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The phrase "affect the rate at" is correct and usable in written English.
It can be used when discussing how one factor influences the speed or frequency of another factor.
Example: "Various environmental conditions can affect the rate at which plants grow."
Alternatives: "influence the speed of" or "impact the frequency of".
Exact(53)
The quantity and composition of what is available can affect the rate at which ecosystem processes progress.
Most studies are deterministic, but uncertainty will also affect the rate at which future levels of climate damage are discounted17,18,19,20.
In these models, global warming and its impacts via climate change don't significantly affect the rate at which the economy grows.
The study, appearing in the Journal of Geophysical Research — Biogeosciences, suggests that the size of water bodies can affect the rate at which greenhouse gases, such as carbon dioxide and methane, move from ponds and lakes into the atmosphere.
I'm not sure how such a shift would affect the rate at which evangelicals and conservative Catholics turn out for Republicans — that would be the big strategic risk, obviously.
There, having a variety of different particle sizes in the sediment of a stream will affect the rate at which it recovers from environmental damage, and also its eventual biological productivity.Similar ideas apply to marine environments.
Similar(7)
Intralocus allelic interactions can affect the rate of evolution at a given locus (Hartl and Ruvolo 2012).
This will affect the rates at which CO2 is assimilated and water is transpired through stomata.
Scientists can now study classes of genes that affect absorption of sugars and carbohydrates, which in turn can change the chemical processes that affect the rates at which trees rot and release stored carbon.
The old wind-chill charts were based on research in Antarctica in 1945, when scientists measured how wind affected the rate at which water froze.
In the purest test of reasoning, pupils were shown a pendulum and asked how to find out what affects the rate at which it swings.
More suggestions(17)
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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