Source code for statsmodels.stats.anova
from statsmodels.compat.python import lrange
import numpy as np
import pandas as pd
from pandas import DataFrame, Index
import patsy
from scipy import stats
from statsmodels.formula.formulatools import (
_has_intercept,
_intercept_idx,
_remove_intercept_patsy,
)
from statsmodels.iolib import summary2
from statsmodels.regression.linear_model import OLS
def _get_covariance(model, robust):
if robust is None:
return model.cov_params()
elif robust == "hc0":
return model.cov_HC0
elif robust == "hc1":
return model.cov_HC1
elif robust == "hc2":
return model.cov_HC2
elif robust == "hc3":
return model.cov_HC3
else: # pragma: no cover
raise ValueError("robust options %s not understood" % robust)
# NOTE: these need to take into account weights !
def anova_single(model, **kwargs):
"""
Anova table for one fitted linear model.
Parameters
----------
model : fitted linear model results instance
A fitted linear model
typ : int or str {1,2,3} or {"I","II","III"}
Type of sum of squares to use.
**kwargs**
scale : float
Estimate of variance, If None, will be estimated from the largest
model. Default is None.
test : str {"F", "Chisq", "Cp"} or None
Test statistics to provide. Default is "F".
Notes
-----
Use of this function is discouraged. Use anova_lm instead.
"""
test = kwargs.get("test", "F")
scale = kwargs.get("scale", None)
typ = kwargs.get("typ", 1)
robust = kwargs.get("robust", None)
if robust:
robust = robust.lower()
endog = model.model.endog
exog = model.model.exog
nobs = exog.shape[0]
response_name = model.model.endog_names
design_info = model.model.data.design_info
exog_names = model.model.exog_names
# +1 for resids
n_rows = (len(design_info.terms) - _has_intercept(design_info) + 1)
pr_test = "PR(>%s)" % test
names = ['df', 'sum_sq', 'mean_sq', test, pr_test]
table = DataFrame(np.zeros((n_rows, 5)), columns=names)
if typ in [1, "I"]:
return anova1_lm_single(model, endog, exog, nobs, design_info, table,
n_rows, test, pr_test, robust)
elif typ in [2, "II"]:
return anova2_lm_single(model, design_info, n_rows, test, pr_test,
robust)
elif typ in [3, "III"]:
return anova3_lm_single(model, design_info, n_rows, test, pr_test,
robust)
elif typ in [4, "IV"]:
raise NotImplementedError("Type IV not yet implemented")
else: # pragma: no cover
raise ValueError("Type %s not understood" % str(typ))
def anova1_lm_single(model, endog, exog, nobs, design_info, table, n_rows, test,
pr_test, robust):
"""
Anova table for one fitted linear model.
Parameters
----------
model : fitted linear model results instance
A fitted linear model
**kwargs**
scale : float
Estimate of variance, If None, will be estimated from the largest
model. Default is None.
test : str {"F", "Chisq", "Cp"} or None
Test statistics to provide. Default is "F".
Notes
-----
Use of this function is discouraged. Use anova_lm instead.
"""
#maybe we should rethink using pinv > qr in OLS/linear models?
effects = getattr(model, 'effects', None)
if effects is None:
q,r = np.linalg.qr(exog)
effects = np.dot(q.T, endog)
arr = np.zeros((len(design_info.terms), len(design_info.column_names)))
slices = [design_info.slice(name) for name in design_info.term_names]
for i,slice_ in enumerate(slices):
arr[i, slice_] = 1
sum_sq = np.dot(arr, effects**2)
#NOTE: assumes intercept is first column
idx = _intercept_idx(design_info)
sum_sq = sum_sq[~idx]
term_names = np.array(design_info.term_names) # want boolean indexing
term_names = term_names[~idx]
index = term_names.tolist()
table.index = Index(index + ['Residual'])
table.loc[index, ['df', 'sum_sq']] = np.c_[arr[~idx].sum(1), sum_sq]
# fill in residual
table.loc['Residual', ['sum_sq','df']] = model.ssr, model.df_resid
if test == 'F':
table[test] = ((table['sum_sq'] / table['df']) /
(model.ssr / model.df_resid))
table[pr_test] = stats.f.sf(table["F"], table["df"],
model.df_resid)
table.loc['Residual', [test, pr_test]] = np.nan, np.nan
table['mean_sq'] = table['sum_sq'] / table['df']
return table
#NOTE: the below is not agnostic about formula...
def anova2_lm_single(model, design_info, n_rows, test, pr_test, robust):
"""
Anova type II table for one fitted linear model.
