Nearly all of the power used by a conventional (non-DAB) radio is used by the audio amplification stage. Using a radio at full volume will drain the batteries several times faster than using it at minimum volume.
For example, I keep a very small transistor radio (powered by two little AA cells) switched on all night, every night, in my bedroom. (I'm an insomniac who also hates silence). The volume is so low that I can only hear the radio if I'm lying on the same side of the bed as the radio. A set of batteries lasts nearly a year. However, if I had the volume turned up to the maximum, I doubt that the batteries would last more than a week. (I probably wouldn't anyway, since my neighbours would kill me for keeping them awake every night!).
With DAB radios, the difference in battery usage (with different volumes) is not as marked because a great deal of the power is used by the parts of the radio that do the same job whatever the volume is. Even so, there would still be a noticeable difference, with the 'maximum volume' radio's batteries expiring first.
Chris