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Empirical null

The nipy.algorithms.statistics.empirical_pvalue module contains a class that fits a Gaussian model to the central part of an histogram, following Schwartzman et al, 2009. This is typically necessary to estimate a FDR when one is not certain that the data behaves as a standard normal under H_0.

The NormalEmpiricalNull class learns its null distribution on the data provided at initialisation. Two different methods can be used to set a threshold from the null distribution: the NormalEmpiricalNull.threshold() method returns the threshold for a given false discovery rate, and thus accounts for multiple comparisons with the given dataset; the NormalEmpiricalNull.uncorrected_threshold() returns the threshold for a given uncorrected p-value, and as such does not account for multiple comparisons.

Example

If we use the empirical normal null estimator on a two Gaussian mixture distribution, with a central Gaussian, and a wide one, it uses the central distribution as a null hypothesis, and returns the threshold following which the data can be claimed to belong to the wide Gaussian:

The threshold evaluated with the NormalEmpiricalNull.threshold() method is around 2.8 (using the default p-value of 0.05). The NormalEmpiricalNull.uncorrected_threshold() returns, for the same p-value, a threshold of 1.9. It is necessary to use a higher p-value with uncorrected comparisons.

Class documentation

class nipy.algorithms.statistics.empirical_pvalue.NormalEmpiricalNull(x)

Class to compute the empirical null normal fit to the data.

The data which is used to estimate the FDR, assuming a Gaussian null from Schwartzmann et al., NeuroImage 44 (2009) 71–82

Methods

fdr([p_values, verbose]) Returns the FDR associated with each p value
fdrcurve
learn
plot
threshold
uncorrected_threshold
__init__(x)

Initialize an empirical null normal object.

Parameters :

x : 1D ndarray

The data used to estimate the empirical null.

fdr(theta)

Given a threshold theta, find the estimated FDR

Parameters :

theta : float or array of shape (n_samples)

values to test

Returns :

afp : value of array of shape(n)

fdrcurve()

Returns the FDR associated with any point of self.x

learn(left=0.20000000000000001, right=0.80000000000000004)

Estimate the proportion, mean and variance of a Gaussian distribution for a fraction of the data

Parameters :

left: float, optional :

Left cut parameter to prevent fitting non-gaussian data

right: float, optional :

Right cut parameter to prevent fitting non-gaussian data

Notes

This method stores the following attributes:

mu = mu p0 = min(1, np.exp(lp0)) sqsigma: standard deviation of the estimated normal

distribution
sigma: np.sqrt(sqsigma) : variance of the estimated
normal distribution
plot(efp=None, alpha=0.050000000000000003, bar=1, mpaxes=None)

Plot the histogram of x

Parameters :

efp : float, optional

The empirical FDR (corresponding to x) if efp==None, the false positive rate threshold plot is not drawn.

alpha : float, optional

The chosen FDR threshold

bar=1 : bool, optional

mpaxes=None: if not None, handle to an axes where the fig :

will be drawn. Avoids creating unnecessarily new figures :

threshold(alpha=0.050000000000000003, verbose=0)

Compute the threshold corresponding to an alpha-level FDR for x

Parameters :

alpha : float, optional

the chosen false discovery rate threshold.

verbose : boolean, optional

the verbosity level, if True a plot is generated.

Returns :

theta: float :

the critical value associated with the provided FDR

uncorrected_threshold(alpha=0.001, verbose=0)

Compute the threshold corresponding to a specificity alpha for x

Parameters :

alpha : float, optional

the chosen false discovery rate (FDR) threshold.

verbose : boolean, optional

the verbosity level, if True a plot is generated.

Returns :

theta: float :

the critical value associated with the provided p-value


Reference: Schwartzmann et al., NeuroImage 44 (2009) 71–82