def SCpm(SC_0=SC_0, i=i, r=r, f=f, m_all_cause=m_all_cause, age_mesh=dm.get_param_age_mesh()):
SC = np.zeros([2, len(age_mesh)])
p = np.zeros(len(age_mesh))
m = np.zeros(len(age_mesh))
SC[:, 0] = SC_0
p[0] = SC_0[1] / (SC_0[0] + SC_0[1])
m[0] = trim(
m_all_cause[age_mesh[0]] - f[age_mesh[0]] * p[0], 0.1 * m_all_cause[age_mesh[0]], 1 - NEARLY_ZERO
) # trim m[0] to avoid numerical instability
for ii, a in enumerate(age_mesh[:-1]):
A = np.array([[-i[a] - m[ii], r[a]], [i[a], -r[a] - m[ii] - f[a]]]) * (age_mesh[ii + 1] - age_mesh[ii])
SC[:, ii + 1] = np.dot(scipy.linalg.expm(A), SC[:, ii])
p[ii + 1] = trim(SC[1, ii + 1] / (SC[0, ii + 1] + SC[1, ii + 1]), NEARLY_ZERO, 1 - NEARLY_ZERO)
m[ii + 1] = trim(
m_all_cause[age_mesh[ii + 1]] - f[age_mesh[ii + 1]] * p[ii + 1],
0.1 * m_all_cause[age_mesh[ii + 1]],
1 - NEARLY_ZERO,
)
SCpm = np.zeros([4, len(age_mesh)])
SCpm[0:2, :] = SC
SCpm[2, :] = p
SCpm[3, :] = m
return SCpm
def setup(dm, key, data_list, rate_stoch=None, emp_prior={}):
""" Generate the PyMC variables for a beta binomial model of
a single rate function
Parameters
----------
dm : dismod3.DiseaseModel
the object containing all the data, priors, and additional
information (like input and output age-mesh)
key : str
the name of the key for everything about this model (priors,
initial values, estimations)
data_list : list of data dicts
the observed data to use in the beta-binomial liklihood function
rate_stoch : pymc.Stochastic, optional
a PyMC stochastic (or deterministic) object, with
len(rate_stoch.value) == len(dm.get_estimation_age_mesh()).
This is used to link beta-binomial stochs into a larger model,
for example.
emp_prior : dict, optional
the empirical prior dictionary, retrieved from the disease model
if appropriate by::
>>> t, r, y, s = type_region_year_sex_from_key(key)
>>> emp_prior = dm.get_empirical_prior(t)
Results
-------
vars : dict
Return a dictionary of all the relevant PyMC objects for the
beta binomial model. vars['rate_stoch'] is of particular
relevance; this is what is used to link the beta-binomial model
into more complicated models, like the generic disease model.
Details
-------
The beta binomial model parameters are the following:
* the mean age-specific rate function
* dispersion of this mean
* the p_i value for each data observation that has a standard
error (data observations that do not have standard errors
recorded are fit as observations of the beta r.v., while
observations with standard errors recorded have a latent
variable for the beta, and an observed binomial r.v.).
