| Title: | Calculation of Evolvability Parameters |
|---|---|
| Description: | Provides tools for calculating evolvability parameters from estimated G-matrices as defined in Hansen and Houle (2008) <doi:10.1111/j.1420-9101.2008.01573.x> and fits phylogenetic comparative models that link the rate of evolution of a trait to the state of another evolving trait (see Hansen et al. 2021 Systematic Biology <doi:10.1093/sysbio/syab079>). The package was released with Bolstad et al. (2014) <doi:10.1098/rstb.2013.0255>, which contains some examples of use. |
| Authors: | Geir H. Bolstad [aut, cre] |
| Maintainer: | Geir H. Bolstad <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 2.0.0 |
| Built: | 2025-01-22 04:26:41 UTC |
| Source: | https://github.com/ghbolstad/evolvability |
This package calculates evolvability parameters from estimated G-matrices as defined in Hansen and Houle (2008) <doi:10.1111/j.1420-9101.2008.01573.x>. It can use both point estimates and posterior/bootstrap distributions of the G-matrices. This package was released with Bolstad et al. (2014) <doi:10.1098/rstb.2013.0255>, which contains some examples of use.
| Package: | evolvability |
| Type: | Package |
| Version: | 1.1.0 |
| Date: | 2015-04-13 |
| License: | GPL (>= 2) |
Geir H. Bolstad [email protected]
Bolstad G. H., Hansen T. F., Pelabon C. Falahati-Anabaran M., Perez-Barrales R. & Armbruster W. S. 2014. Genetic constraints predict evolutionary divergence in Dalechampia blossoms. Phil. Trans. R. Soc. B. 369:20130255. doi:10.1098/rstb.2013.0255
Hansen T. F. & Houle D. (2008) Measuring and comparing evolvability and constraint in multivariate characters. J. Evol. Biol. 21:1201-1219. doi:10.1111/j.1420-9101.2008.01573.x
Almer fits a univariate linear mixed model incorporating a correlated
random effects structure. Can be used to fit phylogenetic mixed models and
animal models. The function is based on the lme4 package and is
very similar to lmer, apart from the A argument.
Almer( formula, data = NULL, A = list(), REML = TRUE, control = lme4::lmerControl(check.nobs.vs.nlev = "ignore", check.nobs.vs.rankZ = "ignore", check.nobs.vs.nRE = "ignore"), start = NULL, verbose = 0L, weights = NULL, na.action = "na.omit", offset = NULL, contrasts = NULL, devFunOnly = FALSE, ... )Almer( formula, data = NULL, A = list(), REML = TRUE, control = lme4::lmerControl(check.nobs.vs.nlev = "ignore", check.nobs.vs.rankZ = "ignore", check.nobs.vs.nRE = "ignore"), start = NULL, verbose = 0L, weights = NULL, na.action = "na.omit", offset = NULL, contrasts = NULL, devFunOnly = FALSE, ... )
formula |
as in |
data |
as in |
A |
an optional named list of sparse matrices. The names must correspond to the names of the random effects in the formula argument. All levels of the random effect should appear as row and column names for the matrices. |
REML |
as in |
control |
as in |
start |
as in |
verbose |
as in |
weights |
as in |
na.action |
as in |
offset |
as in |
contrasts |
as in |
devFunOnly |
as in |
... |
as in |
Almer an object of class merMod.
Geir H. Bolstad
# See the vignette 'Phylogenetic mixed model'.# See the vignette 'Phylogenetic mixed model'.
Almer model fitAlmer_boot performs a parametric bootstrap from an Almer
model fit
Almer_boot(mod, nsim = 1000)Almer_boot(mod, nsim = 1000)
mod |
A fitted object from |
nsim |
The number of simulations. |
Almer_boot a list with entries fixef, vcov, fixef_distribution
and vcov_distribution, where the two first entries includes the means,
standard deviations, and quantiles of the fixed effects means and
(co)variances, respectively, and the two latter includes the complete
bootstrap distribution.
Geir H. Bolstad
# See the vignette 'Phylogenetic mixed model'.# See the vignette 'Phylogenetic mixed model'.
Almer_SE Linear mixed model for response variables with uncertainty
Almer_SE( formula, SE = NULL, maxiter = 100, control = lme4::lmerControl(check.nobs.vs.nlev = "ignore", check.nobs.vs.rankZ = "ignore", check.nobs.vs.nRE = "ignore"), ... )Almer_SE( formula, SE = NULL, maxiter = 100, control = lme4::lmerControl(check.nobs.vs.nlev = "ignore", check.nobs.vs.rankZ = "ignore", check.nobs.vs.nRE = "ignore"), ... )
formula |
as in |
SE |
A vector of standard errors associated with the response variable. NB! Must have column name "SE" in the data. |
maxiter |
The maximum number of iterations. |
control |
as in |
... |
Further optional arguments, see |
Almer_SE returns an object of class merMod.
Geir H. Bolstad
# See the vignette 'Phylogenetic mixed model'.# See the vignette 'Phylogenetic mixed model'.
Almer fitAlmer_sim Simulate responses from an Almer model fit.
Almer_sim(mod, nsim = 1000)Almer_sim(mod, nsim = 1000)
mod |
A fitted object from |
nsim |
The number of simulations. |
This function is only included as the simulate.merMod
function did not seem to work properly when the number of random effect
levels equal the number of observations.
Almer_sim a matrix of simulated responses, columns correspond
to each simulations.
Geir H. Bolstad
# See the vignette 'Phylogenetic mixed model'.# See the vignette 'Phylogenetic mixed model'.
conditinoalG calculates a conditional variance matrix.
conditionalG(G, condition_on = NULL)conditionalG(G, condition_on = NULL)
G |
A variance matrix (must be symmetric and positive definite). |
condition_on |
Either an integer with the column number indicating which trait to condition on or a vector with several column numbers (integers). |
The function calculates a sub-matrix of G conditional on the
traits defined by the the condition_on vector. The function is based
on equation 3 in Hansen et al. (2003).
A matrix that is a sub-matrix of the input matrix conditional on the non-included traits.
Geir H. Bolstad
Hansen TF, Armbruster WS, Carlsson ML & Pélabon C. 2003. Evolvability and genetic constraint in Dalechampia blossoms: genetic correlations and conditional evolvability. J. Exp. Zool. 296B:23-39.
