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Do you report the regression equation for non-significant multiple regressions if you have reported them for significant regressions in the same work?

My paper has six multiple regressions in it. Three are significant and I have reported the regression equation for each of these (e.g., $y=b_1+b_2x+a$); I can't find out if I need to report the equation for non-significant results too.

Would it be best to just report them all or are there rules about it?

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There is no single definitive answer to this question.

Some journals may prefer you to report, some may forbid (or strongly discourage) reporting of such results. Some editors seem to feel that if results are not significant you should not say anything about them beyond that fact.

My own view is that the report/not report decision depends on things beyond significance: Chiefly on what Robert Abelson calls the MAGIC criteria:

Magnitude - how big is the effect? Big effects are more reportable than small ones
Articulation - How precisely stated is it? Precisely stated effects are more reportable
Generality - How general is it? General effects are more reportable than ones that apply only to some tiny group 
Interestingness - Is it interesting? Will people in the field care?
and
Credibility - Is it believable? Credible effects are more reportable; incredible effects require incredible evidence.

For more on this, see my review of Abelson's book

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When you say "equation," do you mean "parameter?"

I will assume you actually mean equations (i.e. six unique fitted models). In a scientific publication, one should describe all the models that were fitted whether they were significant or not. Your methods section ought to say you fitted 6 models, and an interested reader may want to know the results of each. Non-significant findings should never be considered non-reportable findings. Not everything can be significantly different from 0. When we discover such things, they count as knowledge, too.

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