The performance of objective speech and audio quality measures for the prediction of the perceived quality of frequency-compressed speech in hearing aids is investigated in this paper. A number of existing quality measures have been applied to speech signals processed by a hearing aid, which compresses speech spectra along frequency in order to make information contained in higher frequencies audible for listeners with severe high-frequency hearing loss. Quality measures were compared with subjective ratings obtained from normal hearing and hearing impaired children and adults in an earlier study. High correlations were achieved with quality measures computed by quality models that are based on the auditory model of Dau et al., namely, the measure PSM, computed by the quality model PEMO-Q; the measure qc, computed by the quality model proposed by Hansen and Kollmeier; and the linear subcomponent of the HASQI. For the prediction of quality ratings by hearing impaired listeners, extensions of some models incorporating hearing loss were implemented and shown to achieve improved prediction accuracy. Results indicate that these objective quality measures can potentially serve as tools for assisting in initial setting of frequency compression parameters.