def plot_funnel(pi_true, sigma_str):
n = pl.exp(mc.rnormal(10, 2**-2, size=10000))
sigma = float(sigma_str)*pl.ones_like(n)
p = pi_true*pl.ones_like(n)
oln = rate_model.offset_log_normal('funnel', p, sigma, p, pl.sqrt(p*(1-p)/n))
r = oln['p_pred'].value
pl.vlines([pi_true], .1*n.min(), 10*n.max(),
linewidth=2, linestyle='-', color='w', zorder=9)
pl.vlines([pi_true], .1*n.min(), 10*n.max(),
linewidth=1, linestyle='--', color='black', zorder=10)
pl.plot(r, n, 'ko',
mew=0, alpha=.25)
pl.semilogy(schiz['r'], schiz['n'], 'ks', mew=1, mec='white', ms=4,
label='Observed values')
pl.xlabel('Rate (per PY)')
pl.ylabel('Study size (PY)')
pl.xticks([0, .005, .01])
pl.axis([-.0001, .0101, 50., 15000000])
pl.title(r'$\sigma = %s$'%sigma_str)
def age_specific_rate(model,
data_type,
reference_area='all',
reference_sex='total',
reference_year='all',
mu_age=None,
mu_age_parent=None,
sigma_age_parent=None,
rate_type='neg_binom',
lower_bound=None,
interpolation_method='linear',
include_covariates=True,
zero_re=False):
# TODO: expose (and document) interface for alternative rate_type as well as other options,
# record reference values in the model
""" Generate PyMC objects for model of epidemological age-interval data
:Parameters:
- `model` : data.ModelData
- `data_type` : str, one of 'i', 'r', 'f', 'p', or 'pf'
- `reference_area, reference_sex, reference_year` : the node of the model to fit consistently
- `mu_age` : pymc.Node, will be used as the age pattern, set to None if not needed
- `mu_age_parent` : pymc.Node, will be used as the age pattern of the parent of the root area, set to None if not needed
- `sigma_age_parent` : pymc.Node, will be used as the standard deviation of the age pattern, set to None if not needed
- `rate_type` : str, optional. One of 'beta_binom', 'binom', 'log_normal_model', 'neg_binom', 'neg_binom_lower_bound_model', 'neg_binom_model', 'normal_model', 'offest_log_normal', or 'poisson'
- `lower_bound` :
- `interpolation_method` : str, optional, one of 'linear', 'nearest', 'zero', 'slinear', 'quadratic, or 'cubic'
- `include_covariates` : boolean
- `zero_re` : boolean, change one stoch from each set of siblings in area hierarchy to a 'sum to zero' deterministic
:Results:
- Returns dict of PyMC objects, including 'pi', the covariate adjusted predicted values for each row of data
"""
name = data_type
import data
result = data.ModelVars()
if (mu_age_parent != None and pl.any(pl.isnan(mu_age_parent))) \
or (sigma_age_parent != None and pl.any(pl.isnan(sigma_age_parent))):
mu_age_parent = None
sigma_age_parent = None
print 'WARNING: nan found in parent mu/sigma. Ignoring'
ages = pl.array(model.parameters['ages'])
data = model.get_data(data_type)
if lower_bound:
lb_data = model.get_data(lower_bound)
parameters = model.parameters.get(data_type, {})
area_hierarchy = model.hierarchy
vars = dismod3.data.ModelVars()
vars += dict(data=data)
if 'parameter_age_mesh' in parameters:
knots = pl.array(parameters['parameter_age_mesh'])
else:
knots = pl.arange(ages[0], ages[-1] + 1, 5)
smoothing_dict = {
'No Prior': pl.inf,
'Slightly': .5,
'Moderately': .05,
'Very': .005
}
if 'smoothness' in parameters:
smoothing = smoothing_dict[parameters['smoothness']['amount']]
else:
smoothing = 0.
