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Characterizing the long-term variability of active galactic nuclei (AGNs) is a key legacy of the Rossi X-ray Timing Explorer (RXTE). We carry out a spectral analysis on a sample of 20 Seyfert 1 galaxies, which had been observed by the RXTE for at least 100 times. All 18 335 spectra are fitted in a uniform way using a power-law component plus an additional Gaussian line when necessary. For any source in our sample, we confirm that the spectrum softens or the photon index, Γ, increases with increasing the 2–10 keV luminosity, L X. However, different source holds distinct Γ– L X/ L Edd relation, rather than a common one. We also fit the correlation with a function of Γ = C + β × log (flux), where C is a constant and parameter β is the slope of the correlation. In this way, the increase speed of the Γ– L X/ L Edd relation can be depicted with the parameter β. Since our sample contains a large sample of long-term monitored sources, it allows us to explore the second-order effect of spectral evolution. We find that there is an anticorrelation between β and the X-ray Eddington ratio, L X/ L Edd. That is, the increasing tendency of Γ with increasing X-ray luminosity becomes slower when the source has a larger X-ray Eddington ratio. Our results may indicate changes in the geometry and/or the viscosity parameter of the accretion disc at different Eddington ratios. [ABSTRACT FROM AUTHOR]