"""
vars = {}
est_mesh = dm.get_estimate_age_mesh()
if np.any(np.diff(est_mesh) != 1):
raise ValueError, "ERROR: Gaps in estimation age mesh must all equal 1"
# set up age-specific rate function, if it does not yet exist
if not rate_stoch:
param_mesh = dm.get_param_age_mesh()
if emp_prior.has_key("mu"):
initial_value = emp_prior["mu"]
else:
initial_value = dm.get_initial_value(key)
# find the logit of the initial values, which is a little bit
# of work because initial values are sampled from the est_mesh,
# but the logit_initial_values are needed on the param_mesh
logit_initial_value = mc.logit(interpolate(est_mesh, initial_value, param_mesh))
logit_rate = mc.Normal(
"logit(%s)" % key, mu=-5.0 * np.ones(len(param_mesh)), tau=1.0e-2, value=logit_initial_value
)
# logit_rate = [mc.Normal('logit(%s)_%d' % (key, a), mu=-5., tau=1.e-2) for a in param_mesh]
vars["logit_rate"] = logit_rate
@mc.deterministic(name=key)
def rate_stoch(logit_rate=logit_rate):
return interpolate(param_mesh, mc.invlogit(logit_rate), est_mesh)
if emp_prior.has_key("mu"):
@mc.potential(name="empirical_prior_%s" % key)
def emp_prior_potential(f=rate_stoch, mu=emp_prior["mu"], tau=1.0 / np.array(emp_prior["se"]) ** 2):
return mc.normal_like(f, mu, tau)
vars["empirical_prior"] = emp_prior_potential
vars["rate_stoch"] = rate_stoch
# create stochastic variable for over-dispersion "random effect"
mu_od = emp_prior.get("dispersion", 0.001)
dispersion = mc.Gamma("dispersion_%s" % key, alpha=10.0, beta=10.0 / mu_od)
vars["dispersion"] = dispersion
@mc.deterministic(name="alpha_%s" % key)
def alpha(rate=rate_stoch, dispersion=dispersion):
return rate / dispersion ** 2
@mc.deterministic(name="beta_%s" % key)
def beta(rate=rate_stoch, dispersion=dispersion):
return (1.0 - rate) / dispersion ** 2
vars["alpha"] = alpha
vars["beta"] = beta
# create potentials for priors
vars["priors"] = generate_prior_potentials(dm.get_priors(key), est_mesh, rate_stoch, dispersion)
# create latent and observed stochastics for data
vars["data"] = data_list
vars["ab"] = []
vars["latent_p"] = []
vars["observations"] = []
for d in data_list:
# set up observed stochs for all relevant data
id = d["id"]
if d["value"] == MISSING:
print "WARNING: data %d missing value" % id
continue
# ensure all rate data is valid
d_val = dm.value_per_1(d)
d_se = dm.se_per_1(d)
if d_val < 0 or d_val > 1:
print "WARNING: data %d not in range [0,1]" % id
continue
if d["age_start"] < est_mesh[0] or d["age_end"] > est_mesh[-1]:
raise ValueError, "Data %d is outside of estimation range---([%d, %d] is not inside [%d, %d])" % (
d["id"],
d["age_start"],
d["age_end"],
est_mesh[0],
est_mesh[-1],
)
age_indices = indices_for_range(est_mesh, d["age_start"], d["age_end"])
age_weights = d["age_weights"]
@mc.deterministic(name="a_%d^%s" % (id, key))
def a_i(alpha=alpha, age_indices=age_indices, age_weights=age_weights):
return rate_for_range(alpha, age_indices, age_weights)
@mc.deterministic(name="b_%d^%s" % (id, key))
def b_i(beta=beta, age_indices=age_indices, age_weights=age_weights):
return rate_for_range(beta, age_indices, age_weights)
vars["ab"] += [a_i, b_i]
if d_se > 0:
# if the data has a standard error, model it as a realization
# of a beta binomial r.v.
latent_p_i = mc.Beta(
"latent_p_%d^%s" % (id, key), alpha=a_i, beta=b_i, value=trim(d_val, NEARLY_ZERO, 1 - NEARLY_ZERO)
)
vars["latent_p"].append(latent_p_i)
denominator = d_val * (1 - d_val) / d_se ** 2.0
numerator = d_val * denominator
obs_binomial = mc.Binomial(
"data_%d^%s" % (id, key), value=numerator, n=denominator, p=latent_p_i, observed=True
)
vars["observations"].append(obs_binomial)
else:
# if the data is a point estimate with no uncertainty
# recorded, model it as a realization of a beta r.v.
obs_p_i = mc.Beta(
"latent_p_%d" % id, value=trim(d_val, NEARLY_ZERO, 1 - NEARLY_ZERO), alpha=a_i, beta=b_i, observed=True
)
vars["observations"].append(obs_p_i)
return vars
def generate_disease_data(condition, cov):
""" Generate csv files with gold-standard disease data,
and somewhat good, somewhat dense disease data, as might be expected from a
condition that is carefully studied in the literature
"""
age_len = dismod3.MAX_AGE
ages = np.arange(age_len, dtype='float')
# incidence rate
i0 = .005 + .02 * mc.invlogit((ages - 44) / 3)
#i0 = np.maximum(0., .001 * (-.125 + np.ones_like(ages) + (ages / age_len)**2.))
