limix.qtl.iscan

limix.qtl.iscan(G, y, lik='normal', K=None, M=None, idx=None, E0=None, E1=None, verbose=True)

Single-trait association with interaction test via generalized linear mixed models.

The general formulae for normally distributed traits is

\[\begin{split}𝐲 = 𝙼𝛂 + (𝙶⊙𝙴₀)𝛃₀ + (𝙶⊙𝙴₁)𝛃₁ + 𝐮 + 𝛆,\\ \text{where}~~ 𝐮∼𝓝(𝟎, 𝓋₀𝙺) ~~\text{and}~~ 𝛆∼𝓝(𝟎, 𝓋₁𝙸).\end{split}\]

The operator ⊙ works as follows:

\[𝙰⊙𝙱 = [𝙰₀𝙱₀ ~~...~~ 𝙰₀𝙱ₙ ~~ 𝙰₁𝙱₀ ~~...~~ 𝙰₁𝙱ₙ ~~...~~ 𝙰ₘ𝙱ₙ]\]

The covariates is enconded in matrix 𝙼 while the candidate set is enconded in matrix 𝙶. The parameters are the effect sizes 𝛂, 𝛃₀, and 𝛃₁, and the variances 𝓋₀ and 𝓋₁.

It performs likelihood-ratio tests for the following cases, where the first hypothesis is the null one while the second hypothesis is the alternative one:

• H₀ vs H₁: testing for vec(𝛃₀) ≠ 𝟎 while vec(𝛃₁) = 𝟎

• H₀ vs H₂: testing for [vec(𝛃₀) vec(𝛃₁)] ≠ 𝟎

• H₁ vs H₂: testing for vec(𝛃₁) ≠ 𝟎

It also supports generalized linear mixed models (GLMM). In this case, the following likelihoods are implemented:

• Bernoulli

• Probit

• Binomial

• Poisson

Formally, let p(𝜇) be one of the supported probability distributions where 𝜇 is its mean. The H₀ model is defined as follows:

\[yᵢ ∼ p(𝜇ᵢ=g(zᵢ)) ~~\text{for}~~ 𝐳 ∼ 𝓝(𝙼𝛂 + (𝙶⊙𝙴₀)𝛃₀ + (𝙶⊙𝙴₁)𝛃₁, 𝓋₀𝙺 + 𝓋₁𝙸).\]

g(⋅) is the corresponding canonical link function for the Bernoulli, Binomial, and Poisson likelihoods. The Probit likelihood, on the other hand, is a Bernoulli likelihood with probit link function.

Parameters

• G (n×m array_like) – Genetic candidates.

• Y (n×p array_like) – Rows are samples and columns are phenotypes.

• lik (tuple, "normal", "bernoulli", "probit", "binomial", "poisson") – 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 (n×n array_like) – Sample covariance, often the so-called kinship matrix.

• M (n×c array_like) – Covariates matrix.

• idx (list) – List of candidate indices that defines the set of candidates to be used in the tests.

• E0 (array_like) – Matrix representing the first environment.

• E1 (array_like) – Matrix representing the second environment.

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

Returns

result – P-values, log of marginal likelihoods, effect sizes, and associated statistics.

Return type

limix.qtl._result.IScanResult

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.