import numpy as np
import pandas as pd
from scipy import stats
from sklearn.cross_decomposition import CCA
from statsmodels.multivariate.manova import MANOVA
from pgmpy.global_vars import logger
from pgmpy.independencies import IndependenceAssertion
[docs]
def independence_match(X, Y, Z, independencies, **kwargs):
"""
Checks if `X \u27C2 Y | Z` is in `independencies`. This method is implemented to
have an uniform API when the independencies are provided instead of data.
Parameters
----------
X: str
The first variable for testing the independence condition X \u27C2 Y | Z
Y: str
The second variable for testing the independence condition X \u27C2 Y | Z
Z: list/array-like
A list of conditional variable for testing the condition X \u27C2 Y | Z
data: pandas.DataFrame The dataset in which to test the indepenedence condition.
Returns
-------
p-value: float (Fixed to 0 since it is always confident)
"""
return IndependenceAssertion(X, Y, Z) in independencies
[docs]
def chi_square(X, Y, Z, data, boolean=True, **kwargs):
"""
Chi-square conditional independence test.
Tests the null hypothesis that X is independent from Y given Zs.
This is done by comparing the observed frequencies with the expected
frequencies if X,Y were conditionally independent, using a chisquare
deviance statistic. The expected frequencies given independence are
:math:`P(X,Y,Zs) = P(X|Zs)*P(Y|Zs)*P(Zs)`. The latter term can be computed
as :math:`P(X,Zs)*P(Y,Zs)/P(Zs).
Parameters
----------
X: int, string, hashable object
A variable name contained in the data set
Y: int, string, hashable object
A variable name contained in the data set, different from X
Z: list, array-like
A list of variable names contained in the data set, different from X and Y.
This is the separating set that (potentially) makes X and Y independent.
Default: []
data: pandas.DataFrame
The dataset on which to test the independence condition.
boolean: bool
If boolean=True, an additional argument `significance_level` must
be specified. If p_value of the test is greater than equal to
`significance_level`, returns True. Otherwise returns False.
If boolean=False, returns the chi2 and p_value of the test.
Returns
-------
CI Test Results: tuple or bool
If boolean = False, Returns a tuple (chi, p_value, dof). `chi` is the
chi-squared test statistic. The `p_value` for the test, i.e. the
probability of observing the computed chi-square statistic (or an even
higher value), given the null hypothesis that X \u27C2 Y | Zs is True.
If boolean = True, returns True if the p_value of the test is greater
than `significance_level` else returns False.
References
----------
[1] https://en.wikipedia.org/wiki/Chi-squared_test
Examples
--------
>>> import pandas as pd
>>> import numpy as np
>>> data = pd.DataFrame(np.random.randint(0, 2, size=(50000, 4)), columns=list('ABCD'))
>>> data['E'] = data['A'] + data['B'] + data['C']
>>> chi_square(X='A', Y='C', Z=[], data=data, boolean=True, significance_level=0.05)
True
>>> chi_square(X='A', Y='B', Z=['D'], data=data, boolean=True, significance_level=0.05)
True
>>> chi_square(X='A', Y='B', Z=['D', 'E'], data=data, boolean=True, significance_level=0.05)
False
"""
return power_divergence(
X=X, Y=Y, Z=Z, data=data, boolean=boolean, lambda_="pearson", **kwargs
)
[docs]
def g_sq(X, Y, Z, data, boolean=True, **kwargs):
"""
G squared test for conditional independence. Also commonly known as G-test,
likelihood-ratio or maximum likelihood statistical significance test.
Tests the null hypothesis that X is independent of Y given Zs.
Parameters
----------
X: int, string, hashable object
A variable name contained in the data set
Y: int, string, hashable object
A variable name contained in the data set, different from X
Z: list (array-like)
A list of variable names contained in the data set, different from X and Y.
This is the separating set that (potentially) makes X and Y independent.
Default: []
data: pandas.DataFrame
The dataset on which to test the independence condition.
boolean: bool
If boolean=True, an additional argument `significance_level` must be
specified. If p_value of the test is greater than equal to
`significance_level`, returns True. Otherwise returns False. If
boolean=False, returns the chi2 and p_value of the test.
Returns
-------
CI Test Results: tuple or bool
If boolean = False, Returns a tuple (chi, p_value, dof). `chi` is the
chi-squared test statistic. The `p_value` for the test, i.e. the
probability of observing the computed chi-square statistic (or an even
higher value), given the null hypothesis that X \u27C2 Y | Zs is True.
If boolean = True, returns True if the p_value of the test is greater
than `significance_level` else returns False.