Parameters
----------
model : fitted linear model results instance
A fitted linear model
**kwargs**
scale : float
Estimate of variance, If None, will be estimated from the largest
model. Default is None.
test : str {"F", "Chisq", "Cp"} or None
Test statistics to provide. Default is "F".
Notes
-----
Use of this function is discouraged. Use anova_lm instead.
Type II
Sum of Squares compares marginal contribution of terms. Thus, it is
not particularly useful for models with significant interaction terms.
"""
terms_info = design_info.terms[:] # copy
terms_info = _remove_intercept_patsy(terms_info)
names = ['sum_sq', 'df', test, pr_test]
table = DataFrame(np.zeros((n_rows, 4)), columns = names)
cov = _get_covariance(model, None)
robust_cov = _get_covariance(model, robust)
col_order = []
index = []
for i, term in enumerate(terms_info):
# grab all varaibles except interaction effects that contain term
# need two hypotheses matrices L1 is most restrictive, ie., term==0
# L2 is everything except term==0
cols = design_info.slice(term)
L1 = lrange(cols.start, cols.stop)
L2 = []
term_set = set(term.factors)
for t in terms_info: # for the term you have
other_set = set(t.factors)
if term_set.issubset(other_set) and not term_set == other_set:
col = design_info.slice(t)
# on a higher order term containing current `term`
L1.extend(lrange(col.start, col.stop))
L2.extend(lrange(col.start, col.stop))
L1 = np.eye(model.model.exog.shape[1])[L1]
L2 = np.eye(model.model.exog.shape[1])[L2]
if L2.size:
LVL = np.dot(np.dot(L1,robust_cov),L2.T)
from scipy import linalg
orth_compl,_ = linalg.qr(LVL)
r = L1.shape[0] - L2.shape[0]
# L1|2
# use the non-unique orthogonal completion since L12 is rank r
L12 = np.dot(orth_compl[:,-r:].T, L1)
else:
L12 = L1
r = L1.shape[0]
#from IPython.core.debugger import Pdb; Pdb().set_trace()
if test == 'F':
f = model.f_test(L12, cov_p=robust_cov)
table.loc[table.index[i], test] = test_value = f.fvalue
table.loc[table.index[i], pr_test] = f.pvalue
# need to back out SSR from f_test
table.loc[table.index[i], 'df'] = r
col_order.append(cols.start)
index.append(term.name())
table.index = Index(index + ['Residual'])
table = table.iloc[np.argsort(col_order + [model.model.exog.shape[1]+1])]
# back out sum of squares from f_test
ssr = table[test] * table['df'] * model.ssr/model.df_resid
table['sum_sq'] = ssr
# fill in residual
table.loc['Residual', ['sum_sq','df', test, pr_test]] = (model.ssr,
model.df_resid,
np.nan, np.nan)
return table
def anova3_lm_single(model, design_info, n_rows, test, pr_test, robust):
n_rows += _has_intercept(design_info)
terms_info = design_info.terms
names = ['sum_sq', 'df', test, pr_test]
table = DataFrame(np.zeros((n_rows, 4)), columns = names)
cov = _get_covariance(model, robust)
col_order = []
index = []
for i, term in enumerate(terms_info):
# grab term, hypothesis is that term == 0
cols = design_info.slice(term)
L1 = np.eye(model.model.exog.shape[1])[cols]
L12 = L1
r = L1.shape[0]
if test == 'F':
f = model.f_test(L12, cov_p=cov)
table.loc[table.index[i], test] = test_value = f.fvalue
table.loc[table.index[i], pr_test] = f.pvalue
# need to back out SSR from f_test
table.loc[table.index[i], 'df'] = r
#col_order.append(cols.start)
index.append(term.name())
table.index = Index(index + ['Residual'])
#NOTE: Do not need to sort because terms are an ordered dict now
#table = table.iloc[np.argsort(col_order + [model.model.exog.shape[1]+1])]
# back out sum of squares from f_test
ssr = table[test] * table['df'] * model.ssr/model.df_resid
table['sum_sq'] = ssr
# fill in residual
table.loc['Residual', ['sum_sq','df', test, pr_test]] = (model.ssr,
model.df_resid,
np.nan, np.nan)
return table
[docs]
def anova_lm(*args, **kwargs):
"""
Anova table for one or more fitted linear models.
