scipy-yli/yli/regress.py

#   scipy-yli: Helpful SciPy utilities and recipes
#   Copyright © 2022–2023  Lee Yingtong Li (RunasSudo)
#
#   This program is free software: you can redistribute it and/or modify
#   it under the terms of the GNU Affero General Public License as published by
#   the Free Software Foundation, either version 3 of the License, or
#   (at your option) any later version.
#
#   This program is distributed in the hope that it will be useful,
#   but WITHOUT ANY WARRANTY; without even the implied warranty of
#   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
#   GNU Affero General Public License for more details.
#
#   You should have received a copy of the GNU Affero General Public License
#   along with this program.  If not, see <https://www.gnu.org/licenses/>.

import numpy as np
import pandas as pd
import patsy
from scipy import stats
from scipy.special import expit
import statsmodels, statsmodels.base.model, statsmodels.miscmodels.ordinal_model
import statsmodels.api as sm
from statsmodels.iolib.table import SimpleTable
from statsmodels.stats.outliers_influence import variance_inflation_factor

from datetime import datetime
import io
import itertools
import json
import subprocess
import weakref

from .bayes_factors import BayesFactor, bayesfactor_afbf
from .config import config
from .shap import ShapResult
from .sig_tests import ChiSquaredResult, FTestResult
from .utils import Estimate, PValueStyle, as_numeric, check_nan, cols_for_formula, convert_pandas_nullable, fmt_p, formula_factor_ref_category, parse_patsy_term

# TODO: Documentation
# TODO: Bootstrap

def vif(df, formula=None, *, nan_policy='warn'):
	"""
	Calculate the variance inflation factor (VIF) for each variable in *df*
	
	:param df: Data to calculate the VIF for
	:type df: DataFrame
	:param formula: If specified, calculate the VIF only for the variables in the formula
	:type formula: str
	
	:return: The variance inflation factors
	:rtype: Series
	
	**Example:**
	
	.. code-block::
		
		df = pd.DataFrame({
			'D': [68.58, 67.33, 67.33, ...],
			'T1': [14, 10, 10, ...],
			'T2': [46, 73, 85, ...],
			...
		})
		yli.vif(df[['D', 'T1', 'T2', ...]])
	
	.. code-block:: text
		
		D     8.318301
		T1    6.081590
		T2    2.457122
		...
		dtype: float64
	
	The output shows the variance inflation factor for each variable in *df*.
	"""
	
	if formula:
		# Only consider columns in the formula
		df = df[cols_for_formula(formula, df)]
	
	# Check for/clean NaNs
	df = check_nan(df, nan_policy)
	
	# Convert all to float64 otherwise statsmodels chokes with "ufunc 'isfinite' not supported for the input types ..."
	df = pd.get_dummies(df, drop_first=True)  # Convert categorical dtypes
	df = df.astype('float64')  # Convert all other dtypes
	
	# Add intercept column
	orig_columns = list(df.columns)
	df['Intercept'] = [1] * len(df)
	
	vifs = {}
	
	for i, col in enumerate(orig_columns):
		vifs[col] = variance_inflation_factor(df, i)
	
	return pd.Series(vifs)

def regress(
	model_class, df, dep, formula,
	*,
	nan_policy='warn',
	exposure=None, family=None, status=None,
	method=None, maxiter=None, start_params=None, tolerance=None,
	reduced=None,
	bool_baselevels=False, exp=None
):
	"""
	Fit a statsmodels regression model
	
	:param model_class: Type of regression model to fit
	:type model_class: :class:`yli.regress.RegressionModel` subclass
	:param df: Data to perform regression on
	:type df: DataFrame
	:param dep: Column(s) in *df* for the dependent variable (numeric)
	:type dep: str
	:param formula: Patsy formula for the regression model
	:type formula: str
	:param nan_policy: How to handle *nan* values (see :ref:`nan-handling`)
	:type nan_policy: str
	:param exposure: Column in *df* for the exposure variable (numeric, some models only)
	:type exposure: str
	:param family: See :class:`yli.GLM`
	:type family: str
	:param status: See :class:`yli.Cox`
	:type status: str
	:param method: See statsmodels *model.fit*
	:param maxiter: See statsmodels *model.fit*
	:param start_params: See statsmodels *model.fit*
	:param tolerance: See statsmodels *model.fit*
	:param reduced: See :meth:`yli.IntervalCensoredCox`
	:param bool_baselevels: Show reference categories for boolean independent variables even if reference category is *False*
	:type bool_baselevels: bool
	:param exp: Report exponentiated parameters rather than raw parameters, default (*None*) is to autodetect based on *model_class*
	:type exp: bool
	
	:rtype: :class:`yli.regress.RegressionModel`
	
	**Example:** See :class:`yli.OLS`, :class:`yli.Logit`, etc.
	"""
	
	if not any(x.__name__ == 'RegressionModel' for x in model_class.__bases__):
		raise ValueError('model_class must be a RegressionModel')
	
	# Additional columns to check for NaN - exposure, status, etc.
	additional_columns = []
	
	# Build function call arguments
	fit_kwargs = {}
	if exposure is not None:
		additional_columns.append(exposure)
	if family is not None:
		fit_kwargs['family'] = family
	if status is not None:
		additional_columns.append(status)
	if method is not None:
		fit_kwargs['method'] = method
	if maxiter is not None:
		fit_kwargs['maxiter'] = maxiter
	if start_params is not None:
		fit_kwargs['start_params'] = start_params
	if tolerance is not None:
		fit_kwargs['tolerance'] = tolerance
	if reduced is not None:
		fit_kwargs['reduced'] = reduced
	
	# Preprocess data, check for NaN and get design matrices
	df_ref = weakref.ref(df)
	df_clean, dmatrices, dep_categories = df_to_dmatrices(df, dep, formula, nan_policy, additional_columns)
	
	# Add function call arguments for supplemental columns - exposure, status, etc.
	if exposure is not None:
		fit_kwargs['exposure'] = df_clean[exposure]
	if status is not None:
		fit_kwargs['status'] = df_clean[status]
	
	# Fit model
	result = model_class.fit(dmatrices[0], dmatrices[1], **fit_kwargs)
	
	# Fill in general information
	result.df = df_ref
	result.dep = dep
	result.formula = formula
	result.nan_policy = nan_policy
	if exp is not None:
		result.exp = exp
	result.fitted_dt = datetime.now()
	result.nobs = len(dmatrices[0])
	