References
----------
[1] https://en.wikipedia.org/wiki/G-test
Examples
--------
>>> import pandas as pd
>>> import numpy as np
>>> data = pd.DataFrame(np.random.randint(0, 2, size=(50000, 4)), columns=list('ABCD'))
>>> data['E'] = data['A'] + data['B'] + data['C']
>>> g_sq(X='A', Y='C', Z=[], data=data, boolean=True, significance_level=0.05)
True
>>> g_sq(X='A', Y='B', Z=['D'], data=data, boolean=True, significance_level=0.05)
True
>>> g_sq(X='A', Y='B', Z=['D', 'E'], data=data, boolean=True, significance_level=0.05)
False
"""
return power_divergence(
X=X, Y=Y, Z=Z, data=data, boolean=boolean, lambda_="log-likelihood", **kwargs
)
[docs]
def log_likelihood(X, Y, Z, data, boolean=True, **kwargs):
"""
Log likelihood ratio test for conditional independence. Also commonly known
as G-test, G-squared test or maximum likelihood statistical significance
test. Tests the null hypothesis that X is independent of Y given Zs.
Parameters
----------
X: int, string, hashable object
A variable name contained in the data set
Y: int, string, hashable object
A variable name contained in the data set, different from X
Z: list (array-like)
A list of variable names contained in the data set, different from X and Y.
This is the separating set that (potentially) makes X and Y independent.
Default: []
data: pandas.DataFrame
The dataset on which to test the independence condition.
boolean: bool
If boolean=True, an additional argument `significance_level` must be
specified. If p_value of the test is greater than equal to
`significance_level`, returns True. Otherwise returns False. If
boolean=False, returns the chi2 and p_value of the test.
Returns
-------
CI Test Results: tuple or bool
If boolean = False, Returns a tuple (chi, p_value, dof). `chi` is the
chi-squared test statistic. The `p_value` for the test, i.e. the
probability of observing the computed chi-square statistic (or an even
higher value), given the null hypothesis that X \u27C2 Y | Zs is True.
If boolean = True, returns True if the p_value of the test is greater
than `significance_level` else returns False.
References
----------
[1] https://en.wikipedia.org/wiki/G-test
Examples
--------
>>> import pandas as pd
>>> import numpy as np
>>> data = pd.DataFrame(np.random.randint(0, 2, size=(50000, 4)), columns=list('ABCD'))
>>> data['E'] = data['A'] + data['B'] + data['C']
>>> log_likelihood(X='A', Y='C', Z=[], data=data, boolean=True, significance_level=0.05)
True
>>> log_likelihood(X='A', Y='B', Z=['D'], data=data, boolean=True, significance_level=0.05)
True
>>> log_likelihood(X='A', Y='B', Z=['D', 'E'], data=data, boolean=True, significance_level=0.05)
False
"""
return power_divergence(
X=X, Y=Y, Z=Z, data=data, boolean=boolean, lambda_="log-likelihood", **kwargs
)
[docs]
def modified_log_likelihood(X, Y, Z, data, boolean=True, **kwargs):
"""
Modified log likelihood ratio test for conditional independence.
Tests the null hypothesis that X is independent of Y given Zs.
Parameters
----------
X: int, string, hashable object
A variable name contained in the data set
Y: int, string, hashable object
A variable name contained in the data set, different from X
Z: list (array-like)
A list of variable names contained in the data set, different from X and Y.
This is the separating set that (potentially) makes X and Y independent.
Default: []
data: pandas.DataFrame
The dataset on which to test the independence condition.
boolean: bool
If boolean=True, an additional argument `significance_level` must be
specified. If p_value of the test is greater than equal to
`significance_level`, returns True. Otherwise returns False.
If boolean=False, returns the chi2 and p_value of the test.
Returns
-------
CI Test Results: tuple or bool
If boolean = False, Returns a tuple (chi, p_value, dof). `chi` is the
chi-squared test statistic. The `p_value` for the test, i.e. the
probability of observing the computed chi-square statistic (or an even
higher value), given the null hypothesis that X \u27C2 Y | Zs is True.
If boolean = True, returns True if the p_value of the test is greater
than `significance_level` else returns False.
