  Matter-wave interferometers utilizing different isotopes or chemical elements
intrinsically have different sensitivities, and the analysis tools available
until now are insufficient for accurately estimating the atomic phase
difference under many experimental conditions. In this work, we describe and
demonstrate two new methods for extracting the differential phase between
dual-species atom interferometers for precise tests of the weak equivalence
principle. The first method is a generalized Bayesian analysis, which uses
knowledge of the system noise to estimate the differential phase based on a
statistical model. The second method utilizes a mechanical accelerometer to
reconstruct single-sensor interference fringes based on measurements of the
vibration-induced phase. An improved ellipse-fitting algorithm is also
implemented as a third method for comparison. These analysis tools are
investigated using both numerical simulations and experimental data from
simultaneous $^{87}$Rb and $^{39}$K interferometers, and both new techniques
are shown to produce bias-free estimates of the differential phase. We also
report observations of phase correlations between atom interferometers composed
of different chemical species. This correlation enables us to reject
common-mode vibration noise by a factor of 730, and to make preliminary tests
of the weak equivalence principle with a sensitivity of $1.6 \times 10^{-6}$
per measurement with an interrogation time of $T = 10$ ms. We study the level
of vibration rejection by varying the temporal overlap between interferometers
in a symmetric timing sequence. Finally, we discuss the limitations of the new
analysis methods for future applications of differential atom interferometry.
