FaST linear mixed models for genome-wide association studies

Position: 'FAST_LMM.FaST_LMM.FASTLMM'

Type: Class

Model:

$$ Y_i = \sum_{j = 1}^{p} \beta_j X_{i,j} + u_i +\epsilon_i \ \text{where } i = 1, 2,3,\dots,n $$ where $(u_1, u_2, \dots, u_n)^\top \sim MVN(0, \sigma_g^2 K)$ and $\epsilon_i \sim ^{i.i.d} N(0, \sigma_e^2)$, and $K = WW^\top$, $\sigma_e^2 = \delta \sigma_g^2$ .

Parameters:

• lowRank: boolean, whether to use the low rank methods shown in the paper;
• REML: boolean, whether to use the REML methods or MLE methods.

Attributes:

• sigma_g2: float, estimated $\sigma_g^2$
• sigma_e2: float, estimated $\sigma_e^2$
• delta: float, estimated $\frac{\sigma_e^2}{\sigma_g^2}$
• beta: float, estimated $\hat \beta$
• X: np.array, the fixed effects term of shape (n, p)
• y: np.array, the phenotype data of shape (n, 1)
• W: np.array, the random effects indicator matrix of shape (n, sc). If it is set as None, W = 1/ np.sqrt(n) X.
• rank: int, the rank of W
• U: np.array, eigenvalues matrix calculated by using svd(W) of shape (n, rank) if lowRank is set as true.
• S: the eigenvalues array of W of shape (rank,) if lowRank is set as true.

Methods:

fit(X:np.array, y:np.array, W=None):

fitting the model parameters:

• X: np.array with shape of (n, p)
• y: np.array with shape of (n,1), if it is shape of (n,), it will be reshape to (n,1)
• W: np.array, the random effects indicator matrix of shape (n, sc). If it is set as None, W = 1/ np.sqrt(n) X.

return: None

summary:

print the summary statistics

V(W=None, sigma_g2=None, sigma_e2=None):

Get the $\text{var}(y, y)$. If all parameters are None then using the estimated values, otherwise using the parameter values.

V_inv(self, W=None, sigma_g2=None, sigma_e2=None):

Get the $\text{var}(y, y)^{-1}$. If all parameters are None then using the estimated values, otherwise using the parameter values.

plot_likelihood(REML=True):

Plotting the log-likelihood v.s. $log(\delta)$. Parameters is to determine whether to plot log-likelihood or restricted log-likelihood. The restricted log-likelihood is plot when bot REML and self.REML are set as True.

Example

Example is available in ./test_FAST_LMM.py.