# Constructing a G-matrix: G <- matrix(c( 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 1, 3 ), ncol = 4) # Computing a conditional 2x2 sub-matrix by conditioning on # trait 3 and 4: G_sub_conditional <- conditionalG(G, condition_on = c(3, 4)) G_sub_conditional # The average evolvabilities of this matrix can then be # compared can than be compared to the average evolvabilities # of the corresponding unconditional sub-matrix of G: evolvabilityMeans(G_sub_conditional) evolvabilityMeans(G[-c(3, 4), -c(3, 4)])# Constructing a G-matrix: G <- matrix(c( 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 1, 1, 3 ), ncol = 4) # Computing a conditional 2x2 sub-matrix by conditioning on # trait 3 and 4: G_sub_conditional <- conditionalG(G, condition_on = c(3, 4)) G_sub_conditional # The average evolvabilities of this matrix can then be # compared can than be compared to the average evolvabilities # of the corresponding unconditional sub-matrix of G: evolvabilityMeans(G_sub_conditional) evolvabilityMeans(G[-c(3, 4), -c(3, 4)])
G needs to be symmetric and positive definite.
evolvabilityBeta(G, Beta, means = 1)evolvabilityBeta(G, Beta, means = 1)
G |
A variance matrix. |
Beta |
Either a vector or a matrix of unit length selection gradients stacked column wise. |
means |
An optional vector of trait means (for internal mean standardization). |
evolvabilityBeta calculates (unconditional) evolvability (e),
respondability (r), conditional evolvability (c), autonomy (a) and
integration (i) along selection gradients given an additive-genetic variance
matrix as described in Hansen and Houle (2008).
An object of class 'evolvabilityBeta', which is a list
with the following components:
Beta |
The matrix of selection gradients. | |||
e |
The evolvability of each selection gradient. | |||
r |
The respondability of each selection gradient. | |||
c |
The conditional evolvability of each selection gradient. | |||
a |
The autonomy of each selection gradient. | |||
i |
The integration of each selection gradient. |
Geir H. Bolstad
Hansen, T. F. & Houle, D. (2008) Measuring and comparing evolvability and constraint in multivariate characters. J. Evol. Biol. 21:1201-1219.
G <- matrix(c(1, 1, 0, 1, 2, 2, 0, 2, 3), ncol = 3) / 10 Beta <- randomBeta(5, 3) X <- evolvabilityBeta(G, Beta) summary(X)G <- matrix(c(1, 1, 0, 1, 2, 2, 0, 2, 3), ncol = 3) / 10 Beta <- randomBeta(5, 3) X <- evolvabilityBeta(G, Beta) summary(X)
evolvabilityBetaMCMC calculates (unconditional) evolvability (e),
respondability (r), conditional evolvability (c), autonomy (a) and
integration (i) from selection gradients given the posterior distribution of
an additive-genetic variance matrix. These measures and their meanings are
described in Hansen and Houle (2008).
evolvabilityBetaMCMC(G_mcmc, Beta, post.dist = FALSE)evolvabilityBetaMCMC(G_mcmc, Beta, post.dist = FALSE)
G_mcmc |
posterior distribution of a variance matrix in the form of a
table. Each row in the table must be one iteration of the posterior
distribution (or bootstrap distribution). Each iteration of the matrix must
be on the form as given by |
Beta |
either a vector or a matrix of unit length selection gradients stacked column wise. |
post.dist |
logical: should the posterior distribution of the evolvability parameters be saved. |
An object of class 'evolvabilityBetaMCMC', which is a
list with the following components:
eB |
The posterior median and highest posterior density interval of evolvability for each selection gradient. | |||
rB |
The posterior median and highest posterior density interval of respondability for each selection gradient. | |||
cB |
The posterior median and highest posterior density interval of conditional evolvability for each selection gradient. | |||
aB |
The posterior median and highest posterior density interval of autonomy for each selection gradient. | |||
iB |
The posterior median and highest posterior density interval of integration for each selection gradient. | |||
Beta |
The matrix of selection gradients. | |||
summary |
The means of evolvability parameters across all selection gradients. | |||
post.dist |
The full posterior distribution. |
Geir H. Bolstad
Hansen, T. F. & Houle, D. (2008) Measuring and comparing evolvability and constraint in multivariate characters. J. Evol. Biol. 21:1201-1219.
# Simulating a posterior distribution # (or bootstrap distribution) of a G-matrix: G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3) G_mcmc <- sapply(c(G), function(x) rnorm(10, x, 0.01)) G_mcmc <- t(apply(G_mcmc, 1, function(x) { G <- matrix(x, ncol = sqrt(length(x))) G[lower.tri(G)] <- t(G)[lower.tri(G)] c(G) })) # Simulating a posterior distribution # (or bootstrap distribution) of trait means: means <- c(1, 1.4, 2.1) means_mcmc <- sapply(means, function(x) rnorm(10, x, 0.01)) # Mean standardizing the G-matrix: G_mcmc <- meanStdGMCMC(G_mcmc, means_mcmc) # Generating selection gradients in five random directions: Beta <- randomBeta(5, 3) # Calculating evolvability parameters: x <- evolvabilityBetaMCMC(G_mcmc, Beta, post.dist = TRUE) summary(x)# Simulating a posterior distribution # (or bootstrap distribution) of a G-matrix: G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3) G_mcmc <- sapply(c(G), function(x) rnorm(10, x, 0.01)) G_mcmc <- t(apply(G_mcmc, 1, function(x) { G <- matrix(x, ncol = sqrt(length(x))) G[lower.tri(G)] <- t(G)[lower.tri(G)] c(G) })) # Simulating a posterior distribution # (or bootstrap distribution) of trait means: means <- c(1, 1.4, 2.1) means_mcmc <- sapply(means, function(x) rnorm(10, x, 0.01)) # Mean standardizing the G-matrix: G_mcmc <- meanStdGMCMC(G_mcmc, means_mcmc) # Generating selection gradients in five random directions: Beta <- randomBeta(5, 3) # Calculating evolvability parameters: x <- evolvabilityBetaMCMC(G_mcmc, Beta, post.dist = TRUE) summary(x)
evolvabilityBetaMCMC2 calculates (unconditional) evolvability (e),
respondability (r), conditional evolvability (c), autonomy (a) and
integration (i) along a selection gradient estimate with uncertainty.
evolvabilityBetaMCMC2(G_mcmc, Beta_mcmc, post.dist = FALSE)evolvabilityBetaMCMC2(G_mcmc, Beta_mcmc, post.dist = FALSE)
G_mcmc |
A posterior distribution of a variance matrix in the form of a
table. Each row in the table must be one iteration of the posterior
distribution (or bootstrap distribution). Each iteration of the matrix must
be on the form as given by |
|||||||||||||||
Beta_mcmc |
A posterior distribution of a unit length selection gradient where iterations are given row wise. |
|||||||||||||||
post.dist |
logical: should the posterior distribution of the evolvability parameters be saved.
|
Geir H. Bolstad
Hansen, T. F. & Houle, D. (2008) Measuring and comparing evolvability and constraint in multivariate characters. J. Evol. Biol. 21:1201-1219.