if mu_age == None:
vars.update(
age_pattern.age_pattern(name,
ages=ages,
knots=knots,
smoothing=smoothing,
interpolation_method=interpolation_method))
else:
vars.update(dict(mu_age=mu_age, ages=ages))
vars.update(
expert_prior_model.level_constraints(name, parameters, vars['mu_age'],
ages))
vars.update(
expert_prior_model.derivative_constraints(name, parameters,
vars['mu_age'], ages))
if mu_age_parent != None:
# setup a hierarchical prior on the simliarity between the
# consistent estimate here and (inconsistent) estimate for its
# parent in the areas hierarchy
#weight_dict = {'Unusable': 10., 'Slightly': 10., 'Moderately': 1., 'Very': .1}
#weight = weight_dict[parameters['heterogeneity']]
vars.update(
similarity_prior_model.similar('parent_similarity_%s' % name,
vars['mu_age'], mu_age_parent,
sigma_age_parent, 0.))
# also use this as the initial value for the age pattern, if it is not already specified
if mu_age == None:
if isinstance(mu_age_parent, mc.Node): # TODO: test this code
initial_mu = mu_age_parent.value
else:
initial_mu = mu_age_parent
for i, k_i in enumerate(knots):
vars['gamma'][i].value = (pl.log(
initial_mu[k_i - ages[0]])).clip(-12, 6)
age_weights = pl.ones_like(
vars['mu_age'].value
) # TODO: use age pattern appropriate to the rate type
if len(data) > 0:
vars.update(
age_integrating_model.age_standardize_approx(
name, age_weights, vars['mu_age'], data['age_start'],
data['age_end'], ages))
# uncomment the following to effectively remove alleffects
#if 'random_effects' in parameters:
# for i in range(5):
# effect = 'sigma_alpha_%s_%d' % (name, i)
# parameters['random_effects'][effect] = dict(dist='TruncatedNormal', mu=.0001, sigma=.00001, lower=.00009, upper=.00011)
#if 'fixed_effects' in parameters:
# for effect in ['x_sex', 'x_LDI_id_Updated_7July2011']:
# parameters['fixed_effects'][effect] = dict(dist='normal', mu=.0001, sigma=.00001)
if include_covariates:
vars.update(
covariate_model.mean_covariate_model(name,
vars['mu_interval'],
data,
parameters,
model,
reference_area,
reference_sex,
reference_year,
zero_re=zero_re))
else:
vars.update({'pi': vars['mu_interval']})
## ensure that all data has uncertainty quantified appropriately
# first replace all missing se from ci
missing_se = pl.isnan(
data['standard_error']) | (data['standard_error'] < 0)
data['standard_error'][missing_se] = (data['upper_ci'][missing_se] -
data['lower_ci'][missing_se]) / (
2 * 1.96)
# then replace all missing ess with se
missing_ess = pl.isnan(data['effective_sample_size'])
data['effective_sample_size'][missing_ess] = data['value'][
missing_ess] * (1 - data['value'][missing_ess]
) / data['standard_error'][missing_ess]**2
if rate_type == 'neg_binom':
# warn and drop data that doesn't have effective sample size quantified, or is is non-positive
missing_ess = pl.isnan(data['effective_sample_size']) | (
data['effective_sample_size'] < 0)
if sum(missing_ess) > 0:
print 'WARNING: %d rows of %s data has invalid quantification of uncertainty.' % (
sum(missing_ess), name)
data['effective_sample_size'][missing_ess] = 0.0
# warn and change data where ess is unreasonably huge
large_ess = data['effective_sample_size'] >= 1.e10
if sum(large_ess) > 0:
print 'WARNING: %d rows of %s data have effective sample size exceeding 10 billion.' % (
sum(large_ess), name)
data['effective_sample_size'][large_ess] = 1.e10
if 'heterogeneity' in parameters:
lower_dict = {'Slightly': 9., 'Moderately': 3., 'Very': 1.}
lower = lower_dict[parameters['heterogeneity']]
else:
lower = 1.