Examples
--------
>>> import pandas as pd
>>> import numpy as np
>>> data = pd.DataFrame(np.random.randint(0, 2, size=(50000, 4)), columns=list('ABCD'))
>>> data['E'] = data['A'] + data['B'] + data['C']
>>> modified_log_likelihood(X='A', Y='C', Z=[], data=data, boolean=True, significance_level=0.05)
True
>>> modified_log_likelihood(X='A', Y='B', Z=['D'], data=data, boolean=True, significance_level=0.05)
True
>>> modified_log_likelihood(X='A', Y='B', Z=['D', 'E'], data=data, boolean=True, significance_level=0.05)
False
"""
return power_divergence(
X=X,
Y=Y,
Z=Z,
data=data,
boolean=boolean,
lambda_="mod-log-likelihood",
**kwargs,
)
[docs]
def power_divergence(X, Y, Z, data, boolean=True, lambda_="cressie-read", **kwargs):
"""
Computes the Cressie-Read power divergence statistic [1]. The null hypothesis
for the test is X is independent of Y given Z. A lot of the frequency comparision
based statistics (eg. chi-square, G-test etc) belong to power divergence family,
and are special cases of this test.
Parameters
----------
X: int, string, hashable object
A variable name contained in the data set
Y: int, string, hashable object
A variable name contained in the data set, different from X
Z: list, array-like
A list of variable names contained in the data set, different from X and Y.
This is the separating set that (potentially) makes X and Y independent.
Default: []
data: pandas.DataFrame
The dataset on which to test the independence condition.
lambda_: float or string
The lambda parameter for the power_divergence statistic. Some values of
lambda_ results in other well known tests:
"pearson" 1 "Chi-squared test"
"log-likelihood" 0 "G-test or log-likelihood"
"freeman-tuckey" -1/2 "Freeman-Tuckey Statistic"
"mod-log-likelihood" -1 "Modified Log-likelihood"
"neyman" -2 "Neyman's statistic"
"cressie-read" 2/3 "The value recommended in the paper[1]"
boolean: bool
If boolean=True, an additional argument `significance_level` must
be specified. If p_value of the test is greater than equal to
`significance_level`, returns True. Otherwise returns False.
If boolean=False, returns the chi2 and p_value of the test.
Returns
-------
CI Test Results: tuple or bool
If boolean = False, Returns a tuple (chi, p_value, dof). `chi` is the
chi-squared test statistic. The `p_value` for the test, i.e. the
probability of observing the computed chi-square statistic (or an even
higher value), given the null hypothesis that X \u27C2 Y | Zs is True.
If boolean = True, returns True if the p_value of the test is greater
than `significance_level` else returns False.
References
----------
[1] Cressie, Noel, and Timothy RC Read. "Multinomial goodness‐of‐fit tests." Journal of the Royal Statistical Society: Series B (Methodological) 46.3 (1984): 440-464.
Examples
--------
>>> import pandas as pd
>>> import numpy as np
>>> data = pd.DataFrame(np.random.randint(0, 2, size=(50000, 4)), columns=list('ABCD'))
>>> data['E'] = data['A'] + data['B'] + data['C']
>>> chi_square(X='A', Y='C', Z=[], data=data, boolean=True, significance_level=0.05)
True
>>> chi_square(X='A', Y='B', Z=['D'], data=data, boolean=True, significance_level=0.05)
True
>>> chi_square(X='A', Y='B', Z=['D', 'E'], data=data, boolean=True, significance_level=0.05)
False
"""
# Step 1: Check if the arguments are valid and type conversions.
if hasattr(Z, "__iter__"):
Z = list(Z)
else:
raise (f"Z must be an iterable. Got object type: {type(Z)}")
if (X in Z) or (Y in Z):
raise ValueError(
f"The variables X or Y can't be in Z. Found {X if X in Z else Y} in Z."
)
# Step 2: Do a simple contingency test if there are no conditional variables.
if len(Z) == 0:
chi, p_value, dof, expected = stats.chi2_contingency(
data.groupby([X, Y], observed=False).size().unstack(Y, fill_value=0),
lambda_=lambda_,
)
# Step 3: If there are conditionals variables, iterate over unique states and do
# the contingency test.
else:
chi = 0
dof = 0
for z_state, df in data.groupby(Z, observed=True):
# Compute the contingency table
unique_x, x_inv = np.unique(df[X], return_inverse=True)
unique_y, y_inv = np.unique(df[Y], return_inverse=True)
contingency = np.bincount(
x_inv * len(unique_y) + y_inv, minlength=len(unique_x) * len(unique_y)
).reshape(len(unique_x), len(unique_y))
# If all values of a column in the contingency table are zeros, skip the test.
if any(contingency.sum(axis=0) == 0) or any(contingency.sum(axis=1) == 0):
if isinstance(z_state, str):
logger.info(
f"Skipping the test {X} \u27C2 {Y} | {Z[0]}={z_state}. Not enough samples"
)
else:
z_str = ", ".join(
[f"{var}={state}" for var, state in zip(Z, z_state)]
)
logger.info(
f"Skipping the test {X} \u27C2 {Y} | {z_str}. Not enough samples"
)
else:
c, _, d, _ = stats.chi2_contingency(contingency, lambda_=lambda_)
chi += c
dof += d
p_value = 1 - stats.chi2.cdf(chi, df=dof)
# Step 4: Return the values
if boolean:
return p_value >= kwargs["significance_level"]
else:
return chi, p_value, dof
[docs]
def pearsonr(X, Y, Z, data, boolean=True, **kwargs):
"""
Computes Pearson correlation coefficient and p-value for testing non-correlation.