{ # Simulating a posterior distribution # (or bootstrap distribution) of a G-matrix: G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3) G_mcmc <- sapply(c(G), function(x) rnorm(10, x, 0.01)) G_mcmc <- t(apply(G_mcmc, 1, function(x) { G <- matrix(x, ncol = sqrt(length(x))) G[lower.tri(G)] <- t(G)[lower.tri(G)] c(G) })) # Simulating a posterior distribution # (or bootstrap distribution) of trait means: means <- c(1, 1.4, 2.1) means_mcmc <- sapply(means, function(x) rnorm(10, x, 0.01)) # Mean standardizing the G-matrix: G_mcmc <- meanStdGMCMC(G_mcmc, means_mcmc) # Simulating a posterior distribution (or bootstrap distribution) # of a unit length selection gradient: Beta <- randomBeta(1, 3) Beta.mcmc <- sapply(c(Beta), function(x) rnorm(10, x, 0.01)) Beta.mcmc <- t(apply(Beta.mcmc, 1, function(x) x / sqrt(sum(x^2)))) # Running the model: evolvabilityBetaMCMC2(G_mcmc, Beta_mcmc = Beta.mcmc, post.dist = TRUE) }{ # Simulating a posterior distribution # (or bootstrap distribution) of a G-matrix: G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3) G_mcmc <- sapply(c(G), function(x) rnorm(10, x, 0.01)) G_mcmc <- t(apply(G_mcmc, 1, function(x) { G <- matrix(x, ncol = sqrt(length(x))) G[lower.tri(G)] <- t(G)[lower.tri(G)] c(G) })) # Simulating a posterior distribution # (or bootstrap distribution) of trait means: means <- c(1, 1.4, 2.1) means_mcmc <- sapply(means, function(x) rnorm(10, x, 0.01)) # Mean standardizing the G-matrix: G_mcmc <- meanStdGMCMC(G_mcmc, means_mcmc) # Simulating a posterior distribution (or bootstrap distribution) # of a unit length selection gradient: Beta <- randomBeta(1, 3) Beta.mcmc <- sapply(c(Beta), function(x) rnorm(10, x, 0.01)) Beta.mcmc <- t(apply(Beta.mcmc, 1, function(x) x / sqrt(sum(x^2)))) # Running the model: evolvabilityBetaMCMC2(G_mcmc, Beta_mcmc = Beta.mcmc, post.dist = TRUE) }
evolvabilityMeans calculates the average (unconditional) evolvability
(e), respondability (r), conditional evolvability (c), autonomy (a) and
integration (i) of a additive-genetic variance matrix using the approximation
formulas described in Hansen and Houle (2008, 2009).
evolvabilityMeans(G, means = 1)evolvabilityMeans(G, means = 1)
G |
A variance matrix (must be symmetric and positive definite). |
means |
An optional vector of trait means, for mean standardization. |
The equations for calculating the evolvability parameters are
approximations, except for the minimum, maximum and unconditional
evolvability which are exact. The bias of the approximations depends on the
dimensionality of the G-matrix, with higher bias for few dimensions (see
Hansen and Houle 2008). For low dimensional G-matrices, we recommend
estimating the averages of the evolvability parameters using
evolavbilityBetaMCMC over many random selection gradients (
randomBeta). The maximum and minimum evolvability, which
are also the maximum and minimum respondability and conditional
evolvability, equals the largest and smallest eigenvalue of the G-matrix,
respectively.
A vector with the following components:
e_mean |
The average (unconditional) evolvability. | |||
e_min |
The minimum evolvability. | |||
e_max |
The maximum evolvability. | |||
r_mean |
The average respondability. | |||
c_mean |
The average conditional evolvability. | |||
a_mean |
The average autonomy. | |||
i_mean |
The average integration. |
Geir H. Bolstad
Hansen, T. F. & Houle, D. (2008) Measuring and comparing evolvability and
constraint in multivariate characters. J. Evol. Biol. 21:1201-1219.
Hansen, T. F. & Houle, D. (2009) Corrigendum. J. Evol. Biol. 22:913-915.
G <- matrix(c(1, 1, 0, 1, 2, 1, 0, 1, 2), ncol = 3) evolvabilityMeans(G)G <- matrix(c(1, 1, 0, 1, 2, 1, 0, 1, 2), ncol = 3) evolvabilityMeans(G)
evolvabilityMeans calculates the average (unconditional) evolvability
(e), respondability (r), conditional evolvability (c), autonomy (a) and
integration (i) given the posterior distribution of a additive-genetic
variance matrix using the approximation formulas described in Hansen and
Houle (2008, 2009).
evolvabilityMeansMCMC(G_mcmc)evolvabilityMeansMCMC(G_mcmc)
G_mcmc |
the posterior distribution of a variance matrix in the form of
a table. Each row in the table must be one iteration of the posterior
distribution (or bootstrap distribution). Each iteration of the matrix must
be on the form as given by |
The equations for calculating the evolvability parameters are
approximations, except for the minimum, maximum and unconditional
evolvability which are exact. The bias of the approximations depends on the
dimensionality of the G-matrix, with higher bias for few dimensions (see
Hansen and Houle 2008). For low dimensional G-matrices, we recommend
estimating the averages of the evolvability parameters using
evolavbilityBetaMCMC over many random selection gradients (
randomBeta). The maximum and minimum evolvability, which
are also the maximum and minimum respondability and conditional
evolvability, equals the largest and smallest eigenvalue of the G-matrix,
respectively.
An object of class 'evolvabilityMeansMCMC', which is a
list with the following components:
post.dist |
The posterior distribution of the average evolvability parameters. | |||
post.medians |
The posterior medians and HPD interval of the average evolvability parameters. |
Geir H. Bolstad
Hansen, T. F. & Houle, D. (2008) Measuring and comparing evolvability and
constraint in multivariate characters. J. Evol. Biol. 21:1201-1219.
