We investigate the impact of Raman scattering on pure quartic solitons (PQSs) in an optical fiber with quartic dispersion. An analytical expression of the Raman-induced frequency shift (RIFS) of a PQS is obtained by using the variational approach with the Gaussian function ansatz. We find the RIFS of a PQS is inversely proportional to the sixth power of pulse width, when the fiber is short enough. The RIFS of a PQS is more sensitive to the pulse width, compared with that of a conventional soliton which is inversely proportional to the fourth power of pulse width. The theoretical predictions show good agreement with numerical results. In addition, we also discuss the RIFS of the other three typical pulses with the same peak power and pulse width as the PQS. These results provide a thorough understanding of the role of higher-order nonlinear effects on the propagation dynamics of PQSs.