# algorithms.statistics.mixed_effects_stat¶

## Module: `algorithms.statistics.mixed_effects_stat`¶

Inheritance diagram for `nipy.algorithms.statistics.mixed_effects_stat`:

Module for computation of mixed effects statistics with an EM algorithm. i.e. solves problems of the form y = X beta + e1 + e2, where X and Y are known, e1 and e2 are centered with diagonal covariance. V1 = var(e1) is known, and V2 = var(e2) = lambda identity. the code estimates beta and lambda using an EM algorithm. Likelihood ratio tests can then be used to test the columns of beta.

Author: Bertrand Thirion, 2012.

```>>> N, P = 15, 500
>>> V1 = np.random.randn(N, P) ** 2
>>> effects = np.ones(P)
>>> Y = generate_data(np.ones(N), effects, .25, V1)
>>> T1 = one_sample_ttest(Y, V1, n_iter=5)
>>> T2 = t_stat(Y)
>>> assert(T1.std() < T2.std())
```

## `MixedEffectsModel`¶

class nipy.algorithms.statistics.mixed_effects_stat.MixedEffectsModel(X, n_iter=5, verbose=False)

Bases: `object`

Class to handle multiple one-sample mixed effects models

__init__(X, n_iter=5, verbose=False)

Set the effects and first-level variance, and initialize related quantities

Parameters:
X: array of shape(n_samples, n_effects),

the design matrix

n_iter: int, optional,

number of iterations of the EM algorithm

verbose: bool, optional, verbosity mode
fit(Y, V1)

Launches the EM algorithm to estimate self

Parameters:
Y, array of shape (n_samples, n_tests) or (n_samples)

the estimated effects

V1, array of shape (n_samples, n_tests) or (n_samples)

first-level variance

Returns:
self
log_like(Y, V1)

Compute the log-likelihood of (Y, V1) under the model

Parameters:
Y, array of shape (n_samples, n_tests) or (n_samples)

the estimated effects

V1, array of shape (n_samples, n_tests) or (n_samples)

first-level variance

Returns:
logl: array of shape self.n_tests,

the log-likelihood of the model

predict(Y, V1)

Return the log_likelihood of the data.See the log_like method

score(Y, V1)

Return the log_likelihood of the data. See the log_like method

## Functions¶

nipy.algorithms.statistics.mixed_effects_stat.check_arrays(Y, V1)

Check that the given data can be used for the models

Parameters:
Y: array of shape (n_samples, n_tests) or (n_samples)

the estimated effects

V1: array of shape (n_samples, n_tests) or (n_samples)

first-level variance

nipy.algorithms.statistics.mixed_effects_stat.generate_data(X, beta, V2, V1)

Generate a group of individuals from the provided parameters

Parameters:
X: array of shape (n_samples, n_reg),

the design matrix of the model

beta: float or array of shape (n_reg, n_tests),

the associated effects

V2: float or array of shape (n_tests),

group variance

V1: array of shape(n_samples, n_tests),

the individual variances

Returns:
Y: array of shape(n_samples, n_tests)

the individual data related to the two-level normal model

nipy.algorithms.statistics.mixed_effects_stat.mfx_stat(Y, V1, X, column, n_iter=5, return_t=True, return_f=False, return_effect=False, return_var=False, verbose=False)

Run a mixed-effects model test on the column of the design matrix

Parameters:
Y: array of shape (n_samples, n_tests)

the data

V1: array of shape (n_samples, n_tests)

first-level variance associated with the data

X: array of shape(n_samples, n_regressors)

the design matrix of the model

column: int,

index of the column of X to be tested

n_iter: int, optional,

number of iterations of the EM algorithm

return_t: bool, optional,

should one return the t test (True by default)

return_f: bool, optional,

should one return the F test (False by default)

return_effect: bool, optional,

should one return the effect estimate (False by default)

return_var: bool, optional,

should one return the variance estimate (False by default)

verbose: bool, optional, verbosity mode
Returns:
(tstat, fstat, effect, var): tuple of arrays of shape (n_tests),

those required by the input return booleans

nipy.algorithms.statistics.mixed_effects_stat.one_sample_ftest(Y, V1, n_iter=5, verbose=False)

Returns the mixed effects F-stat for each row of the X (one sample test) This uses the Formula in Roche et al., NeuroImage 2007

Parameters:
Y: array of shape (n_samples, n_tests)

the data

V1: array of shape (n_samples, n_tests)

first-level variance ssociated with the data

n_iter: int, optional,

number of iterations of the EM algorithm

verbose: bool, optional, verbosity mode
Returns:
fstat, array of shape (n_tests),

statistical values obtained from the likelihood ratio test

sign, array of shape (n_tests),

sign of the mean for each test (allow for post-hoc signed tests)

nipy.algorithms.statistics.mixed_effects_stat.one_sample_ttest(Y, V1, n_iter=5, verbose=False)

Returns the mixed effects t-stat for each row of the X (one sample test) This uses the Formula in Roche et al., NeuroImage 2007

Parameters:
Y: array of shape (n_samples, n_tests)

the observations

V1: array of shape (n_samples, n_tests)

first-level variance associated with the observations

n_iter: int, optional,

number of iterations of the EM algorithm

verbose: bool, optional, verbosity mode
Returns:
tstat: array of shape (n_tests),

statistical values obtained from the likelihood ratio test

nipy.algorithms.statistics.mixed_effects_stat.t_stat(Y)

Returns the t stat of the sample on each row of the matrix

Parameters:
Y, array of shape (n_samples, n_tests)
Returns:
t_variates, array of shape (n_tests)
nipy.algorithms.statistics.mixed_effects_stat.two_sample_ftest(Y, V1, group, n_iter=5, verbose=False)

Returns the mixed effects t-stat for each row of the X (one sample test) This uses the Formula in Roche et al., NeuroImage 2007

Parameters:
Y: array of shape (n_samples, n_tests)

the data

V1: array of shape (n_samples, n_tests)

first-level variance associated with the data

group: array of shape (n_samples)

a vector of indicators yielding the samples membership

n_iter: int, optional,

number of iterations of the EM algorithm

verbose: bool, optional, verbosity mode
Returns:
tstat: array of shape (n_tests),

statistical values obtained from the likelihood ratio test

nipy.algorithms.statistics.mixed_effects_stat.two_sample_ttest(Y, V1, group, n_iter=5, verbose=False)

Returns the mixed effects t-stat for each row of the X (one sample test) This uses the Formula in Roche et al., NeuroImage 2007

Parameters:
Y: array of shape (n_samples, n_tests)

the data

V1: array of shape (n_samples, n_tests)

first-level variance associated with the data

group: array of shape (n_samples)

a vector of indicators yielding the samples membership

n_iter: int, optional,

number of iterations of the EM algorithm

verbose: bool, optional, verbosity mode
Returns:
tstat: array of shape (n_tests),

statistical values obtained from the likelihood ratio test