	# Process categorical terms, etc.
	raw_terms = result.terms
	result.terms = {}
	
	for raw_name, raw_term in raw_terms.items():
		# Rename terms
		if raw_name == '(Intercept)':
			# Already processed
			result.terms[raw_name] = raw_term
		elif raw_name == '(Cutoffs)':
			result.terms[raw_name] = raw_term
			
			if dep_categories is not None:
				# Need to convert factorised names back into original names
				old_cutoffs = raw_term.categories
				raw_term.categories = {}
				
				for cutoff_raw_name, cutoff_term in old_cutoffs.items():
					bits = cutoff_raw_name.split('/')
					term = dep_categories[round(float(bits[0]))] + '/' + dep_categories[round(float(bits[1]))]
					result.terms[raw_name].categories[term] = cutoff_term
		else:
			# Parse if required
			factor, column, contrast = parse_patsy_term(formula, df_clean, raw_name)
			
			if contrast is not None:
				# Categorical term
				
				if bool_baselevels is False and contrast == 'True' and set(df[column].unique()) == set([True, False]):
					# Treat as single term
					result.terms[column] = raw_term
				else:
					# Add a new categorical term if not exists
					if column not in result.terms:
						ref_category = formula_factor_ref_category(formula, df_clean, factor)
						result.terms[column] = CategoricalTerm({}, ref_category)
					
					result.terms[column].categories[contrast] = raw_term
			else:
				# Single term
				result.terms[column] = raw_term
	
	return result

def df_to_dmatrices(df, dep, formula, nan_policy, additional_columns=[]):
	# Check for/clean NaNs in input columns
	columns = cols_for_formula(dep, df) + cols_for_formula(formula, df)
	
	if additional_columns:
		columns += additional_columns
	
	df = df[columns]
	df = check_nan(df, nan_policy)
	
	# Ensure numeric type for dependent variable
	if '+' in dep:
		dep_categories = None
	else:
		df[dep], dep_categories = as_numeric(df[dep])
	
	# Convert pandas nullable types for independent variables as this breaks statsmodels
	df = convert_pandas_nullable(df)
	
	# Construct design matrix from formula
	dmatrices = patsy.dmatrices(dep + ' ~ ' + formula, df, return_type='dataframe')
	
	return df, dmatrices, dep_categories

class RegressionModel:
	# TODO: Documentation
	
	def __init__(self):
		# Model configuration (all set in yli.regress)
		self.df = None
		self.dep = None
		self.formula = None
		self.nan_policy = None
		self.exp = False
		self.cov_type = None
		
		# Summary information
		self.fitted_dt = None  # Set in yli.regress
		self.nobs = None  # Set in yli.regress
		self.nevents = None
		self.dof_model = None
		self.dof_resid = None
		self.rsquared = None
		self.ll_model = None
		self.ll_null = None
		self.ll_saturated = None
		self.f_statistic = None
		
		# Parameters
		self.terms = None
		self.vcov = None
		
		# Comments
		self.comments = []
	
	@property
	def model_long_name(self):
		return None
	
	@property
	def model_short_name(self):
		return None
	
	def _header_table(self, html):
		"""Return the entries for the header table"""
		
		# Left column
		left_col = []
		
		left_col.append(('Dep. Variable:', self.dep))
		left_col.append(('Model:', self.model_short_name))
		left_col.append(('Date:', self.fitted_dt.strftime('%Y-%m-%d')))
		left_col.append(('Time:', self.fitted_dt.strftime('%H:%M:%S')))
		if self.cov_type:
			left_col.append(('Std. Errors:', 'Non-Robust' if self.cov_type == 'nonrobust' else self.cov_type.upper() if self.cov_type.startswith('hc') else self.cov_type))
		
		# Right column
		right_col = []
		
		right_col.append(('No. Observations:', format(self.nobs, '.0f')))
		if self.nevents:
			right_col.append(('No. Events:', format(self.nevents, '.0f')))
		if self.dof_model:
			right_col.append(('Df. Model:', format(self.dof_model, '.0f')))
		if self.dof_resid:
			right_col.append(('Df. Residuals:', format(self.dof_resid, '.0f')))
		if self.rsquared:
			right_col.append(('<i>R</i><sup>2</sup>:' if html else 'R²:', format(self.rsquared, '.2f')))
		elif self.ll_null:
			right_col.append(('Pseudo <i>R</i><sup>2</sup>:' if html else 'Pseudo R²:', format(self.pseudo_rsquared, '.2f')))
		if self.f_statistic:
			# Report the F test if available
			f_result = self.ftest()
			
			if html:
				right_col.append(('<i>F</i>:', format(f_result.statistic, '.2f')))
				right_col.append(('<i>p</i> (<i>F</i>):', fmt_p(f_result.pvalue, PValueStyle.VALUE_ONLY | PValueStyle.HTML)))
			else:
				right_col.append(('F:', format(f_result.statistic, '.2f')))
				right_col.append(('p (F):', fmt_p(f_result.pvalue, PValueStyle.VALUE_ONLY)))
		else:
			# Otherwise report likelihood ratio test as overall test
			right_col.append(('LL-Model:', format(self.ll_model, '.2f')))
			
			if self.ll_null:
				lrtest_result = self.lrtest_null()
				
				right_col.append(('LL-Null:', format(self.ll_null, '.2f')))
				if html:
					right_col.append(('<i>p</i> (LR):', fmt_p(lrtest_result.pvalue, PValueStyle.VALUE_ONLY | PValueStyle.HTML)))
				else:
					right_col.append(('p (LR):', fmt_p(lrtest_result.pvalue, PValueStyle.VALUE_ONLY)))
		
		return left_col, right_col
	
	def __repr__(self):
		if config.repr_is_summary:
			return self.summary()
		return super().__repr__()
	
	def _repr_html_(self):
		# Render header table
		left_col, right_col = self._header_table(html=True)
		
		out = '<table><caption>{} Results</caption>'.format(self.model_long_name)
		for left_cell, right_cell in itertools.zip_longest(left_col, right_col):
			out += '<tr><th>{}</th><td>{}</td><th>{}</th><td>{}</td></tr>'.format(
				left_cell[0] if left_cell else '',
				left_cell[1] if left_cell else '',
				right_cell[0] if right_cell else '',
				right_cell[1] if right_cell else ''
			)
		out += '</table>'
		
		# Render coefficients table
		out += '<table><tr><th></th><th style="text-align:center">{}</th><th colspan="3" style="text-align:center">({:g}% CI)</th><th style="text-align:center"><i>p</i></th></tr>'.format('exp(<i>β</i>)' if self.exp else '<i>β</i>', (1-config.alpha)*100)
		
		for term_name, term in self.terms.items():
			if isinstance(term, SingleTerm):
				# Single term
				
				# Exponentiate beta if requested
				beta = term.beta
				if self.exp:
					beta = np.exp(beta)
				
				out += '<tr><th>{}</th><td>{:.2f}</td><td style="padding-right:0">({:.2f}</td><td>–</td><td style="padding-left:0">{:.2f})</td><td style="text-align:left">{}</td></tr>'.format(term_name, beta.point, beta.ci_lower, beta.ci_upper, fmt_p(term.pvalue, PValueStyle.TABULAR | PValueStyle.HTML))
			elif isinstance(term, CategoricalTerm):
				# Categorical term
				out += '<tr><th>{}</th><td></td><td style="padding-right:0"></td><td></td><td style="padding-left:0"></td><td></td></tr>'.format(term_name)
				
