code: def test_formula(cov_struct):
      np.random.seed(3423)
  
      y = np.random.normal(size=100)
      x = np.random.normal(size=(100, 2))
      groups = np.kron(np.arange(25), np.ones(4))
  
      model1 = QIF(y, x, groups=groups, cov_struct=cov_struct)
      result1 = model1.fit()
  
      df = pd.DataFrame(
          {"y": y, "x1": x[:, 0], "x2": x[:, 1], "groups": groups}
      )
  
      model2 = QIF.from_formula(
          "y ~ 0 + x1 + x2",
          groups="groups",
          cov_struct=cov_struct,
          data=df,
      )
      result2 = model2.fit()
  
      assert_allclose(result1.params, result2.params)
      assert_allclose(result1.bse, result2.bse)
  
      if not isinstance(cov_struct, QIFIndependence):
          _ = result2.bic
          _ = result2.aic
  
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 3 for calling a global function with example usage: test_formula(cov_struct) without return types