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mtag.py
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mtag.py
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#!/usr/bin/env python
'''
'''
from __future__ import division
import numpy as np
import pandas as pd
import scipy.optimize
import argparse
import itertools
import time
import os, re
import joblib
import sys, gzip, bz2
import logging
from argparse import Namespace
from ldsc_mod.ldscore import sumstats as sumstats_sig
from ldsc_mod.ldscore import allele_info
import ldsc_mod.munge_sumstats as munge_sumstats
__version__ = '1.0.7'
borderline = "<><><<>><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>"
header ="\n"
header += borderline +"\n"
header += "<>\n"
header += "<> MTAG: Multi-trait Analysis of GWAS \n"
header += "<> Version: {}\n".format(str(__version__))
header += "<> (C) 2017 Omeed Maghzian, Raymond Walters, and Patrick Turley\n"
header += "<> Harvard University Department of Economics / Broad Institute of MIT and Harvard\n"
header += "<> GNU General Public License v3\n"
header += borderline + "\n"
header += "<> Note: It is recommended to run your own QC on the input before using this program. \n"
header += "<> Software-related correspondence: [email protected] \n"
header += "<> All other correspondence: [email protected] \n"
header += borderline +"\n"
header += "\n\n"
pd.set_option('display.max_rows', 500)
pd.set_option('display.width', 800)
pd.set_option('precision', 12)
pd.set_option('max_colwidth', 800)
pd.set_option('colheader_justify', 'left')
np.set_printoptions(linewidth=800)
np.set_printoptions(precision=3)
## General helper functions
def safely_create_folder(folder_path):
try:
os.makedirs(folder_path)
except OSError:
if not os.path.isdir(folder_path):
raise
class DisableLogger():
'''
For disabling the logging module when calling munge_sumstats
'''
def __enter__(self):
logging.disable(logging.CRITICAL)
def __exit__(self, a, b, c):
logging.disable(logging.NOTSET)
## Read / Write functions
def _read_SNPlist(file_path, SNP_index):
# TODO Add more possible ways of reading SNPlists
snplist = pd.read_csv(file_path, header=0, index_col=False)
if SNP_index not in snplist.columns:
raise ValueError("SNPlist read from {} does include --snp_name {} in its columns.".format(file_path, SNP_index))
return pd.read_csv(file_path, header=0, index_col=False)
def _read_GWAS_sumstats(GWAS_file_name, chunksize):
'''
read GWAS summary statistics from file that is in one of the acceptable formats.
'''
# TODO read more file types
(openfunc, compression) = munge_sumstats.get_compression(GWAS_file_name)
dat_gen = pd.read_csv(GWAS_file_name, index_col=False, header=0,delim_whitespace=True, compression=compression, na_values=['.','NA'],
iterator=True, chunksize=chunksize)
dat_gen = list(dat_gen)
dat_gen_unfiltered = pd.concat(dat_gen, axis=0).reset_index(drop=True)
return dat_gen_unfiltered, dat_gen
def _read_matrix(file_path):
'''
For reading 2-dimensional matrices. These files must be in .npy form or whitespace delimited .csv files
'''
ext = file_path[-4:]
if ext == '.npy':
return np.load(file_path)
if ext == '.txt':
return np.loadtxt(file_path)
else:
raise ValueError('{} is not one of the acceptable file paths for reading in matrix-valued objects.'.format(ext))
## LDSC related functions
def sec_to_str(t):
'''Convert seconds to days:hours:minutes:seconds'''
[d, h, m, s, n] = reduce(lambda ll, b : divmod(ll[0], b) + ll[1:], [(t, 1), 60, 60, 24])
f = ''
if d > 0:
f += '{D}d:'.format(D=d)
if h > 0:
f += '{H}h:'.format(H=h)
if m > 0:
f += '{M}m:'.format(M=m)
f += '{S}s'.format(S=s)
return f
class Logger_to_Logging(object):
"""
Logger class that write uses logging module and is needed to use munge_sumstats or ldsc from the LD score package.
"""
def __init__(self):
logging.info('created Logger instance to pass through ldsc.')
super(Logger_to_Logging, self).__init__()
def log(self,x):
logging.info(x)
def _perform_munge(args, GWAS_df, GWAS_dat_gen,p):
original_cols = GWAS_df.columns
merge_alleles = None
out=None
zz= args.z_name if args.z_name is not None else 'z'
ignore_list = ""
if args.info_min is None:
ignore_list += "info"
a1_munge = None if args.a1_name == "a1" else args.a1_name
a2_munge = None if args.a2_name == "a2" else args.a2_name
eaf_munge = None if args.eaf_name == "freq" else args.eaf_name
# sumstats is set to null because generator passed manually
argnames = Namespace(sumstats=None,N=None,N_cas=None,N_con=None,out=out,maf_min=args.maf_min_list[p], info_min =args.info_min_list[p],daner=False, no_alleles=False, merge_alleles=merge_alleles,n_min=args.n_min_list[p],chunksize=args.chunksize, snp=args.snp_name,N_col=args.n_name, N_cas_col=None, N_con_col = None, a1=a1_munge, a2=a2_munge, p=None,frq=eaf_munge,signed_sumstats=zz+',0', info=None,info_list=None, nstudy=None,nstudy_min=None,ignore=ignore_list,a1_inc=False, keep_maf=True, daner_n=False, keep_str_ambig=True, input_datgen=GWAS_dat_gen, cnames=list(original_cols))
logging.info(borderline)
logging.info('Munging Trait {} {}'.format(p+1,borderline[:-17]))
logging.info(borderline)
munged_results = munge_sumstats.munge_sumstats(argnames, write_out=False, new_log=False)
GWAS_df = GWAS_df.merge(munged_results, how='inner',left_on =args.snp_name,right_on='SNP',suffixes=('','_ss'))
GWAS_df = GWAS_df[original_cols]
logging.info(borderline)
logging.info('Munging of Trait {} complete. SNPs remaining:\t {}'.format(p+1, len(GWAS_df)))
logging.info(borderline+'\n')
return GWAS_df, munged_results
def _quick_mode(ndarray,axis=0):
'''
From stackoverflow: Efficient calculation of the mode of an array. Scipy.stats.mode is way too slow
'''
if ndarray.size == 1:
return (ndarray[0],1)
elif ndarray.size == 0:
raise Exception('Attempted to find mode on an empty array!')
