  We calculate how strongly one can put constraints on alternative theories of
gravity such as Brans-Dicke and massive graviton theories with LISA. We
consider inspiral gravitational waves from a compact binary composed of a
neutron star (NS) and an intermediate mass black hole (IMBH) in Brans-Dicke
(BD) theory and that composed of 2 super massive black holes (SMBHs) in massive
graviton theories. We use the restricted 2PN waveforms including the effects of
spins. We also take both precession and eccentricity of the orbit into account.
For simplicity, we set the fiducial value for the spin of one of the binary
constituents to zero so that we can apply the approximation called
\textit{simple precession}. We perform the Monte Carlo simulations of $10^4$
binaries, estimating the determination accuracy of binary parameters including
the BD parameter $\omega_{\mathrm{BD}}$ and the Compton wavelength of graviton
$\lambda_g$ for each binary using the Fisher matrix method. We find that
including both the spin-spin coupling $\sigma$ and the eccentricity $e$ into
the binary parameters reduces the determination accuracy by an order of
magnitude for the Brans-Dicke case, whilst it has less influence on massive
graviton theories. On the other hand, including precession enhances the
constraint on $\omega_{\mathrm{BD}}$ only 20$%$ but it increases the constraint
on $\lambda_g$ by an order of magnitude. Using a $(1.4+1000)M_{\odot}$ NS/BH
binary of SNR=$\sqrt{200}$, one can put a constraint
$\omega_{\mathrm{BD}}>6944$, whilst using a $(10^7+10^6)M_{\odot}$ BH/BH binary
at 3Gpc, one can put $\lambda_g>3.06\times10^{21}$cm, on average. The latter is
4 orders of magnitude stronger than the one obtained from the solar system
experiment. These results indicate that the effects of precession and
eccentricity must be taken carefully in the parameter estimation analysis.