Parameters
----------
args : fitted linear model results instance
One or more fitted linear models
scale : float
Estimate of variance, If None, will be estimated from the largest
model. Default is None.
test : str {"F", "Chisq", "Cp"} or None
Test statistics to provide. Default is "F".
typ : str or int {"I","II","III"} or {1,2,3}
The type of Anova test to perform. See notes.
robust : {None, "hc0", "hc1", "hc2", "hc3"}
Use heteroscedasticity-corrected coefficient covariance matrix.
If robust covariance is desired, it is recommended to use `hc3`.
Returns
-------
anova : DataFrame
When args is a single model, return is DataFrame with columns:
sum_sq : float64
Sum of squares for model terms.
df : float64
Degrees of freedom for model terms.
F : float64
F statistic value for significance of adding model terms.
PR(>F) : float64
P-value for significance of adding model terms.
When args is multiple models, return is DataFrame with columns:
df_resid : float64
Degrees of freedom of residuals in models.
ssr : float64
Sum of squares of residuals in models.
df_diff : float64
Degrees of freedom difference from previous model in args
ss_dff : float64
Difference in ssr from previous model in args
F : float64
F statistic comparing to previous model in args
PR(>F): float64
P-value for significance comparing to previous model in args
Notes
-----
Model statistics are given in the order of args. Models must have been fit
using the formula api.
See Also
--------
model_results.compare_f_test, model_results.compare_lm_test
Examples
--------
>>> import statsmodels.api as sm
>>> from statsmodels.formula.api import ols
>>> moore = sm.datasets.get_rdataset("Moore", "carData", cache=True) # load
>>> data = moore.data
>>> data = data.rename(columns={"partner.status" :
... "partner_status"}) # make name pythonic
>>> moore_lm = ols('conformity ~ C(fcategory, Sum)*C(partner_status, Sum)',
... data=data).fit()
>>> table = sm.stats.anova_lm(moore_lm, typ=2) # Type 2 Anova DataFrame
>>> print(table)
"""
typ = kwargs.get('typ', 1)
### Farm Out Single model Anova Type I, II, III, and IV ###
if len(args) == 1:
model = args[0]
return anova_single(model, **kwargs)
if typ not in [1, "I"]:
raise ValueError("Multiple models only supported for type I. "
"Got type %s" % str(typ))
test = kwargs.get("test", "F")
scale = kwargs.get("scale", None)
n_models = len(args)
pr_test = "Pr(>%s)" % test
names = ['df_resid', 'ssr', 'df_diff', 'ss_diff', test, pr_test]
table = DataFrame(np.zeros((n_models, 6)), columns=names)
if not scale: # assume biggest model is last
scale = args[-1].scale
table["ssr"] = [mdl.ssr for mdl in args]
table["df_resid"] = [mdl.df_resid for mdl in args]
table.loc[table.index[1:], "df_diff"] = -np.diff(table["df_resid"].values)
table["ss_diff"] = -table["ssr"].diff()
if test == "F":
table["F"] = table["ss_diff"] / table["df_diff"] / scale
table[pr_test] = stats.f.sf(table["F"], table["df_diff"],
table["df_resid"])
# for earlier scipy - stats.f.sf(np.nan, 10, 2) -> 0 not nan
table.loc[table['F'].isnull(), pr_test] = np.nan
return table
def _not_slice(slices, slices_to_exclude, n):
ind = np.array([True]*n)
for term in slices_to_exclude:
s = slices[term]
ind[s] = False
return ind
def _ssr_reduced_model(y, x, term_slices, params, keys):
"""
Residual sum of squares of OLS model excluding factors in `keys`
Assumes x matrix is orthogonal
Parameters
----------
y : array_like
dependent variable
x : array_like
independent variables
term_slices : a dict of slices
term_slices[key] is a boolean array specifies the parameters
associated with the factor `key`
params : ndarray
OLS solution of y = x * params
keys : keys for term_slices
factors to be excluded
Returns
-------
rss : float
residual sum of squares
df : int
degrees of freedom
"""
ind = _not_slice(term_slices, keys, x.shape[1])