try:
axis = [i for i in range(ndarray.ndim)][axis]
except IndexError:
raise Exception('Axis %i out of range for array with %i dimension(s)' % (axis,ndarray.ndim))
srt = np.sort(ndarray,axis=axis)
dif = np.diff(srt,axis=axis)
shape = [i for i in dif.shape]
shape[axis] += 2
indices = np.indices(shape)[axis]
index = tuple([slice(None) if i != axis else slice(1,-1) for i in range(dif.ndim)])
indices[index][dif == 0] = 0
indices.sort(axis=axis)
bins = np.diff(indices,axis=axis)
location = np.argmax(bins,axis=axis)
mesh = np.indices(bins.shape)
index = tuple([slice(None) if i != axis else 0 for i in range(dif.ndim)])
index = [mesh[i][index].ravel() if i != axis else location.ravel() for i in range(bins.ndim)]
counts = bins[tuple(index)].reshape(location.shape)
index[axis] = indices[tuple(index)]
modals = srt[tuple(index)].reshape(location.shape)
return (modals, counts)
def load_and_merge_data(args):
'''
TODO Add description
Parses file names from MTAG command line arguments and returns the relevant used for method.
'''
GWAS_input_files = args.sumstats.split(',')
P = len(GWAS_input_files) # of phenotypes/traits
if args.n_min is not None:
args.n_min_list = [float(x) for x in args.n_min.split(',')]
if len(args.n_min_list) == 1:
args.n_min_list = args.n_min_list * P
else:
args.n_min_list = [None]*P
if args.maf_min is not None:
args.maf_min_list = [float(x) for x in args.maf_min.split(',')]
if len(args.maf_min_list) == 1:
args.maf_min_list = args.maf_min_list * P
else:
args.maf_min_list = [None]*P
if args.info_min is not None:
args.info_min_list = [float(x) for x in args.info_min.split(',')]
if len(args.info_min_list) == 1:
args.info_min_list = args.info_min_list * P
else:
args.info_min_list = [None]*P
GWAS_d = dict()
sumstats_format = dict()
for p, GWAS_input in enumerate(GWAS_input_files):
GWAS_d[p], gwas_dat_gen = _read_GWAS_sumstats(GWAS_input, args.chunksize)
# add suffix
logging.info('Read in Trait {} summary statistics ({} SNPs) from {} ...'.format(p+1,len(GWAS_d[p]), GWAS_input))
# perform munge sumstats
GWAS_d[p], sumstats_format[p] = _perform_munge(args, GWAS_d[p], gwas_dat_gen, p)
GWAS_d[p] = GWAS_d[p].add_suffix(p)
# convert Alleles to uppercase
for col in [col+str(p) for col in [args.a1_name, args.a2_name]]:
GWAS_d[p][col] = GWAS_d[p][col].str.upper()
GWAS_d[p] =GWAS_d[p].rename(columns={x+str(p):x for x in GWAS_d[p].columns})
GWAS_d[p] = GWAS_d[p].rename(columns={args.snp_name+str(p):args.snp_name})
# Drop SNPs that are missing or duplicated
missing_snps = GWAS_d[p][args.snp_name].isin(['NA','.'])
M0 = len(GWAS_d[p])
GWAS_d[p] = GWAS_d[p][np.logical_not(missing_snps)]
if M0-len(GWAS_d[p]) > 0:
logging.info('Trait {}: Dropped {} SNPs for missing values in the "snp_name" column'.format(p+1, M0-len(GWAS_d[p])))
# drop snps that are duplicated
M0 = len(GWAS_d[p])
GWAS_d[p] = GWAS_d[p].drop_duplicates(subset=args.snp_name, keep='first')
if M0-len(GWAS_d[p]) > 0:
logging.info('Trait {}: Dropped {} SNPs for duplicate values in the "snp_name" column'.format(p+1, M0-len(GWAS_d[p])))
## Merge summary statistics of GWA studies by snp index
for p in range(P):
if p == 0:
GWAS_all = GWAS_d[p]
else:
GWAS_all = GWAS_all.merge(GWAS_d[p], how = 'inner', on=args.snp_name)
M_0 = len(GWAS_all)
logging.info('Trait {} summary statistics: \t {} SNPs remaining merging with previous traits.'.format(p+1, M_0))
if True:
snps_to_flip = np.logical_and(GWAS_all[args.a1_name+str(0)] == GWAS_all[args.a2_name+str(p)], GWAS_all[args.a2_name+str(0)] == GWAS_all[args.a1_name+str(p)])
GWAS_all['flip_snps'+str(p)]= snps_to_flip
logging.debug('Columns after merging :{}'.format(GWAS_all.columns))
logging.debug(GWAS_all.head(15))
snps_to_keep = np.logical_or(np.logical_and(GWAS_all[args.a1_name+str(0)]==GWAS_all[args.a1_name+str(p)], GWAS_all[args.a2_name+str(0)]==GWAS_all[args.a2_name+str(p)]), snps_to_flip)
GWAS_all = GWAS_all[snps_to_keep]
if len(GWAS_all) < M_0:
logging.info('Dropped {} SNPs due to inconsistent allele pairs from phenotype {}. {} SNPs remain.'.format(M_0 - len(GWAS_all),p+1, len(GWAS_all)))
if np.sum(snps_to_flip) > 0:
zz= args.z_name if args.z_name is not None else 'z'
freq_name = args.eaf_name if args.eaf_name is not None else 'freq'
GWAS_all.loc[snps_to_flip, zz+str(p)] = -1*GWAS_all.loc[snps_to_flip, zz+str(p)]
