In biology, a substitution model , also called models of sequence evolution , are Markov models that describe changes over evolutionary time. These models describe evolutionary changes in macromolecules, such as DNA sequences or protein sequences , that can be represented as sequence of symbols (e.g., A, C, G, and T in the case of DNA or the 20 "standard" proteinogenic amino acids in the case of proteins ). Substitution models are used to calculate the likelihood of phylogenetic trees using multiple sequence alignment data. Thus, substitution models are central to maximum likelihood estimation of phylogeny as well as Bayesian inference in phylogeny . Estimates of evolutionary distances (numbers of substitutions that have occurred since a pair of sequences diverged from a common ancestor) are typically calculated using substitution models (evolutionary distances are used input for distance methods such as neighbor joining ). Substitution models are also central to phylogenetic invariants because they are necessary to predict site pattern frequencies given a tree topology. Substitution models are also necessary to simulate sequence data for a group of organisms related by a specific tree.
106-471: K80 or K-80 may refer to: K-80 (Kansas highway) , a state highway in Kansas See also [ edit ] Substitution model [REDACTED] Topics referred to by the same term This disambiguation page lists articles associated with the same title formed as a letter–number combination. If an internal link led you here, you may wish to change
212-607: A {\displaystyle a} through e {\displaystyle e} , which are expressed relative to the fixed f = r G T = 1 {\displaystyle f=r_{GT}=1} in this example) and three equilibrium base frequency parameters (as described above, only three π i {\displaystyle \pi _{i}} values need to be specified because π → {\displaystyle {\vec {\pi }}} must sum to 1). The alternative notation also makes it easier to understand
318-1276: A π C b π G c π T a π A − ( a π A + d π G + e π T ) d π G e π T b π A d π C − ( b π A + d π C + f π T ) f π T c π A e π C f π G − ( c π A + e π C + f π G ) ) {\displaystyle Q={\begin{pmatrix}{-(a\pi _{C}+b\pi _{G}+c\pi _{T})}&a\pi _{C}&b\pi _{G}&c\pi _{T}\\a\pi _{A}&{-(a\pi _{A}+d\pi _{G}+e\pi _{T})}&d\pi _{G}&e\pi _{T}\\b\pi _{A}&d\pi _{C}&{-(b\pi _{A}+d\pi _{C}+f\pi _{T})}&f\pi _{T}\\c\pi _{A}&e\pi _{C}&f\pi _{G}&{-(c\pi _{A}+e\pi _{C}+f\pi _{G})}\end{pmatrix}}} The Q {\displaystyle Q} matrix
424-400: A Student's t distribution with n − 1 degrees of freedom when the hypothesized mean μ 0 {\displaystyle \mu _{0}} is correct. Again, the degrees-of-freedom arises from the residual vector in the denominator. When the results of structural equation models (SEM) are presented, they generally include one or more indices of overall model fit,
530-501: A and b in the model where x i is given, but e i and hence Y i are random. Let a ^ {\displaystyle {\widehat {a}}} and b ^ {\displaystyle {\widehat {b}}} be the least-squares estimates of a and b . Then the residuals are constrained to lie within the space defined by the two equations One says that there are n − 2 degrees of freedom for error. Notationally,
636-409: A binary alphabet to score the following phenotypic traits "has feathers", "lays eggs", "has fur", "is warm-blooded", and "capable of powered flight". In this toy example hummingbirds would have sequence 11011 (most other birds would have the same string), ostriches would have the sequence 11010, cattle (and most other land mammals ) would have 00110, and bats would have 00111. The likelihood of
742-468: A fossil record may make it possible to determine the number of years between an ancestral species and a descendant species. Because some species evolve at faster rates than others, these two measures of branch length are not always in direct proportion. The expected number of substitutions per site per year is often indicated with the Greek letter mu (μ). A model is said to have a strict molecular clock if
848-421: A four nucleotide alphabet (A, C, G, and U). However, substitution models can be used for alphabets of any size; the alphabet is the 20 proteinogenic amino acids for proteins and the sense codons (i.e., the 61 codons that encode amino acids in the standard genetic code ) for aligned protein-coding gene sequences. In fact, substitution models can be developed for any biological characters that can be encoded using
954-471: A function of a number of parameters which are estimated for every data set analyzed, preferably using maximum likelihood . This has the advantage that the model can be adjusted to the particularities of a specific data set (e.g. different composition biases in DNA). Problems can arise when too many parameters are used, particularly if they can compensate for each other (this can lead to non-identifiability ). Then it
1060-484: A given position, conditional on there being a base i in that position at time 0. When the model is time reversible, this can be performed between any two sequences, even if one is not the ancestor of the other, if you know the total branch length between them. The asymptotic properties of P ij (t) are such that P ij (0) = δ ij , where δ ij is the Kronecker delta function. That is, there
1166-421: A model is time-reversible, which species was the ancestral species is irrelevant. Instead, the phylogenetic tree can be rooted using any of the species, re-rooted later based on new knowledge, or left unrooted. This is because there is no 'special' species, all species will eventually derive from one another with the same probability. A model is time reversible if and only if it satisfies the property (the notation
