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The phrase "a second criterion for" is correct and usable in written English.
You can use it when discussing additional standards or benchmarks that are being considered in a particular context or evaluation.
Example: "In addition to the first criterion for selection, we have established a second criterion for assessing the candidates' qualifications."
Alternatives: "an additional standard for" or "a further benchmark for".
Exact(7)
A second criterion for 85th percentile driver is given as k = 42 m3/2/s2.
To avoid false positive results, peak shape was established as a second criterion for diagnosing clonal rearrangement.
We computed it on the basis of the χ2 distances and used it as a second criterion for ranking and filtering.
A second criterion for inclusion within the IMPACT trial is a score of 27 or more on the self-report Moods and Feelings Questionnaire (MFQ).
A second criterion for the selection of genes was presence of their transcripts in a GenChip hybridization experiment using RNA isolated from a glucose fed-batch fermentation (Pel et al. 2007).
The average SV of all measurements annotated with a particular annotation value i is used as a second criterion for filtering the annotation values (Tabs. 4 and 5 in the Additional file 1).
Similar(51)
The normality of residual distributions was a third criterion for model selection.
The knowledge of this size effect could be adopted as a useful design guideline and can be taken as a first criterion for the adoption of these systems.
We consider the average inventory size of agents equalling 1 as a first criterion for reached consensus among agents (cf. Fig. 2).
A third criterion for choosing a utility instrument is the generation of an appropriate range for the utility values.
As a third criterion for validation of the tasks, we calculated correlations between task performance and two standard measures of user satisfaction (table 5, row 5 8).
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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