params1 = params[ind]
ssr = np.subtract(y, x[:, ind].dot(params1))
ssr = ssr.T.dot(ssr)
df_resid = len(y) - len(params1)
return ssr, df_resid
[docs]
class AnovaRM:
"""
Repeated measures Anova using least squares regression
The full model regression residual sum of squares is
used to compare with the reduced model for calculating the
within-subject effect sum of squares [1].
Currently, only fully balanced within-subject designs are supported.
Calculation of between-subject effects and corrections for violation of
sphericity are not yet implemented.
Parameters
----------
data : DataFrame
depvar : str
The dependent variable in `data`
subject : str
Specify the subject id
within : list[str]
The within-subject factors
between : list[str]
The between-subject factors, this is not yet implemented
aggregate_func : {None, 'mean', callable}
If the data set contains more than a single observation per subject
and cell of the specified model, this function will be used to
aggregate the data before running the Anova. `None` (the default) will
not perform any aggregation; 'mean' is s shortcut to `numpy.mean`.
An exception will be raised if aggregation is required, but no
aggregation function was specified.
Returns
-------
results : AnovaResults instance
Raises
------
ValueError
If the data need to be aggregated, but `aggregate_func` was not
specified.
Notes
-----
This implementation currently only supports fully balanced designs. If the
data contain more than one observation per subject and cell of the design,
these observations need to be aggregated into a single observation
before the Anova is calculated, either manually or by passing an aggregation
function via the `aggregate_func` keyword argument.
Note that if the input data set was not balanced before performing the
aggregation, the implied heteroscedasticity of the data is ignored.
References
----------
.. [*] Rutherford, Andrew. Anova and ANCOVA: a GLM approach. John Wiley & Sons, 2011.
"""
def __init__(self, data, depvar, subject, within=None, between=None,
aggregate_func=None):
self.data = data
self.depvar = depvar
self.within = within
if 'C' in within:
raise ValueError("Factor name cannot be 'C'! This is in conflict "
"with patsy's contrast function name.")
self.between = between
if between is not None:
raise NotImplementedError('Between subject effect not '
'yet supported!')
self.subject = subject
if aggregate_func == 'mean':
self.aggregate_func = pd.Series.mean
else:
self.aggregate_func = aggregate_func
if not data.equals(data.drop_duplicates(subset=[subject] + within)):
if self.aggregate_func is not None:
self._aggregate()
else:
msg = ('The data set contains more than one observation per '
'subject and cell. Either aggregate the data manually, '
'or pass the `aggregate_func` parameter.')
raise ValueError(msg)
self._check_data_balanced()
def _aggregate(self):
self.data = (self.data
.groupby([self.subject] + self.within,
as_index=False)[self.depvar]
.agg(self.aggregate_func))
def _check_data_balanced(self):
"""raise if data is not balanced
This raises a ValueError if the data is not balanced, and
returns None if it is balance
Return might change
"""
factor_levels = 1
for wi in self.within:
factor_levels *= len(self.data[wi].unique())
cell_count = {}
for index in range(self.data.shape[0]):
key = []
for col in self.within:
key.append(self.data[col].iloc[index])
key = tuple(key)
if key in cell_count:
cell_count[key] = cell_count[key] + 1
else:
cell_count[key] = 1
error_message = "Data is unbalanced."
if len(cell_count) != factor_levels:
raise ValueError(error_message)
count = cell_count[key]
for key in cell_count:
if count != cell_count[key]:
raise ValueError(error_message)
if self.data.shape[0] > count * factor_levels:
raise ValueError('There are more than 1 element in a cell! Missing'
' factors?')