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#17327907770001272-453: A phylogenetic tree can then be calculated using those binary sequences and an appropriate substitution model. The existence of these morphological models make it possible to analyze data matrices with fossil taxa, either using the morphological data alone or a combination of morphological and molecular data (with the latter scored as missing data for the fossil taxa). There is an obvious similarity between use of molecular or phenotypic data in
1378-437: A phylogenetic tree is expressed as the expected number of substitutions per site; if the evolutionary model indicates that each site within an ancestral sequence will typically experience x substitutions by the time it evolves to a particular descendant's sequence then the ancestor and descendant are considered to be separated by branch length x . Sometimes a branch length is measured in terms of geological years. For example,
1484-439: A rate matrix, Q , which describes the rate at which bases of one type change into bases of another type; element Q i j {\displaystyle Q_{ij}} for i ≠ j is the rate at which base i goes to base j . The diagonals of the Q matrix are chosen so that the rows sum to zero: The equilibrium row vector π must be annihilated by the rate matrix Q : The transition matrix function
1590-526: A set of exchangeability parameters ( r i j {\displaystyle r_{ij}} ) for any alphabet of k {\displaystyle k} character states. These values can then be used to populate the Q {\displaystyle Q} matrix by setting the off-diagonal elements as shown above (the general notation would be Q i j = r i j π j {\displaystyle Q_{ij}=r_{ij}\pi _{j}} ), setting
1696-458: A specific alphabet (e.g., amino acid sequences combined with information about the conformation of those amino acids in three-dimensional protein structures ). The majority of substitution models used for evolutionary research assume independence among sites (i.e., the probability of observing any specific site pattern is identical regardless of where the site pattern is in the sequence alignment). This simplifies likelihood calculations because it
1802-462: A specific multinomial distribution for site pattern frequencies. If we consider a multiple sequence alignment of four DNA sequences there are 256 possible site patterns so there are 255 degrees of freedom for the site pattern frequencies. However, it is possible to specify the expected site pattern frequencies using five degrees of freedom if using the Jukes-Cantor model of DNA evolution, which
1908-443: A sum-of-squares is the degrees-of-freedom of the corresponding component vectors. The three-population example above is an example of one-way Analysis of Variance . The model, or treatment, sum-of-squares is the squared length of the second vector, with 2 degrees of freedom. The residual, or error, sum-of-squares is with 3( n −1) degrees of freedom. Of course, introductory books on ANOVA usually state formulae without showing
2014-411: Is a function from the branch lengths (in some units of time, possibly in substitutions), to a matrix of conditional probabilities. It is denoted P ( t ) {\displaystyle P(t)} . The entry in the i column and the j row, P i j ( t ) {\displaystyle P_{ij}(t)} , is the probability, after time t , that there is a base j at
2120-429: Is a simple substitution model that allows one to calculate the expected site pattern frequencies only the tree topology and the branch lengths (given four taxa an unrooted bifurcating tree has five branch lengths). Substitution models also make it possible to simulate sequence data using Monte Carlo methods . Simulated multiple sequence alignments can be used to assess the performance of phylogenetic methods and generate
2226-614: Is beneficial because one can use reduced alphabets for amino acids. For example, one can use k = 6 {\displaystyle k=6} and encode amino acids by recoding the amino acids using the six categories proposed by Margaret Dayhoff . Reduced amino acid alphabets are viewed as a way to reduce the impact of compositional variation and saturation. Importantly, evolutionary patterns can vary among genomic regions and thus different genomic regions can fit with different substitution models. Actually, ignoring heterogeneous evolutionary patterns along sequences can lead to biases in
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#17327907770002332-498: Is enough data available to create empirical models with any number of parameters, including empirical codon models. Because of the problems mentioned above, the two approaches are often combined, by estimating most of the parameters once on large-scale data, while a few remaining parameters are then adjusted to the data set under consideration. The following sections give an overview of the different approaches taken for DNA, protein or codon-based models. The first models of DNA evolution
2438-456: Is explained below) or, equivalently, the detailed balance property, for every i , j , and t . Time-reversibility should not be confused with stationarity . A model is stationary if Q does not change with time. The analysis below assumes a stationary model. Stationary, neutral, independent, finite sites models (assuming a constant rate of evolution) have two parameters, π , an equilibrium vector of base (or character) frequencies and
2544-409: Is generally not useful for these procedures. However, these procedures are still linear in the observations, and the fitted values of the regression can be expressed in the form where y ^ {\displaystyle {\hat {y}}} is the vector of fitted values at each of the original covariate values from the fitted model, y is the original vector of responses, and H