# remission rate
#r = 0. * ages
r = .1 * np.ones_like(ages)
# excess-mortality rate
#f_init = .085 * (ages / 100) ** 2.5
SMR = 3. * np.ones_like(ages) - ages / age_len
# all-cause mortality-rate
mort = dismod3.get_disease_model('all-cause_mortality')
#age_intervals = [[a, a+9] for a in range(0, dismod3.MAX_AGE-4, 10)] + [[0, 100] for ii in range(1)]
age_intervals = [[a, a] for a in range(0, dismod3.MAX_AGE, 1)]
# TODO: take age structure from real data
sparse_intervals = dict([[
region,
random.sample(age_intervals,
(ii**3 * len(age_intervals)) / len(countries_for)**3 / 1)
] for ii, region in enumerate(countries_for)])
dense_intervals = dict(
[[region, random.sample(age_intervals,
len(age_intervals) / 2)]
for ii, region in enumerate(countries_for)])
gold_data = []
noisy_data = []
for ii, region in enumerate(sorted(countries_for)):
if region == 'world':
continue
print region
sys.stdout.flush()
# introduce unexplained regional variation
#i = i0 * (1 + float(ii) / 21)
# or not
i = i0
for year in [1990, 2005]:
for sex in ['male', 'female']:
param_type = 'all-cause_mortality'
key = dismod3.gbd_key_for(param_type, region, year, sex)
m_all_cause = mort.mortality(key, mort.data)
# calculate excess-mortality rate from smr
f = (SMR - 1.) * m_all_cause
## compartmental model (bins S, C, D, M)
import scipy.linalg
from dismod3 import NEARLY_ZERO
from dismod3.utils import trim
SCDM = np.zeros([4, age_len])
p = np.zeros(age_len)
m = np.zeros(age_len)
SCDM[0, 0] = 1.
SCDM[1, 0] = 0.
SCDM[2, 0] = 0.
SCDM[3, 0] = 0.
p[0] = SCDM[1, 0] / (SCDM[0, 0] + SCDM[1, 0] + NEARLY_ZERO)
m[0] = trim(m_all_cause[0] - f[0] * p[0], NEARLY_ZERO,
1 - NEARLY_ZERO)
for a in range(age_len - 1):
A = [[-i[a] - m[a], r[a], 0., 0.],
[i[a], -r[a] - m[a] - f[a], 0., 0.],
[m[a], m[a], 0., 0.], [0., f[a], 0., 0.]]
SCDM[:, a + 1] = np.dot(scipy.linalg.expm(A), SCDM[:, a])
p[a + 1] = SCDM[1, a + 1] / (SCDM[0, a + 1] +
SCDM[1, a + 1] + NEARLY_ZERO)
m[a + 1] = m_all_cause[a + 1] - f[a + 1] * p[a + 1]
# duration = E[time in bin C]
hazard = r + m + f
pr_not_exit = np.exp(-hazard)
X = np.empty(len(hazard))
X[-1] = 1 / hazard[-1]
for ii in reversed(range(len(X) - 1)):
X[ii] = (pr_not_exit[ii] *
(X[ii + 1] + 1)) + (1 / hazard[ii] *
(1 - pr_not_exit[ii]) -
pr_not_exit[ii])
country = countries_for[region][0]
params = dict(age_intervals=age_intervals,
condition=condition,
gbd_region=region,
country=country,
year=year,
sex=sex,
effective_sample_size=1000)
params['age_intervals'] = [[0, 99]]
generate_and_append_data(gold_data, 'prevalence data', p,
**params)
generate_and_append_data(gold_data, 'incidence data', i,
**params)
generate_and_append_data(gold_data, 'excess-mortality data', f,
**params)
generate_and_append_data(gold_data, 'remission data', r,
**params)
generate_and_append_data(gold_data, 'duration data', X,
**params)
# TODO: use this approach to age standardize all gold data, and then change it to get iX as a direct sum
params['age_intervals'] = [[0, 99]]
iX = i * X * (1 - p) * regional_population(key)
generate_and_append_data(gold_data, 'incidence_x_duration', iX,
**params)
params['effective_sample_size'] = 1000
params['cov'] = 0.
