# Heritability estimationΒΆ

We provide heritability estimation for Normal, Bernoulli, Probit, Binomial, and Poisson phenotypes. A standard LMM is used for Normal traits:

$π² = πΌπ + π― + π,$

where

$π― βΌ π(π, πβπΊ) ~~\text{and}~~ π βΌ π(π, πβπΈ).$

A GLMM is used to model the other type of traits:

$π³ = πΌπ + π― + π, ~~\text{where}~~ yα΅’|π³ βΌ π΄π‘ππ΅ππ(πα΅’=g(zα΅’))$

and π― and π are defined as before.

In both cases, the parameters are the same: π, πβ, and πβ. They are fitted via restricted maximum likelihood for LMM and via maximum likelihood for GLMM. The covariance-matrix πΊ given by the user is normalised before the model is fitted as follows:

K = K / K.diagonal().mean()

limix.her.estimate(y, lik, K, M=None, verbose=True)[source]

Estimate the so-called narrow-sense heritability.

It supports Normal, Bernoulli, Probit, Binomial, and Poisson phenotypes.

Parameters
• y (array_like) β Array of trait values of n individuals.

• lik (tuple, "normal", "bernoulli", "probit", "binomial", "poisson") β Sample likelihood describing the residual distribution. Either a tuple or a string specifying the likelihood is required. The Normal, Bernoulli, Probit, and Poisson likelihoods can be selected by providing a string. Binomial likelihood on the other hand requires a tuple because of the number of trials: ("binomial", array_like). Defaults to "normal".

• K (nΓn array_like) β Sample covariance, often the so-called kinship matrix. It might be, for example, the estimated kinship relationship between the individuals. The provided matrix will be normalised as K = K / K.diagonal().mean().

• M (nΓc array_like, optional) β Covariates matrix. If an array is passed, it will used as is; no normalisation will be performed. If None is passed, an offset will be used as the only covariate. Defaults to None.

• verbose (bool, optional) β True to display progress and summary; False otherwise.

Returns

Estimated heritability.

Return type

float

Examples

>>> from numpy import dot, exp, sqrt
>>> from numpy.random import RandomState
>>> from limix.her import estimate
>>>
>>> random = RandomState(0)
>>>
>>> G = random.randn(150, 200) / sqrt(200)
>>> K = dot(G, G.T)
>>> z = dot(G, random.randn(200)) + random.randn(150)
>>> y = random.poisson(exp(z))
>>>
>>> print(estimate(y, 'poisson', K, verbose=False))
0.18311439918863426


Notes

It will raise a ValueError exception if non-finite values are passed. Please, refer to the limix.qc.mean_impute() function for missing value imputation.