code: def vfunc(pars_1, pars_2):
      
      cart_1 = dir2cart([pars_1["dec"], pars_1["inc"], pars_1["r"]])
      cart_2 = dir2cart([pars_2['dec'], pars_2['inc'], pars_2["r"]])
      Sw = pars_1['k'] * pars_1['r'] + pars_2['k'] * pars_2['r']  # k1*r1+k2*r2
      xhat_1 = pars_1['k'] * cart_1[0] + pars_2['k'] * cart_2[0]  # k1*x1+k2*x2
      xhat_2 = pars_1['k'] * cart_1[1] + pars_2['k'] * cart_2[1]  # k1*y1+k2*y2
      xhat_3 = pars_1['k'] * cart_1[2] + pars_2['k'] * cart_2[2]  # k1*z1+k2*z2
      Rw = np.sqrt(xhat_1**2 + xhat_2**2 + xhat_3**2)
      return 2 * (Sw - Rw)
docs: Calculate the Watson Vw test statistic. Calculated as 2*(Sw-Rw)
  
      Parameters
      ----------
      pars_1 : dictionary of Fisher statistics from population 1
      pars_2 : dictionary of Fisher statistics from population 2
  
      Returns
      -------
      Vw : Watson's Vw statistic