params['age_intervals'] = age_intervals
generate_and_append_data(noisy_data, 'prevalence data', p,
**params)
generate_and_append_data(noisy_data, 'excess-mortality data',
f, **params)
generate_and_append_data(noisy_data, 'remission data', r,
**params)
generate_and_append_data(noisy_data, 'incidence data', i,
**params)
col_names = sorted(data_dict_for_csv(gold_data[0]).keys())
f_file = open(OUTPUT_PATH + '%s_gold.tsv' % condition, 'w')
csv_f = csv.writer(f_file, dialect='excel-tab')
csv_f.writerow(col_names)
for d in gold_data:
dd = data_dict_for_csv(d)
csv_f.writerow([dd[c] for c in col_names])
f_file.close()
f_name = OUTPUT_PATH + '%s_data.tsv' % condition
f_file = open(f_name, 'w')
csv_f = csv.writer(f_file, dialect='excel-tab')
csv_f.writerow(col_names)
for d in noisy_data:
dd = data_dict_for_csv(d)
csv_f.writerow([dd[c] for c in col_names])
f_file.close()
# upload data file
from dismod3.disease_json import dismod_server_login, twc, DISMOD_BASE_URL
dismod_server_login()
twc.go(DISMOD_BASE_URL + 'dismod/data/upload/')
twc.formvalue(1, 'tab_separated_values', open(f_name).read())
# TODO: find or set the model number for this model, set the
# expert priors and covariates, merge the covariate data into the
# model, and add the "ground truth" to the disease json
try:
url = twc.submit()
except Exception, e:
print e
def generate_disease_data(condition, cov):
""" Generate csv files with gold-standard disease data,
and somewhat good, somewhat dense disease data, as might be expected from a
condition that is carefully studied in the literature
"""
age_len = dismod3.MAX_AGE
ages = np.arange(age_len, dtype='float')
# incidence rate
i0 = .005 + .02 * mc.invlogit((ages - 44) / 3)
#i0 = np.maximum(0., .001 * (-.125 + np.ones_like(ages) + (ages / age_len)**2.))
# remission rate
#r = 0. * ages
r = .1 * np.ones_like(ages)
# excess-mortality rate
#f_init = .085 * (ages / 100) ** 2.5
SMR = 3. * np.ones_like(ages) - ages / age_len
# all-cause mortality-rate
mort = dismod3.get_disease_model('all-cause_mortality')
#age_intervals = [[a, a+9] for a in range(0, dismod3.MAX_AGE-4, 10)] + [[0, 100] for ii in range(1)]
age_intervals = [[a, a] for a in range(0, dismod3.MAX_AGE, 1)]
# TODO: take age structure from real data
sparse_intervals = dict([[region, random.sample(age_intervals, (ii**3 * len(age_intervals)) / len(countries_for)**3 / 1)] for ii, region in enumerate(countries_for)])
dense_intervals = dict([[region, random.sample(age_intervals, len(age_intervals)/2)] for ii, region in enumerate(countries_for)])
gold_data = []
noisy_data = []
for ii, region in enumerate(sorted(countries_for)):
if region == 'world':
continue
print region
sys.stdout.flush()
# introduce unexplained regional variation
#i = i0 * (1 + float(ii) / 21)
# or not
i = i0
for year in [1990, 2005]:
for sex in ['male', 'female']:
param_type = 'all-cause_mortality'
key = dismod3.gbd_key_for(param_type, region, year, sex)
m_all_cause = mort.mortality(key, mort.data)
# calculate excess-mortality rate from smr
f = (SMR - 1.) * m_all_cause
## compartmental model (bins S, C, D, M)
import scipy.linalg
from dismod3 import NEARLY_ZERO
from dismod3.utils import trim
SCDM = np.zeros([4, age_len])
p = np.zeros(age_len)
m = np.zeros(age_len)
SCDM[0,0] = 1.