				# Render reference category
				if term.ref_category is not None:
					out += '<tr><td style="text-align:right;font-style:italic">{}</td><td>Ref.</td><td style="padding-right:0"></td><td></td><td style="padding-left:0"></td><td></td></tr>'.format(term.ref_category)
				
				# Loop over terms
				for sub_term_name, sub_term in term.categories.items():
					# Exponentiate beta if requested
					beta = sub_term.beta
					if self.exp:
						beta = np.exp(beta)
					
					out += '<tr><td style="text-align:right;font-style:italic">{}</td><td>{:.2f}</td><td style="padding-right:0">({:.2f}</td><td>–</td><td style="padding-left:0">{:.2f})</td><td style="text-align:left">{}</td></tr>'.format(sub_term_name, beta.point, beta.ci_lower, beta.ci_upper, fmt_p(sub_term.pvalue, PValueStyle.TABULAR | PValueStyle.HTML))
			else:
				raise Exception('Attempt to render unknown term type')
		
		out += '</table>'
		
		# TODO: Have a detailed view which shows SE, t/z, etc.
		
		if self.comments:
			out += '<ol>'
			for comment in self.comments:
				out += '<li>{}</li>'.format(comment)
			out += '</ol>'
		
		return out
	
	def summary(self):
		"""
		Return a stringified summary of the regression model
		
		:rtype: str
		"""
		
		# Render header table
		left_col, right_col = self._header_table(html=False)
		
		# Ensure equal sizes for SimpleTable
		if len(right_col) > len(left_col):
			left_col.extend([('', '')] * (len(right_col) - len(left_col)))
		elif len(left_col) > len(right_col):
			right_col.extend([('', '')] * (len(left_col) - len(right_col)))
		
		table1 = SimpleTable(np.concatenate([left_col, right_col], axis=1), title='{} Results'.format(self.model_long_name))
		table1.insert_stubs(2, [' | '] * len(left_col))
		
		# Get rid of last line (merge with next table)
		table1_lines = table1.as_text().splitlines(keepends=False)
		out = '\n'.join(table1_lines[:-1]) + '\n'
		
		# Render coefficients table
		table_data = []
		
		for term_name, term in self.terms.items():
			if isinstance(term, SingleTerm):
				# Single term
				
				# Exponentiate beta if requested
				beta = term.beta
				if self.exp:
					beta = np.exp(beta)
				
				# Add some extra padding
				table_data.append([term_name + '  ', format(beta.point, '.2f'), '({:.2f}'.format(beta.ci_lower), '-', '{:.2f})'.format(beta.ci_upper), '  ' + fmt_p(term.pvalue, PValueStyle.TABULAR)])
			elif isinstance(term, CategoricalTerm):
				# Categorical term
				table_data.append([term_name + '  ', '', '', '', '', ''])
				
				# Render reference category
				if term.ref_category is not None:
					table_data.append(['{}  '.format(term.ref_category), 'Ref.', '', '', '', ''])
				
				# Loop over terms
				for sub_term_name, sub_term in term.categories.items():
					# Exponentiate beta if requested
					beta = sub_term.beta
					if self.exp:
						beta = np.exp(beta)
					
					table_data.append([sub_term_name + '  ', format(beta.point, '.2f'), '({:.2f}'.format(beta.ci_lower), '-', '{:.2f})'.format(beta.ci_upper), '  ' + fmt_p(sub_term.pvalue, PValueStyle.TABULAR)])
			else:
				raise Exception('Attempt to render unknown term type')
		
		table2 = SimpleTable(data=table_data, headers=['', 'exp(β)' if self.exp else 'β', '', '\ue000', '', ' p'])  # U+E000 is in Private Use Area, mark middle of CI column
		table2_text = table2.as_text().replace('    \ue000   ', '({:g}% CI)'.format((1-config.alpha)*100))  # Render heading in the right spot
		table2_lines = table2_text.splitlines(keepends=False)
		
		# Render divider line between 2 tables
		max_table_len = max(len(table1_lines[-1]), len(table2_lines[-1]))
		out += '=' * max_table_len + '\n'
		
		out += '\n'.join(table2_lines[1:])
		
		if self.comments:
			out += '\n'
			for i, comment in enumerate(self.comments):
				out += '\n{}. {}'.format(i + 1, comment)
		
		return out
	
	def terms_flat(self):
		"""
		Iterate over each :class:`SingleTerm` in *self.terms*, recursively stepping through :class:`CategoricalTerm`\ s
		"""
		
		for t in self.terms.values():
			if isinstance(t, CategoricalTerm):
				for t in t.categories.values():
					yield t
			else:
				yield t
	
	# --------------------
	# Post hoc tests, etc.
	
	def bayesfactor_beta_zero(self, term):
		"""
		Compute a Bayes factor testing the hypothesis that the given beta is 0
		
		Uses the R *BFpack* library.
		
		Requires the regression to be from statsmodels.
		The term must be specified as the *raw name* from the statsmodels regression, available via :attr:`SingleTerm.raw_name`.
		
		:param term: Raw name of the term to be tested
		:type term: str
		
		:rtype: :class:`yli.bayes_factors.BayesFactor`
		"""
		
		# FIXME: Allow specifying our renamed terms
		
		# Get parameters required for AFBF
		raw_params = {}
		for t in self.terms_flat():
			raw_params[t.raw_name.replace('[', '_').replace(']', '_')] = t.beta.point
		
		# Compute BF matrix
		bf01 = bayesfactor_afbf(pd.Series(raw_params), self.vcov, self.nobs, '{} = 0'.format(term.replace('[', '_').replace(']', '_')))
		bf01 = BayesFactor(bf01.factor, '0', '{} = 0'.format(term), '1', '{} ≠ 0'.format(term))
		
		if bf01.factor >= 1:
			return bf01
		else:
			return bf01.invert()
	
	def deviance_chi2(self):
		"""
		Perform the deviance *χ*:sup:`2` test for goodness of fit
		
		:rtype: :class:`yli.sig_tests.ChiSquaredResult`
		"""
		
		deviance_model = 2 * (self.ll_saturated - self.ll_model)
		pvalue = 1 - stats.chi2.cdf(deviance_model, df=self.dof_resid)
		
		return ChiSquaredResult(deviance_model, int(self.dof_resid), pvalue)
	
	def ftest(self):
		"""
		Perform the *F* test that all slopes are 0
		
		:rtype: :class:`yli.sig_tests.FTestResult`
		"""
		
		pvalue = 1 - stats.f(self.dof_model, self.dof_resid).cdf(self.f_statistic)
		return FTestResult(self.f_statistic, self.dof_model, self.dof_resid, pvalue)
	
	def lrtest_null(self):
		"""
		Compute the likelihood ratio test comparing the model to the null model
		
		:rtype: :class:`LikelihoodRatioTestResult`
		"""
		
		statistic = -2 * (self.ll_null - self.ll_model)
		pvalue = 1 - stats.chi2.cdf(statistic, self.dof_model)
		LikelihoodRatioTestResult
		return LikelihoodRatioTestResult(statistic, self.dof_model, pvalue)
	
	@property
	def pseudo_rsquared(self):
		"""McFadden's pseudo *R*:sup:`2` statistic"""
		
		return 1 - self.ll_model/self.ll_null
	
	@property
	def pseudo_rsquared_adj(self):
		"""McFadden's pseudo *R*:sup:`2` statistic, adjusted for number of parameters"""
		
		return 1 - (self.ll_model - self.dof_model) / self.ll_null
	
	def shap(self, **kwargs):
		"""
		Compute SHAP values for the model
		
		Uses the Python *shap* library.
		