# special case, treat pf data as poisson
if data_type == 'pf':
lower = 1.e12
vars.update(
covariate_model.dispersion_covariate_model(
name, data, lower, lower * 9.))
vars.update(
rate_model.neg_binom_model(name, vars['pi'], vars['delta'],
data['value'],
data['effective_sample_size']))
elif rate_type == 'log_normal':
# warn and drop data that doesn't have effective sample size quantified
missing = pl.isnan(
data['standard_error']) | (data['standard_error'] < 0)
if sum(missing) > 0:
print 'WARNING: %d rows of %s data has no quantification of uncertainty.' % (
sum(missing), name)
data['standard_error'][missing] = 1.e6
# TODO: allow options for alternative priors for sigma
vars['sigma'] = mc.Uniform('sigma_%s' % name,
lower=.0001,
upper=1.,
value=.01)
#vars['sigma'] = mc.Exponential('sigma_%s'%name, beta=100., value=.01)
vars.update(
rate_model.log_normal_model(name, vars['pi'], vars['sigma'],
data['value'],
data['standard_error']))
elif rate_type == 'normal':
# warn and drop data that doesn't have standard error quantified
missing = pl.isnan(
data['standard_error']) | (data['standard_error'] < 0)
if sum(missing) > 0:
print 'WARNING: %d rows of %s data has no quantification of uncertainty.' % (
sum(missing), name)
data['standard_error'][missing] = 1.e6
vars['sigma'] = mc.Uniform('sigma_%s' % name,
lower=.0001,
upper=.1,
value=.01)
vars.update(
rate_model.normal_model(name, vars['pi'], vars['sigma'],
data['value'], data['standard_error']))
elif rate_type == 'binom':
missing_ess = pl.isnan(data['effective_sample_size']) | (
data['effective_sample_size'] < 0)
if sum(missing_ess) > 0:
print 'WARNING: %d rows of %s data has invalid quantification of uncertainty.' % (
sum(missing_ess), name)
data['effective_sample_size'][missing_ess] = 0.0
vars += rate_model.binom(name, vars['pi'], data['value'],
data['effective_sample_size'])
elif rate_type == 'beta_binom':
vars += rate_model.beta_binom(name, vars['pi'], data['value'],
data['effective_sample_size'])
elif rate_type == 'poisson':
missing_ess = pl.isnan(data['effective_sample_size']) | (
data['effective_sample_size'] < 0)
if sum(missing_ess) > 0:
print 'WARNING: %d rows of %s data has invalid quantification of uncertainty.' % (
sum(missing_ess), name)
data['effective_sample_size'][missing_ess] = 0.0
vars += rate_model.poisson(name, vars['pi'], data['value'],
data['effective_sample_size'])
elif rate_type == 'offset_log_normal':
vars['sigma'] = mc.Uniform('sigma_%s' % name,
lower=.0001,
upper=10.,
value=.01)
vars += rate_model.offset_log_normal(name, vars['pi'],
vars['sigma'], data['value'],
data['standard_error'])
else:
raise Exception, 'rate_model "%s" not implemented' % rate_type
else:
if include_covariates:
vars.update(
covariate_model.mean_covariate_model(name, [],
data,
parameters,
model,
reference_area,
reference_sex,
reference_year,
zero_re=zero_re))
if include_covariates:
vars.update(
expert_prior_model.covariate_level_constraints(
name, model, vars, ages))
if lower_bound and len(lb_data) > 0:
vars['lb'] = age_integrating_model.age_standardize_approx(
'lb_%s' % name, age_weights, vars['mu_age'], lb_data['age_start'],
lb_data['age_end'], ages)
if include_covariates:
vars['lb'].update(
covariate_model.mean_covariate_model('lb_%s' % name,
vars['lb']['mu_interval'],
lb_data,
parameters,
model,
