code: def test_qif_numdiff(fam, cov_struct):
      # Test the analytic scores against numeric derivatives
  
      np.random.seed(234234)
      n = 200
      q = 4
      x = np.random.normal(size=(n, 3))
      if isinstance(fam, families.Gaussian):
          e = np.kron(np.random.normal(size=n // q), np.ones(q))
          e = np.sqrt(0.5) * e + np.sqrt(
              1 - 0.5**2
          ) * np.random.normal(size=n)
          y = x.sum(1) + e
      elif isinstance(fam, families.Poisson):
          y = np.random.poisson(5, size=n)
      elif isinstance(fam, families.Binomial):
          y = np.random.randint(0, 2, size=n)
      g = np.kron(np.arange(n // q), np.ones(q)).astype(int)
  
      model = QIF(y, x, groups=g, family=fam, cov_struct=cov_struct)
  
      for _ in range(5):
          pt = np.random.normal(size=3)
  
          # Check the Jacobian of the vector of estimating equations.
          _, grad, _, _, gn_deriv = model.objective(pt)
  
          def llf_gn(params):
              return model.objective(params)[3]
  
          gn_numdiff = approx_fprime(pt, llf_gn, 1e-7)
          assert_allclose(gn_deriv, gn_numdiff, 1e-4)
  
          # Check the gradient of the QIF
          def llf(params):
              return model.objective(params)[0]
  
          grad_numdiff = approx_fprime(pt, llf, 1e-7)
          assert_allclose(grad, grad_numdiff, 1e-4)
  
file_path: statsmodels/statsmodels/genmod/tests/test_qif.py
imports: import numpy
  from numpy.testing import assert_allclose
  import pandas
  import pytest
  from statsmodels.genmod.qif import QIF, QIFIndependence, QIFExchangeable, QIFAutoregressive
  from statsmodels.tools.numdiff import approx_fprime
  from statsmodels.genmod import families
label_desc: Use image node_id 1 for calling a global function with example usage: test_qif_numdiff(fam, cov_struct) and returns: model, model