		:param kwargs: Keyword arguments to pass to *shap.LinearExplainer*
		
		:rtype: :class:`yli.shap.ShapResult`
		
		**Reference:** Lundberg SM, Lee SI. A unified approach to interpreting model predictions. In: Guyon I, Von Luxburg U, Bengio S, et al., editors. *Advances in Neural Information Processing Systems*; 2017 Dec 4–9; Long Beach, CA. https://proceedings.neurips.cc/paper/2017/hash/8a20a8621978632d76c43dfd28b67767-Abstract.html
		"""
		
		import shap
		
		xdata = ShapResult._get_xdata(self)
		
		# Combine terms into single list
		params = []
		for term in self.terms.values():
			if isinstance(term, SingleTerm):
				params.append(term.beta.point)
			else:
				params.extend(s.beta.point for s in term.categories.values())
		
		explainer = shap.LinearExplainer((np.array(params[1:]), params[0]), xdata, **kwargs)  # FIXME: Assumes zeroth term is intercept
		shap_values = explainer.shap_values(xdata).astype('float')
		
		return ShapResult(weakref.ref(self), shap_values, list(xdata.columns))

class LikelihoodRatioTestResult(ChiSquaredResult):
	"""
	Result of a likelihood ratio test for regression
	
	See :meth:`RegressionModel.lrtest_null`.
	"""
	
	def __init__(self, statistic, dof, pvalue):
		super().__init__(statistic, dof, pvalue)
	
	def _repr_html_(self):
		return 'LR({}) = {:.2f}; <i>p</i> {}'.format(self.dof, self.statistic, fmt_p(self.pvalue, PValueStyle.RELATION | PValueStyle.HTML))
	
	def summary(self):
		"""
		Return a stringified summary of the likelihood ratio test
		
		:rtype: str
		"""
		
		return 'LR({}) = {:.2f}; p {}'.format(self.dof, self.statistic, fmt_p(self.pvalue, PValueStyle.RELATION))

class SingleTerm:
	"""A term in a :class:`RegressionModel` which is a single term"""
	
	def __init__(self, raw_name, beta, pvalue):
		#: Raw name of the term (*str*)
		self.raw_name = raw_name
		#: :class:`yli.utils.Estimate` of the coefficient
		self.beta = beta
		#: *p* value for the coefficient (*float*)
		self.pvalue = pvalue

class CategoricalTerm:
	"""A group of terms in a :class:`RegressionModel` corresponding to a categorical variable"""
	
	def __init__(self, categories, ref_category):
		#: Terms for each of the categories, excluding the reference category (*dict* of :class:`SingleTerm`)
		self.categories = categories
		#: Name of the reference category (*str*)
		self.ref_category = ref_category

def raw_terms_from_statsmodels_result(raw_result, *, wrapped=True):
	if wrapped:
		zipped_iter = zip(raw_result.model.exog_names, raw_result.params.values, raw_result.conf_int(config.alpha).values, raw_result.pvalues.values)
	else:
		zipped_iter = zip(raw_result.model.exog_names, raw_result.params, raw_result.conf_int(config.alpha), raw_result.pvalues)
	
	return {
		('(Intercept)' if raw_name == 'Intercept' else raw_name): SingleTerm(
			raw_name=raw_name,
			beta=Estimate(raw_param, raw_ci[0], raw_ci[1]),
			pvalue=raw_p
		)
		for raw_name, raw_param, raw_ci, raw_p in zipped_iter
	}

# ------------------------
# Concrete implementations

class Cox(RegressionModel):
	# TODO: Documentation
	
	@property
	def model_long_name(self):
		return 'Cox Regression'
	
	@property
	def model_short_name(self):
		return 'Cox'
	
	@classmethod
	def fit(cls, data_dep, data_ind, *, status=None):
		# Drop explicit intercept term
		if 'Intercept' in data_ind:
			del data_ind['Intercept']
		
		result = cls()
		result.exp = True
		result.cov_type = 'nonrobust'
		
		# Perform regression
		raw_result = sm.PHReg(endog=data_dep, exog=data_ind, status=status, missing='raise').fit(disp=False)
		
		result.ll_model = raw_result.llf
		
		result.terms = raw_terms_from_statsmodels_result(raw_result, wrapped=False)
		result.vcov = raw_result.cov_params()
		
		return result

class GLM(RegressionModel):
	# TODO: Documentation
	
	@property
	def model_long_name(self):
		return 'Generalised Linear Model'
	
	@property
	def model_short_name(self):
		return 'GLM'
	
	@classmethod
	def fit(cls, data_dep, data_ind, family, method='irls', maxiter=None, start_params=None):
		result = cls()
		result.exp = True
		result.cov_type = 'nonrobust'
		
		if family == 'binomial':
			sm_family = sm.families.Binomial()
		elif family == 'gamma':
			sm_family = sm.families.Gamma()
		elif family == 'gaussian':
			sm_family = sm.families.Gaussian()
		elif family == 'inverse_gaussian':
			sm_family = sm.families.InverseGaussian()
		elif family == 'negative_binomial':
			sm_family = sm.families.NegativeBinomial()
		elif family == 'poisson':
			sm_family = sm.families.Poisson()
		elif family == 'tweedie':
			sm_family = sm.families.Tweedie()
		else:
			raise ValueError('Unknown GLM family')
		
		# Perform regression
		raw_result = sm.GLM(endog=data_dep, exog=data_ind, family=sm_family, missing='raise').fit(disp=False, method=method, start_params=start_params)
		
		result.dof_model = raw_result.df_model
		result.dof_resid = raw_result.df_resid
		result.ll_model = raw_result.llf
		result.ll_null = raw_result.llnull
		
		result.terms = raw_terms_from_statsmodels_result(raw_result)
		result.vcov = raw_result.cov_params()
		
		return result

class IntervalCensoredCox(RegressionModel):
	"""
	Interval-censored Cox regression
	
	Uses hpstat *intcox* command.
	