Should be used only on continuous data. In case when :math:`Z != \null` uses
linear regression and computes pearson coefficient on residuals.
Parameters
----------
X: str
The first variable for testing the independence condition X \u27C2 Y | Z
Y: str
The second variable for testing the independence condition X \u27C2 Y | Z
Z: list/array-like
A list of conditional variable for testing the condition X \u27C2 Y | Z
data: pandas.DataFrame
The dataset in which to test the indepenedence condition.
boolean: bool
If boolean=True, an additional argument `significance_level` must
be specified. If p_value of the test is greater than equal to
`significance_level`, returns True. Otherwise returns False.
If boolean=False, returns the pearson correlation coefficient and p_value
of the test.
Returns
-------
CI Test results: tuple or bool
If boolean=True, returns True if p-value >= significance_level, else False. If
boolean=False, returns a tuple of (Pearson's correlation Coefficient, p-value)
References
----------
[1] https://en.wikipedia.org/wiki/Pearson_correlation_coefficient
[2] https://en.wikipedia.org/wiki/Partial_correlation#Using_linear_regression
"""
# Step 1: Test if the inputs are correct
if not hasattr(Z, "__iter__"):
raise ValueError(f"Variable Z. Expected type: iterable. Got type: {type(Z)}")
else:
Z = list(Z)
if not isinstance(data, pd.DataFrame):
raise ValueError(
f"Variable data. Expected type: pandas.DataFrame. Got type: {type(data)}"
)
# Step 2: If Z is empty compute a non-conditional test.
if len(Z) == 0:
coef, p_value = stats.pearsonr(data.loc[:, X], data.loc[:, Y])
# Step 3: If Z is non-empty, use linear regression to compute residuals and test independence on it.
else:
X_coef = np.linalg.lstsq(data.loc[:, Z], data.loc[:, X], rcond=None)[0]
Y_coef = np.linalg.lstsq(data.loc[:, Z], data.loc[:, Y], rcond=None)[0]
residual_X = data.loc[:, X] - data.loc[:, Z].dot(X_coef)
residual_Y = data.loc[:, Y] - data.loc[:, Z].dot(Y_coef)
coef, p_value = stats.pearsonr(residual_X, residual_Y)
if boolean:
if p_value >= kwargs["significance_level"]:
return True
else:
return False
else:
return coef, p_value
def _get_predictions(X, Y, Z, data, **kwargs):
"""
Function to get predictions using XGBoost for `ci_pillai`.
"""
# Step 0: Check if XGboost is installed.
try:
from xgboost import XGBClassifier, XGBRegressor
except ImportError as e:
raise ImportError(
e.message
+ ". xgboost is required for using pillai_trace test. Please install using: pip install xgboost"
)
# Step 1: Check if any of the conditional variables are categorical
if any(data.loc[:, Z].dtypes == "category"):
enable_categorical = True
else:
enable_categorical = False
# Step 2: Check variable type of X, choose estimator, and compute predictions.
if data.loc[:, X].dtype == "category":
clf_x = XGBClassifier(
enable_categorical=enable_categorical,
seed=kwargs.get("seed"),
random_state=kwargs.get("seed"),
)
x, x_cat_index = pd.factorize(data.loc[:, X])
clf_x.fit(data.loc[:, Z], x)
pred_x = clf_x.predict_proba(data.loc[:, Z])
else:
clf_x = XGBRegressor(
enable_categorical=enable_categorical,
seed=kwargs.get("seed"),
random_state=kwargs.get("seed"),
)
x = data.loc[:, X]
x_cat_index = None
clf_x.fit(data.loc[:, Z], x)
pred_x = clf_x.predict(data.loc[:, Z])
# Step 3: Check variable type of Y, choose estimator, and compute predictions.
if data.loc[:, Y].dtype == "category":
clf_y = XGBClassifier(
enable_categorical=enable_categorical,
seed=kwargs.get("seed"),
random_state=kwargs.get("seed"),
)
y, y_cat_index = pd.factorize(data.loc[:, Y])
clf_y.fit(data.loc[:, Z], y)
pred_y = clf_y.predict_proba(data.loc[:, Z])
else:
clf_y = XGBRegressor(
enable_categorical=enable_categorical,
seed=kwargs.get("seed"),
random_state=kwargs.get("seed"),
)
y = data.loc[:, Y]
y_cat_index = None
clf_y.fit(data.loc[:, Z], y)
pred_y = clf_y.predict(data.loc[:, Z])
# Step 4: Return the predictions.
return (pred_x, pred_y, x_cat_index, y_cat_index)
[docs]
def pillai_trace(X, Y, Z, data, boolean=True, **kwargs):
"""
A mixed-data residualization based conditional independence test[1].