GWAS_all.loc[snps_to_flip, freq_name + str(p)] = 1. - GWAS_all.loc[snps_to_flip, freq_name + str(p)]
store_allele = GWAS_all.loc[snps_to_flip, args.a1_name+str(p)]
GWAS_all.loc[snps_to_flip, args.a1_name+str(p)] = GWAS_all.loc[snps_to_flip, args.a2_name+str(p)]
GWAS_all.loc[snps_to_flip, args.a2_name+str(p)] = store_allele
logging.info('Flipped the signs of of {} SNPs to make them consistent with the effect allele orderings of the first trait.'.format(np.sum(snps_to_flip)))
# tag strand ambiguous SNPs
# logging.info(GWAS_all.head(15))
STRAND_AMBIGUOUS_SET = [x for x in allele_info.STRAND_AMBIGUOUS.keys() if allele_info.STRAND_AMBIGUOUS[x]]
GWAS_all['strand_ambig'] = (GWAS_all[args.a1_name+str(0)].str.upper() + GWAS_all[args.a2_name+str(0)].str.upper()).isin(STRAND_AMBIGUOUS_SET)
if args.drop_ambig_snps:
M_0 = len(GWAS_all)
GWAS_all = GWAS_all[np.logical_not(GWAS_all['strand_ambig'])]
logging.info('Dropped {} SNPs due to strand ambiguity, {} SNPs remain.'.format(M_0-len(GWAS_all),len(GWAS_all)))
logging.info('... Merge of GWAS summary statistics complete. Number of SNPs:\t {}'.format(len(GWAS_all)))
GWAS_orig_cols = GWAS_all.columns
## Parses include files
if args.include is not None:
for j, include_file in enumerate(args.include.split(',')):
if j == 0:
snps_include = _read_SNPlist(include_file, args.snp_name)
else:
snps_include = snps_include.merge(_read_SNPlist(include_file,args.snp_name),how='outer', on=args.snp_name)
GWAS_all = GWAS_all.merge(snps_include, how="left", on = args.snp_name, indicator="included_merge", suffixes=('','_incl'))
GWAS_all = GWAS_all.loc[GWAS_all['included_merge']=='both']
GWAS_all = GWAS_all.loc[:,GWAS_orig_cols]
logging.info('(--include) Number of SNPs remaining after restricting to SNPs in the union of {include_path}: \t {M} remain'.format(include_path=args.include,M=len(GWAS_all)))
## Parses exclude files
if args.exclude is not None:
for exclude_file in args.exclude.split(','):
snps_exclude = _read_SNPlist(exclude_file, args.snp_name)
GWAS_all = GWAS_all.merge(snps_exclude, how="left", on = args.snp_name, indicator="excluded_merge", suffixes=('','_incl'))
GWAS_all = GWAS_all.loc[GWAS_all['excluded_merge']=='left_only']
GWAS_all = GWAS_all.loc[:,GWAS_orig_cols]
logging.info('(-exclude) Number of SNPs remaining after excluding to SNPs in {exclude_path}: \t {M} remain'.format(exclude_path=exclude_file,M=len(GWAS_all)))
## Parse chromosomes
if args.only_chr is not None:
chr_toInclude = args.only_chr.split(',')
chr_toInclude = [int(c) for c in chr_toInclude]
GWAS_all = GWAS_all[GWAS_all[args.chr_name+str(0)].isin(chr_toInclude)]
## add information to Namespace
args.P = P
return GWAS_all, args
def ldsc_matrix_formatter(result_rg, output_var):
''' Key Arguments:
result_rg - matrix w/ RG objects obtained from estimate_rg (w/ None's on the diagonal)
output_var - interested variable in the form of '.[VAR_NAME]'
'''
output_mat = np.empty_like(result_rg, dtype=float)
(nrow, ncol) = result_rg.shape
for i in range(nrow):
for j in range(ncol):
if result_rg[i, j] is None:
output_mat[i, j] = None
else:
exec('output_mat[i, j] = result_rg[i, j]{}'.format(output_var))
return(output_mat)
def estimate_sigma(data_df, args):
sigma_hat = np.empty((args.P,args.P))
args.munge_out = args.out+'_ldsc_temp/'
# Creates data files for munging
# Munge data
ignore_list = ""
if args.info_min is None:
ignore_list += "info"
gwas_ss_df = dict()
for p in range(args.P):
logging.info('Preparing phenotype {} to estimate sigma'.format(p))
ld_ss_name = {args.snp_name : 'SNP',
args.a1_name + str(p): 'A1',
args.a2_name + str(p): 'A2',
args.z_name + str(p): 'Z',
args.n_name + str(p): 'N',
args.eaf_name + str(p): 'FRQ'}
# single_colnames = [col for col in data_df.columns if col[-1] == str(p) or col in args.snp_name]
gwas_ss_df[p] = data_df[ld_ss_name.keys()].copy()
# gwas_filtered_df= gwas_filtered_df.rename(columns={args.snp_name:args.snp_name+str(p)})
gwas_ss_df[p] = gwas_ss_df[p].rename(columns=ld_ss_name)
## remove phenotype index from names
# run ldsc
h2_files = None
rg_files = args.sumstats
rg_out = '{}_rg_misc'.format(args.out)
rg_mat = True
