[Joseph Coveney]

i m waiting your answer Joseph Coveney

I don't have much to offer on this topic that Nick hasn't already said here and that he and others haven't already said in your other thread related to what seems to be the same problem.

Both of your peer outcome variables* seem to be so-called limited dependent variables (LDVs), ordered categorical in particular, with both floors and ceilings. You might want to Google for methods suitable for assessing association with LDVs.

Stas Kolenikov's polychoric has a limit of I think 10 categories before it automatically switches over to providing the polyserial correlation, but you can use gsem to compute the polychoric correlation in such cases. I illustrate its use in your case below (the syntax has changed slightly since that post years ago, because Stata has since changed the manner in which to reference constant equations in estimation commands).

version 18.0

clear *

quietly input byte(EMOTIV ados_todtoddler_totale)
<redacted for brevity>
end

rename (EMOTIV ados_todtoddler_totale ) (emo ado)
contract _all, freq(count)

gsem (emo@1 ado@1 <- F, oprobit) [fweight=count], nocnsreport nodvheader nolog
nlcom rho:_b[/var(F)] / (1 + _b[/var(F)]) // <= here

estimates store Full
quietly gsem (emo@1 ado@1 <- , oprobit) [fweight=count]
lrtest Full

exit

There are assumptions underlying the polychoric correlation coefficient whose plausibility and sensitivity to violation need to be assessed before it can be considered suitable for a given set of data, of course.

I'm not sure what's convinced you that the relationship between whatever these two variables measure is nonmonotonic—I'm not sold on what I see with lowess on the subset of 90 nonmissing observations that you provide above, but you might have seen something more compelling with the complete dataset or have reasons based in theory or prior experience.

* One's variable name suggests that it's the total score on the Toddler Module of the ADOS-2 instrument. Google suggests from the other's that it's some kind of electroencephalographic measurement.