highly statistically significant even though they both are a defining characteristic of mammals.

“For example, Pagel's (1994) correlation test would find the phylogenetic codistribution of milk production and middle ear bones highly statistically significant even though they both are a defining characteristic of mammals, an inference so obviously dubious that even Darwin (1872) warned against it.”

• cites Pagel (1994) sourceNot checked — no full text of the source
• cites Darwin (1872)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Standard linear regression assumes that the errors of the fitted model are independent and identically distributed, not that the data are independent.

“Rather, standard linear regression assumes that the errors of the fitted model are independent and identically distributed (i.i.d.).”

Nonindependence can create apparent correlations between traits that are in truth evolving independently.

“This nonindependence can create apparent correlations between traits that, are in truth, evolving independently.”

Susceptibility to singular events is a recurring problem in comparative biology that links several seemingly unrelated controversies.

“Through three Case Studies, we demonstrate that the susceptibility to singular events is indeed a recurring problem in comparative biology that links several seemingly unrelated controversies.”

Felsenstein recovered a significant regression slope using Ordinary Least Squares despite there being no evolutionary covariance between the traits.

“He then evolved traits according to a BM process along the phylogeny; he recovered a significant regression slope using Ordinary Least Squares (OLS) despite there being no evolutionary covariance between the traits.”

• cites Felsenstein 1985 sourceNot checked — no full text of the source

Traits measured on one species are not independent of traits measured on others.

“In statistical terms, this means that traits measured on one species will not be independent of traits measured on others.”

• AssumptionRests on an assumption: Descent with modification causes closely related species to be similar to one another in many traits.

The phylogeny imposes a challenge to the inference of meaningful associations between traits not because it renders errors nonindependent, but because its structure allows ancient, potentially unknowable causal factors to drive widespread associations between traits.

“Nevertheless, the example illustrates that the phylogeny imposes a challenge to the inference of meaningful associations between traits not because it renders errors nonindependent, but because the structure of the phylogeny allows for ancient, potentially unknowable causal factors (which may be few or even singular) to drive widespread associations between traits.”

The IC method is based on the Brownian motion model of trait evolution.

“As stated above, the IC method is based on the BM model of trait evolution.”

• cites Felsenstein 1985 sourceNot checked — no full text of the source

If all phylogenetic signal in a data set is present in the predictor trait and the errors are i.i.d., then no phylogenetic correction is needed.

“if all of the phylogenetic signal in a data set is present in the predictor trait and the errors are i.i.d., then there is no need for any phylogenetic correction”

• cites Rohlf 2001 sourceNot checked — no full text of the source
• cites Rohlf 2006 sourceNot checked — no full text of the source

The intermediate result likely only occurs because the PE hypothesis places Phrynosoma in the same regime as Holbrookia maculata, which does not share the extreme shift found in Phrynosoma.

“However, the intermediate result likely only occurs because the PE hypothesis places Phrynosoma in the same regime as Holbrookia maculata, which does not share the extreme shift that is found in Phrynosoma.”

By combining phylogenetic natural history approaches with a priori hypotheses, the authors can account for singular evolutionary events that are not well-accounted for by the generating model.

“By combining phylogenetic natural history approaches with our a priori hypotheses, we show that we can account for singular evolutionary events that are not well-accounted for by our generating model.”

Simpson's paradox is not really paradoxical when considered from the standpoint of Bayesian Networks.

“However, as Pearl (2014) has convincingly demonstrated, Simpson's paradox is not really paradoxical at all when considered from the standpoint of Bayesian Networks.”

• cites Pearl (2014) sourceNot checked — no full text of the source

If migratory behavior causes changes in body size, then selecting for an increase in body size will not result in a lineage changing its migratory status at all.

“In contrast, if migratory behavior causes changes in body size, then selecting for an increase in body size will not result in a lineage changing their migratory status at all.”

• AssumptionThe authors state this as an assumption: Migratory behavior causes changes in body size (in the illustrative example).

Darwin's scenario, the singular origin of two coextensive traits on the phylogeny, represents a boundary case for finding the correlation between discrete characters.

“Darwin's scenario-the singular origin of two coextensive traits on the phylogeny-represents a boundary case to finding the correlation between discrete characters.”

The precise assumptions of PCMs remain poorly understood to this day.

“and remain poorly understood to this day (Hansen and Orzack 2005;Hansen and Bartoszek 2012).”

• cites Hansen and Orzack 2005 sourceNot checked — no full text of the source
• cites Hansen and Bartoszek 2012 sourceNot checked — no full text of the source

Researchers' prior ideas about how traits map unto trees likely inform which traits they choose to test for correlated evolution.

“Rather, researchers' prior ideas about how traits map unto trees likely inform which traits they choose to test for correlated evolution.”

Graphical modeling would improve inference in cases where considering phylogeny reverses the sign of the relationship between two variables.

“One clear case where such graphical modeling would improve inference are cases where considering phylogeny reverses the sign of the relationship between two variables.”

Under Darwin's scenario, including additional taxa in the analysis will increase the support for the dependent model simply as a consequence of increasing the total length of the tree.

“Under Darwin's scenario, including additional taxa in the analysis will increase the support for the dependent model simply as a consequence of increasing the total length of the tree (i.e., because T increases and the difference between ln(T) and ln(t) will get larger as long as the additional sampled taxa do not break Darwin's scenario).”

Under Darwin's scenario, the results of the analysis are predetermined.

“In other words, the results of the analysis are predetermined.”

The traits used in comparative analyses are not chosen independently with respect to their phylogenetic distribution.

“As Maddison and FitzJohn (2015) point out, the traits we use in comparative analyses are not chosen independently with respect to their phylogenetic distribution (as we assumed in our analysis).”