args_ldsc_rg = Namespace(out=rg_out, bfile=None,l2=None,extract=None,keep=None, ld_wind_snps=None,ld_wind_kb=None, ld_wind_cm=None,print_snps=None, annot=None,thin_annot=False,cts_bin=None, cts_break=None,cts_names=None, per_allele=False, pq_exp=None, no_print_annot=False,maf=None,h2=h2_files, rg=rg_files,ref_ld=None,ref_ld_chr=args.ld_ref_panel, w_ld=None,w_ld_chr=args.ld_ref_panel,overlap_annot=False,no_intercept=False, intercept_h2=None, intercept_gencov=None,M=None,two_step=None, chisq_max=None,print_cov=False,print_delete_vals=False,chunk_size=50, pickle=False,invert_anyway=False,yes_really=False,n_blocks=200,not_M_5_50=False,return_silly_things=False,no_check_alleles=False,print_coefficients=False,samp_prev=None,pop_prev=None, frqfile=None, h2_cts=None, frqfile_chr=None,print_all_cts=False, sumstats_frames=[ gwas_ss_df[i] for i in range(args.P)], rg_mat=rg_mat)
if args.no_overlap:
sigma_hat = np.zeros((args.P, args.P))
for t in range(args.P):
args_ldsc_rg.sumstats_frames = [gwas_ss_df[t]]
rg_results_t = sumstats_sig.estimate_rg(args_ldsc_rg, Logger_to_Logging())
sigma_hat[t,t] = ldsc_matrix_formatter(rg_results_t, '.gencov.intercept')[0]
else:
rg_results = sumstats_sig.estimate_rg(args_ldsc_rg, Logger_to_Logging())
sigma_hat = ldsc_matrix_formatter(rg_results, '.gencov.intercept')
# if args.no_overlap:
# T = sigma_hat.shape[0]
# sigma_hat = sigma_hat * np.eye(T)
# logging.info(type(sigma_hat))
logging.info(sigma_hat)
return sigma_hat
def _posDef_adjustment(mat, scaling_factor=0.99,max_it=1000):
'''
Checks whether the provided is pos semidefinite. If it is not, then it performs the the adjustment procedure descried in 1.2.2 of the Supplementary Note
scaling_factor: the multiplicative factor that all off-diagonal elements of the matrix are scaled by in the second step of the procedure.
max_it: max number of iterations set so that
'''
logging.info('Checking for positive definiteness ..')
assert mat.ndim == 2
assert mat.shape[0] == mat.shape[1]
is_pos_semidef = lambda m: np.all(np.linalg.eigvals(m) >= 0)
if is_pos_semidef(mat):
return mat
else:
logging.info('matrix is not positive definite, performing adjustment..')
P = mat.shape[0]
for i in range(P):
for j in range(i,P):
if np.abs(mat[i,j]) > np.sqrt(mat[i,i] * mat[j,j]):
mat[i,j] = scaling_factor*np.sign(mat[i,j])*np.sqrt(mat[i,i] * mat[j,j])
mat[j,i] = mat[i,j]
n=0
while not is_pos_semidef(mat) and n < max_it:
dg = np.diag(mat)
mat = scaling_factor * mat
mat[np.diag_indices(P)] = dg
n += 1
if n == max_it:
logging.info('Warning: max number of iterations reached in adjustment procedure. Sigma matrix used is still non-positive-definite.')
else:
logging.info('Completed in {} iterations'.format(n))
return mat
def extract_gwas_sumstats(DATA, args):
'''
Output:
-------
All matrices are of the shape MxP, where M is the number of SNPs used in MTAG and P is the number of summary statistics results used. Columns are ordered according to the initial ordering of GWAS input files.
results_template = pd.Dataframe of snp_name chr bpos a1 a2
Zs: matriix of Z scores
Ns: matrix of sample sizes
Fs: matrix of allele frequencies
'''
n_cols = [args.n_name +str(p) for p in range(args.P)]
Ns = DATA.filter(items=n_cols).as_matrix()
# Apply sample-size specific filters
N_passFilter = np.ones(len(Ns), dtype=bool)
N_nearMode = np.ones_like(Ns, dtype=bool)
if args.homogNs_frac is not None or args.homogNs_dist is not None:
N_modes, _ = _quick_mode(Ns)
assert len(N_modes) == Ns.shape[1]
if args.homogNs_frac is not None:
logging.info('--homogNs_frac {} is on, filtering SNPs ...'.format(args.homogNs_frac))
assert args.homogNs_frac >= 0.
homogNs_frac_list = [float(x) for x in args.homogNs_frac.split(',')]
if len(homogNs_frac_list) == 1:
homogNs_frac_list = homogNs_frac_list*args.P
for p in range(args.P):
N_nearMode[:,p] = np.abs((Ns[:,p] - N_modes[p])) / N_modes[p] <= homogNs_frac_list[p]
elif args.homogNs_dist is not None:
logging.info('--homogNs_dist {} is on, filtering SNPs ...'.format(args.homogNs_dist))
homogNs_dist_list = [float(x) for x in args.homogNs_dist.split(',')]
if len(homogNs_dist_list) == 1:
homogNs_dist_list = homogNs_dist_list*args.P
assert np.all(np.array(homogNs_dist_list) >=0)
for p in range(args.P):
N_nearMode[:,p] = np.abs(Ns[:,p] - N_modes[p]) <= homogNs_dist_list[p]
else:
raise ValueError('Cannot specify both --homogNs_frac and --homogNs_dist at the same time.')
