  We discuss the relation between the fluctuation-dissipation relation derived
by Chatelain and Ricci-Tersenghi [C.Chatelain, J.Phys. A {\bf 36}, 10739
(2003); F. Ricci-Tersenghi, Phys.Rev.E 68, 065104(R) (2003)] and that by
Lippiello-Corberi-Zannetti [E. Lippiello, F. Corberi and M. Zannetti Phys. Rev.
E {\bf 72}, 056103 (2005)]. In order to do that, we re-derive the
fluctuation-dissipation relation for systems of discrete variables evolving in
discrete time via a stochastic non-equilibrium Markov process. The calculation
is carried out in a general formalism comprising the Chatelain, Ricci-Tersenghi
result and that by Lippiello-Corberi-Zannetti as special cases. The
applicability, generality, and experimental feasibility of the two approaches
is thoroughly discussed. Extending the analytical calculation to the variance
of the response function we show the vantage of field-free numerical methods
with respect to the standard method where the perturbation is applied. We also
show that the signal to noise ratio is better (by a factor $\sqrt 2$) in the
algorithm of Lippiello-Corberi-Zannetti with respect to that of Chatelain-Ricci
Tersenghi.