reference_area,
reference_sex,
reference_year,
zero_re=zero_re))
else:
vars['lb'].update({'pi': vars['lb']['mu_interval']})
vars['lb'].update(
covariate_model.dispersion_covariate_model(
'lb_%s' % name, lb_data, 1e12, 1e13) # treat like poisson
)
## ensure that all data has uncertainty quantified appropriately
# first replace all missing se from ci
missing_se = pl.isnan(
lb_data['standard_error']) | (lb_data['standard_error'] <= 0)
lb_data['standard_error'][missing_se] = (
lb_data['upper_ci'][missing_se] -
lb_data['lower_ci'][missing_se]) / (2 * 1.96)
# then replace all missing ess with se
missing_ess = pl.isnan(lb_data['effective_sample_size'])
lb_data['effective_sample_size'][missing_ess] = lb_data['value'][
missing_ess] * (1 - lb_data['value'][missing_ess]
) / lb_data['standard_error'][missing_ess]**2
# warn and drop lb_data that doesn't have effective sample size quantified
missing_ess = pl.isnan(lb_data['effective_sample_size']) | (
lb_data['effective_sample_size'] <= 0)
if sum(missing_ess) > 0:
print 'WARNING: %d rows of %s lower bound data has no quantification of uncertainty.' % (
sum(missing_ess), name)
lb_data['effective_sample_size'][missing_ess] = 1.0
vars['lb'].update(
rate_model.neg_binom_lower_bound_model(
'lb_%s' % name, vars['lb']['pi'], vars['lb']['delta'],
lb_data['value'], lb_data['effective_sample_size']))
result[data_type] = vars
return result
def age_specific_rate(
model,
data_type,
reference_area="all",
reference_sex="total",
reference_year="all",
mu_age=None,
mu_age_parent=None,
sigma_age_parent=None,
rate_type="neg_binom",
lower_bound=None,
interpolation_method="linear",
include_covariates=True,
zero_re=False,
):
# TODO: expose (and document) interface for alternative rate_type as well as other options,
# record reference values in the model
""" Generate PyMC objects for model of epidemological age-interval data
:Parameters:
- `model` : data.ModelData
- `data_type` : str, one of 'i', 'r', 'f', 'p', or 'pf'
- `reference_area, reference_sex, reference_year` : the node of the model to fit consistently
- `mu_age` : pymc.Node, will be used as the age pattern, set to None if not needed
- `mu_age_parent` : pymc.Node, will be used as the age pattern of the parent of the root area, set to None if not needed
- `sigma_age_parent` : pymc.Node, will be used as the standard deviation of the age pattern, set to None if not needed
- `rate_type` : str, optional. One of 'beta_binom', 'binom', 'log_normal_model', 'neg_binom', 'neg_binom_lower_bound_model', 'neg_binom_model', 'normal_model', 'offest_log_normal', or 'poisson'
- `lower_bound` :
- `interpolation_method` : str, optional, one of 'linear', 'nearest', 'zero', 'slinear', 'quadratic, or 'cubic'
- `include_covariates` : boolean
- `zero_re` : boolean, change one stoch from each set of siblings in area hierarchy to a 'sum to zero' deterministic
:Results:
- Returns dict of PyMC objects, including 'pi', the covariate adjusted predicted values for each row of data
"""
name = data_type
import data
result = data.ModelVars()
if (mu_age_parent != None and pl.any(pl.isnan(mu_age_parent))) or (
sigma_age_parent != None and pl.any(pl.isnan(sigma_age_parent))
):
mu_age_parent = None
sigma_age_parent = None
print "WARNING: nan found in parent mu/sigma. Ignoring"
ages = pl.array(model.parameters["ages"])
