Each step in the erectile process can -reasonably- be assessed directly. A good history and physical exam is the best place to start, with hormone profiling (Testosterone, prolactin, sometimes with screening chemical and thyroid) often points initially to where the problem is. The penile circulation can be measured with penile duplex (color) Doppler exam, done after erection is induced, and the venous side (or corporal competence) inferred from those results but directly measured with DICC (Infusion corpus cavernosometry). The site and size of true venous leak require corpus cavernosography (xray) for demonstration. If arterial insufficiency is found, and arterial repair is contemplated, pelvic/penile arteriograms are required to definitively demonstrate the arterial vessels involved.
Finally, the function of the pelvic and penile nerves can be measured directly with evoked potential and conduction studies. The competency of the corpora can be assessed with compliance and conduction studies, most of which are more commonly used in Europe than in the US, and many of which are still considered investigation.
Erections which occur during sleep (during REM sleep) have been considered to demonstrate erectile ability in the absence of mental or psychological inhibition, and does provide some useful data, but is not 100% accurate in excluding organic erectile dysfunction. For example, erectile dysfunction due to circulatory steal syndrome (arterial insufficiency which is marginal and only above threshold for dysfunction when other demands, such as pelvic motion or different position occur), or to the use of certain antihypertensive medications, can give a perfectly normal set of sleep erections (NPT).