  Axion-like particles (ALPs) may undergo mixing with photons in the presence
of astrophysical magnetic fields, leading to alterations in the observed high
energy $\gamma$-ray spectra. In this study, we investigate the ALP-photon
oscillation effect using the spectra of the blazar Mrk 421 over 15 observation
periods measured by Major Atmospheric Gamma Imaging Cherenkov Telescopes
(MAGIC) and Fermi Large Area Telescope (Fermi-LAT). Compared with previous
studies, we generate mock data under the ALP hypothesis and employ the ${\rm
CL_s}$ method to set constraints on the ALP parameters. This method is widely
utilized in high energy experiments and avoids the exclusion of specific
parameter regions where distinguishing between the null and ALP hypotheses is
challenging. We find that the ALP-photon coupling $g_{a\gamma}$ is constrained
to be smaller than $\sim 2\times10^{-11}$ GeV$^{-1}$ for ALP masses ranging
from $10^{-9}$ eV to $10^{-7}$ eV at the 95\% confidence level. We also present
the constraints derived from the TS distribution under the null hypothesis,
which is commonly utilized in previous astrophysical ALP studies. Our results
reveal that the combined constraints of all the periods obtained from both
methods are consistent. However, the ${\rm CL_s}$ method remains effective in
cases where the latter method fails to provide constraints for specific
observation periods.
