Phylogenetic comparative methods (PCMs) use information on the historical relationships of lineages (

phylogenies A phylogenetic tree or phylogeny is a graphical representation which shows the evolutionary history between a set of species or taxa during a specific time.Felsenstein J. (2004). ''Inferring Phylogenies'' Sinauer Associates: Sunderland, MA. In o ...

) to test evolutionary hypotheses. The comparative method has a long history in evolutionary biology; indeed,

Charles Darwin Charles Robert Darwin ( ; 12 February 1809 – 19 April 1882) was an English Natural history#Before 1900, naturalist, geologist, and biologist, widely known for his contributions to evolutionary biology. His proposition that all speci ...

used differences and similarities between species as a major source of evidence in ''

The Origin of Species ''On the Origin of Species'' (or, more completely, ''On the Origin of Species by Means of Natural Selection, or the Preservation of Favoured Races in the Struggle for Life'')The book's full original title was ''On the Origin of Species by M ...

''. However, the fact that closely related lineages share many traits and trait combinations as a result of the process of descent with modification means that lineages are not independent. This realization inspired the development of explicitly phylogenetic comparative methods. Initially, these methods were primarily developed to control for phylogenetic history when testing for

adaptation In biology, adaptation has three related meanings. Firstly, it is the dynamic evolutionary process of natural selection that fits organisms to their environment, enhancing their evolutionary fitness. Secondly, it is a state reached by the p ...

; however, in recent years the use of the term has broadened to include any use of phylogenies in statistical tests. Although most studies that employ PCMs focus on extant organisms, many methods can also be applied to extinct taxa and can incorporate information from the

fossil record A fossil (from Classical Latin , ) is any preserved remains, impression, or trace of any once-living thing from a past geological age. Examples include bones, shells, exoskeletons, stone imprints of animals or microbes, objects preserved ...

. PCMs can generally be divided into two types of approaches: those that infer the evolutionary history of some character (

phenotypic In genetics, the phenotype () is the set of observable characteristics or traits of an organism. The term covers the organism's morphology (physical form and structure), its developmental processes, its biochemical and physiological propert ...

or genetic) across a phylogeny and those that infer the process of evolutionary branching itself ( diversification rates), though there are some approaches that do both simultaneously. Typically the tree that is used in conjunction with PCMs has been estimated independently (see

computational phylogenetics Computational phylogenetics, phylogeny inference, or phylogenetic inference focuses on computational and optimization algorithms, Heuristic (computer science), heuristics, and approaches involved in Phylogenetics, phylogenetic analyses. The goal i ...

) such that both the relationships between lineages and the length of branches separating them is assumed to be known.

Applications

Phylogenetic comparative approaches can complement other ways of studying adaptation, such as studying natural populations, experimental studies, and mathematical models. Interspecific comparisons allow researchers to assess the generality of evolutionary phenomena by considering independent evolutionary events. Such an approach is particularly useful when there is little or no variation within species. And because they can be used to explicitly model evolutionary processes occurring over very long time periods, they can provide insight into macroevolutionary questions, once the exclusive domain of

paleontology Paleontology, also spelled as palaeontology or palæontology, is the scientific study of the life of the past, mainly but not exclusively through the study of fossils. Paleontologists use fossils as a means to classify organisms, measure ge ...

Phylogenetic comparative methods are commonly applied to such questions as: * What is the

slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...

of an

allometric Allometry (Ancient Greek "other", "measurement") is the study of the relationship of body size to shape, anatomy, physiology and behaviour, first outlined by Otto Snell in 1892, by D'Arcy Thompson in 1917 in ''On Growth and Form'' and by Juli ...

scaling relationship? → ''Example: how does brain mass vary in relation to body mass?'' * Do different

clades In biology, a clade (), also known as a monophyletic group or natural group, is a group of organisms that is composed of a common ancestor and all of its descendants. Clades are the fundamental unit of cladistics, a modern approach to taxonomy ...

of organisms differ with respect to some

trait? → ''Example: do canids have larger hearts than felids?'' * Do groups of species that share a

behavioral Behavior (American English) or behaviour (British English) is the range of actions of individuals, organisms, systems or artificial entities in some environment. These systems can include other systems or organisms as well as the inanimate p ...

or ecological feature (e.g.,

social system In sociology, a social system is the patterned network of relationships constituting a coherent whole that exist between individuals, groups, and institutions. It is the formal Social structure, structure of role and status that can form in a smal ...

, diet) differ in average phenotype? → ''Example: do carnivores have larger

home range A home range is the area in which an animal lives and moves on a periodic basis. It is related to the concept of an animal's territory which is the area that is actively defended. The concept of a home range was introduced by W. H. Burt in 1943. ...

s than herbivores?'' * What was the

ancestral An ancestor, also known as a forefather, fore-elder, or a forebear, is a parent or ( recursively) the parent of an antecedent (i.e., a grandparent, great-grandparent, great-great-grandparent and so forth). ''Ancestor'' is "any person from ...

state of a trait? → ''Example: where did

endothermy An endotherm (from Greek ἔνδον ''endon'' "within" and θέρμη ''thermē'' "heat") is an organism that maintains its body at a metabolically favorable temperature, largely by the use of heat released by its internal bodily functions inste ...