Hansen, T. F. & Houle, D. (2009) Corrigendum. J. Evol. Biol. 22:913-915.
# Simulating a posterior distribution # (or bootstrap distribution) of a G-matrix: G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3) G_mcmc <- sapply(c(G), function(x) rnorm(10, x, 0.01)) G_mcmc <- t(apply(G_mcmc, 1, function(x) { G <- matrix(x, ncol = sqrt(length(x))) G[lower.tri(G)] <- t(G)[lower.tri(G)] c(G) })) # Simulating a posterior distribution # (or bootstrap distribution) of trait means: means <- c(1, 1.4, 2.1) means_mcmc <- sapply(means, function(x) rnorm(10, x, 0.01)) # Mean standardizing the G-matrix: G_mcmc <- meanStdGMCMC(G_mcmc, means_mcmc) # Estimating average evolvability paramters: evolvabilityMeansMCMC(G_mcmc)# Simulating a posterior distribution # (or bootstrap distribution) of a G-matrix: G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3) G_mcmc <- sapply(c(G), function(x) rnorm(10, x, 0.01)) G_mcmc <- t(apply(G_mcmc, 1, function(x) { G <- matrix(x, ncol = sqrt(length(x))) G[lower.tri(G)] <- t(G)[lower.tri(G)] c(G) })) # Simulating a posterior distribution # (or bootstrap distribution) of trait means: means <- c(1, 1.4, 2.1) means_mcmc <- sapply(means, function(x) rnorm(10, x, 0.01)) # Mean standardizing the G-matrix: G_mcmc <- meanStdGMCMC(G_mcmc, means_mcmc) # Estimating average evolvability paramters: evolvabilityMeansMCMC(G_mcmc)
GLS utilizes lm.fit and Cholesky decomposition to fit a
generalized least squares regression
GLS(y, X, R = NULL, L = NULL, coef_only = FALSE)GLS(y, X, R = NULL, L = NULL, coef_only = FALSE)
y |
response variable |
X |
design matrix |
R |
residual covariance or correlation matrix (can be sparse), ignored
if |
L |
lower triangular matrix of the Cholesky decomposition of |
coef_only |
reduces the output of the model to the estimated coefficients (and the generalized residual sums of squares) only. |
Note that the size of R does not matter (i.e. if R is
multiplied by a scalar, the results don't change). Note also that the
R-squared is estimated as 1-GSSE/GSST, where GSSE is the generalized
residual sum of squares (i.e. the objective function score of the model)
and GSST is the generalized total sum of squares (i.e. the objective
function score of the model when only the intercept is included in the
model)
GLS a list of
coef: a table of estimates and standard errors
R2: the R-squared of the model fit
sigma2: the residual variance
GSSE: the generalized residual sum of squares (objective function
score)
coef_vcov: the error variance matrix of the estimates
Geir H. Bolstad
macro_pred Macroevolutionary predictions
macro_pred(y, V, useLFO = TRUE)macro_pred(y, V, useLFO = TRUE)
y |
vector of species means |
V |
phylogenetic variance matrix, must have same order as |
useLFO |
excludes the focal species when calculating the corresponding species' mean. The correct way is to use TRUE, but in practice it has little effect and FALSE will speed up the model fit. |
macro_pred Gives a vector of macroevolutionary predictions for each
species based on the other species given the phylogeny and a phylogenetic
variance matrix.
macro_pred returns a of macroevolutionary predictions at the
tips.
Geir H. Bolstad
# Trait values y <- rnorm(3) # A variance matrix (the diagonal must be the same order as y). V <- matrix(c(1.0, 0.5, 0.2, 0.5, 1.0, 0.4, 0.2, 0.4, 1.0), ncol = 3) # Macroevolutionary predictions (output in the same order as y). macro_pred(y, V)# Trait values y <- rnorm(3) # A variance matrix (the diagonal must be the same order as y). V <- matrix(c(1.0, 0.5, 0.2, 0.5, 1.0, 0.4, 0.2, 0.4, 1.0), ncol = 3) # Macroevolutionary predictions (output in the same order as y). macro_pred(y, V)
meanStdG mean standardizes a variance matrix (e.g. a G-matrix).
meanStdG(G, means)meanStdG(G, means)
G |
A variance matrix. |
means |
A vector of trait means. |
A mean standardized variance matrix.
Geir H. Bolstad
G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3) means <- c(1, 1.4, 2.1) meanStdG(G, means)G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3) means <- c(1, 1.4, 2.1) meanStdG(G, means)
meanStdGMCMC mean standardizes the posterior distribution of a
variance matrix (e.g. a G-matrix)
meanStdGMCMC(G_mcmc, means_mcmc)meanStdGMCMC(G_mcmc, means_mcmc)
G_mcmc |
A posterior distribution of a variance matrix in the form of a
table. Each row in the table must be one iteration of the posterior
distribution (or bootstrap distribution). Each iteration of the matrix must
be on the form as given by |
means_mcmc |
A posterior distribution of a vector of means in the form
of a table. Each row in the table must be one iteration of the posterior
distribution (or bootstrap distribution). A posterior distribution of a
mean vector in the slot |
The posterior distribution of a mean standardized variance matrix.
Geir H. Bolstad
# Simulating a posterior distribution # (or bootstrap distribution) of a G-matrix: G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3) G_mcmc <- sapply(c(G), function(x) rnorm(10, x, 0.01)) G_mcmc <- t(apply(G_mcmc, 1, function(x) { G <- matrix(x, ncol = sqrt(length(x))) G[lower.tri(G)] <- t(G)[lower.tri(G)] c(G) })) # Simulating a posterior distribution # (or bootstrap distribution) of trait means: means <- c(1, 1.4, 2.1) means_mcmc <- sapply(means, function(x) rnorm(10, x, 0.01)) # Mean standardizing the G-matrix: meanStdGMCMC(G_mcmc, means_mcmc)# Simulating a posterior distribution # (or bootstrap distribution) of a G-matrix: G <- matrix(c(1, 1, 0, 1, 4, 1, 0, 1, 2), ncol = 3) G_mcmc <- sapply(c(G), function(x) rnorm(10, x, 0.01)) G_mcmc <- t(apply(G_mcmc, 1, function(x) { G <- matrix(x, ncol = sqrt(length(x))) G[lower.tri(G)] <- t(G)[lower.tri(G)] c(G) })) # Simulating a posterior distribution # (or bootstrap distribution) of trait means: means <- c(1, 1.4, 2.1) means_mcmc <- sapply(means, function(x) rnorm(10, x, 0.01)) # Mean standardizing the G-matrix: meanStdGMCMC(G_mcmc, means_mcmc)
phylH calculates the phylogenetic heritability from an Almer
model fit and provides associated uncertainty using parametric
bootstrapping.
phylH(mod, numerator, residual = "Residual", nsim = 10)phylH(mod, numerator, residual = "Residual", nsim = 10)
mod |
An object of class |
numerator |
The name of phylogenetic effect level |
residual |
name of the residual effect level |
nsim |
number of bootstraps |
phylH returns a list with the REML estimate, the 95%
confidence interval from the parametric bootstrap, and the bootstrap
samples.