2650-409: Is likely necessary to adjust the model to these circumstances. Degrees of freedom (statistics) In statistics , the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into
2756-404: Is no change in base composition between a sequence and itself. At the other extreme, lim t → ∞ P i j ( t ) = π j , {\displaystyle \lim _{t\rightarrow \infty }P_{ij}(t)=\pi _{j}\,,} or, in other words, as time goes to infinity the probability of finding base j at a position given there
2862-510: Is normalized so − ∑ i = 1 4 π i Q i i = 1 {\displaystyle -\sum _{i=1}^{4}\pi _{i}Q_{ii}=1} . This notation is easier to understand than the notation originally used by Tavaré , because all model parameters correspond either to "exchangeability" parameters ( a {\displaystyle a} through f {\displaystyle f} , which can also be written using
2968-409: Is not possible to estimate all entries of the substitution matrix from the current data set only. On the downside, the parameters estimated from the training data might be too generic and therefore have a poor fit to any particular dataset. A potential solution for that problem is to estimate some parameters from the data using maximum likelihood (or some other method). In studies of protein evolution
3074-516: Is often the case that the data set is too small to yield enough information to estimate all parameters accurately. Empirical models are created by estimating many parameters (typically all entries of the rate matrix as well as the character frequencies, see the GTR model above) from a large data set. These parameters are then fixed and will be reused for every data set. This has the advantage that those parameters can be estimated more accurately. Normally, it
3180-482: Is often unrealistic, especially across long periods of evolution. For example, even though rodents are genetically very similar to primates , they have undergone a much higher number of substitutions in the estimated time since divergence in some regions of the genome . This could be due to their shorter generation time , higher metabolic rate , increased population structuring, increased rate of speciation , or smaller body size . When studying ancient events like
3286-415: Is only necessary to calculate the probability of all site patterns that appear in the alignment then use those values to calculate the overall likelihood of the alignment (e.g., the probability of three "GGGG" site patterns given some model of DNA sequence evolution is simply the probability of a single "GGGG" site pattern raised to the third power). This means that substitution models can be viewed as implying
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3392-616: Is presented here; the geometry of linear models is discussed in more complete detail by Christensen (2002). Suppose independent observations are made for three populations, X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} , Y 1 , … , Y n {\displaystyle Y_{1},\ldots ,Y_{n}} and Z 1 , … , Z n {\displaystyle Z_{1},\ldots ,Z_{n}} . The restriction to three groups and equal sample sizes simplifies notation, but
3498-481: Is the hat matrix or, more generally, smoother matrix. For statistical inference, sums-of-squares can still be formed: the model sum-of-squares is ‖ H y ‖ 2 {\displaystyle \|Hy\|^{2}} ; the residual sum-of-squares is ‖ y − H y ‖ 2 {\displaystyle \|y-Hy\|^{2}} . However, because H does not correspond to an ordinary least-squares fit (i.e.
3604-485: Is the exchangeability of nucleotides i {\displaystyle i} and j {\displaystyle j} and π j {\displaystyle \pi _{j}} is the equilibrium frequency of the j t h {\displaystyle j^{th}} nucleotide. The matrix shown above uses the letters a {\displaystyle a} through f {\displaystyle f} for
3710-463: Is the matrix Q multiplied by itself enough times to give its n power. If Q is diagonalizable , the matrix exponential can be computed directly: let Q = U Λ U be a diagonalization of Q , with where Λ is a diagonal matrix and where { λ i } {\displaystyle \lbrace \lambda _{i}\rbrace } are the eigenvalues of Q , each repeated according to its multiplicity. Then where
3816-791: Is the mean of all 3 n observations. In vector notation this decomposition can be written as The observation vector, on the left-hand side, has 3 n degrees of freedom. On the right-hand side, the first vector has one degree of freedom (or dimension) for the overall mean. The second vector depends on three random variables, X ¯ − M ¯ {\displaystyle {\bar {X}}-{\bar {M}}} , Y ¯ − M ¯ {\displaystyle {\bar {Y}}-{\bar {M}}} and Z ¯ − M ¯ {\displaystyle {\overline {Z}}-{\overline {M}}} . However, these must sum to 0 and so are constrained;
3922-631: Is the ratio, after scaling by the degrees of freedom. If there is no difference between population means this ratio follows an F -distribution with 2 and 3 n − 3 degrees of freedom. In some complicated settings, such as unbalanced split-plot designs, the sums-of-squares no longer have scaled chi-squared distributions. Comparison of sum-of-squares with degrees-of-freedom is no longer meaningful, and software may report certain fractional 'degrees of freedom' in these cases. Such numbers have no genuine degrees-of-freedom interpretation, but are simply providing an approximate chi-squared distribution for
4028-436: Is to reduce the number of codons by forbidding the stop (or nonsense ) codons. This is a biologically reasonable assumption because including the stop codons would mean that one is calculating the probability of finding sense codon j {\displaystyle j} after time t {\displaystyle t} given that the ancestral codon is i {\displaystyle i} would involve
