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"established associations" is a correct and usable phrase in written English.
It is often used to refer to well-known or recognized connections between things or ideas. Example: The study aims to explore the established associations between diet and health outcomes.
Exact(57)
The scheme was wholly financed by the employer, who paid insurance contributions, assessed on the degree of risk involved in the employee's occupation, to statutorily established associations.
This update takes place regardless of whether there are established associations.
Although this method can provide an association recovery, all the established associations are lost by the collision.
Similarly, association findings of genome-wide association studies have identified also established associations with IL10, IL23R, CCR1, STAT4, KLRC4, GIMAP2/GIMAP4, and UBAC2 genes in Behçet's disease patients of different ethnicities.
Two previously established associations, namely with the MHC and IL23R loci, were confirmed.
Similarly for CD, the well established associations with NOD2, IL23R and the recently implicated IL12B and STAT3 were all confirmed.
We herein first confirm two previously established associations with the risk to develop AS, respectively with the MHC and IL23R loci.
However, established associations explain only a part of the heritability.
Similar(3)
These results complement the well-established associations between Parental Education and Cognitive Skills, such as Literacy and Numeracy.
Despite well-established associations between cardiorespiratory fitness (CRF) and long-term cardiovascular disease risk, CRF remains underused in clinical practice.
There are well-established associations between overweight or obesity and type 2 diabetes (1, 2).
More suggestions(15)
established partnerships
organized associations
maintained associations
provided associations
tested associations
established governments
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incorporated associations
established trade unions
acknowledged associations
formation associations
established partnership
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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