def __init__(self, y, M, E, W=None):
from numpy import asarray, atleast_2d, sqrt
from numpy_sugar import ddot
self._y = atleast_2d(asarray(y, float).ravel()).T
self._E = atleast_2d(asarray(E, float).T).T
if W is None:
self._W = self._E
elif isinstance(W, tuple):
# W must be an eigen-decomposition of ЮџєЮџєрхђ
self._W = ddot(W[0], sqrt(W[1]))
else:
self._W = atleast_2d(asarray(W, float).T).T
self._M = atleast_2d(asarray(M, float).T).T
nsamples = len(self._y)
if nsamples != self._M.shape[0]:
raise ValueError("Number of samples mismatch between y and M.")
if nsamples != self._E.shape[0]:
raise ValueError("Number of samples mismatch between y and E.")
if nsamples != self._W.shape[0]:
raise ValueError("Number of samples mismatch between y and W.")
self._lmm = None
self._rhos = [0.0, 0.1**2, 0.2**2, 0.3**2, 0.4**2, 0.5**2, 0.5, 0.999]
def _score_stats(self, g, rhos):
"""
Let ЮЎ║ be the optimal covariance matrix under the null hypothesis.
For a given ¤Ђ, the score-based test statistic is given by
ЮЉёрхе = ┬йЮљ▓рхђЮЎ┐рхе(РѕѓЮЎ║рхе)ЮЎ┐рхеЮљ▓,
where
РѕѓЮЎ║рхе = ЮЎ│(¤ЂЮЪЈЮЪЈрхђ + (1-¤Ђ)ЮЎ┤ЮЎ┤рхђ)ЮЎ│
and ЮЎ│ = diag(Юља).
"""
from numpy import zeros
from numpy_sugar import ddot
Q = zeros(len(rhos))
DPy = ddot(g, self._P(self._y))
s = DPy.sum()
l = s * s
DPyE = DPy.T @ self._E
r = DPyE @ DPyE.T
for i, rho in enumerate(rhos):
Q[i] = (rho * l + (1 - rho) * r) / 2
return Q
def _score_stats_null_dist(self, g):
"""
Under the null hypothesis, the score-based test statistic follows a weighted sum
of random variables:
ЮЉё Рѕ╝ РѕЉрхбЮюєрхб¤Є┬▓(1),
where Ююєрхб are the non-zero eigenvalues of ┬йРѕџЮЎ┐(РѕѓЮЎ║)РѕџЮЎ┐.
Note that
РѕѓЮЎ║рхе = ЮЎ│(¤ЂЮЪЈЮЪЈрхђ + (1-¤Ђ)ЮЎ┤ЮЎ┤рхђ)ЮЎ│ = (¤ЂЮљаЮљархђ + (1-¤Ђ)ЮЎ┤╠ЃЮЎ┤╠Ѓрхђ)
for ЮЎ┤╠Ѓ = ЮЎ│ЮЎ┤.
By using SVD decomposition, one can show that the non-zero eigenvalues of ЮџЄЮџЄрхђ
are equal to the non-zero eigenvalues of ЮџЄрхђЮџЄ.
Therefore, Ююєрхб are the non-zero eigenvalues of
┬й[Рѕџ¤ЂЮља Рѕџ(1-¤Ђ)ЮЎ┤╠Ѓ]ЮЎ┐[Рѕџ¤ЂЮља Рѕџ(1-¤Ђ)ЮЎ┤╠Ѓ]рхђ.
"""
from math import sqrt
from numpy import empty
from numpy.linalg import eigvalsh
from numpy_sugar import ddot
Et = ddot(g, self._E)
Pg = self._P(g)
PEt = self._P(Et)
gPg = g.T @ Pg
EtPEt = Et.T @ PEt
gPEt = g.T @ PEt
n = Et.shape[1] + 1
F = empty((n, n))
lambdas = []
for i in range(len(self._rhos)):
rho = self._rhos[i]
F[0, 0] = rho * gPg
F[0, 1:] = sqrt(rho) * sqrt(1 - rho) * gPEt
F[1:, 0] = F[0, 1:]
F[1:, 1:] = (1 - rho) * EtPEt
lambdas.append(eigvalsh(F) / 2)
return lambdas
def score_2dof_inter(self, X):
from numpy import empty
from numpy_sugar import ddot
Q_rho = self._score_stats(X.ravel(), [0])
g = X.ravel()
Et = ddot(g, self._E)
PEt = self._P(Et)
EtPEt = Et.T @ PEt
gPEt = g.T @ PEt
n = Et.shape[1] + 1
F = empty((n, n))
F[0, 0] = 0
F[0, 1:] = gPEt
F[1:, 0] = F[0, 1:]
F[1:, 1:] = EtPEt
F /= 2
return davies_pvalue(Q_rho[0], F)
def score_2dof_inter(self, X):
"""
Interaction test.
Parameters
----------
X : 1d-array
Genetic variant.
Returns
-------
float
P-value.
"""
from numpy import empty
from numpy_sugar import ddot
Q_rho = self._score_stats(X.ravel(), [0])
g = X.ravel()
Et = ddot(g, self._E)
PEt = self._P(Et)
EtPEt = Et.T @ PEt
gPEt = g.T @ PEt
n = Et.shape[1] + 1
F = empty((n, n))
F[0, 0] = 0
F[0, 1:] = gPEt
F[1:, 0] = F[0, 1:]
F[1:, 1:] = EtPEt
F /= 2
return davies_pvalue(Q_rho[0], F)