marker_h2 | R Documentation |
Given a genetic relatedness matrix and phenotypic observations at individual
plant or plot level, this function computes REML-estimates of the genetic and
residual variance and their standard errors, using the AI-algorithm (Gilmour et al. 1995).
Based on this, heritability estimates and confidence intervals are given
(the estimator h_r^2
in Kruijer et al.).
marker_h2(data.vector, geno.vector, covariates = NULL, K, alpha = 0.05,
eps = 1e-06, max.iter = 100, fix.h2 = FALSE, h2 = 0.5)
data.vector |
A vector of phenotypic observations. Needs to be of type numeric. May contain missing values. |
geno.vector |
A vector of genotype labels, either a factor or character. This vector should
correspond to |
covariates |
A data-frame or matrix with optional covariates, the rows corresponding to
the phenotypic observations in |
K |
A genetic relatedness or kinship matrix, typically marker-based.
Must have row- and column-names corresponding to the levels of |
alpha |
Confidence level, for the 1-alpha confidence intervals. |
eps |
Numerical precision, used as convergence criterion in the AI-algorithm. |
max.iter |
Maximal number of iterations in the AI-algorithm. |
fix.h2 |
Compute the log-likelihood and inverse AI-matrix for a fixed heritability value. Default is |
h2 |
When |
Given phenotypic observations Y_{ij}
for genotypes i=1,...,n
and replicates
j = 1,...,n_i
, the mixed model
Y_{ij} = \mu + G_i + E_{ij}
is assumed. The vector of additive genetic effects (G_1,...,G_n)'
follows a
multivariate normal distribution with mean zero and covariance \sigma_A^2 K
,
where \sigma_A^2
is the additive genetic variance, and K
is a genetic relatedness matrix derived from a dense set of markers.
The errors E_{ij}
are independent and normally distributed with variance \sigma_E^2
.
Under certain assumptions (see Speed et al. 2012) the marker- or chip-heritability h^2 = \sigma_A^2 / (\sigma_A^2 + \sigma_E^2)
equals the narrow-sense heritability.
It is assumed that the genetic relatedness matrix K
is scaled such that trace(P K P) = n - 1
, where
P
is the projection matrix I_n - 1_n 1_n' / n
, for the identity matrix I_n
and 1_n
being a column vector of ones.
If this is not the case, K
is automatically scaled prior to fitting the mixed model.
The model can optionally include a term X_{ij} \beta
, where X_{ij}
is the row vector with observations on k
extra covariates and the vector \beta
contains their effects.
In this case the argument covariates
should be the (N x k) matrix or
data-frame with rows X_{ij}
(N being the total number of observations).
Observations where either Y_{ij}
or any of the covariates is missing are discarded.
Confidence intervals for heritability are constructed using the delta-method and the inverse AI-matrix.
The delta-method can be applied either directly to the
function (\sigma_A^2,\sigma_E^2) -> \sigma_A^2 / (\sigma_A^2 + \sigma_E^2)
or to the function
(\sigma_A^2,\sigma_E^2) -> log(\sigma_A^2 / \sigma_E^2)
. In the latter case,
a confidence interval for log(\sigma_A^2 / \sigma_E^2)
is obtained, which is back-transformed to a confidence interval for heritability.
This approach (proposed in Kruijer et al.) has the advantage that intervals are always contained in the unit interval.
The AI-algorithm is run for max.iter
iterations. If by then there is no convergence a warning is printed and the current estimates are returned.
A list with the following components:
va: REML-estimate of the (additive) genetic variance.
ve: REML-estimate of the residual variance.
h2: Plug-in estimate of heritability: va / (va + ve)
.
conf.int1: 1-alpha confidence interval for heritability.
conf.int2: 1-alpha confidence interval for heritability, obtained by application of the delta method on a logarithmic scale.
inv.ai: The inverse of the average information (AI) matrix.
loglik: The log-likelihood.
Willem Kruijer.
Gilmour et al. Gilmour, A.R., R. Thompson and B.R. Cullis (1995) Average Information REML: An Efficient Algorithm for Variance Parameter Estimation in Linear Mixed Models. Biometrics, volume 51, number 4, 1440-1450.
Kruijer, W. et al. (2015) Marker-based estimation of heritability in immortal populations. Genetics, Vol. 199(2), p. 1-20.
Speed, D., G. Hemani, M. R. Johnson, and D.J. Balding (2012) Improved heritability estimation from genome-wide snps. the American journal of human genetics 91: 1011-1021.
For marker-based estimation of heritability using genotypic means, see
marker_h2_means
.
data(LD)
data(K_atwell)
# Heritability estimation for all observations:
#out <- marker_h2(data.vector=LD$LD,geno.vector=LD$genotype,
# covariates=LD[,4:8],K=K_atwell)
# Heritability estimation for a randomly chosen subset of 20 accessions:
set.seed(123)
sub.set <- which(LD$genotype %in% sample(levels(LD$genotype),20))
out <- marker_h2(data.vector=LD$LD[sub.set],geno.vector=LD$genotype[sub.set],
covariates=LD[sub.set,4:8],K=K_atwell)
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