Geir H. Bolstad
# See the vignette 'Phylogenetic mixed model'.# See the vignette 'Phylogenetic mixed model'.
plot method for class 'rate_gls'.
## S3 method for class 'rate_gls' plot( x, scale = "SD", print_param = TRUE, digits_param = 2, digits_rsquared = 1, main = "GLS regression", xlab = "x", ylab = "Response", col = "grey", cex.legend = 1, ... )## S3 method for class 'rate_gls' plot( x, scale = "SD", print_param = TRUE, digits_param = 2, digits_rsquared = 1, main = "GLS regression", xlab = "x", ylab = "Response", col = "grey", cex.legend = 1, ... )
x |
An object of class |
scale |
The scale of the y-axis, either the variance scale ('VAR'), that is y^2, or the standard deviation scale ('SD'), that is abs(y). |
print_param |
logical: if parameter estimates should be printed in the plot or not. |
digits_param |
The number of significant digits displayed for the parameters in the plots. |
digits_rsquared |
The number of decimal places displayed for the r-squared. |
main |
as in |
xlab |
as in |
ylab |
as in |
col |
as in |
cex.legend |
A character expansion factor relative to current par("cex") for the printed parameters. |
... |
Additional arguments passed to |
Plots the gls rate regression fitted by the rate_gls
function. The regression line gives the expected variance or standard
deviation (depending on scale). The regression is linear on the variance
scale.
plot returns a plot of the gls rate regression
Geir H. Bolstad
# See the vignette 'Analyzing rates of evolution'.# See the vignette 'Analyzing rates of evolution'.
plot method for class 'simulate_rate'.
## S3 method for class 'simulate_rate' plot( x, response = "rate_y", xlab = "Simulation timesteps", ylab = "Evolutionary rate of y", ... )## S3 method for class 'simulate_rate' plot( x, response = "rate_y", xlab = "Simulation timesteps", ylab = "Evolutionary rate of y", ... )
x |
An object of class |
response |
The variable for the y-axis of the plot, can be 'rate_y', 'y', or 'x'. |
xlab |
A label for the x axis. |
ylab |
A label for the y axis. |
... |
Additional arguments passed to |
No plot is returned if model = 'recent_evol'.
plot A plot of the evolution of the traits x or y, or the
evolution of the evolutionary rate of y (i.e. ) in the
simulation.
Geir H. Bolstad
# See the vignette 'Analyzing rates of evolution'.# See the vignette 'Analyzing rates of evolution'.
randomBeta generates unit length vectors (selection gradients)
uniformly distributed in a k-dimensional hypersphere.
randomBeta(n = 1, k = 2)randomBeta(n = 1, k = 2)
n |
Number of selection gradients/vectors. |
k |
Number of dimensions. |
randomBeta exploits the spherical symmetry of a multidimensional
Gaussian density function. Each element of each vector is randomly sampled
from a univariate Gaussian distribution with zero mean and unit variance. The
vector is then divided by its norm to standardize it to unit length.
randomBeta returns a matrix where the vectors are stacked column
wise.
Geir H. Bolstad
# Two vectors of dimension 3: randomBeta(n = 2, k = 3)# Two vectors of dimension 3: randomBeta(n = 2, k = 3)
rate_gls fits a generalized least squares model to estimate parameters
of an evolutionary model of two traits x and y, where the evolutionary rate
of y depends on the value of x. Three models are implemented. In the two
first, 'predictor_BM' and 'predictor_gBM', the evolution of y follows a
Brownian motion with variance linear in x, while the evolution of x either
follows a Brownian motion or a geometric Brownian motion, respectively. In
the third model, 'recent_evol', the residuals of the macroevolutionary
predictions of y have variance linear in x. It is highly recommended to read
Hansen et al. (in review) and vignette("Analyzing_rates_of_evolution")
before fitting these models.
rate_gls( x, y, species, tree, model = "predictor_BM", startv = list(a = NULL, b = NULL), maxiter = 100, silent = FALSE, useLFO = TRUE, tol = 0.001 )rate_gls( x, y, species, tree, model = "predictor_BM", startv = list(a = NULL, b = NULL), maxiter = 100, silent = FALSE, useLFO = TRUE, tol = 0.001 )
x |
The explanatory variable, which must be equal to the length of
|
y |
The trait values of response variable. Note that the algorithm mean centers y. |
species |
A vector with the names of the species, must be equal in
length and in the same order as |
tree |
An object of class |
model |
The acronym of the evolutionary model to be fitted. There are three options: 'predictor_BM', 'predictor_gBM' or 'recent_evol' (see details). |
startv |
A vector of optional starting values for the a and b parameters. |
maxiter |
The maximum number of iterations for updating the GLS. |
silent |
logical: if the function should not print the generalized sum of squares for each iteration. |
useLFO |
logical: whether the focal species should be left out when
calculating the corresponding species' means. Note that this is only
relevant for the 'recent_evol' model. The most correct is to use
|
tol |
tolerance for convergence. If the change in 'a' and 'b' is below this limit between the two last iteration, convergence is reached. The change is measured in proportion to the standard deviation of the response for 'a' and the ratio of the standard deviation of the response to the standard deviation of the predictor for 'b'. |
rate_gls is an iterative generalized least squares (GLS)
model fitting a regression where the response variable is a vector of
squared mean-centered y-values for the 'predictor_BM' and
'predictor_gBM' models and squared deviation from the evolutionary
predictions (see macro_pred) for the 'recent_evol' model.