	**Example:**
	
	.. code-block::
		
		df = pd.DataFrame(...)
		yli.regress(yli.IntervalCensoredCox, df, 'Left_Time + Right_Time', 'A + B + C + D + E + F + G + H')
	
	.. code-block:: text
		
		              Interval-Censored Cox Regression Results             
		===================================================================
		Dep. Variable: Left_Time + Right_Time  |  No. Observations:    1124
		        Model:  Interval-Censored Cox  |        No. Events:     133
		         Date:             2023-04-17  |         Df. Model:       8
		         Time:               22:30:40  |         Pseudo R²:    0.00
		  Std. Errors:                    OPG  |          LL-Model: -613.21
		                                       |           LL-Null: -615.81
		                                       |            p (LR):    0.82
		===================================================================
		    exp(β)   (95% CI)         p   
		----------------------------------
		A     0.83 (0.37 - 1.88)    0.66  
		B     1.08 (0.81 - 1.46)    0.60  
		C     0.49 (0.24 - 1.00)    0.052 
		D     0.79 (0.42 - 1.50)    0.48  
		E     0.87 (0.40 - 1.85)    0.71  
		F     0.64 (0.28 - 1.45)    0.29  
		G     1.07 (0.44 - 2.62)    0.88  
		H     1.23 (0.48 - 3.20)    0.67  
		----------------------------------
	
	The output summarises the result of the regression.
	
	**Reference:** Zeng D, Mao L, Lin DY. Maximum likelihood estimation for semiparametric transformation models with interval-censored data. *Biometrika*. 2016;103(2):253–71. `doi:10.1093/biomet/asw013 <https://doi.org/10.1093/biomet/asw013>`_
	"""
	
	@property
	def model_long_name(self):
		return 'Interval-Censored Cox Regression'
	
	@property
	def model_short_name(self):
		return 'Interval-Censored Cox'
	
	def __init__(self):
		super().__init__()
		
		# TODO: Documentation
		self.lambda_ = None
	
	@classmethod
	def fit(cls, data_dep, data_ind, *, reduced=False, maxiter=None, tolerance=None):
		if len(data_dep.columns) != 2:
			raise ValueError('IntervalCensoredCox requires left and right times')
		
		# Drop explicit intercept term
		if 'Intercept' in data_ind:
			del data_ind['Intercept']
		
		result = cls()
		result.exp = True
		result.cov_type = 'OPG'
		result.nevents = np.isfinite(data_dep.iloc[:, 1]).sum()
		result.dof_model = len(data_ind.columns)
		
		# Prepare arguments
		intcox_args = [config.hpstat_path, 'intcox', '-', '--output', 'json']
		if reduced:
			intcox_args.append('--reduced')
		if maxiter:
			intcox_args.append('--max-iterations')
			intcox_args.append(str(maxiter))
		if tolerance:
			intcox_args.append('--param-tolerance')
			intcox_args.append(str(tolerance))
		
		# Export data to CSV
		csv_buf = io.StringIO()
		data_dep.join(data_ind).to_csv(csv_buf, index=False)
		csv_str = csv_buf.getvalue()
		
		# Run intcens binary
		proc = subprocess.run(intcox_args, input=csv_str, stdout=subprocess.PIPE, stderr=None, encoding='utf-8', check=True)
		raw_result = json.loads(proc.stdout)
		
		from IPython.display import clear_output
		clear_output(wait=True)
		
		z_critical = -stats.norm.ppf(config.alpha / 2)
		result.terms = {raw_name: SingleTerm(
			raw_name=raw_name,
			beta=Estimate(raw_param, raw_param - z_critical * raw_se, raw_param + z_critical * raw_se),
			pvalue=stats.norm.cdf(-np.abs(raw_param) / raw_se) * 2
		) for raw_name, raw_param, raw_se in zip(data_ind.columns, raw_result['params'], raw_result['params_se'])}
		
		result.cumulative_hazard = pd.Series(data=raw_result['cumulative_hazard'], index=raw_result['cumulative_hazard_times'])
		
		result.ll_model = raw_result['ll_model']
		result.ll_null = raw_result['ll_null']
		
		return result
	
	def survival_function(self):
		# TODO: Documentation
		
		return np.exp(-self.cumulative_hazard)
	
	def plot_survival_function(self, ax=None):
		# TODO: Documentation
		
		import matplotlib.pyplot as plt
		
		if ax is None:
			fig, ax = plt.subplots()
		
		sf = self.survival_function()
		
		# Draw straight lines
		# np.concatenate(...) to force starting drawing from time 0, survival 100%
		xpoints = np.concatenate([[0], sf.index]).repeat(2)[1:]
		ypoints = np.concatenate([[1], sf]).repeat(2)[:-1]
		
		ax.plot(xpoints, ypoints)
		
		ax.set_xlabel('Analysis time')
		ax.set_ylabel('Survival probability')
		ax.set_xlim(left=0)
		ax.set_ylim(0, 1)
	
	def plot_loglog_survival(self, ax=None):
		# TODO: Documentation
		
		import matplotlib.pyplot as plt
		
		if ax is None:
			fig, ax = plt.subplots()
		
		sf = self.survival_function()
		
		# Draw straight lines
		xpoints = np.log(sf.index).to_numpy().repeat(2)[1:]
		ypoints = (-np.log(-np.log(sf))).to_numpy().repeat(2)[:-1]
		
		ax.plot(xpoints, ypoints)
		
		ax.set_xlabel('ln(Analysis time)')
		ax.set_ylabel('−ln(−ln(Survival probability))')

class Logit(RegressionModel):
	"""
	Logistic regression
	
	**Example:**
	
	.. code-block::
		
		df = pd.DataFrame({
			'Unhealthy': [False, False, False, ...],
			'Fibrinogen': [2.52, 2.46, 2.29, ...],
			'GammaGlobulin': [38, 36, 36, ...]
		})
		yli.regress(yli.Logit, df, 'Unhealthy', 'Fibrinogen + GammaGlobulin')
	
	.. code-block:: text
		
		             Logistic Regression Results              
		======================================================
		Dep. Variable:  Unhealthy  |  No. Observations:     32
		        Model:      Logit  |         Df. Model:      2
		         Date: 2022-10-18  |     Df. Residuals:     29
		         Time:   19:00:34  |         Pseudo R²:   0.26
		  Std. Errors: Non-Robust  |          LL-Model: -11.47
		                           |           LL-Null: -15.44
		                           |            p (LR):  0.02*
		======================================================
		                exp(β)   (95% CI)          p   
		-----------------------------------------------
		  (Intercept)     0.00 (0.00 -  0.24)    0.03* 
		   Fibrinogen     6.80 (1.01 - 45.79)    0.049*
		GammaGlobulin     1.17 (0.92 -  1.48)    0.19  
		-----------------------------------------------
	