SCDM[1,0] = 0.
SCDM[2,0] = 0.
SCDM[3,0] = 0.
p[0] = SCDM[1,0] / (SCDM[0,0] + SCDM[1,0] + NEARLY_ZERO)
m[0] = trim(m_all_cause[0] - f[0] * p[0], NEARLY_ZERO, 1-NEARLY_ZERO)
for a in range(age_len - 1):
A = [[-i[a]-m[a], r[a] , 0., 0.],
[ i[a] , -r[a]-m[a]-f[a], 0., 0.],
[ m[a], m[a] , 0., 0.],
[ 0., f[a], 0., 0.]]
SCDM[:,a+1] = np.dot(scipy.linalg.expm(A), SCDM[:,a])
p[a+1] = SCDM[1,a+1] / (SCDM[0,a+1] + SCDM[1,a+1] + NEARLY_ZERO)
m[a+1] = m_all_cause[a+1] - f[a+1] * p[a+1]
# duration = E[time in bin C]
hazard = r + m + f
pr_not_exit = np.exp(-hazard)
X = np.empty(len(hazard))
X[-1] = 1 / hazard[-1]
for ii in reversed(range(len(X)-1)):
X[ii] = (pr_not_exit[ii] * (X[ii+1] + 1)) + (1 / hazard[ii] * (1 - pr_not_exit[ii]) - pr_not_exit[ii])
country = countries_for[region][0]
params = dict(age_intervals=age_intervals, condition=condition, gbd_region=region,
country=country, year=year, sex=sex, effective_sample_size=1000)
params['age_intervals'] = [[0,99]]
generate_and_append_data(gold_data, 'prevalence data', p, **params)
generate_and_append_data(gold_data, 'incidence data', i, **params)
generate_and_append_data(gold_data, 'excess-mortality data', f, **params)
generate_and_append_data(gold_data, 'remission data', r, **params)
generate_and_append_data(gold_data, 'duration data', X, **params)
# TODO: use this approach to age standardize all gold data, and then change it to get iX as a direct sum
params['age_intervals'] = [[0,99]]
iX = i * X * (1-p) * regional_population(key)
generate_and_append_data(gold_data, 'incidence_x_duration', iX, **params)
params['effective_sample_size'] = 1000
params['cov'] = 0.
params['age_intervals'] = age_intervals
generate_and_append_data(noisy_data, 'prevalence data', p, **params)
generate_and_append_data(noisy_data, 'excess-mortality data', f, **params)
generate_and_append_data(noisy_data, 'remission data', r, **params)
generate_and_append_data(noisy_data, 'incidence data', i, **params)
col_names = sorted(data_dict_for_csv(gold_data[0]).keys())
f_file = open(OUTPUT_PATH + '%s_gold.tsv' % condition, 'w')
csv_f = csv.writer(f_file, dialect='excel-tab')
csv_f.writerow(col_names)
for d in gold_data:
dd = data_dict_for_csv(d)
csv_f.writerow([dd[c] for c in col_names])
f_file.close()
f_name = OUTPUT_PATH + '%s_data.tsv' % condition
f_file = open(f_name, 'w')
csv_f = csv.writer(f_file, dialect='excel-tab')
csv_f.writerow(col_names)
for d in noisy_data:
dd = data_dict_for_csv(d)
csv_f.writerow([dd[c] for c in col_names])
f_file.close()
# upload data file
from dismod3.disease_json import dismod_server_login, twc, DISMOD_BASE_URL
dismod_server_login()
twc.go(DISMOD_BASE_URL + 'dismod/data/upload/')
twc.formvalue(1, 'tab_separated_values', open(f_name).read())
# TODO: find or set the model number for this model, set the
# expert priors and covariates, merge the covariate data into the
# model, and add the "ground truth" to the disease json
try:
url = twc.submit()
except Exception, e:
print e
## compartmental model (bins S, C, D, M)
import scipy.linalg
from dismod3 import NEARLY_ZERO
from dismod3.utils import trim
SCDM = np.zeros([4, age_len])
p = np.zeros(age_len)
m = np.zeros(age_len)
SCDM[0, 0] = 1.
