import numpy as np
import pandas as pd
from scipy import stats
from pgmpy.independencies import IndependenceAssertion
from pgmpy.global_vars import logger
[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):
r"""
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 freeman_tuckey(X, Y, Z, data, boolean=True, **kwargs):
"""
Freeman Tuckey test for conditional independence [1].
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] Read, Campbell B. "Freeman—Tukey chi-squared goodness-of-fit statistics." Statistics & probability letters 18.4 (1993): 271-278.
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']
>>> freeman_tuckey(X='A', Y='C', Z=[], data=data, boolean=True, significance_level=0.05)
True
>>> freeman_tuckey(X='A', Y='B', Z=['D'], data=data, boolean=True, significance_level=0.05)
True
>>> freeman_tuckey(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_="freeman-tukey", **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 neyman(X, Y, Z, data, boolean=True, **kwargs):
"""
Neyman's test for conditional independence[1].
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/Neyman%E2%80%93Pearson_lemma
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']
>>> neyman(X='A', Y='C', Z=[], data=data, boolean=True, significance_level=0.05)
True
>>> neyman(X='A', Y='B', Z=['D'], data=data, boolean=True, significance_level=0.05)
True
>>> neyman(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_="neyman", **kwargs
)
[docs]def cressie_read(X, Y, Z, data, boolean=True, **kwargs):
"""
Cressie Read statistic for conditional independence[1].
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] 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']
>>> cressie_read(X='A', Y='C', Z=[], data=data, boolean=True, significance_level=0.05)
True
>>> cressie_read(X='A', Y='B', Z=['D'], data=data, boolean=True, significance_level=0.05)
True
>>> cressie_read(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_="cressie-read", **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]).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):
try:
c, _, d, _ = stats.chi2_contingency(
df.groupby([X, Y]).size().unstack(Y, fill_value=0), lambda_=lambda_
)
chi += c
dof += d
except ValueError:
# If one of the values is 0 in the 2x2 table.
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"
)
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):
r"""
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