	The output summarises the results of the regression.
	Note that the parameter estimates are automatically exponentiated.
	For example, the adjusted odds ratio for unhealthiness per unit increase in fibrinogen is 6.80, with 95% confidence interval 1.01–45.79, and is significant with *p* value 0.049.
	"""
	
	@property
	def model_long_name(self):
		return 'Logistic Regression'
	
	@property
	def model_short_name(self):
		return 'Logit'
	
	@classmethod
	def fit(cls, data_dep, data_ind):
		result = cls()
		result.exp = True
		result.cov_type = 'nonrobust'
		
		# Perform regression
		raw_result = sm.Logit(endog=data_dep, exog=data_ind, missing='raise').fit(disp=False)
		
		result.dof_model = raw_result.df_model
		result.dof_resid = raw_result.df_resid
		result.ll_model = raw_result.llf
		result.ll_null = raw_result.llnull
		
		result.terms = raw_terms_from_statsmodels_result(raw_result)
		result.vcov = raw_result.cov_params()
		
		return result

class OLS(RegressionModel):
	"""
	Ordinary least squares linear regression
	
	**Example:**
	
	.. code-block::
		
		df = pd.DataFrame(...)
		yli.regress(yli.OLS, df, 'LNC', 'D + T1 + T2 + S + PR + NE + CT + BW + N + PT')
	
	.. code-block:: text
		
		       Ordinary Least Squares Regression Results       
		=======================================================
		Dep. Variable:        LNC  |  No. Observations:      32
		        Model:        OLS  |         Df. Model:      10
		         Date: 2023-04-16  |     Df. Residuals:      21
		         Time:   23:34:01  |                R²:    0.86
		  Std. Errors: Non-Robust  |                 F:   13.28
		                           |             p (F): <0.001*
		=======================================================
		                β        (95% CI)         p   
		----------------------------------------------
		(Intercept)   -10.63 (-22.51 - 1.24)    0.08  
		          D     0.23   (0.05 - 0.41)    0.02* 
		         T1     0.01  (-0.04 - 0.05)    0.82  
		         T2     0.01  (-0.00 - 0.02)    0.24  
		          S     0.00   (0.00 - 0.00)   <0.001*
		         PR    -0.11  (-0.28 - 0.07)    0.21  
		         NE     0.26   (0.09 - 0.42)    0.004*
		         CT     0.12  (-0.03 - 0.26)    0.12  
		         BW     0.04  (-0.18 - 0.26)    0.73  
		          N    -0.01  (-0.03 - 0.00)    0.14  
		         PT    -0.22  (-0.49 - 0.05)    0.10  
		----------------------------------------------
	
	The output summarises the results of the regression.
	For example, the adjusted mean difference in "LNC" per unit increase in "D" is 0.23, with 95% confidence interval 0.05–0.41, and is significant with *p* value 0.02.
	"""
	
	@property
	def model_long_name(self):
		return 'Ordinary Least Squares Regression'
	
	@property
	def model_short_name(self):
		return 'OLS'
	
	@classmethod
	def fit(cls, data_dep, data_ind):
		result = cls()
		result.cov_type = 'nonrobust'
		
		# Perform regression
		raw_result = sm.OLS(endog=data_dep, exog=data_ind, missing='raise', hasconst=True).fit()
		
		result.dof_model = raw_result.df_model
		result.dof_resid = raw_result.df_resid
		result.rsquared = raw_result.rsquared
		result.ll_model = raw_result.llf
		result.f_statistic = raw_result.fvalue
		
		result.terms = raw_terms_from_statsmodels_result(raw_result)
		result.vcov = raw_result.cov_params()
		
		return result

class OrdinalLogit(RegressionModel):
	"""
	Ordinal logistic (or probit) regression
	
	The implementation calls statsmodels' native *OrderedModel*, but substitutes an alternative parameterisation used by R and Stata.
	In the native statsmodels implementation, the first cutoff parameter is the true cutoff, but further cutoff parameter are log differences between consecutive cutoffs.
	In this parameterisation, cutoff terms are represented directly in the model.
	
	**Example:**
	
	.. code-block::
		
		df = pd.DataFrame(...)
		yli.regress(yli.OrdinalLogit, df, 'apply', 'pared + public + gpa', exp=False)
	
	.. code-block:: text
		
		           Ordinal Logistic Regression Results            
		==========================================================
		Dep. Variable:         apply  |  No. Observations:     400
		        Model: Ordinal Logit  |         Df. Model:       5
		         Date:    2022-12-02  |     Df. Residuals:     395
		         Time:      21:30:38  |         Pseudo R²:    0.03
		  Std. Errors:    Non-Robust  |          LL-Model: -358.51
		                              |           LL-Null: -370.60
		                              |            p (LR): <0.001*
		============================================================
		                                β      (95% CI)         p   
		------------------------------------------------------------
		                      pared    1.05  (0.53 - 1.57)   <0.001*
		                     public   -0.06 (-0.64 - 0.53)    0.84  
		                        gpa    0.62  (0.10 - 1.13)    0.02* 
		                  (Cutoffs)                                 
		   unlikely/somewhat likely    2.20  (0.68 - 3.73)    0.005*
		somewhat likely/very likely    4.30  (2.72 - 5.88)   <0.001*
		------------------------------------------------------------
	
	The output summarises the result of the regression.
	The parameters shown under "(Cutoffs)" are the cutoff values in the latent variable parameterisation of ordinal regression.
	Note that because `exp=False` was passed, the parameter estimates are not automatically exponentiated.
	"""
	
	@property
	def model_long_name(self):
		return 'Ordinal Logistic Regression'
	
	@property
	def model_short_name(self):
		return 'Ordinal Logit'
	
	@classmethod
	def fit(cls, data_dep, data_ind):
		result = cls()
		result.cov_type = 'nonrobust'
		
		# Drop explicit intercept term
		if 'Intercept' in data_ind:
			del data_ind['Intercept']
		
		# Perform regression
		raw_result = _SMOrdinalLogit(endog=data_dep, exog=data_ind, missing='raise').fit(disp=False)
		
		result.dof_model = raw_result.df_model
		result.dof_resid = raw_result.df_resid
		result.ll_model = raw_result.llf
		result.ll_null = raw_result.llnull
		
		raw_terms = raw_terms_from_statsmodels_result(raw_result)
		result.terms = {}
		
		# Group ordinal regression cutoffs
		for raw_name, raw_term in raw_terms.items():
			if '/' in raw_name:
				if '(Cutoffs)' not in result.terms:
					result.terms['(Cutoffs)'] = CategoricalTerm({}, None)
				
				result.terms['(Cutoffs)'].categories[raw_name] = raw_term
			else:
				result.terms[raw_name] = raw_term
		
		return result
	
	def brant(self):
		"""
		Perform the Brant test for the parallel regression assumption in ordinal regression
		
		:rtype: :class:`BrantResult`
		
		**Example:**
		
		.. code-block::
			
			df = pd.DataFrame(...)
			model = yli.regress(yli.OrdinalLogit, df, 'apply', 'pared + public + gpa', exp=False)
			model.brant()
		
		.. code-block:: text
			
			          χ²  df     p   
			Omnibus  4.34  3   0.23  
			pared    0.13  1   0.72  
			public   3.44  1   0.06  
			gpa      0.18  1   0.67  
		
		The output shows the result of the Brant test. For example, for the omnibus test of the parallel regression assumption across all independent variables, the *χ*:sup:`2` statistic is 4.34, the *χ*:sup:`2` distribution has 3 degrees of freedom, and the test is not significant, with *p* value 0.23.
		