SCDM[1, 0] = 0.
SCDM[2, 0] = NEARLY_ZERO
SCDM[3, 0] = NEARLY_ZERO
p[0] = SCDM[1, 0] / (SCDM[0, 0] + SCDM[1, 0] + NEARLY_ZERO)
m[0] = trim(m_all_cause[0] - f[0] * p[0], NEARLY_ZERO,
1 - NEARLY_ZERO)
for a in range(age_len - 1):
A = [[-i[a] - m[a], r[a], 0., 0.],
[i[a], -r[a] - m[a] - f[a], 0., 0.], [m[a], m[a], 0., 0.],
[0., f[a], 0., 0.]]
SCDM[:, a + 1] = np.dot(scipy.linalg.expm(A), SCDM[:, a])
p[a + 1] = SCDM[1, a + 1] / (SCDM[0, a + 1] + SCDM[1, a + 1] +
NEARLY_ZERO)
m[a + 1] = trim(m_all_cause[a + 1] - f[a + 1] * p[a + 1],
.1 * m_all_cause[a + 1], 1 - NEARLY_ZERO)
# duration = E[time in bin C]
pr_exit = 1 - r - m - f
## compartmental model (bins S, C, D, M)
import scipy.linalg
from dismod3 import NEARLY_ZERO
from dismod3.utils import trim
SCDM = np.zeros([4, age_len])
p = np.zeros(age_len)
m = np.zeros(age_len)
SCDM[0,0] = 1.
SCDM[1,0] = 0.
SCDM[2,0] = NEARLY_ZERO
SCDM[3,0] = NEARLY_ZERO
p[0] = SCDM[1,0] / (SCDM[0,0] + SCDM[1,0] + NEARLY_ZERO)
m[0] = trim(m_all_cause[0] - f[0] * p[0], NEARLY_ZERO, 1-NEARLY_ZERO)
for a in range(age_len - 1):
A = [[-i[a]-m[a], r[a] , 0., 0.],
[ i[a] , -r[a]-m[a]-f[a], 0., 0.],
[ m[a], m[a] , 0., 0.],
[ 0., f[a], 0., 0.]]
SCDM[:,a+1] = np.dot(scipy.linalg.expm(A), SCDM[:,a])
p[a+1] = SCDM[1,a+1] / (SCDM[0,a+1] + SCDM[1,a+1] + NEARLY_ZERO)
m[a+1] = trim(m_all_cause[a+1] - f[a+1] * p[a+1], .1*m_all_cause[a+1], 1-NEARLY_ZERO)
# duration = E[time in bin C]
pr_exit = 1 - r - m - f
# excess-mortality rate
f = .085 * (ages / 100) ** 2.5
truth[key % 'excess-mortality'] = f
## compartmental model (bins S, C, D, M)
SCDM = np.zeros([4, age_len])
SCDM[0,0] = 1.
for a in range(age_len - 1):
A = [[-i[a]-m[a], r[a] , 0., 0.],
[ i[a] , -r[a]-m[a]-f[a], 0., 0.],
[ m[a], m[a] , 0., 0.],
[ 0., f[a], 0., 0.]]
SCDM[:,a+1] = trim(np.dot(scipy.linalg.expm2(A), SCDM[:,a]), 0, 1)
S = SCDM[0,:]
C = SCDM[1,:]
# prevalence = # with condition / (# with condition + # without)
p = C / (S + C + NEARLY_ZERO)
truth[key % 'prevalence'] = p
truth[key % 'relative-risk'] = (m + f) / m
# duration = E[time in bin C]
pr_exit = 1 - r - m - f
X = np.empty(len(pr_exit))
t = 1.
for a in xrange(len(X) - 1, -1, -1):
X[a] = t * pr_exit[a]