4134-453: Is typically set to a value of 1 to increase the readability of the exchangeability parameter estimates (since it allows users to express those values relative to chosen exchangeability parameter). The practice of expressing the exchangeability parameters in relative terms is not problematic because the Q {\displaystyle Q} matrix is normalized. Normalization allows t {\displaystyle t} (time) in
4240-470: The Cambrian explosion under a molecular clock assumption, poor concurrence between cladistic and phylogenetic data is often observed. There has been some work on models allowing variable rate of evolution. Models that can take into account variability of the rate of the molecular clock between different evolutionary lineages in the phylogeny are called “relaxed” in opposition to “strict”. In such models
4346-482: The null distribution for certain statistical tests in the fields of molecular evolution and molecular phylogenetics. Examples of these tests include tests of model fit and the "SOWH test" that can be used to examine tree topologies. The fact that substitution models can be used to analyze any biological alphabet has made it possible to develop models of evolution for phenotypic datasets (e.g., morphological and behavioural traits). Typically, "0" is. used to indicate
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4452-701: The sample mean . The random vector can be decomposed as the sum of the sample mean plus a vector of residuals: The first vector on the right-hand side is constrained to be a multiple of the vector of 1's, and the only free quantity is X ¯ {\displaystyle {\bar {X}}} . It therefore has 1 degree of freedom. The second vector is constrained by the relation ∑ i = 1 n ( X i − X ¯ ) = 0 {\textstyle \sum _{i=1}^{n}(X_{i}-{\bar {X}})=0} . The first n − 1 components of this vector can be anything. However, once you know
4558-415: The transition rate, one for the rate of transversions that conserve the strong/weak properties of nucleotides ( A ↔ T {\displaystyle A\leftrightarrow T} and C ↔ G {\displaystyle C\leftrightarrow G} , designated β {\displaystyle \beta } by Kimura ), and one for rate of transversions that conserve
4664-534: The ( n − 1)-dimensional orthogonal complement of this subspace, and has n − 1 degrees of freedom. In statistical testing applications, often one is not directly interested in the component vectors, but rather in their squared lengths. In the example above, the residual sum-of-squares is If the data points X i {\displaystyle X_{i}} are normally distributed with mean 0 and variance σ 2 {\displaystyle \sigma ^{2}} , then
4770-425: The 4 frequency parameters must sum to 1, there are only 3 free frequency parameters. The total of 9 free parameters is often further reduced to 8 parameters plus μ {\displaystyle \mu } , the overall number of substitutions per unit time. When measuring time in substitutions ( μ {\displaystyle \mu } =1) only 8 free parameters remain. In general, to compute
4876-411: The GTR model can be applied to biological alphabets with a larger state-space (e.g., amino acids or codons ). It is possible to write a set of equilibrium state frequencies as π 1 {\displaystyle \pi _{1}} , π 2 {\displaystyle \pi _{2}} , ... π k {\displaystyle \pi _{k}} and
4982-693: The SYM model and the full GTR (or REV ) model (where all exchangeability parameters are free). The equilibrium base frequencies are typically treated in two different ways: 1) all π i {\displaystyle \pi _{i}} values are constrained to be equal (i.e., π A = π C = π G = π T = 0.25 {\displaystyle \pi _{A}=\pi _{C}=\pi _{G}=\pi _{T}=0.25} ); or 2) all π i {\displaystyle \pi _{i}} values are treated as free parameters. Although
5088-414: The absence of a trait and "1" is used to indicate the presence of a trait, although it is also possible to score characters using multiple states. Using this framework, we might encode a set of phenotypes as binary strings (this could be generalized to k -state strings for characters with more than two states) before analyses using an appropriate mode. This can be illustrated using a "toy" example: we can use
5194-407: The amino/keto properties of nucleotides ( A ↔ C {\displaystyle A\leftrightarrow C} and G ↔ T {\displaystyle G\leftrightarrow T} , designated γ {\displaystyle \gamma } by Kimura ). In 1981, Joseph Felsenstein proposed a four-parameter model (F81 ) in which the substitution rate corresponds to
5300-414: The analysis, sometimes called knowns, and the number of parameters that are uniquely estimated, sometimes called unknowns. For example, in a one-factor confirmatory factor analysis with 4 items, there are 10 knowns (the six unique covariances among the four items and the four item variances) and 8 unknowns (4 factor loadings and 4 error variances) for 2 degrees of freedom. Degrees of freedom are important to
5406-423: The capital letter Y is used in specifying the model, while lower-case y in the definition of the residuals; that is because the former are hypothesized random variables and the latter are actual data. We can generalise this to multiple regression involving p parameters and covariates (e.g. p − 1 predictors and one mean (=intercept in the regression)), in which case the cost in degrees of freedom of
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#17327907770005512-454: The corresponding sum-of-squares. The details of such approximations are beyond the scope of this page. Several commonly encountered statistical distributions ( Student's t , chi-squared , F ) have parameters that are commonly referred to as degrees of freedom . This terminology simply reflects that in many applications where these distributions occur, the parameter corresponds to the degrees of freedom of an underlying random vector, as in