data = model.get_data(data_type)
if lower_bound:
lb_data = model.get_data(lower_bound)
parameters = model.parameters.get(data_type, {})
area_hierarchy = model.hierarchy
vars = dismod3.data.ModelVars()
vars += dict(data=data)
if "parameter_age_mesh" in parameters:
knots = pl.array(parameters["parameter_age_mesh"])
else:
knots = pl.arange(ages[0], ages[-1] + 1, 5)
smoothing_dict = {"No Prior": pl.inf, "Slightly": 0.5, "Moderately": 0.05, "Very": 0.005}
if "smoothness" in parameters:
smoothing = smoothing_dict[parameters["smoothness"]["amount"]]
else:
smoothing = 0.0
if mu_age == None:
vars.update(
age_pattern.age_pattern(
name, ages=ages, knots=knots, smoothing=smoothing, interpolation_method=interpolation_method
)
)
else:
vars.update(dict(mu_age=mu_age, ages=ages))
vars.update(expert_prior_model.level_constraints(name, parameters, vars["mu_age"], ages))
vars.update(expert_prior_model.derivative_constraints(name, parameters, vars["mu_age"], ages))
if mu_age_parent != None:
# setup a hierarchical prior on the simliarity between the
# consistent estimate here and (inconsistent) estimate for its
# parent in the areas hierarchy
# weight_dict = {'Unusable': 10., 'Slightly': 10., 'Moderately': 1., 'Very': .1}
# weight = weight_dict[parameters['heterogeneity']]
vars.update(
similarity_prior_model.similar(
"parent_similarity_%s" % name, vars["mu_age"], mu_age_parent, sigma_age_parent, 0.0
)
)
# also use this as the initial value for the age pattern, if it is not already specified
if mu_age == None:
if isinstance(mu_age_parent, mc.Node): # TODO: test this code
initial_mu = mu_age_parent.value
else:
initial_mu = mu_age_parent
for i, k_i in enumerate(knots):
vars["gamma"][i].value = (pl.log(initial_mu[k_i - ages[0]])).clip(-12, 6)
age_weights = pl.ones_like(vars["mu_age"].value) # TODO: use age pattern appropriate to the rate type
if len(data) > 0:
vars.update(
age_integrating_model.age_standardize_approx(
name, age_weights, vars["mu_age"], data["age_start"], data["age_end"], ages
)
)
# uncomment the following to effectively remove alleffects
# if 'random_effects' in parameters:
# for i in range(5):
# effect = 'sigma_alpha_%s_%d' % (name, i)
# parameters['random_effects'][effect] = dict(dist='TruncatedNormal', mu=.0001, sigma=.00001, lower=.00009, upper=.00011)
# if 'fixed_effects' in parameters:
# for effect in ['x_sex', 'x_LDI_id_Updated_7July2011']:
# parameters['fixed_effects'][effect] = dict(dist='normal', mu=.0001, sigma=.00001)
if include_covariates:
vars.update(
covariate_model.mean_covariate_model(
name,
vars["mu_interval"],
data,
parameters,
model,
reference_area,
reference_sex,
reference_year,
zero_re=zero_re,
)
)
else:
vars.update({"pi": vars["mu_interval"]})
## ensure that all data has uncertainty quantified appropriately
# first replace all missing se from ci
missing_se = pl.isnan(data["standard_error"]) | (data["standard_error"] < 0)
data["standard_error"][missing_se] = (data["upper_ci"][missing_se] - data["lower_ci"][missing_se]) / (2 * 1.96)
# then replace all missing ess with se
missing_ess = pl.isnan(data["effective_sample_size"])
data["effective_sample_size"][missing_ess] = (
data["value"][missing_ess] * (1 - data["value"][missing_ess]) / data["standard_error"][missing_ess] ** 2
)
if rate_type == "neg_binom":