# report restrictions
mode_restrictions = 'Sample size restrictions close to mode:\n'
for p in range(Ns.shape[1]):
mode_restrictions +="Phenotype {}: \t {} SNPs pass modal sample size filter \n".format(p+1,np.sum(N_nearMode[:,p]))
mode_restrictions+="Intersection of SNPs that pass modal sample size filter for all traits:\t {}".format(np.sum(np.all(N_nearMode, axis=1)))
logging.info(mode_restrictions)
N_passFilter = np.logical_and(N_passFilter, np.all(N_nearMode,axis=1))
if args.n_max is not None:
n_max_restrictions = "--n_max used, removing SNPs with sample size greater than {}".format(args.n_max)
N_passMax = Ns <= args.n_max
for p in range(Ns.shape[1]):
n_max_restrictions += "Phenotype {}: \t {} SNPs pass modal sample size filter".format(p+1,np.sum(N_passMax[:,p]))
n_max_restrictions += "Intersection of SNPs that pass maximum sample size filter for all traits:\t {}".format(np.sum(np.all(N_passMax, axis=1)))
logging.info(n_max_restrictions)
N_passFilter = np.logical_and(N_passFilter, np.all(N_passMax,axis=1))
Ns = Ns[N_passFilter]
DATA = DATA[N_passFilter].reset_index()
if args.z_name is not None:
z_cols = [args.z_name +str(p) for p in range(args.P)]
Zs = DATA.filter(items=z_cols).as_matrix()
else:
Zs = DATA.filter(regex='^[zZ].').as_matrix()
args.z_name = 'z'
if args.eaf_name is not None:
f_cols = [args.eaf_name + str(p) for p in range(args.P)]
Fs =DATA.filter(items=f_cols).as_matrix()
else:
orig_case_cols = DATA.columns
DATA.columns = map(str.upper, DATA.columns)
Fs = DATA.filter(regex='^/MAF|FREQ|FRQ/.').as_matrix()
args.eaf_name = 'freq'
DATA.columns = orig_case_cols
assert Zs.shape[1] == Ns.shape[1] == Fs.shape[1]
results_template = DATA[[args.snp_name]].copy()
# results_template = pd.DataFrame(index=np.arange(len(DATA)))
# results_template.loc[:,args.snp_name] = DATA[args.snp_name]
# args.chr args.bpos args.alelle_names
for col in [args.chr_name, args.bpos_name, args.a1_name, args.a2_name]:
results_template.loc[:,col] = DATA[col+str(0)]
# TODO: non-error form of integer conversion
# results_template[args.chr_name] = results_template[args.chr_name].astype(int)
# results_template[args.bpos_name] = results_template[args.bpos_name].astype(int)
return Zs, Ns, Fs, results_template, DATA
###########################################
## OMEGA ESTIMATION
##########################################
def jointEffect_probability(Z_score, omega_hat, sigma_hat,N_mats, S=None):
''' For each SNP m in each state s , computes the evaluates the multivariate normal distribution at the observed row of Z-scores
Calculate the distribution of (Z_m | s ) for all s in S, m in M. --> M x|S| matrix
The output is a M x n_S matrix of joint probabilities
'''
DTYPE = np.float64
(M,P) = Z_score.shape
if S is None: # 2D dimensional form
assert omega_hat.ndim == 2
omega_hat = omega_hat.reshape(1,P,P)
S = np.ones((1,P),dtype=bool)
(n_S,_) = S.shape
jointProbs = np.empty((M,n_S))
xRinvs = np.zeros([M,n_S,P], dtype=DTYPE)
logSqrtDetSigmas = np.zeros([M,n_S], dtype=DTYPE)
Ls = np.zeros([M,n_S,P,P], dtype=DTYPE)
cov_s = np.zeros([M,n_S,P,P], dtype=DTYPE)
Zs_rep = np.einsum('mp,s->msp',Z_score,np.ones(n_S)) # functionally equivalent to repmat
cov_s = np.einsum('mpq,spq->mspq',N_mats,omega_hat) + sigma_hat
Ls = np.linalg.cholesky(cov_s)
Rs = np.transpose(Ls, axes=(0,1,3,2))
xRinvs = np.linalg.solve(Ls, Zs_rep)
logSqrtDetSigmas = np.sum(np.log(np.diagonal(Rs,axis1=2,axis2=3)),axis=2).reshape(M,n_S)
quadforms = np.sum(xRinvs**2,axis=2).reshape(M,n_S)
jointProbs = np.exp(-0.5 * quadforms - logSqrtDetSigmas - P * np.log(2 * np.pi) / 2)
if n_S == 1:
jointProbs = jointProbs.flatten()
return jointProbs
def gmm_omega(Zs, Ns, sigma_LD):
logging.info('Using GMM estimator of Omega ..')
