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The phrase "a slight correlation with" is correct and usable in written English.
It can be used when discussing the relationship or connection between two variables or factors, indicating that the relationship is not strong but still present.
Example: "The study found a slight correlation with increased exercise and improved mental health among participants."
Alternatives: "a minor association with" or "a weak link to".
Exact(12)
There is a slight correlation with the key velocity, suggesting that the phenomenon may be correlated to acoustical nonlinearities (e.g., string termination yielding), similar to those appearing in other instruments such as the kantele [4].
The activity for CO and C3H8 oxidation presented a slight correlation with the surface area of the aged catalysts.
Body mass index was the only maternal or fetal variable that showed even a slight correlation with laceration length (R = 0.30, P =.04).
The Spearman ρ indicates a slight correlation with r = 0.161, while the non-linear Kendall τ-B is calculated with r = 0.141 - both with an error chance around 10.5%.
The results showed a slight correlation with those of qPCR.
Newborn birth length in the lower quartile also had a slight correlation with smoking during pregnancy.
Similar(48)
However, in this study the level of YKL-40 was found to be lying in the normal range and only a slight correlation was found with the liver status as demonstrated by liver transaminases.
Only patients in stage I showed a slight correlation of IL-8 level with the circulatory percentage of CD16+ and CD8+ (PC = 0.75) and no other significant correlation.
Corresponding with this finding, in our sample there is only a slight correlation between the two variables (r = 0.28).
In the early 1980's, a C.D.C. study showed a slight correlation between carpet cleaning and Kawasaki.
There's more than a slight correlation.
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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