• cites Maddison and FitzJohn (2015) sourceNot checked — no full text of the source

Working through the logic of comparative analyses using graphical models leads to essentially the same line of reasoning as Westoby et al. (1995a,b), who challenged the growing consensus that phylogeny needed to be included in any interspecific comparison.

“By working through the logic of comparative analyses using graphical models we have come to essentially the same line of reasoning of Westoby et al. (1995a,b), who, in the early days of PCMs, challenged the growing consensus that phylogeny needed to be included in any interspecific comparison-a consensus which has only gotten stronger as the years passed by (also see McNab 2003 for a related crit”

• cites Westoby et al. (1995a,b)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites McNab 2003 sourceNot checked — no full text of the source

Pagel's correlation test for Darwin's scenario can essentially be reduced to the difference in probability between choosing the same branch twice versus choosing the branch only once.

“Pagel's correlation test for Darwin's scenario can essentially be reduced to the difference in probability between choosing the same branch twice vs. choosing the branch only once.”

Darwin's scenario-the singular origin of two coextensive traits on the phylogeny-represents a boundary case to finding the correlation between discrete characters.

“Darwin's scenario-the singular origin of two coextensive traits on the phylogeny-represents a boundary case to finding the correlation between discrete characters.”

Under network (B), increasing body size will only have a direct effect of decreasing species abundance, which is reflected in the estimate of the slope.

“Here, increasing body size will only have a direct effect of decreasing species abundance, which is reflected in the estimate of the slope.”

• AssumptionThe authors state this as an assumption: Migratory behavior causes changes in body size (in the illustrative example).

The authors agree with Maddison and FitzJohn (2015) that ascertainment bias is likely prevalent in empirical studies.

“We agree with Maddison and FitzJohn (2015) that this type of ascertainment bias is likely prevalent in empirical studies, even if it is usually more subtle than testing for a correlation between milk and middle ear bones.”

• cites Maddison and FitzJohn (2015) sourceNot checked — no full text of the source

Nee et al. (1991) found that in British birds the direction of correlation between body size and abundance flipped back and forth depending on how the data were aggregated.

“This is precisely what Nee et al. (1991) found looking at the relationship between body size and abundance in British birds; depending on how they aggregated the data (means of species, means of genera, means of tribes, etc.) the direction of correlation flipped back and forth.”

• cites Nee et al. (1991) sourceNot checked — no full text of the source

The appropriate way to analyze data depends crucially on what one assumes is causing what.

“In fact, Pearl shows that the appropriate way to analyze the data depends crucially on what one assumes is causing what.”

• cites Pearl (2014) sourceNot checked — no full text of the source

Many PCMs, including those that form the bedrock of the field, are susceptible to being misled by singular evolutionary events.

“We have shown in the three Case Studies that many PCMs, including those that form the bedrock of our field, are susceptible to being misled by singular evolutionary events.”

For even a modestly sized tree, this coincidence is so unlikely that the alternative hypothesis of correlated evolution is preferred over the null.

“For even a modestly sized tree, this coincidence is so unlikely that the alternative hypothesis of correlated evolution is preferred over the null.”

To estimate the net effect of B on N in network (A), the appropriate method of analysis is OLS regression.

“To estimate the net effect of B on N in network (A), the appropriate method of analysis is OLS regression (black line).”

• AssumptionThe authors state this as an assumption: Body size influences migratory behavior (in the illustrative example).

Pagel's correlation test for Darwin's scenario can essentially be reduced to the difference in probability between choosing the same branch twice vs. choosing the branch only once.

“Pagel's correlation test for Darwin's scenario can essentially be reduced to the difference in probability between choosing the same branch twice vs. choosing the branch only once.”

The use of phylogeny in interspecific comparisons is an assumption that depends on the precise question being asked and the hypothesized causal network.

“We emphasize that this implies that the use of phylogeny in interspecific comparisons is an assumption that depends on the precise question being asked and the hypothesized causal network.”

In cases where phylogeny does matter, one must specify the generating model for unobserved states in causal graphs.

“In cases where phylogeny does matter, we must specify the generating model for unobserved states in our causal graphs.”

It is therefore not completely unreasonable that Pagel's test tells us that these traits have evolved in an entirely correlated fashion.

“It is therefore not completely unreasonable that Pagel's test tells us that these traits have evolved in an entirely correlated fashion.”

The asymmetry that ancestors are causes of descendants and not vice versa results in a probabilistic Bayesian Network (a type of directed acyclic graph, or DAG) that predicts a specific set of conditional probabilities among the data.

“This asymmetry results in a what is known as a probabilistic Bayesian Network (a type of directed acyclic graph, or DAG) that predicts a specific set of conditional probabilities among the data.”

• AssumptionThe authors state this as an assumption: Ancestors are causes of descendants, and not vice versa.

If only one event occurs, the causally linked model is only distinguishable from graph (D) proportional to the probability that events K and L occur on the same branch.

“If only one event occurs however, this model is only distinguishable from graph (D) proportional to the probability that events K and L occur on the same branch (see Case Study III).”

Differences likely reflect the fact that the authors have not explicitly made the assumption that P(N_x(t)=1)=P(N_y(t)=1)≈1 when fitting the model with diversitree.

“Differences likely reflect the fact that we have not explicitly made the assumption that P(N x (t) = 1) = P(N y (t) = 1) ≈ 1 when we fit the model with diversitree.”

To estimate the effect of B in both networks, one must use different analytical approaches.

“However, to estimate the effect of B in both networks, one must use different analytical approaches.”

BM and other continuous Gaussian or Markov processes are only a few of the many types of processes that may generate change on a phylogeny.

“However, BM and other continuous Gaussian or Markov processes are only a few of the many types of processes that may generate change on a phylogeny.”