Uses XGBoost estimator to compute LS residuals[2], and then does an
association test (Pillai's Trace) on the residuals.
Parameters
----------
X: str
The first variable for testing the independence condition X \u27C2 Y | Z
Y: str
The second variable for testing the independence condition X \u27C2 Y | Z
Z: list/array-like
A list of conditional variable for testing the condition X \u27C2 Y | Z
data: pandas.DataFrame
The dataset in which to test the indepenedence condition.
boolean: bool
If boolean=True, an additional argument `significance_level` must
be specified. If p_value of the test is greater than equal to
`significance_level`, returns True. Otherwise returns False.
If boolean=False, returns the pearson correlation coefficient and p_value
of the test.
Returns
-------
CI Test results: tuple or bool
If boolean=True, returns True if p-value >= significance_level, else False. If
boolean=False, returns a tuple of (Pearson's correlation Coefficient, p-value)
References
----------
[1] Ankan, Ankur, and Johannes Textor. "A simple unified approach to testing high-dimensional conditional independences for categorical and ordinal data." Proceedings of the AAAI Conference on Artificial Intelligence.
[2] Li, C.; and Shepherd, B. E. 2010. Test of Association Between Two Ordinal Variables While Adjusting for Covariates. Journal of the American Statistical Association.
[3] Muller, K. E. and Peterson B. L. (1984) Practical Methods for computing power in testing the multivariate general linear hypothesis. Computational Statistics & Data Analysis.
"""
# Step 1: Test if the inputs are correct
if not hasattr(Z, "__iter__"):
raise ValueError(f"Variable Z. Expected type: iterable. Got type: {type(Z)}")
else:
Z = list(Z)
if not isinstance(data, pd.DataFrame):
raise ValueError(
f"Variable data. Expected type: pandas.DataFrame. Got type: {type(data)}"
)
# Step 1.1: If no conditional variables are specified, use a constant value.
if len(Z) == 0:
Z = ["cont_Z"]
data = data.assign(cont_Z=np.ones(data.shape[0]))
# Step 2: Get the predictions
pred_x, pred_y, x_cat_index, y_cat_index = _get_predictions(X, Y, Z, data, **kwargs)
# Step 3: Compute the residuals
if data.loc[:, X].dtype == "category":
x = pd.get_dummies(data.loc[:, X]).loc[
:, x_cat_index.categories[x_cat_index.codes]
]
# Drop last column to avoid multicollinearity
res_x = (x - pred_x).iloc[:, :-1]
else:
res_x = data.loc[:, X] - pred_x
if data.loc[:, Y].dtype == "category":
y = pd.get_dummies(data.loc[:, Y]).loc[
:, y_cat_index.categories[y_cat_index.codes]
]
# Drop last column to avoid multicollinearity
res_y = (y - pred_y).iloc[:, :-1]
else:
res_y = data.loc[:, Y] - pred_y
# Step 4: Compute Pillai's trace.
if isinstance(res_x, pd.Series):
res_x = res_x.to_frame()
if isinstance(res_y, pd.Series):
res_y = res_y.to_frame()
cca = CCA(scale=False, n_components=min(res_x.shape[1], res_y.shape[1]))
res_x_c, res_y_c = cca.fit_transform(res_x, res_y)
cancor = []
for i in range(min(res_x.shape[1], res_y.shape[1])):
cancor.append(np.corrcoef(res_x_c[:, [i]].T, res_y_c[:, [i]].T)[0, 1])
coef = (np.array(cancor) ** 2).sum()
# Step 5: Compute p-value using f-approximation [3].
s = min(res_x.shape[1], res_y.shape[1])
df1 = res_x.shape[1] * res_y.shape[1]
df2 = s * (data.shape[0] - 1 + s - res_x.shape[1] - res_y.shape[1])
f_stat = (coef / df1) * (df2 / (s - coef))
p_value = 1 - stats.f.cdf(f_stat, df1, df2)
# Step 6: Return
if boolean:
if p_value >= kwargs["significance_level"]:
return True
else:
return False
else:
return coef, p_value
