Firth regression in python

firth_regression.py

#!/usr/bin/env python
'''Python implementation of Firth regression by John Lees
See https://www.ncbi.nlm.nih.gov/pubmed/12758140'''

def firth_likelihood(beta, logit):
    return -(logit.loglike(beta) + 0.5*np.log(np.linalg.det(-logit.hessian(beta))))

# Do firth regression
# Note information = -hessian, for some reason available but not implemented in statsmodels
def fit_firth(y, X, start_vec=None, step_limit=1000, convergence_limit=0.0001):

    logit_model = smf.Logit(y, X)
    
    if start_vec is None:
        start_vec = np.zeros(X.shape[1])
    
    beta_iterations = []
    beta_iterations.append(start_vec)
    for i in range(0, step_limit):
        pi = logit_model.predict(beta_iterations[i])
        W = np.diagflat(np.multiply(pi, 1-pi))
        var_covar_mat = np.linalg.pinv(-logit_model.hessian(beta_iterations[i]))

        # build hat matrix
        rootW = np.sqrt(W)
        H = np.dot(np.transpose(X), np.transpose(rootW))
        H = np.matmul(var_covar_mat, H)
        H = np.matmul(np.dot(rootW, X), H)

        # penalised score
        U = np.matmul(np.transpose(X), y - pi + np.multiply(np.diagonal(H), 0.5 - pi))
        new_beta = beta_iterations[i] + np.matmul(var_covar_mat, U)

        # step halving
        j = 0
        while firth_likelihood(new_beta, logit_model) > firth_likelihood(beta_iterations[i], logit_model):
            new_beta = beta_iterations[i] + 0.5*(new_beta - beta_iterations[i])
            j = j + 1
            if (j > step_limit):
                sys.stderr.write('Firth regression failed\n')
                return None

        beta_iterations.append(new_beta)
        if i > 0 and (np.linalg.norm(beta_iterations[i] - beta_iterations[i-1]) < convergence_limit):
            break

    return_fit = None
    if np.linalg.norm(beta_iterations[i] - beta_iterations[i-1]) >= convergence_limit:
        sys.stderr.write('Firth regression failed\n')
    else:
        # Calculate stats
        fitll = -firth_likelihood(beta_iterations[-1], logit_model)
        intercept = beta_iterations[-1][0]
        beta = beta_iterations[-1][1:].tolist()
        bse = np.sqrt(np.diagonal(np.linalg.pinv(-logit_model.hessian(beta_iterations[-1]))))
        
        return_fit = intercept, beta, bse, fitll

    return return_fit

if __name__ == "__main__":

    import sys
    import warnings
    import math
    import statsmodels
    import numpy as np
    from scipy import stats
    import statsmodels.api as smf
  
    # create X and y here. Make sure X has an intercept term (column of ones)
    # ...

    # How to call and calculate p-values
    (intercept, beta, bse, fitll) = fit_firth(y, X)
    beta = [intercept] + beta
    
    # Wald test
    waldp = []
    for beta_val, bse_val in zip(beta, bse):
        waldp.append(2 * (1 - stats.norm.cdf(abs(beta_val/bse_val))))

    # LRT
    lrtp = []
    for beta_idx, (beta_val, bse_val) in enumerate(zip(beta, bse)):
        null_X = np.delete(X, beta_idx, axis=1)
        (null_intercept, null_beta, null_bse, null_fitll) = fit_firth(y, null_X)
        lrstat = -2*(null_fitll - fitll)
        lrt_pvalue = 1
        if lrstat > 0: # non-convergence
            lrt_pvalue = stats.chi2.sf(lrstat, 1)
        lrtp.append(lrt_pvalue)

@IRambags actually sorry, I'm not sure what's causing the error with fitting the null. What length are beta and bse in the loop?

both len(beta) and len(bse) equal 2

Does running np.delete(X, 0, axis=1) or np.delete(X, 1, axis=1) work?

Unfortunately not, both give the same ValueError as before. Does it matter that I work in jupyter i.e. iPython?

Does it matter that I work in jupyter i.e. iPython?

I wouldn't have thought so.

Double check the shape of X being passed. You can also remove the necessary column using other indexing techniques if this isn't working for you https://numpy.org/doc/stable/reference/arrays.indexing.html

I indeed now just went for null_X = X.drop('const', 1) and this seems to work. Very strange functioning of the np.delete() though. Thanks for your quick replies and support!

Hi all, I'm working on an sklearn-compatible implementation based on logistf here: https://github.com/jzluo/firthlogist
I would greatly appreciate any feedback on it!
There are a few implementation differences compared to this gist. Most notably, the full hat matrix is not calculated which gives significant memory savings, and penalized likelihood ratio tests are used as in logistf for p-values (some more detail here).

Hi all, I'm working on an sklearn-compatible implementation based on logistf here: https://github.com/jzluo/firthlogist I would greatly appreciate any feedback on it! There are a few implementation differences compared to this gist. Most notably, the full hat matrix is not calculated which gives significant memory savings, and penalized likelihood ratio tests are used as in logistf for p-values (some more detail here).

@jzluo Amazing! Looks like a nice improvement over this code 🙂

Thank you for your code!
I am running your code by I am receiving an error at line:
-> (intercept, beta, bse, fitll) = fit_firth(y, X) with the following message "Unable to coerce to Series, length must be 1: given 11871"

I have included an extract of X: X.shape -> (11871, 4)
is_male age genotype intercept term
0 0 28.92 0 1
1 0 70.95 0 1
2 0 29.92 0 1
.... ... ... ... ...
11869 0 74.95 0
11870 0 73.95 0

and y: y.shape -> (11871, 1)
Experimental Group
0 0
1 0
... ...
11869 1
11870 1

Thank you for your help it is much appreciated