		**Reference:** Brant R. Assessing proportionality in the proportional odds model for ordinal logistic regression. *Biometrics*. 1990;46(4):1171–8. `doi:10.2307/2532457 <https://doi.org/10.2307/2532457>`_
		"""
		
		df = self.df()
		if df is None:
			raise Exception('Referenced DataFrame has been dropped')
		dep = self.dep
		
		# Precompute design matrices
		_, dmatrices, dep_categories = df_to_dmatrices(df, dep, self.formula, 'omit')
		s_dep = dmatrices[0][dmatrices[0].columns[0]]  # df[dep] as series - must get this from dmatrices to account for as_numeric
		dmatrix_right = dmatrices[1]
		dmatrix_right.reset_index(drop=True, inplace=True)  # Otherwise this confuses matrix multiplication
		
		# Fit individual logistic regressions
		logit_models = []
		for upper_limit in sorted(s_dep.unique())[:-1]:
			dep_dichotomous = (s_dep <= upper_limit).astype(int).reset_index(drop=True)
			logit_result = sm.Logit(dep_dichotomous, dmatrix_right).fit(disp=False)
			
			if not logit_result.mle_retvals['converged']:
				raise Exception('Maximum likelihood estimation failed to converge for {} <= {}. Check raw_result.mle_retvals.'.format(dep, upper_limit))
			
			if pd.isna(logit_result.bse).any():
				raise Exception('Regression returned NaN standard errors for {} <= {}.'.format(dep, upper_limit))
			
			logit_models.append(logit_result)
		
		logit_betas = np.array([model._results.params for model in logit_models]).T
		logit_pihat = np.array([expit(-model.fittedvalues) for model in logit_models]).T  # Predicted probabilities
		
		# vcov is the variance-covariance matrix of all individually fitted betas across all terms
		
		#                  |     model 1     |     model 2     |     model 3     | ...
		#                  | term 1 | term 2 | term 1 | term 2 | term 1 | term 2 | ...
		# model 1 | term 1 |
		#         | term 2 |
		# model 2 | term 1 |
		#         | term 2 |
		#   ...
		
		n_terms = len(dmatrix_right.columns) - 1  # number of beta terms (excluding intercept)
		n_betas = len(logit_models) * n_terms
		vcov = np.zeros((n_betas, n_betas))
		
		# Populate the variance-covariance matrix for comparisons between individually fitted models
		for j in range(0, len(logit_models) - 1):
			for l in range(j + 1, len(logit_models)):
				Wjj = np.diag(logit_pihat[:,j] - logit_pihat[:,j] * logit_pihat[:,j])
				Wjl = np.diag(logit_pihat[:,l] - logit_pihat[:,j] * logit_pihat[:,l])
				Wll = np.diag(logit_pihat[:,l] - logit_pihat[:,l] * logit_pihat[:,l])
				
				matrix_result = np.linalg.inv(dmatrix_right.T @ Wjj @ dmatrix_right) @ dmatrix_right.T @ Wjl @ dmatrix_right @ np.linalg.inv(dmatrix_right.T @ Wll @ dmatrix_right)
				j_vs_l_vcov = np.asarray(matrix_result)[1:,1:]  # Asymptotic covariance for j,l
				
				vcov[j*n_terms:(j+1)*n_terms, l*n_terms:(l+1)*n_terms] = j_vs_l_vcov
				vcov[l*n_terms:(l+1)*n_terms, j*n_terms:(j+1)*n_terms] = j_vs_l_vcov
		
		# Populate the variance-covariance matrix within each individually fitted model
		for i in range(len(logit_models)):
			vcov[i*n_terms:(i+1)*n_terms, i*n_terms:(i+1)*n_terms] = logit_models[i]._results.cov_params()[1:,1:]
		
		# ------------------
		# Perform Wald tests
		
		beta_names = ['{}_{}'.format(raw_name, i) for i in range(len(logit_models)) for raw_name in dmatrix_right.columns[1:]]
		wald_results = {}
		
		# Omnibus test
		constraints = [' = '.join('{}_{}'.format(raw_name, i) for i in range(len(logit_models))) for raw_name in dmatrix_right.columns[1:]]
		constraint = ', '.join(constraints)
		dof = (len(logit_models) - 1) * (len(dmatrix_right.columns) - 1)  # df = (number of levels minus 2) * (number of terms excluding intercept)
		wald_result = _wald_test(beta_names, logit_betas[1:].ravel('F'), constraint, vcov, dof)
		wald_results['Omnibus'] = ChiSquaredResult(wald_result.statistic, wald_result.df_denom, wald_result.pvalue)
		
		# Individual terms
		for raw_name in dmatrix_right.columns[1:]:
			constraint = ' = '.join('{}_{}'.format(raw_name, i) for i in range(len(logit_models)))
			dof = len(logit_models) - 1  # df = (number of levels minus 2)
			wald_result = _wald_test(beta_names, logit_betas[1:].ravel('F'), constraint, vcov, dof)
			wald_results[raw_name] = ChiSquaredResult(wald_result.statistic, wald_result.df_denom, wald_result.pvalue)
		
		return BrantResult(wald_results)

class _SMOrdinalLogit(statsmodels.miscmodels.ordinal_model.OrderedModel):
	def __init__(self, endog, exog, **kwargs):
		if 'distr' not in kwargs:
			kwargs['distr'] = 'logit'
		
		super().__init__(endog, exog, **kwargs)
	
	def transform_threshold_params(self, params):
		th_params = params[-(self.k_levels - 1):]
		thresh = np.concatenate(([-np.inf], th_params, [np.inf]))
		return thresh
	
	def transform_reverse_threshold_params(self, params):
		return params[:-1]

class PenalisedLogit(RegressionModel):
	"""
	Firth penalised logistic regression
	
	Uses the R *logistf* library.
	