# warn and drop data that doesn't have effective sample size quantified, or is is non-positive
missing_ess = pl.isnan(data["effective_sample_size"]) | (data["effective_sample_size"] < 0)
if sum(missing_ess) > 0:
print "WARNING: %d rows of %s data has invalid quantification of uncertainty." % (
sum(missing_ess),
name,
)
data["effective_sample_size"][missing_ess] = 0.0
# warn and change data where ess is unreasonably huge
large_ess = data["effective_sample_size"] >= 1.0e10
if sum(large_ess) > 0:
print "WARNING: %d rows of %s data have effective sample size exceeding 10 billion." % (
sum(large_ess),
name,
)
data["effective_sample_size"][large_ess] = 1.0e10
if "heterogeneity" in parameters:
lower_dict = {"Slightly": 9.0, "Moderately": 3.0, "Very": 1.0}
lower = lower_dict[parameters["heterogeneity"]]
else:
lower = 1.0
# special case, treat pf data as poisson
if data_type == "pf":
lower = 1.0e12
vars.update(covariate_model.dispersion_covariate_model(name, data, lower, lower * 9.0))
vars.update(
rate_model.neg_binom_model(
name, vars["pi"], vars["delta"], data["value"], data["effective_sample_size"]
)
)
elif rate_type == "log_normal":
# warn and drop data that doesn't have effective sample size quantified
missing = pl.isnan(data["standard_error"]) | (data["standard_error"] < 0)
if sum(missing) > 0:
print "WARNING: %d rows of %s data has no quantification of uncertainty." % (sum(missing), name)
data["standard_error"][missing] = 1.0e6
# TODO: allow options for alternative priors for sigma
vars["sigma"] = mc.Uniform("sigma_%s" % name, lower=0.0001, upper=1.0, value=0.01)
# vars['sigma'] = mc.Exponential('sigma_%s'%name, beta=100., value=.01)
vars.update(
rate_model.log_normal_model(name, vars["pi"], vars["sigma"], data["value"], data["standard_error"])
)
elif rate_type == "normal":
# warn and drop data that doesn't have standard error quantified
missing = pl.isnan(data["standard_error"]) | (data["standard_error"] < 0)
if sum(missing) > 0:
print "WARNING: %d rows of %s data has no quantification of uncertainty." % (sum(missing), name)
data["standard_error"][missing] = 1.0e6
vars["sigma"] = mc.Uniform("sigma_%s" % name, lower=0.0001, upper=0.1, value=0.01)
vars.update(rate_model.normal_model(name, vars["pi"], vars["sigma"], data["value"], data["standard_error"]))
elif rate_type == "binom":
missing_ess = pl.isnan(data["effective_sample_size"]) | (data["effective_sample_size"] < 0)
if sum(missing_ess) > 0:
print "WARNING: %d rows of %s data has invalid quantification of uncertainty." % (
sum(missing_ess),
name,
)
data["effective_sample_size"][missing_ess] = 0.0
vars += rate_model.binom(name, vars["pi"], data["value"], data["effective_sample_size"])
elif rate_type == "beta_binom":
vars += rate_model.beta_binom(name, vars["pi"], data["value"], data["effective_sample_size"])
elif rate_type == "poisson":
missing_ess = pl.isnan(data["effective_sample_size"]) | (data["effective_sample_size"] < 0)
if sum(missing_ess) > 0:
print "WARNING: %d rows of %s data has invalid quantification of uncertainty." % (
sum(missing_ess),
name,
)
data["effective_sample_size"][missing_ess] = 0.0
vars += rate_model.poisson(name, vars["pi"], data["value"], data["effective_sample_size"])
elif rate_type == "offset_log_normal":
vars["sigma"] = mc.Uniform("sigma_%s" % name, lower=0.0001, upper=10.0, value=0.01)
