For a compiled sample of 120 reverberation-mapped AGNs, the bivariate correlations of the broad-line region (BLR) size (R_BLR_) with the continuum luminosity at 5100{AA} (L_5100_) and the dimensionless accretion rates (dM/dt) are investigated. Using our recently calibrated virial factor f, and the velocity tracer from the H{beta} full width at half-maximum (FWHM(H{beta})) or the line dispersion ({sigma}_H{beta}_) measured in the mean spectra, three kinds of supermassive black hole (SMBH) masses and dM/dt are calculated. An extended R_BLR_(H{beta})-L_5100_ relation including dM/dt is found to be stronger than the canonical R_BLR_(H{beta})-L_5100_ relation, showing smaller scatters. The observational parameters, R_Fe_ (the ratio of optical FeII to H{beta} line flux) and the line profile parameter D_H{beta}_ (D_H{beta}_=FWHM(H{beta})/{sigma}_H{beta}_), have relations with three kinds of dM/dt. Using R_Fe_ and D_H{beta}_ to substitute dM/dt, extended empirical R_BLR_(H{beta})-L_5100_ relations are presented. R_Fe_ is a better 'fix' for the R_BLR_(H{beta})-L_5100_ offset than the H{beta} shape D_H{beta}_. The extended empirical R_BLR_(H{beta})-L_5100_ relation including R_Fe_ can be used to calculate R_BLR_, and thus the single-epoch SMBH mass M_BH_. Our measured accretion rate dependence is not consistent with the simple model of the accretion disc instability leading the BLR formation. The BLR may instead form from the inner edge of the torus, or from some other means in which BLR size is positively correlated with accretion rate and the SMBH mass.