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"the correlation with" is a valid phrase in English and can be used in written English.
It is used to describe the relationship between two things. For example: "There is a strong correlation with age and experience."
Dictionary
the correlation with
noun
A reciprocal, parallel or complementary relationship between two or more comparable objects
Exact(58)
For none of the limbs, we found a significant difference between the correlation with actual length and the correlation with the most highly correlated nonspecifying variable (ps > 0.05).
Then he checked the correlation with eudaemonic happiness.
In the uneven distribution of comforts, the correlation with apartheid was unstated but hard to miss.
Early figures for 2009 appear to confirm the correlation with the recession.
This wasn't nearly as true for goals scored, where the correlation with points was almost the same, home or away.
The correlation with the won-loss record might be stronger for turnover margin than for any statistic other than points.
Self-observation in the form of a sleep diary will enable them to see the correlation with their ability to focus.
The sample size is small – 58 retailers – and the correlation with the official figures for retail sales has tended to be poor.
THE correlation with custody is so strong, Dr. Brinig said, that she has changed her view about the best way to preserve marriages and protect children.
The correlation with latitude was unchanged even after excluding data from Asian and African countries that might have been skewed by abortion or the killing of baby girls.
The exciting thing is the correlation with entrepreneurship and these labs.
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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