N_mats = np.sqrt(np.einsum('mp,mq->mpq', Ns,Ns))
Z_outer = np.einsum('mp,mq->mpq',Zs, Zs)
return np.mean((Z_outer - sigma_LD) / N_mats, axis=0)
def numerical_omega(args, Zs,N_mats,sigma_LD,omega_start):
M,P = Zs.shape
solver_options = dict()
solver_options['fatol'] = 1.0e-8
solver_options['xatol'] = args.tol
solver_options['disp'] = False
solver_options['maxiter'] = P*250 if args.perfect_gencov else P*(P+1)*500
if args.perfect_gencov:
x_start = np.log(np.diag(omega_start))
else:
x_start = flatten_out_omega(omega_start)
opt_results = scipy.optimize.minimize(_omega_neglogL,x_start,args=(Zs,N_mats,sigma_LD,args),method='Nelder-Mead',options=solver_options)
if args.perfect_gencov:
return np.sqrt(np.outer(np.exp(opt_results.x), np.exp(opt_results.x))), opt_results
else:
return rebuild_omega(opt_results.x), opt_results
def _omega_neglogL(x,Zs,N_mats,sigma_LD,args):
if args.perfect_gencov:
omega_it = np.sqrt(np.outer(np.exp(x),np.exp(x)))
else:
omega_it = rebuild_omega(x)
joint_prob = jointEffect_probability(Zs,omega_it,sigma_LD,N_mats)
return - np.sum(np.log(joint_prob))
def flatten_out_omega(omega_est):
# stacks the lower part of the cholesky decomposition ROW_WISE [(0,0) (1,0) (1,1) (2,0) (2,1) (2,2) ...]
P_c = len(omega_est)
x_chol = np.linalg.cholesky(omega_est)
# transform components of cholesky decomposition for better optimization
lowTr_ind = np.tril_indices(P_c)
x_chol_trf = np.zeros((P_c,P_c))
for i in range(P_c):
for j in range(i): # fill in lower triangular components not on diagonal
x_chol_trf[i,j] = x_chol[i,j]/np.sqrt(x_chol[i,i]*x_chol[j,j])
x_chol_trf[np.diag_indices(P_c)] = np.log(np.diag(x_chol)) # replace with log transformation on the diagonal
return tuple(x_chol_trf[lowTr_ind])
def rebuild_omega(chol_elems, s=None):
'''Rebuild state-dependent Omega given combination of causal states
cholX_elements are the elements (entered row-wise) of the lower triangular cholesky decomposition of Omega_s
'''
if s is None:
P = int((-1 + np.sqrt(1.+ 8.*len(chol_elems)))/2.)
s = np.ones(P,dtype=bool)
P_c = P
else:
P_c = int(np.sum(s))
P = s.shape[1] if s.ndim == 2 else len(s)
cholL = np.zeros((P_c,P_c))
cholL[np.tril_indices(P_c)] = np.array(chol_elems)
cholL[np.diag_indices(P_c)] = np.exp(np.diag(cholL)) # exponentiate the diagnoal so cholL unique
for i in range(P_c):
for j in range(i): # multiply by exponentiated diags
cholL[i,j] = cholL[i,j]*np.sqrt(cholL[i,i]*cholL[j,j])
omega_c = np.dot(cholL, cholL.T)
# Expand to include zeros of matrix
omega = np.zeros((P,P))
s_caus_ind = np.argwhere(np.outer(s, s))
omega[(s_caus_ind[:,0],s_caus_ind[:,1])] = omega_c.flatten()
return omega
def estimate_omega(args,Zs,Ns,sigma_LD, omega_in=None):
# start_time =time.time()
logging.info('Beginning estimation of Omega ...')
M,P = Zs.shape
N_mats = np.sqrt(np.einsum('mp, mq -> mpq',Ns, Ns))
if args.perfect_gencov and args.equal_h2:
logging.info('--perfect_gencov and --equal_h2 option used')
return np.ones((P,P))
if args.numerical_omega:
if omega_in is None: # omega_in serves as starting point
omega_in = np.zeros((P,P))
omega_in[np.diag_indices(P)] = np.diag(gmm_omega(Zs,Ns,sigma_LD))
omega_hat = omega_in
omega_hat, opt_results = numerical_omega(args, Zs,N_mats, sigma_LD,omega_hat)
numerical_msg = "\n Numerical optimization of Omega complete:"
numerical_msg += "\nSuccessful termination? {}".format("Yes" if opt_results.success else "No")
numerical_msg += "\nTermination message:\t{}".format(opt_results.message)
numerical_msg += "\nCompleted in {} iterations".format(opt_results.nit)
logging.info(numerical_msg)
return omega_hat
if args.perfect_gencov:
omega_hat = _posDef_adjustment(gmm_omega(Zs,Ns,sigma_LD))
return np.sqrt(np.outer(np.diag(omega_hat), np.diag(omega_hat)))
# else: gmm_omega (default)
return _posDef_adjustment(gmm_omega(Zs,Ns,sigma_LD))
########################
## MTAG CALCULATION ####
########################
def mtag_analysis(Zs, Ns, omega_hat, sigma_LD):
logging.info('Beginning MTAG calculations...')
M,P = Zs.shape
W_N = np.einsum('mp,pq->mpq',np.sqrt(Ns),np.eye(P))
W_N_inv = np.linalg.inv(W_N)
Sigma_N = np.einsum('mpq,mqr->mpr',np.einsum('mpq,qr->mpr',W_N_inv,sigma_LD),W_N_inv)
mtag_betas = np.zeros((M,P))
mtag_se =np.zeros((M,P))
for p in range(P):
# Note that in the code, what I call "gamma should really be omega", but avoid the latter term due to possible confusion with big Omega
gamma_k = omega_hat[:,p]
tau_k_2 = omega_hat[p,p]
om_min_gam = omega_hat - np.outer(gamma_k,gamma_k)/tau_k_2
xx = om_min_gam + Sigma_N
inv_xx = np.linalg.inv(xx)
yy = gamma_k/tau_k_2
W_inv_Z = np.einsum('mqp,mp->mq',W_N_inv,Zs)
beta_denom = np.einsum('mp,p->m',np.einsum('q,mqp->mp',yy,inv_xx),yy)
mtag_betas[:,p] = np.einsum('mp,mp->m',np.einsum('q,mqp->mp',yy,inv_xx), W_inv_Z) / beta_denom
var_denom = np.einsum('mq,q->m',np.einsum('p,mpq->mq',yy,inv_xx),yy)
mtag_var_p = 1. / var_denom
mtag_se[:,p] = np.sqrt(mtag_var_p)
logging.info(' ... Completed MTAG calculations.')
