This is a classic case of spelling difference between American English and British English.
Etiology or aetiology is most commonly used as a medical term for the cause of a certain disease. It is also the name of the field of medicine focused on finding the cause of conditions or diseases.
Outside the United States we find aetiology, aetiologies, aetiologic, aetiological, and aetiologically.
Using as a case study Robert Burton’s 1621 book, The Anatomy of Melancholy, Kendler observed that before the middle of the 19th century, mental disorders were not thought to have a single, clear aetiology; on the contrary, Burton provides a mind-boggling list of possible causes for melancholy. [Practical Ethics]
In their paper, published in BMC Veterinary Research, they explain that C. pseudotuberculosis is the aetiological agent of caseous lymphadenitis, a common disease in small ruminant populations throughout the world and responsible for a significant economic impact for producers. [The Pig Site]
And inside the United States we have etiology, etiologies, etiologic, etiological, and etiologically.