A number of connections between the first nonlinear dielectric increment and dipole correlation functions have been proposed over a large span of years. These range from the Langevin-Debye approach, the use of cavity and reaction fields, a cavity free formulation, to modified Langevin-Debye approaches. Comparisons of the predictions of a number of these approaches, together with results of molecular dynamics simulations and an experimental result, are given. The relations include those by Booth and by Kielich based on the use of cavity and reaction fields as traditionally used in the determination of such connections but also including a partial correction by Booth for some ignored nonlinear effects, those by Sandberg and Edholm and by Jha and Freed based on the use of a Langevin-Debye type of approach, and one based on a cavity free non-Langevin-Debye formulation that automatically includes all nonlinear effects to the appropriate order. The local structures of water used for the determination of the pertinent correlation functions are chosen to be given by the Bernal-Fowler model, by a modification of the Berna-Fowler model, and by the Onsager approximation. In the limit of epsilon>>epsilon(infinity), the cavity free connection gives results for the first nonlinear decrements 36% larger than the decrement obtained by Booth, irrespective of the model used for the dipole correlation functions. The inclusion of epsilon(infinity) is found to decrease the decrements by about 4%. Using parameters deduced from the requirement that the Kirkwood-Frohlich connection give the experimental value of epsilon, the Booth uncorrected expression for the dielectric decrement using the modified Bernal-Fowler model is found to give good agreement with the simulations of Yeh and Berkowitz, while the cavity free result is too large. Using the Bernal-Fowler model for the local structure of water, the cavity free expression gives good agreement with the simulation results, the partially corrected Booth expression gives reasonable agreement, while the Booth uncorrected expression is too small. Comparisons to the experimental value of the nonlinear coefficient of (1.00+/-0.15)x10(-15) m(2)/V(2) as found by Kołodziej et al. are also made. Using the Bernal-Fowler model, the calculated nonlinear coefficients divided by 10(-15) m(2)/V(2) are as follows: Booth, 0.82; Booth including partial corrections for nonlinear effects on the cavity and reaction fields, 0.99; Kielich, 0.83; cavity free, 1.12; and modified Langevin-Debye, 2.93. The partially corrected Booth value shows the best fit, with the cavity free value next best but still within the cited error range. If the slope of the line given by Kołodziej et al. is used as the measure of the nonlinear increment, the experimental value is 1.10x10(-15) m(2)/V(2), similar to the corrected Booth and remarkably similar to the cavity free result.