5618-492: The course of developing what became known as Student's t-distribution . The term itself was popularized by English statistician and biologist Ronald Fisher , beginning with his 1922 work on chi squares. In equations, the typical symbol for degrees of freedom is ν (lowercase Greek letter nu ). In text and tables, the abbreviation "d.f." is commonly used. R. A. Fisher used n to symbolize degrees of freedom but modern usage typically reserves n for sample size. When reporting
5724-422: The data while keeping the exchangeability matrix fixed. Beyond the common practice of estimating amino acid frequencies from the data, methods to estimate exchangeability parameters or adjust the Q {\displaystyle Q} matrix for protein evolution in other ways have been proposed. With the large-scale genome sequencing still producing very large amounts of DNA and protein sequences, there
5830-404: The data. It is also necessary because the patterns of DNA sequence evolution often differ among organisms and among genes within organisms. The later may reflect optimization by the action of selection for specific purposes (e.g. fast expression or messenger RNA stability) or it might reflect neutral variation in the patterns of substitution. Thus, depending on the organism and the type of gene, it
5936-497: The degrees of freedom arises from the residual sum-of-squares in the numerator, and in turn the n − 1 degrees of freedom of the underlying residual vector { X i − X ¯ } {\displaystyle \{X_{i}-{\bar {X}}\}} . In the application of these distributions to linear models, the degrees of freedom parameters can take only integer values. The underlying families of distributions allow fractional values for
6042-456: The degrees of freedom is equal to the number of independent scores ( N ) minus the number of parameters estimated as intermediate steps (one, namely, the sample mean) and is therefore equal to N − 1 {\textstyle N-1} . Mathematically, degrees of freedom is the number of dimensions of the domain of a random vector , or essentially the number of "free" components (how many components need to be known before
6148-433: The degrees-of-freedom parameters, which can arise in more sophisticated uses. One set of examples is problems where chi-squared approximations based on effective degrees of freedom are used. In other applications, such as modelling heavy-tailed data, a t or F -distribution may be used as an empirical model. In these cases, there is no particular degrees of freedom interpretation to the distribution parameters, even though
6254-420: The diagonal elements Q i i {\displaystyle Q_{ii}} to the negative sum of the off-diagonal elements on the same row, and normalizing. Obviously, k = 20 {\displaystyle k=20} for amino acids and k = 61 {\displaystyle k=61} for codons (assuming the standard genetic code ). However, the generality of this notation
6360-757: The diagonal matrix e is given by Generalised time reversible (GTR) is the most general neutral, independent, finite-sites, time-reversible model possible. It was first described in a general form by Simon Tavaré in 1986. The GTR model is often called the general time reversible model in publications; it has also been called the REV model. The GTR parameters for nucleotides consist of an equilibrium base frequency vector, π → = ( π 1 , π 2 , π 3 , π 4 ) {\displaystyle {\vec {\pi }}=(\pi _{1},\pi _{2},\pi _{3},\pi _{4})} , giving
6466-412: The encoded amino acid (synonymous substitutions). Most of the work on substitution models has focused on DNA/ RNA and protein sequence evolution. Models of DNA sequence evolution, where the alphabet corresponds to the four nucleotides (A, C, G, and T), are probably the easiest models to understand. DNA models can also be used to examine RNA virus evolution; this reflects the fact that RNA also has
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#17327907770006572-439: The equilibrium amino acid frequencies π → = ( π A , π R , π N , . . . π V ) {\displaystyle {\vec {\pi }}=(\pi _{A},\pi _{R},\pi _{N},...\pi _{V})} (using the one-letter IUPAC codes for amino acids to indicate their equilibrium frequencies) are often estimated from
6678-830: The equilibrium base frequencies can be constrained in other ways most constraints that link some but not all π i {\displaystyle \pi _{i}} values are unrealistic from a biological standpoint. The possible exception is enforcing strand symmetry (i.e., constraining π A = π T {\displaystyle \pi _{A}=\pi _{T}} and π C = π G {\displaystyle \pi _{C}=\pi _{G}} but allowing π A + π T ≠ π C + π G {\displaystyle \pi _{A}+\pi _{T}\neq \pi _{C}+\pi _{G}} ). The alternative notation also makes it straightforward to see how
6784-614: The equilibrium frequency of the target nucleotide. Hasegawa, Kishino, and Yano unified the two last models to a five-parameter model (HKY ). After these pioneering efforts, many additional sub-models of the GTR model were introduced into the literature (and common use) in the 1990s. Other models that move beyond the GTR model in specific ways were also developed and refined by several researchers. Almost all DNA substitution models are mechanistic models (as described above). The small number of parameters that one needs to estimate for these models makes it feasible to estimate those parameters from
6890-444: The estimate of a parameter is called the degrees of freedom. In general, the degrees of freedom of an estimate of a parameter are equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself. For example, if the variance is to be estimated from a random sample of N {\textstyle N} independent scores, then