Note that the algorithm mean centers x in the 'predictor_BM' and
'recent_evol' analyses, while it mean standardized x (i.e. divided
x by its mean) in the 'predictor_gBM'. The evolutionary parameters a
and b are inferred from the intercept and the slope of the GLS fit. Again,
it is highly recommended to read Hansen et al. (in review) and
vignette("Analyzing_rates_of_evolution") before fitting these
models. In Hansen et al. (2021) the three models 'predictor_BM',
'predictor_gBM' and 'recent_evol' are referred to as 'Model 1', 'Model 2'
and 'Model 3', respectively.
An object of class 'rate_gls', which is a list with the
following components:
model |
The name of the model ('predictor_BM', 'predictor_gBM' or 'recent_evolution'). | |||
param |
The focal parameter estimates and their standard errors, where 'a' and 'b' are parameters of the evolutionary models, and 'sigma(x)^2' is the BM-rate parameter of x for the 'predictor_BM' model, the BM-rate parameter for log x for the 'predictor_gBM' model, and the variance of x for the 'recent_evolution' model. | |||
Rsquared |
The generalized R squared of the GLS model fit. | |||
a_all_iterations |
The values for the parameter a in all iterations. | |||
b_all_iterations |
The values for the parameter b in all iterations. | |||
R |
The residual variance matrix. | |||
Beta |
The intercept and slope of GLS regression (response is y2 and explanatory variable is x). | |||
Beta_vcov |
The error variance matrix of
Beta.
|
|||
tree |
The phylogenetic tree. | |||
data |
The data used in the GLS regression. | |||
convergence |
Whether the algorithm converged or not. | |||
additional_param |
Some additional parameter estimates. | |||
call |
The function call. |
Geir H. Bolstad
Hansen TF, Bolstad GH, Tsuboi M. 2021. Analyzing disparity and rates of morphological evolution with model-based phylogenetic comparative methods. *Systematic Biology*. syab079. doi:10.1093/sysbio/syab079
# Also see vignette("Analyzing_rates_of_evolution"). ## Not run: # Generating a tree with 500 species set.seed(102) tree <- ape::rtree(n = 500) tree <- ape::chronopl(tree, lambda = 1, age.min = 1) ### model = 'predictor_BM' ### sim_data <- simulate_rate(tree, startv_x = 0, sigma_x = 0.25, a = 1, b = 1, model = "predictor_BM" ) head(sim_data$tips) gls_mod <- rate_gls( x = sim_data$tips$x, y = sim_data$tips$y, species = sim_data$tips$species, tree, model = "predictor_BM" ) gls_mod$param par(mfrow = c(1, 2)) # Response shown on the standard deviation scale (default): plot(gls_mod, scale = "SD", cex.legend = 0.8) # Response shown on the variance scale, where the regression is linear: plot(gls_mod, scale = "VAR", cex.legend = 0.8) par(mfrow = c(1, 1)) # Parametric bootstrapping to get the uncertainty of the parameter estimates # taking the complete process into account. # (this takes some minutes) gls_mod_boot <- rate_gls_boot(gls_mod, n = 1000) gls_mod_boot$summary ### model = 'predictor_gBM' ### sim_data <- simulate_rate(tree, startv_x = 1, sigma_x = 1, a = 1, b = 1, model = "predictor_gBM" ) head(sim_data$tips) gls_mod <- rate_gls( x = sim_data$tips$x, y = sim_data$tips$y, species = sim_data$tips$species, tree, model = "predictor_gBM" ) gls_mod$param plot(gls_mod) par(mfrow = c(1, 2)) # Response shown on the standard deviation scale (default): plot(gls_mod, scale = "SD", cex.legend = 0.8) # Response shown on the variance scale, where the regression is linear: plot(gls_mod, scale = "VAR", cex.legend = 0.8) # is linear. par(mfrow = c(1, 1)) # Parametric bootstrapping to get the uncertainty of the parameter estimates # taking the complete process into account. (This takes some minutes.) gls_mod_boot <- rate_gls_boot(gls_mod, n = 1000) gls_mod_boot$summary ### model = 'recent_evol' ### sim_data <- simulate_rate(tree, startv_x = 0, sigma_x = 1, a = 1, b = 1, sigma_y = 1, model = "recent_evol" ) head(sim_data$tips) gls_mod <- rate_gls( x = sim_data$tips$x, y = sim_data$tips$y, species = sim_data$tips$species, tree, model = "recent_evol", useLFO = FALSE ) # useLFO = TRUE is somewhat slower, and although more correct it should give # very similar estimates in most situations. gls_mod$param par(mfrow = c(1, 2)) # Response shown on the standard deviation scale (default): plot(gls_mod, scale = "SD", cex.legend = 0.8) # Response shown on the variance scale, where the regression is linear: plot(gls_mod, scale = "VAR", cex.legend = 0.8) # linear. par(mfrow = c(1, 1)) # Parametric bootstrapping to get the uncertainty of the parameter estimates # taking the complete process into account. Note that x is considered as # fixed effect. (This takes a long time.) gls_mod_boot <- rate_gls_boot(gls_mod, n = 1000, useLFO = FALSE) gls_mod_boot$summary ## End(Not run)# Also see vignette("Analyzing_rates_of_evolution"). ## Not run: # Generating a tree with 500 species set.seed(102) tree <- ape::rtree(n = 500) tree <- ape::chronopl(tree, lambda = 1, age.min = 1) ### model = 'predictor_BM' ### sim_data <- simulate_rate(tree, startv_x = 0, sigma_x = 0.25, a = 1, b = 1, model = "predictor_BM" ) head(sim_data$tips) gls_mod <- rate_gls( x = sim_data$tips$x, y = sim_data$tips$y, species = sim_data$tips$species, tree, model = "predictor_BM" ) gls_mod$param par(mfrow = c(1, 2)) # Response shown on the standard deviation scale (default): plot(gls_mod, scale = "SD", cex.legend = 0.8) # Response shown on the variance scale, where the regression is linear: plot(gls_mod, scale = "VAR", cex.legend = 0.8) par(mfrow = c(1, 1)) # Parametric bootstrapping to get the uncertainty of the parameter estimates # taking the complete process into account. # (this takes some minutes) gls_mod_boot <- rate_gls_boot(gls_mod, n = 1000) gls_mod_boot$summary ### model = 'predictor_gBM' ### sim_data <- simulate_rate(tree, startv_x = 1, sigma_x = 1, a = 1, b = 1, model = "predictor_gBM" ) head(sim_data$tips) gls_mod <- rate_gls( x = sim_data$tips$x, y = sim_data$tips$y, species = sim_data$tips$species, tree, model = "predictor_gBM" ) gls_mod$param plot(gls_mod) par(mfrow = c(1, 2)) # Response shown