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The phrase "associated with a fall" is correct and usable in written English.
It can be used in contexts discussing causes, effects, or relationships related to a fall, such as in medical, safety, or psychological discussions.
Example: "The study found that several factors were associated with a fall, including age, medication use, and environmental hazards."
Alternatives: "linked to a fall" or "related to a fall".
Exact(60)
Except for the first campaign which saw an increase in emergency numbers, the other two were associated with a fall in total numbers of emergencies, though the magnitude of the fall was less than for non-emergencies.
The fracture site associated with a fall is strongly dependent on the site of impact and the orientation of the fall [55,56].
Neither incident was associated with a fall.
B.P. was associated with a fall from the Nipissing II high-water-level phase.
"If you get rising income inequality, that's usually associated with a fall in social mobility," Cassidy responds.
But that decline, though of course welcome, has – unlike here – been associated with a fall in labour participation.
In standard theory, such increases would be associated with a fall in the return to capital and an increase in wages.
In one year – 2008 – treatment with the pesticide was associated with a fall in yield, possibly due to heavy rain causing the chemical to leach away.
An increase of 5 mm Hg in daytime mean diastolic blood pressure was associated with a fall in birth weight of 68.5 g.
Rather than reflecting buoyant job creation, too much of the recent decline in the unemployment rate has been associated with a fall in labour force participation to a level last seen 35 years ago.
This paper shows that job loss is associated with a fall in subjective well-being (SWB).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com