6996-465: The estimation of evolutionary parameters, including the K a /K s ratio . In this regard, the use of mixture models in phylogenentic frameworks is convenient to better mimic the molecular evolution observed in real data. A main difference in evolutionary models is how many parameters are estimated every time for the data set under consideration and how many of them are estimated once on a large data set. Mechanistic models describe all substitutions as
7102-457: The evolutionary distance between those sequences is t {\displaystyle t} whereas p C A ( t ) {\displaystyle p_{\mathrm {CA} }(t)} is the probability of observing C in sequence 1 and A in sequence 2 at the same evolutionary distance). An arbitrarily chosen exchangeability parameters (e.g., f = r G T {\displaystyle f=r_{GT}} )
7208-420: The exchangeability parameters in the interest of readability, but those parameters could also be to written in a systematic manner using the r i j {\displaystyle r_{ij}} notation (e.g., a = r A C {\displaystyle a=r_{AC}} , b = r A G {\displaystyle b=r_{AG}} , and so forth). Note that
7314-400: The expected number of substitutions per year μ is constant regardless of which species' evolution is being examined. An important implication of a strict molecular clock is that the number of expected substitutions between an ancestral species and any of its present-day descendants must be independent of which descendant species is examined. Note that the assumption of a strict molecular clock
7420-425: The field of cladistics and analyses of morphological characters using a substitution model. However, there has been a vociferous debate in the systematics community regarding the question of whether or not cladistic analyses should be viewed as "model-free". The field of cladistics (defined in the strictest sense) favor the use of the maximum parsimony criterion for phylogenetic inference. Many cladists reject
7526-420: The first n − 1 components, the constraint tells you the value of the n th component. Therefore, this vector has n − 1 degrees of freedom. Mathematically, the first vector is the oblique projection of the data vector onto the subspace spanned by the vector of 1's. The 1 degree of freedom is the dimension of this subspace. The second residual vector is the least-squares projection onto
7632-422: The fit is p , leaving n - p degrees of freedom for errors The demonstration of the t and chi-squared distributions for one-sample problems above is the simplest example where degrees-of-freedom arise. However, similar geometry and vector decompositions underlie much of the theory of linear models , including linear regression and analysis of variance . An explicit example based on comparison of three means
7738-444: The frequency at which each base occurs at each site, and the rate matrix Because the model must be time reversible and must approach the equilibrium nucleotide (base) frequencies at long times, each rate below the diagonal equals the reciprocal rate above the diagonal multiplied by the equilibrium ratio of the two bases. As such, the nucleotide GTR requires 6 substitution rate parameters and 4 equilibrium base frequency parameters. Since
7844-648: The genome, it is more common to work with a codon substitution model (a codon is three bases and codes for one amino acid in a protein). There are 4 3 = 64 {\displaystyle 4^{3}=64} codons, resulting in 2078 free parameters. However, the rates for transitions between codons which differ by more than one base are often assumed to be zero, reducing the number of free parameters to only 20 × 19 × 3 2 + 63 − 1 = 632 {\displaystyle {{20\times 19\times 3} \over 2}+63-1=632} parameters. Another common practice
7950-535: The ideas are easily generalized. The observations can be decomposed as where X ¯ , Y ¯ , Z ¯ {\displaystyle {\bar {X}},{\bar {Y}},{\bar {Z}}} are the means of the individual samples, and M ¯ = ( X ¯ + Y ¯ + Z ¯ ) / 3 {\displaystyle {\bar {M}}=({\bar {X}}+{\bar {Y}}+{\bar {Z}})/3}
8056-428: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=K80&oldid=1082554200 " Category : Letter–number combination disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Substitution model Phylogenetic tree topologies are often
8162-413: The mathematics, the model does not care which sequence is the ancestor and which is the descendant so long as all other parameters (such as the number of substitutions per site that is expected between the two sequences) are held constant. When an analysis of real biological data is performed, there is generally no access to the sequences of ancestral species, only to the present-day species. However, when
8268-489: The matrix exponentiation P ( t ) = e Q t {\displaystyle P(t)=e^{Qt}} to be expressed in units of expected substitutions per site (standard practice in molecular phylogenetics). This is the equivalent to the statement that one is setting the mutation rate μ {\displaystyle \mu } to 1) and reducing the number of free parameters to eight. Specifically, there are five free exchangeability parameters (
8374-413: The models described in those papers, leaving the reader to wonder which models were actually tested. A common way to think of degrees of freedom is as the number of independent pieces of information available to estimate another piece of information. More concretely, the number of degrees of freedom is the number of independent observations in a sample of data that are available to estimate a parameter of
8480-494: The most common of which is a χ statistic. This forms the basis for other indices that are commonly reported. Although it is these other statistics that are most commonly interpreted, the degrees of freedom of the χ are essential to understanding model fit as well as the nature of the model itself. Degrees of freedom in SEM are computed as a difference between the number of unique pieces of information that are used as input into