on the standard deviation scale (default): plot(gls_mod, scale = "SD", cex.legend = 0.8) # Response shown on the variance scale, where the regression is linear: plot(gls_mod, scale = "VAR", cex.legend = 0.8) # is linear. par(mfrow = c(1, 1)) # Parametric bootstrapping to get the uncertainty of the parameter estimates # taking the complete process into account. (This takes some minutes.) gls_mod_boot <- rate_gls_boot(gls_mod, n = 1000) gls_mod_boot$summary ### model = 'recent_evol' ### sim_data <- simulate_rate(tree, startv_x = 0, sigma_x = 1, a = 1, b = 1, sigma_y = 1, model = "recent_evol" ) head(sim_data$tips) gls_mod <- rate_gls( x = sim_data$tips$x, y = sim_data$tips$y, species = sim_data$tips$species, tree, model = "recent_evol", useLFO = FALSE ) # useLFO = TRUE is somewhat slower, and although more correct it should give # very similar estimates in most situations. gls_mod$param par(mfrow = c(1, 2)) # Response shown on the standard deviation scale (default): plot(gls_mod, scale = "SD", cex.legend = 0.8) # Response shown on the variance scale, where the regression is linear: plot(gls_mod, scale = "VAR", cex.legend = 0.8) # linear. par(mfrow = c(1, 1)) # Parametric bootstrapping to get the uncertainty of the parameter estimates # taking the complete process into account. Note that x is considered as # fixed effect. (This takes a long time.) gls_mod_boot <- rate_gls_boot(gls_mod, n = 1000, useLFO = FALSE) gls_mod_boot$summary ## End(Not run)
rate_gls model fitrate_gls_boot performs a parametric bootstrap of a
rate_gls model fit.
rate_gls_boot( object, n = 10, useLFO = TRUE, silent = FALSE, maxiter = 100, tol = 0.001 )rate_gls_boot( object, n = 10, useLFO = TRUE, silent = FALSE, maxiter = 100, tol = 0.001 )
object |
The output from |
n |
The number of bootstrap samples |
useLFO |
logical: when calculating the mean vector of the traits in the 'recent_evol' analysis, should the focal species be left out when calculating the corresponding species' mean. The correct way is to use TRUE, but in practice it has little effect and FALSE will speed up the model fit (particularly useful when bootstrapping). |
silent |
logical: whether or not the bootstrap iterations should be printed. |
maxiter |
The maximum number of iterations for updating the GLS. |
tol |
tolerance for convergence. If the change in 'a' and 'b' is below this limit between the two last iteration, convergence is reached. The change is measured in proportion to the standard deviation of the response for 'a' and the ratio of the standard deviation of the response to the standard deviation of the predictor for 'b'. |
A list where the first slot is a table with the original estimates and SE from the GLS fit in the two first columns followed by the bootstrap estimate of the SE and the 2.5%, 50% and 97.5% quantiles of the bootstrap distribution. The second slot contains the complete distribution.
Geir H. Bolstad
# See the vignette 'Analyzing rates of evolution' and in the help # page of rate_gls.# See the vignette 'Analyzing rates of evolution' and in the help # page of rate_gls.
rate_gls fitrate_gls_sim responses from the models defined by an object of class
'rate_gls'.
rate_gls_sim(object, nsim = 10)rate_gls_sim(object, nsim = 10)
object |
The fitted object from |
nsim |
The number of simulations. |
rate_gls_sim simply passes the estimates in an object of
class 'rate_gls' to the function simulate_rate for
simulating responses of the evolutionary process. It is mainly intended for
internal use in rate_gls_boot.
An object of class 'simulate_rate', which is a list
with the following components:
tips
|
A data frame of x and y values for the tips. | |||
percent_negative_roots |
The percent of iterations with negative roots in the rates of y (not given for model = 'recent_evol'). | |||
compl_dynamics |
A list with the output of the complete dynamics (not given for model = 'recent_evol'). |
Geir H. Bolstad
# See the vignette 'Analyzing rates of evolution'.# See the vignette 'Analyzing rates of evolution'.
G1 and G2 need to be symmetric and positive definite.
responsediffBeta(G1, G2, Beta, means1 = 1, means2 = 1)responsediffBeta(G1, G2, Beta, means1 = 1, means2 = 1)
G1 |
A variance matrix. |
G2 |
A variance matrix. |
Beta |
Either a vector or a matrix of unit length selection gradients stacked column wise. |
means1 |
An optional vector of trait means of G1 (for internal mean standardization). |
means2 |
An optional vector of trait means of G2 (for internal mean standardization). |
responsdiffBeta calculates the response difference along selection
gradients between two additive-genetic variance matrices as described in
Hansen and Houle (2008).
An object of class 'responsdiffBeta', which is a list
with the following components:
Beta |
The matrix of selection gradients. | |||
d |
The response difference of each selection gradient. |
Geir H. Bolstad
Hansen, T. F. & Houle, D. (2008) Measuring and comparing evolvability and constraint in multivariate characters. J. Evol. Biol. 21:1201-1219.
G1 <- matrix(c(1, 1, 0, 1, 2, 2, 0, 2, 3), ncol = 3) / 10 G2 <- matrix(c(1, 2, 0,2, 1, 1,0, 1, 3), ncol = 3) Beta <- randomBeta(5, 3) X <- responsediffBeta(G1, G2, Beta) summary(X)G1 <- matrix(c(1, 1, 0, 1, 2, 2, 0, 2, 3), ncol = 3) / 10 G2 <- matrix(c(1, 2, 0,2, 1, 1,0, 1, 3), ncol = 3) Beta <- randomBeta(5, 3) X <- responsediffBeta(G1, G2, Beta) summary(X)
round_and_format rounds and formats a numeric vector. This is useful for
providing output for tables or plots in a standardized format.
round_and_format( x, digits = 2, sign_digits = NULL, scientific = FALSE, trim = TRUE )round_and_format( x, digits = 2, sign_digits = NULL, scientific = FALSE, trim = TRUE )
x |
A numeric vector. |
digits |
Number of decimal places. |
sign_digits |
Number of significant digits (if given this overrides
|
scientific |
logical: whether encoding should be in scientific notation or not. |
trim |
logical: if leading blanks for justification to common width should be excluded or not. |
Rounded and formatted values as characters.