vars += rate_model.offset_log_normal(name, vars["pi"], vars["sigma"], data["value"], data["standard_error"])
else:
raise Exception, 'rate_model "%s" not implemented' % rate_type
else:
if include_covariates:
vars.update(
covariate_model.mean_covariate_model(
name, [], data, parameters, model, reference_area, reference_sex, reference_year, zero_re=zero_re
)
)
if include_covariates:
vars.update(expert_prior_model.covariate_level_constraints(name, model, vars, ages))
if lower_bound and len(lb_data) > 0:
vars["lb"] = age_integrating_model.age_standardize_approx(
"lb_%s" % name, age_weights, vars["mu_age"], lb_data["age_start"], lb_data["age_end"], ages
)
if include_covariates:
vars["lb"].update(
covariate_model.mean_covariate_model(
"lb_%s" % name,
vars["lb"]["mu_interval"],
lb_data,
parameters,
model,
reference_area,
reference_sex,
reference_year,
zero_re=zero_re,
)
)
else:
vars["lb"].update({"pi": vars["lb"]["mu_interval"]})
vars["lb"].update(
covariate_model.dispersion_covariate_model("lb_%s" % name, lb_data, 1e12, 1e13) # treat like poisson
)
## ensure that all data has uncertainty quantified appropriately
# first replace all missing se from ci
missing_se = pl.isnan(lb_data["standard_error"]) | (lb_data["standard_error"] <= 0)
lb_data["standard_error"][missing_se] = (lb_data["upper_ci"][missing_se] - lb_data["lower_ci"][missing_se]) / (
2 * 1.96
)
# then replace all missing ess with se
missing_ess = pl.isnan(lb_data["effective_sample_size"])
lb_data["effective_sample_size"][missing_ess] = (
lb_data["value"][missing_ess]
* (1 - lb_data["value"][missing_ess])
/ lb_data["standard_error"][missing_ess] ** 2
)
# warn and drop lb_data that doesn't have effective sample size quantified
missing_ess = pl.isnan(lb_data["effective_sample_size"]) | (lb_data["effective_sample_size"] <= 0)
if sum(missing_ess) > 0:
print "WARNING: %d rows of %s lower bound data has no quantification of uncertainty." % (
sum(missing_ess),
name,
)
lb_data["effective_sample_size"][missing_ess] = 1.0
vars["lb"].update(
rate_model.neg_binom_lower_bound_model(
"lb_%s" % name,
vars["lb"]["pi"],
vars["lb"]["delta"],
lb_data["value"],
lb_data["effective_sample_size"],
)
)
result[data_type] = vars
return result
pl.title(r'$\sigma = %s$'%sigma_str)
pl.figure(figsize=(11, 8.5), dpi=120)
pl.subplots_adjust(wspace=.4)
pl.subplot(2,2,1)
plot_funnel(.004, '0.5')
pl.subplot(2,2,2)
plot_funnel(.004, '0.1')
pl.subplot(2,1,2)
r = pl.array(schiz['r'])
n = pl.array(schiz['n'])
pi = mc.Uniform('pi', 0, 1, value=.001)
sigma = mc.Uniform('sigma', 0, 100, value=.0001)
oln = rate_model.offset_log_normal('funnel', pi*pl.ones_like(n), sigma*pl.ones_like(n), r, pl.sqrt(r*(1-r)/n))
mcmc = mc.MCMC([pi, sigma, oln])
mcmc.sample(20000, 10000, 10, verbose=False, progress_bar=False)
sorted_indices = r.argsort().argsort()
jitter = mc.rnormal(0, .1**-2, len(oln['p_pred'].trace()))
for i,s_i in enumerate(sorted_indices):
pl.plot(s_i+jitter, oln['p_pred'].trace()[:, i], 'ko', mew=0, alpha=.25, zorder=-99)
pl.errorbar(sorted_indices, r, yerr=1.96*pl.sqrt(r*(1-r)/n), fmt='ws', mew=1, mec='white', ms=5, elinewidth=3, capsize=0)
pl.errorbar(sorted_indices, r, yerr=1.96*pl.sqrt(r*(1-r)/n), fmt='ks', mew=1, mec='white', ms=5)
pl.xticks([])
pl.ylabel('Rate (per PY)')
pl.axis([-.5, 15.5,-.0001,.0121])