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$\begingroup$ There is a legitimate question of whether, and to what degree, these curvilinear fits improve on a model of independence. Now that you have established (at least visually) that the residuals are nicely behaved--symmetric, short-tailed, no outliers--why not formally test your regression against the intercept-only null? $\endgroup$

whuber
– whuber ♦

2026-03-31 19:06:17 +00:00
Commented 10 hours ago
$\begingroup$ @whuber, what is the best tool/function to test this? $\endgroup$

denis
– denis

2026-03-31 20:49:00 +00:00
Commented 9 hours ago
$\begingroup$ For these data, the standard linear regression F test would work fine. If you want a nonparametric solution (requiring fewer assumptions), a permutation test inspired by the standard test would be excellent. $\endgroup$

whuber
– whuber ♦

2026-03-31 21:15:33 +00:00
Commented 8 hours ago
$\begingroup$ @whuber I am happy to regard these tests as suggestions for the OP. My point is only to guess that different techniques will confirm the general shape. $\endgroup$

Nick Cox
– Nick Cox

2026-03-31 22:00:57 +00:00
Commented 7 hours ago
$\begingroup$ Well, sure--but that doesn't imply that the "general shape" is an indicator of non-independence. The issue is whether this shape might have arisen through chance by sampling a pair of independent variables. $\endgroup$

whuber
– whuber ♦

2026-03-31 22:05:31 +00:00
Commented 7 hours ago

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