	**Example:**
	
	.. code-block::
		
		df = pd.DataFrame({
			'Pred': [1] * 20 + [0] * 220,
			'Outcome': [1] * 40 + [0] * 200
		})
		yli.regress(yli.PenalisedLogit, df, 'Outcome', 'Pred', exp=False)
	
	.. code-block:: text
		
		           Penalised Logistic Regression Results            
		============================================================
		Dep. Variable:         Outcome  |  No. Observations:     240
		        Model: Penalised Logit  |         Df. Model:       1
		         Date:      2022-10-19  |         Pseudo R²:    0.37
		         Time:        07:50:40  |          LL-Model:  -66.43
		  Std. Errors:      Non-Robust  |           LL-Null: -105.91
		                                |            p (LR): <0.001*
		============================================================
		                β      (95% CI)          p   
		---------------------------------------------
		(Intercept)   -2.28 (-2.77 - -1.85)   <0.001*
		       Pred    5.99  (3.95 - 10.85)   <0.001*
		---------------------------------------------
	
	The output summarises the result of the regression.
	The summary table shows that a penalised method was applied.
	Note that because `exp=False` was passed, the parameter estimates are not automatically exponentiated.
	"""
	
	@property
	def model_long_name(self):
		return 'Penalised Logistic Regression'
	
	@property
	def model_short_name(self):
		return 'Penalised Logit'
	
	@classmethod
	def fit(cls, data_dep, data_ind):
		result = cls()
		result.cov_type = 'nonrobust'
		
		import rpy2.robjects as ro
		import rpy2.robjects.packages
		import rpy2.robjects.pandas2ri
		
		df = data_dep.join(data_ind)
		
		# Drop explicit intercept term
		if 'Intercept' in df:
			del df['Intercept']
		
		# Import logistf
		ro.packages.importr('logistf')
		
		with ro.conversion.localconverter(ro.default_converter + ro.pandas2ri.converter):
			with ro.local_context() as lc:
				# Convert DataFrame to R
				lc['df'] = df
				
				# Transfer other parameters to R
				lc['formula_'] = df.columns[0] + ' ~ ' + ' + '.join(df.columns[1:])
				lc['alpha'] = config.alpha
				
				# Fit the model
				model = ro.r('logistf(formula_, data=df, alpha=alpha)')
				
				result.dof_model = model['df'][0]
				result.ll_model = model['loglik'][0]
				result.ll_null = model['loglik'][1]
				
				result.terms = {t: SingleTerm(t, Estimate(b, ci0, ci1), p) for t, b, ci0, ci1, p in zip(model['terms'], model['coefficients'], model['ci.lower'], model['ci.upper'], model['prob'])}
		
		return result

class Poisson(RegressionModel):
	"""
	Poisson regression
	
	**Example:**
	
	.. code-block::
		
		df = pd.DataFrame(...)
		yli.regress(yli.Poisson, df, 'num_awards', 'C(prog) + math')
	
	.. code-block:: text
		
		               Poisson Regression Results              
		=======================================================
		Dep. Variable: num_awards  |  No. Observations:     200
		        Model:    Poisson  |         Df. Model:       3
		         Date: 2023-04-22  |     Df. Residuals:     196
		         Time:   16:58:21  |         Pseudo R²:    0.21
		  Std. Errors: Non-Robust  |          LL-Model: -182.75
		                           |           LL-Null: -231.86
		                           |            p (LR): <0.001*
		=======================================================
		              exp(β)   (95% CI)         p   
		--------------------------------------------
		(Intercept)     0.01 (0.00 - 0.02)   <0.001*
		       prog                                 
		          1     Ref.                        
		          2     2.96 (1.46 - 5.97)    0.002*
		          3     1.45 (0.61 - 3.44)    0.40  
		       math     1.07 (1.05 - 1.10)   <0.001*
		--------------------------------------------
	
	The output summarises the results of the regression.
	For example, the adjusted incidence rate ratio in "num_awards" per unit increase in "math" is 1.07, with 95% confidence interval 1.05–1.10, and is significant with *p* value < 0.001.
	"""
	
	@property
	def model_long_name(self):
		return 'Poisson Regression'
	
	@property
	def model_short_name(self):
		return 'Poisson'
	
	@classmethod
	def fit(cls, data_dep, data_ind, exposure=None, method='newton', maxiter=None, start_params=None):
		result = cls()
		result.exp = True
		result.cov_type = 'nonrobust'
		
		# Perform regression
		raw_result = sm.Poisson(endog=data_dep, exog=data_ind, exposure=exposure, missing='raise').fit(disp=False, method=method, start_params=start_params)
		
		result.dof_model = raw_result.df_model
		result.dof_resid = raw_result.df_resid
		result.ll_model = raw_result.llf
		result.ll_null = raw_result.llnull
		
		# Compute saturated log-likelihood
		result.ll_saturated = float(sm.families.Poisson().loglike(data_dep + 1e-10, data_dep + 1e-10))
		
		result.terms = raw_terms_from_statsmodels_result(raw_result)
		result.vcov = raw_result.cov_params()
		
		return result

# ------------------
# Brant test helpers

class _Dummy: pass

def _wald_test(param_names, params, formula, vcov, df):
	# Hack! Piggyback off statsmodels to compute a Wald test
	
	lmr = statsmodels.base.model.LikelihoodModelResults(model=None, params=None)
	lmr.model = _Dummy()
	lmr.model.data = _Dummy()
	lmr.model.data.cov_names = param_names
	lmr.params = params
	lmr.df_resid = df
	
	return lmr.wald_test(formula, cov_p=vcov, use_f=False, scalar=True)

class BrantResult:
	"""
	Result of a Brant test for ordinal regression
	
	See :meth:`OrdinalLogit.brant`.
	"""
	
	def __init__(self, tests):
		#: Results for Brant test on each coefficient (*Dict[str, ChiSquaredResult]*)
		self.tests = tests
	
	def __repr__(self):
		if config.repr_is_summary:
			return self.summary()
		return super().__repr__()
	
	def _repr_html_(self):
		out = '<table><caption>Brant Test Results</caption><thead><tr><th></th><th style="text-align:center"><i>χ</i><sup>2</sup></th><th style="text-align:center">df</th><th style="text-align:center"><i>p</i></th></thead><tbody>'
		
		for raw_name, test in self.tests.items():
			out += '<tr><th>{}</th><td>{:.2f}</td><td>{:.0f}</td><td style="text-align:left">{}</td></tr>'.format(raw_name, test.statistic, test.dof, fmt_p(test.pvalue, PValueStyle.TABULAR | PValueStyle.HTML))
		
		out += '</tbody></table>'
		return out
	
	def summary(self):
		"""
		Return a stringified summary of the *χ*:sup:`2` test
		
		:rtype: str
		"""
		
		table = pd.DataFrame([
			['{:.2f}'.format(test.statistic), '{:.0f}'.format(test.dof), fmt_p(test.pvalue, PValueStyle.TABULAR)]
			for test in self.tests.values()
		], index=self.tests.keys(), columns=['χ² ', 'df', 'p   '])
		
		return str(table)