return mtag_betas, mtag_se
#################
## SAVING RESULTS ##
#########################
def save_mtag_results(args,results_template,Zs,Ns, Fs,mtag_betas,mtag_se):
'''
Output will be of the form:
snp_name z n maf mtag_beta mtag_se mtag_zscore mtag_pval
'''
p_values = lambda z: 2*(scipy.stats.norm.cdf(-1.*np.abs(z)))
M,P = mtag_betas.shape
if args.std_betas:
logging.info('Outputting standardized betas..')
for p in range(P):
logging.info('Writing Phenotype {} to file ...'.format(p+1))
out_df = results_template.copy()
out_df[args.z_name] = Zs[:,p]
out_df[args.n_name] = Ns[:,p]
out_df[args.eaf_name] = Fs[:,p]
if args.std_betas:
weights = np.ones(M,dtype=float)
else:
weights = np.sqrt( 2*Fs[:,p]*(1. - Fs[:,p]))
out_df['mtag_beta'] = mtag_betas[:,p] / weights
out_df['mtag_se'] = mtag_se[:,p] / weights
out_df['mtag_z'] = mtag_betas[:,p]/mtag_se[:,p]
out_df['mtag_pval'] = p_values(out_df['mtag_z'])
if P == 1:
out_path = args.out +'_trait.txt'
else:
out_path = args.out +'_trait_' + str(p+1) + '.txt'
out_df.to_csv(out_path,sep='\t', index=False)
if not args.equal_h2:
omega_out = "\nEstimated Omega:\n"
omega_out += str(args.omega_hat)
omega_out += '\n'
np.savetxt(args.out +'_omega_hat.txt',args.omega_hat, delimiter ='\t')
else:
omega_out = "Omega hat not computed because --equal_h2 was used.\n"
sigma_out = "\nEstimated Sigma:\n"
sigma_out += str(args.sigma_hat)
sigma_out += '\n'
np.savetxt( args.out +'_sigma_hat.txt',args.sigma_hat, delimiter ='\t')
summary_df = pd.DataFrame(index=np.arange(1,P+1))
input_phenotypes = [ '...'+f[-16:] if len(f) > 20 else f for f in args.sumstats.split(',')]
for p in range(P):
summary_df.loc[p+1,'Trait'] = input_phenotypes[p]
summary_df.loc[p+1, 'N (max)'] = np.max(Ns[:,p])
summary_df.loc[p+1, 'N (mean)'] = np.mean(Ns[:,p])
summary_df.loc[p+1, '# SNPs used'] = int(len(Zs[:,p]))
summary_df.loc[p+1, 'GWAS mean chi^2'] = np.mean(np.square(Zs[:,p])) / args.sigma_hat[p,p]
Z_mtag = mtag_betas[:,p]/mtag_se[:,p]
summary_df.loc[p+1, 'MTAG mean chi^2'] = np.mean(np.square(Z_mtag))
summary_df.loc[p+1, 'GWAS equiv. (max) N'] = int(summary_df.loc[p+1, 'N (max)']*(summary_df.loc[p+1, 'MTAG mean chi^2'] - 1) / (summary_df.loc[p+1, 'GWAS mean chi^2'] - 1))
summary_df['N (max)'] = summary_df['N (max)'].astype(int)
summary_df['N (mean)'] = summary_df['N (mean)'].astype(int)
summary_df['# SNPs used'] = summary_df['# SNPs used'].astype(int)
summary_df['GWAS equiv. (max) N'] = summary_df['GWAS equiv. (max) N'].astype(int)
final_summary = "\nSummary of MTAG results:\n"
final_summary +="------------------------\n"
final_summary += str(summary_df.round(3))+'\n'
final_summary += omega_out
final_summary += sigma_out
logging.info(final_summary)
logging.info(' ')
logging.info('MTAG results saved to file.')
'''
Functions for maxFDR parallelization
'''
create_S = lambda P: np.asarray(list(itertools.product([False,True], repeat=P)))
# def _FDR_par(func_args):
# '''
# FDR methods for parallelization
# '''
# probs, t,omega_hat, sigma_hat,S,Ns, coords = func_args
# if coords[0] % 1000 == 0:
# logging.info('Calculating for {}: {}'.format(coords, probs))
# return -1.*compute_fdr(probs, t, omega_hat, sigma_hat, S, Ns) , coords
def MTAG_var_Z_jt_c(t, Omega, Omega_c, sigma_LD, Ns):
'''
Omega: full Omega matrix
Omega_c: conditional Omega
Sigma_LD
N_mean: vector of length of "sample sizes" (1/c**2).
This formula only works with constant N, etc.
