Psychometric functions were measured for the discrimination of the interaural phase difference (IPD) of the envelope of a sinusoidally amplitude-modulated (SAM) 4-kHz pure tone for modulation frequencies of 128 and 300 Hz and modulation depths (m) of 0.2, 0.6, 0.9, and 1.0. Contrary to recent modeling assumptions, it was found that a constant change in normalized interaural envelope correlation, with or without additional model stages to simulate peripheral auditory processing, did not produce a constant level of performance. Rather, in some cases, performance could range from chance to near perfect across modulation depths for a given change in normalized interaural envelope correlation. This was also true for the maximum change in normalized interaural envelope correlation computed across the cross-correlation functions for the stimuli to be discriminated. The change in the interaural time difference (ITD) computed from the IPD accounted for discriminability across modulation depths better than the change in normalized interaural envelope correlation, although ITD could not account for all the data, particularly those for lower values of m.