Software and Hardware Volume Settings

I’ve often wondered, in terms of audio quality is it better to turn the volume up in software, in the operating system, or on the hardware side (e.g., powered speakers or a headphone amp)? I found this clear and concise answer on superuser.com:

Ideally, you should output audio from your computer at full volume, so as to get the highest resolution (bit depth) possible, and then have an analogue volume control as one of the last things in front of the speakers. Assuming all the devices in your signal path are of more or less comparable quality (i.e., you’re not pairing a cheap low-end amplifier with a high-end digital source and DAC), that should give the best audio quality.

The relationship between software volume and bit depth is explained as follows:

Reducing volume in software is basically equivalent to reducing the bit depth. In digital audio, the signal is split up into distinct samples (taken thousands of times per second), and bit depth is the number of bits that are used to describe each sample. Attenuating a signal is done by multiplying each sample by a number less than one, with the result being that you’re no longer using the full resolution to describe the audio, resulting in reduced dynamic range and signal-to-noise ratio.