'''
T = Ns.shape[1]
W_N = np.einsum('mp,pq->mpq',np.sqrt(Ns),np.eye(T))
W_N_inv = np.linalg.inv(W_N)
Sigma_j = np.einsum('mpq,mqr->mpr',np.einsum('mpq,qr->mpr',W_N_inv,sigma_LD),W_N_inv)
gamma_k = Omega[:,t]
tau_k_2 = Omega[t,t]
om_min_gam = Omega - np.outer(gamma_k, gamma_k) / tau_k_2
xx = om_min_gam + Sigma_j
inv_xx = np.linalg.inv(xx)
# num_L / R are the same due to symmetry
num_L = np.einsum('p,mpq->mq', gamma_k / tau_k_2, inv_xx)
num_R = np.einsum('mpq,q->mp', inv_xx, gamma_k / tau_k_2)
numer = np.einsum('mp,mp->m', num_L, np.einsum('mpq,mq->mp', Omega_c + Sigma_j, num_R))
denom = np.einsum('p,mp->m', gamma_k / tau_k_2, np.einsum('mpq,q->mp', inv_xx, gamma_k /tau_k_2))
return numer / denom
def simplex_walk(num_dims, samples_per_dim):
"""
A generator that returns lattice points on an n-simplex.
"""
max_ = samples_per_dim + num_dims - 1
for c in itertools.combinations(range(max_), num_dims):
#print(c)
c = list(c)
yield np.array([(y - x - 1.) / (samples_per_dim - 1.)
for x, y in itertools.izip([-1] + c, c + [max_])])
def scale_omega(gen_corr_mat, priors, S=None):
assert gen_corr_mat.shape[0] == gen_corr_mat.shape[1]
T = gen_corr_mat.shape[1]
omega = np.zeros_like(gen_corr_mat)
if S is None:
S = create_S(T)
n_S = len(S)
for p1 in range(T):
for p2 in range(T):
# indices of states that are casual for both traits p1 and p2.
caus_state = np.arange(n_S)[np.logical_and(S[:, p1], S[:, p2])]
# print(np.sum(priors[caus_state]))
omega[p1,p2] = gen_corr_mat[p1,p2] / np.sum(priors[caus_state])
return omega
def compute_fdr(prob, t, omega, sigma, S, Ns,N_counts, p_threshold):
z_threshold = scipy.stats.norm.isf(p_threshold / 2.) # magnitude of z-score needed for statistical significance
n_S, T = S.shape
omega_TT = scale_omega(omega, prob, S)
if not is_pos_semidef(omega_TT):
return np.inf
Omega_s = np.einsum('st,sr->str',S,S) * omega_TT
Prob_signif_cond_t = np.zeros(n_S)
power_state_t = np.zeros_like(Prob_signif_cond_t)
for k in range(len(S)):
sd = np.sqrt(MTAG_var_Z_jt_c(t, omega, Omega_s[k,:,:], sigma, Ns)) #ZZZ generalize to take in Omega_s rather than one state at a time .
Prob_signif_cond_t[k] = np.sum(2*scipy.stats.norm.sf(z_threshold, loc=0, scale = sd)*N_counts) / float(np.sum(N_counts)) # produces m FDR estimates: take average by weighting each unique sample size row with the counts of SNPs with that sample size (weighted average, denominator will be equal to M)
power_state_t[k] = Prob_signif_cond_t[k] * float(prob[k])
FDR_val = np.sum(power_state_t[~S[:,t]]) / np.sum(power_state_t)
return FDR_val
def is_pos_semidef(m):
if m.shape[0] == 2 and m.shape[1] == 2:
return np.sqrt(m[0, 0]*m[1,1]) >= np.abs(m[0, 1])
else:
eigs = np.linalg.eigvals(m)
return np.all(eigs >= 0)
def neglogL_single_SS(x, beta, se, transformed=True):
'''
Returns the negative loglikelihood of betas from a spike-slab
distribution. Used in the numerical optimziation of the `ss_estimation`.
Arguments:
----------
x: 2-tuple (pi_null, tau). If transformed, `x` consists of bijective transformations of pi_null and tau so that the image of the mapping is all real numbers.
betas: The Mx1 vector of betas
se: The Mx1 vector of standard errors. Must allign with the reported betas.
transformed: boolean, default True,
If True, will perform inverse transformation on pi_null, tau so that they return to their "correct" domain.
'''
if transformed:
prob_null = 1.0 / (1.0 + np.exp(-1 * x[0]))
tau = np.exp(-x[1])
else:
prob_null, tau = x
causal_pdf = scipy.stats.norm.pdf(beta, loc=0,scale=np.sqrt(tau**2 + se**2))
noncausal_pdf = scipy.stats.norm.pdf(beta,loc=0, scale = se)
return -1. * np.sum(np.log( (1.0-prob_null)*causal_pdf + prob_null * noncausal_pdf))
def cback_print(x):
logging.info(x)
def _optim_ss(f_args):
beta_t, se_t, starting_params, solver_opts = f_args
start_pi, start_tau = starting_params
x_0 = ( 1.0/(1.0 + np.exp(-start_pi)), -np.log(start_tau) )
# beta_t, se_t = f_args
optim_results = scipy.optimize.minimize(neglogL_single_SS, x_0, args=(beta_t, se_t,True), method='Nelder-Mead', options=solver_opts, callback=None)
t_pi, t_tau = optim_results.x
pi_null = 1.0 / (1.0 + np.exp(-1 * t_pi))
tau = np.exp(-t_tau)
return pi_null, tau
def ss_estimation(args, betas, se, max_iter=1000, tol=1.0e-10,
starting_params =(0.5, 1.0e-3),
callback=False):
'''
Numerically fit the distribution of betas and standard errors to a spike slab distribution.
Arguments:
----------
betas: The Mx1 vector of betas
se: The Mx1 vector of standard errors. Must allign with the reported betas.
max_iter: int,
Maximum number of iterations
tol: float,
Tolerance used in numerical optimization (for both fatol, xatol)
starting_params: 2-tuple: (pi_0, tau_0)
Starting parameters for optimization. Default is 0.5, 1.0e-3
callback: boolean ,default False
If True, the parameters values will be printed at each step of optimization.
'''