8586-402: The notation r i j {\displaystyle r_{ij}} ) or to equilibrium nucleotide frequencies π → = ( π A , π C , π G , π T ) {\displaystyle {\vec {\pi }}=(\pi _{A},\pi _{C},\pi _{G},\pi _{T})} . Note that
8692-726: The nucleotides in a different order (e.g., some authors choose to group two purines together and the two pyrimidines together; see also models of DNA evolution ). These differences in notation make it important to be clear regarding the order of the states when writing the Q {\displaystyle Q} matrix. The value of this notation is that instantaneous rate of change from nucleotide i {\displaystyle i} to nucleotide j {\displaystyle j} can always be written as r i j π j {\displaystyle r_{ij}\pi _{j}} , where r i j {\displaystyle r_{ij}}
8798-1570: The nucleotides in the Q {\displaystyle Q} matrix have been written in alphabetical order. In other words, the transition probability matrix for the Q {\displaystyle Q} matrix above would be: P ( t ) = e Q t = ( p A A ( t ) p A C ( t ) p A G ( t ) p A T ( t ) p C A ( t ) p C C ( t ) p C G ( t ) p C T ( t ) p G A ( t ) p G C ( t ) p G G ( t ) p G T ( t ) p T A ( t ) p T C ( t ) p T G ( t ) p T T ( t ) ) {\displaystyle P(t)=e^{Qt}={\begin{pmatrix}p_{\mathrm {AA} }(t)&p_{\mathrm {AC} }(t)&p_{\mathrm {AG} }(t)&p_{\mathrm {AT} }(t)\\p_{\mathrm {CA} }(t)&p_{\mathrm {CC} }(t)&p_{\mathrm {CG} }(t)&p_{\mathrm {CT} }(t)\\p_{\mathrm {GA} }(t)&p_{\mathrm {GC} }(t)&p_{\mathrm {GG} }(t)&p_{\mathrm {GT} }(t)\\p_{\mathrm {TA} }(t)&p_{\mathrm {TC} }(t)&p_{\mathrm {TG} }(t)&p_{\mathrm {TT} }(t)\end{pmatrix}}} Some publications write
8904-419: The number of components in the vector. That smaller dimension is the number of degrees of freedom for error , also called residual degrees of freedom . Perhaps the simplest example is this. Suppose are random variables each with expected value μ , and let be the "sample mean." Then the quantities are residuals that may be considered estimates of the errors X i − μ . The sum of
9010-576: The number of parameters, you count the number of entries above the diagonal in the matrix, i.e. for n trait values per site n 2 − n 2 {\displaystyle {{n^{2}-n} \over 2}} , and then add n-1 for the equilibrium frequencies, and subtract 1 because μ {\displaystyle \mu } is fixed. You get For example, for an amino acid sequence (there are 20 "standard" amino acids that make up proteins ), you would find there are 208 parameters. However, when studying coding regions of
9116-435: The ordering of the nucleotide subscripts for exchangeability parameters is irrelevant (e.g., r A C = r C A {\displaystyle r_{AC}=r_{CA}} ) but the transition probability matrix values are not (i.e., p A C ( t ) {\displaystyle p_{\mathrm {AC} }(t)} is the probability of observing A in sequence 1 and C in sequence 2 when
9222-486: The parameter of interest; thus, branch lengths and any other parameters describing the substitution process are often viewed as nuisance parameters . However, biologists are sometimes interested in the other aspects of the model. For example, branch lengths, especially when those branch lengths are combined with information from the fossil record and a model to estimate the timeframe for evolution. Other model parameters have been used to gain insights into various aspects of
9328-404: The parameters of chi-squared and other distributions that arise in associated statistical testing problems. While introductory textbooks may introduce degrees of freedom as distribution parameters or through hypothesis testing, it is the underlying geometry that defines degrees of freedom, and is critical to a proper understanding of the concept. Although the basic concept of degrees of freedom
9434-431: The population from which that sample is drawn. For example, if we have two observations, when calculating the mean we have two independent observations; however, when calculating the variance, we have only one independent observation, since the two observations are equally distant from the sample mean. In fitting statistical models to data, the vectors of residuals are constrained to lie in a space of smaller dimension than
9540-402: The position that maximum parsimony is based on a substitution model and (in many cases) they justify the use of parsimony using the philosophy of Karl Popper . However, the existence of "parsimony-equivalent" models (i.e., substitution models that yield the maximum parsimony tree when used for analyses) makes it possible to view parsimony as a substitution model. Typically, a branch length of
9646-408: The possibility of passing through a state with a premature stop codon. An alternative (and commonly used ) way to write the instantaneous rate matrix ( Q {\displaystyle Q} matrix) for the nucleotide GTR model is: Q = ( − ( a π C + b π G + c π T )
9752-436: The preceding ANOVA example. Another simple example is: if X i ; i = 1 , … , n {\displaystyle X_{i};i=1,\ldots ,n} are independent normal ( μ , σ 2 ) {\displaystyle (\mu ,\sigma ^{2})} random variables, the statistic follows a chi-squared distribution with n − 1 degrees of freedom. Here,
9858-407: The process of evolution. The K a /K s ratio (also called ω in codon substitution models) is a parameter of interest in many studies. The K a /K s ratio can be used to examine the action of natural selection on protein-coding regions, it provides information about the relative rates of nucleotide substitutions that change amino acids (non-synonymous substitutions) to those that do not change