Geir H. Bolstad
simulate_rate Simulates three different evolutionary rates models. In
the two first, 'predictor_BM' and 'predictor_gBM', the evolution of y follows
a Brownian motion with variance linear in x, while the evolution of x either
follows a Brownian motion or a geometric Brownian motion, respectively. In
the third model, 'recent_evol', the residuals of the macroevolutionary
predictions of y have variance linear in x.
simulate_rate( tree, startv_x = NULL, sigma_x = NULL, a, b, sigma_y = NULL, x = NULL, model = "predictor_BM" )simulate_rate( tree, startv_x = NULL, sigma_x = NULL, a, b, sigma_y = NULL, x = NULL, model = "predictor_BM" )
tree |
A |
startv_x |
The starting value for x (usually 0 for 'predictor_BM' and 'recent_evol', and 1 for 'predictor_gBM'). |
sigma_x |
The evolutionary rate of x. |
a |
A parameter of the evolutionary rate of y. |
b |
A parameter of the evolutionary rate of y. |
sigma_y |
The evolutionary rate for the macroevolution of y (Brownian motion process) used in the 'recent_evolution' model. |
x |
Optional fixed values of x (only for the 'recent_evol' model), must equal number of tips in the phylogeny, must correspond to the order of the tip labels. |
model |
Either a Brownian motion model of x 'predictor_BM', geometric Brownian motion model of x 'predictor_gBM', or 'recent_evol'. |
See the vignette 'Analyzing rates of evolution' for an explanation
of the evolutionary models. The data of the tips can be analyzed with the
function rate_gls. Note that a large part of parameter space
will cause negative roots in the rates of y (i.e. negative a+bx). In these
cases the rates are set to 0. A warning message is given if the number of
such instances is larger than 0.1%. For model 1 and 2, it is possible to
set ‘a=’scaleme'', if this chosen then 'a' will be given the lowest
possible value constrained by a+bx>0.
An object of class 'simulate_rate', which is a list
with the following components:
tips
|
A data frame of x and y values for the tips. | |||
percent_negative_roots |
The percent of iterations with negative roots in the rates of y (not given for model = 'recent_evol'). | |||
compl_dynamics |
A list with the output of the complete dynamics (not given for model = 'recent_evol'). |
Geir H. Bolstad
# Also see the vignette 'Analyzing rates of evolution'. ## Not run: # Generating a tree with 50 species set.seed(102) tree <- ape::rtree(n = 50) tree <- ape::chronopl(tree, lambda = 1, age.min = 1) ### model = 'predictor_BM' ### sim_data <- simulate_rate(tree, startv_x = 0, sigma_x = 0.25, a = 1, b = 1, model = "predictor_BM") head(sim_data$tips) par(mfrow = c(1, 3)) plot(sim_data) plot(sim_data, response = "y") plot(sim_data, response = "x") par(mfrow = c(1, 1)) ### model = 'predictor_gBM' ### sim_data <- simulate_rate(tree, startv_x = 1, sigma_x = 1, a = 1, b = 0.1, model = "predictor_gBM") head(sim_data$tips) par(mfrow = c(1, 3)) plot(sim_data) plot(sim_data, response = "y") plot(sim_data, response = "x") par(mfrow = c(1, 1)) ### model = 'recent_evol' ### sim_data <- simulate_rate(tree, startv_x = 0, sigma_x = 1, a = 1, b = 1, sigma_y = 1, model = "recent_evol" ) head(sim_data$tips) ## End(Not run)# Also see the vignette 'Analyzing rates of evolution'. ## Not run: # Generating a tree with 50 species set.seed(102) tree <- ape::rtree(n = 50) tree <- ape::chronopl(tree, lambda = 1, age.min = 1) ### model = 'predictor_BM' ### sim_data <- simulate_rate(tree, startv_x = 0, sigma_x = 0.25, a = 1, b = 1, model = "predictor_BM") head(sim_data$tips) par(mfrow = c(1, 3)) plot(sim_data) plot(sim_data, response = "y") plot(sim_data, response = "x") par(mfrow = c(1, 1)) ### model = 'predictor_gBM' ### sim_data <- simulate_rate(tree, startv_x = 1, sigma_x = 1, a = 1, b = 0.1, model = "predictor_gBM") head(sim_data$tips) par(mfrow = c(1, 3)) plot(sim_data) plot(sim_data, response = "y") plot(sim_data, response = "x") par(mfrow = c(1, 1)) ### model = 'recent_evol' ### sim_data <- simulate_rate(tree, startv_x = 0, sigma_x = 1, a = 1, b = 1, sigma_y = 1, model = "recent_evol" ) head(sim_data$tips) ## End(Not run)
summary method for class 'evolvabilityBeta'.
## S3 method for class 'evolvabilityBeta' summary(object, ...)## S3 method for class 'evolvabilityBeta' summary(object, ...)
object |
An object of class |
... |
Additional arguments. |
A list with the following components:
Averages |
The averages of the evolvability parameters over all selection gradients. | |||
Minimum |
The minimum of the evolvability parameters over all selection gradients. | |||
Maximum |
The maximum of the evolvability parameters over all selection gradients. |
Geir H. Bolstad
summary method for class 'evolvabilityBetaMCMC'.
## S3 method for class 'evolvabilityBetaMCMC' summary(object, ...)## S3 method for class 'evolvabilityBetaMCMC' summary(object, ...)
object |
an object of class |
... |
additional arguments affecting the summary produced. |
A list with the following components:
Averages |
The averages of the evolvability parameters over all selection gradients. | |||
Minimum |
The minimum (given by the posterior median) of the evolvability parameters over all selection gradients. | |||
Maximum |
The maximum (given by the posterior median) of the evolvability parameters over all selection gradients. |
Geir H. Bolstad
summary method for class 'responsediffBeta'.
## S3 method for class 'responsediffBeta' summary(object, ...)## S3 method for class 'responsediffBeta' summary(object, ...)
object |
An object of class |
... |
Additional arguments. |
A list with the following components:
Averages |
The averages of the response differences over all selection gradients. | |||
Minimum |
The minimum of the response differences over all selection gradients. | |||
Maximum |
The maximum of the response differences over all selection gradients. |
Geir H. Bolstad