The phrase "associated not only" is correct and usable in written English.
It can be used when you want to indicate that something is linked or connected to multiple elements or factors, typically followed by "but also" to introduce additional elements.
Example: "The study found that the disease is associated not only with genetic factors but also with environmental influences."
Alternatives: "linked not just to" or "connected not only to".
Low selenium levels were associated not only with an increased risk of osteoarthritis, but more severe disease, too.
Five behaviors in elderly men are associated not only with living into extreme old age, a new study has found, but also with good health and independent functioning.
Stern was closely associated not only with Bellow but also with Philip Roth his bad luck, in a way, as such company would make most writers seem minor.
History shows that trends of rising inequality are associated not only with economic changes, but also with the eroding strength of what the economist Frank Levy of M.I.T. calls "the nation's equalizing institutions".
In the past few years, several studies have concluded that elevated mercury levels may be associated not only with neurological problems but with cardiovascular disease among adults as well.
All turned 21 in the early to mid-50s, a time when homosexuality became the number one enemy within - associated not only with perversion, but, thanks to the defection of Guy Burgess and Donald MacLean, the dreaded spectre of communism.
In large gateway cities, immigration has been associated not only with a decrease in crime but also with economic revitalization and reductions in concentrated poverty.
These are associated not only with global patient movement but also with respiratory motion.
These areas are associated not only with memory, but with behavior and reasoning.
HAM1101 was associated not only with well-maintained glucose formation but also with a higher direct, aerobic glucose oxidation rate.
The dimensional change was associated not only with amorphous ettringite formation, but also with the subsequent formation of a C-S-H gel.