[docs]
def fit(self):
"""estimate the model and compute the Anova table
Returns
-------
AnovaResults instance
"""
y = self.data[self.depvar].values
# Construct OLS endog and exog from string using patsy
within = ['C(%s, Sum)' % i for i in self.within]
subject = 'C(%s, Sum)' % self.subject
factors = within + [subject]
x = patsy.dmatrix('*'.join(factors), data=self.data)
term_slices = x.design_info.term_name_slices
for key in term_slices:
ind = np.array([False]*x.shape[1])
ind[term_slices[key]] = True
term_slices[key] = np.array(ind)
term_exclude = [':'.join(factors)]
ind = _not_slice(term_slices, term_exclude, x.shape[1])
x = x[:, ind]
# Fit OLS
model = OLS(y, x)
results = model.fit()
if model.rank < x.shape[1]:
raise ValueError('Independent variables are collinear.')
for i in term_exclude:
term_slices.pop(i)
for key in term_slices:
term_slices[key] = term_slices[key][ind]
params = results.params
df_resid = results.df_resid
ssr = results.ssr
columns = ['F Value', 'Num DF', 'Den DF', 'Pr > F']
anova_table = pd.DataFrame(np.zeros((0, 4)), columns=columns)
for key in term_slices:
if self.subject not in key and key != 'Intercept':
# Independen variables are orthogonal
ssr1, df_resid1 = _ssr_reduced_model(
y, x, term_slices, params, [key])
df1 = df_resid1 - df_resid
msm = (ssr1 - ssr) / df1
if (key == ':'.join(factors[:-1]) or
(key + ':' + subject not in term_slices)):
mse = ssr / df_resid
df2 = df_resid
else:
ssr1, df_resid1 = _ssr_reduced_model(
y, x, term_slices, params,
[key + ':' + subject])
df2 = df_resid1 - df_resid
mse = (ssr1 - ssr) / df2
F = msm / mse
p = stats.f.sf(F, df1, df2)
term = key.replace('C(', '').replace(', Sum)', '')
anova_table.loc[term, 'F Value'] = F
anova_table.loc[term, 'Num DF'] = df1
anova_table.loc[term, 'Den DF'] = df2
anova_table.loc[term, 'Pr > F'] = p
return AnovaResults(anova_table)
class AnovaResults:
"""
Anova results class
Attributes
----------
anova_table : DataFrame
"""
def __init__(self, anova_table):
self.anova_table = anova_table
def __str__(self):
return self.summary().__str__()
def summary(self):
"""create summary results
Returns
-------
summary : summary2.Summary instance
"""
summ = summary2.Summary()
summ.add_title('Anova')
summ.add_df(self.anova_table)
return summ
if __name__ == "__main__":
import pandas
from statsmodels.formula.api import ols
# in R
#library(car)
#write.csv(Moore, "moore.csv", row.names=FALSE)
moore = pandas.read_csv('moore.csv', skiprows=1,
names=['partner_status','conformity',
'fcategory','fscore'])
moore_lm = ols('conformity ~ C(fcategory, Sum)*C(partner_status, Sum)',
data=moore).fit()
mooreB = ols('conformity ~ C(partner_status, Sum)', data=moore).fit()
# for each term you just want to test vs the model without its
# higher-order terms
# using Monette-Fox slides and Marden class notes for linear algebra /
# orthogonal complement
# https://netfiles.uiuc.edu/jimarden/www/Classes/STAT324/
table = anova_lm(moore_lm, typ=2)
Last update:
Apr 12, 2024