evolve in the lineage that led to mammals?'' → ''Example: where, when, and why did

placenta The placenta (: placentas or placentae) is a temporary embryonic and later fetal organ that begins developing from the blastocyst shortly after implantation. It plays critical roles in facilitating nutrient, gas, and waste exchange between ...

s and

viviparity In animals, viviparity is development of the embryo inside the body of the mother, with the maternal circulation providing for the metabolic needs of the embryo's development, until the mother gives birth to a fully or partially developed juv ...

evolve?'' * Does a trait exhibit significant phylogenetic signal in a particular group of organisms? Do certain types of traits tend to "follow phylogeny" more than others? → ''Example: are behavioral traits more labile during evolution?'' * Do species differences in life history traits trade-off, as in the so-called fast-slow continuum? → ''Example: why do small-bodied species have shorter life spans than their larger relatives?''

Phylogenetically independent contrasts

Felsenstein proposed the first general statistical method in 1985 for incorporating phylogenetic information, i.e., the first that could use any arbitrary

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

(branching order) and a specified set of branch lengths. The method is now recognized as an

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

that implements a special case of what are termed phylogenetic generalized least-squares models. The

logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

of the method is to use phylogenetic information (and an assumed

Brownian motion Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...

like model of trait evolution) to transform the original tip data (mean values for a set of species) into values that are

statistically independent Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two event (probability theory), events are independent, statistically independent, or stochastically independent if, informally s ...

and identically distributed. The algorithm involves computing values at

internal node In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected Node (computer science), nodes. Each node in the tree can be connected to many children (depending on the type of ...

s as an intermediate step, but they are generally not used for inferences by themselves. An exception occurs for the basal (root) node, which can be interpreted as an estimate of the ancestral value for the entire tree (assuming that no directional

evolutionary trends Evolution is the change in the heritable Phenotypic trait, characteristics of biological populations over successive generations. It occurs when evolutionary processes such as natural selection and genetic drift act on genetic variation, re ...

.g., Cope's rule">Cope's_rule.html" ;"title=".g., Cope's rule">.g., Cope's rulehave occurred) or as a phylogenetically weighted estimate of the mean for the entire set of tip species (terminal taxa). The value at the root is equivalent to that obtained from the "squared-change parsimony" algorithm and is also the maximum likelihood estimate under Brownian motion. The independent contrasts algebra can also be used to compute a standard error or confidence interval.

Phylogenetic generalized least squares (PGLS)

Probably the most commonly used PCM is phylogenetic generalized least squares (PGLS). This approach is used to test whether there is a relationship between two (or more) variables while accounting for the fact that lineage are not independent. The method is a special case of

generalized least squares In statistics, generalized least squares (GLS) is a method used to estimate the unknown parameters in a Linear regression, linear regression model. It is used when there is a non-zero amount of correlation between the Residual (statistics), resi ...

(GLS) and as such the PGLS estimator is also

unbiased Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is inaccurate, closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individ ...

consistent In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...

, efficient, and asymptotically normal. In many statistical situations where GLS (or,

ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression In statistics, linear regression is a statistical model, model that estimates the relationship ...

LS is used residual errors ''ε'' are assumed to be independent and identically distributed random variables that are assumed to be normal :

\varepsilon \mid X\sim \mathcal(0, \sigma^2I_n).

whereas in PGLS the errors are assumed to be distributed as :

\varepsilon \mid X\sim \mathcal(0, \mathbf).

where ''V'' is a matrix of expected variance and covariance of the residuals given an evolutionary model and a phylogenetic tree. Therefore, it is the structure of residuals and not the variables themselves that show

phylogenetic signal In biology, phylogenetics () is the study of the evolutionary history of life using observable characteristics of organisms (or genes), which is known as phylogenetic inference. It infers the relationship among organisms based on empirical data ...

. This has long been a source of confusion in the scientific literature. A number of models have been proposed for the structure of ''V'' such as

Ornstein-Uhlenbeck, and Pagel's λ model. (When a Brownian motion model is used, PGLS is identical to the independent contrasts estimator.). In PGLS, the parameters of the evolutionary model are typically co-estimated with the regression parameters. PGLS can only be applied to questions where the

dependent variable A variable is considered dependent if it depends on (or is hypothesized to depend on) an independent variable. Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical functio ...

is continuously distributed; however, the phylogenetic tree can also be incorporated into the residual distribution of

generalized linear models In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a ''link function'' and by ...

, making it possible to generalize the approach to a broader set of distributions for the response.

Phylogenetically informed Monte Carlo computer simulations

Martins and Garland proposed in 1991 that one way to account for phylogenetic relations when conducting statistical analyses was to use computer simulations to create many data sets that are consistent with the null hypothesis under test (e.g., no correlation between two traits, no difference between two ecologically defined groups of species) but that mimic evolution along the relevant phylogenetic tree. If such data sets (typically 1,000 or more) are analyzed with the same statistical procedure that is used to analyze a real data set, then results for the simulated data sets can be used to create phylogenetically correct (or "PC") null distributions of the test statistic (e.g., a correlation coefficient, t, F). Such simulation approaches can also be combined with such methods as phylogenetically independent contrasts or PGLS (see above). Phylogenetic Pseudoreplication