9964-493: The rate can be assumed to be correlated or not between ancestors and descendants and rate variation among lineages can be drawn from many distributions but usually exponential and lognormal distributions are applied. There is a special case, called “local molecular clock” when a phylogeny is divided into at least two partitions (sets of lineages) and a strict molecular clock is applied in each, but with different rates. Many useful substitution models are time-reversible ; in terms of
10070-408: The residual sum of squares has a scaled chi-squared distribution (scaled by the factor σ 2 {\displaystyle \sigma ^{2}} ), with n − 1 degrees of freedom. The degrees-of-freedom, here a parameter of the distribution, can still be interpreted as the dimension of an underlying vector subspace. Likewise, the one-sample t -test statistic, follows
10176-408: The residuals (unlike the sum of the errors) is necessarily 0. If one knows the values of any n − 1 of the residuals, one can thus find the last one. That means they are constrained to lie in a space of dimension n − 1. One says that there are n − 1 degrees of freedom for errors. An example which is only slightly less simple is that of least squares estimation of
10282-612: The results of statistical tests , the degrees of freedom are typically noted beside the test statistic as either subscript or in parentheses . Geometrically, the degrees of freedom can be interpreted as the dimension of certain vector subspaces. As a starting point, suppose that we have a sample of independent normally distributed observations, This can be represented as an n -dimensional random vector : Since this random vector can lie anywhere in n -dimensional space, it has n degrees of freedom. Now, let X ¯ {\displaystyle {\bar {X}}} be
10388-522: The sub-models of the GTR model, which simply correspond to cases where exchangeability and/or equilibrium base frequency parameters are constrained to take on equal values. A number of specific sub-models have been named, largely based on their original publications: There are 203 possible ways that the exchangeability parameters can be restricted to form sub-models of GTR, ranging from the JC69 and F81 models (where all exchangeability parameters are equal) to
10494-412: The terminology may continue to be used. Many non-standard regression methods, including regularized least squares (e.g., ridge regression ), linear smoothers , smoothing splines , and semiparametric regression , are not based on ordinary least squares projections, but rather on regularized ( generalized and/or penalized) least-squares, and so degrees of freedom defined in terms of dimensionality
10600-550: The understanding of model fit if for no other reason than that, all else being equal, the fewer degrees of freedom, the better indices such as χ will be. It has been shown that degrees of freedom can be used by readers of papers that contain SEMs to determine if the authors of those papers are in fact reporting the correct model fit statistics. In the organizational sciences, for example, nearly half of papers published in top journals report degrees of freedom that are inconsistent with
10706-430: The vector is fully determined). The term is most often used in the context of linear models ( linear regression , analysis of variance ), where certain random vectors are constrained to lie in linear subspaces , and the number of degrees of freedom is the dimension of the subspace . The degrees of freedom are also commonly associated with the squared lengths (or "sum of squares" of the coordinates) of such vectors, and
10812-446: The vector therefore must lie in a 2-dimensional subspace, and has 2 degrees of freedom. The remaining 3 n − 3 degrees of freedom are in the residual vector (made up of n − 1 degrees of freedom within each of the populations). In statistical testing problems, one usually is not interested in the component vectors themselves, but rather in their squared lengths, or Sum of Squares. The degrees of freedom associated with
10918-432: The vectors, but it is this underlying geometry that gives rise to SS formulae, and shows how to unambiguously determine the degrees of freedom in any given situation. Under the null hypothesis of no difference between population means (and assuming that standard ANOVA regularity assumptions are satisfied) the sums of squares have scaled chi-squared distributions, with the corresponding degrees of freedom. The F-test statistic
11024-400: Was a base i at that position originally goes to the equilibrium probability that there is base j at that position, regardless of the original base. Furthermore, it follows that π P ( t ) = π {\displaystyle \pi P(t)=\pi } for all t . The transition matrix can be computed from the rate matrix via matrix exponentiation : where Q
11130-465: Was proposed Jukes and Cantor in 1969. The Jukes-Cantor (JC or JC69) model assumes equal transition rates as well as equal equilibrium frequencies for all bases and it is the simplest sub-model of the GTR model. In 1980, Motoo Kimura introduced a model with two parameters (K2P or K80 ): one for the transition and one for the transversion rate. A year later, Kimura introduced a second model (K3ST, K3P, or K81 ) with three substitution types: one for
11236-408: Was recognized as early as 1821 in the work of German astronomer and mathematician Carl Friedrich Gauss , its modern definition and usage was first elaborated by English statistician William Sealy Gosset in his 1908 Biometrika article "The Probable Error of a Mean", published under the pen name "Student". While Gosset did not actually use the term 'degrees of freedom', he explained the concept in
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