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  • $\begingroup$ You can check whether 0 is in the confidence interval of beta. $\endgroup$ Commented Mar 14, 2024 at 9:30
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    $\begingroup$ okay thanks. i have one followup which i'm going to edit the original question to include. how can i compare the fit provided by a Deming regression to that by an OLS. The answer is probably compare the loglik of the best OLS regression (X~Y or Y~X) to the loglik of the total least squares fit. but why are they comparable, it seems like the OLS only considers the loglik of Y|X whereas shouldn't total least squares give the loglik of the joint distribution (X,Y)? $\endgroup$ Commented Mar 14, 2024 at 10:15
  • $\begingroup$ @AFriendlyFish Do you mean to ask something like, “How do I know if the OLS does a batter job of doing what it does than Deming regression does at what it does?” $\endgroup$ Commented Mar 14, 2024 at 14:44
  • $\begingroup$ actually my question is even simpler, are they equivalent? I know the estimated slope will be different, but will the actual significance of a purported relationship be the same? I know regressing Y on X and X on Y which minimize the vertical and horizontal distances yield the same p-value, but what if you minimize both simultaneously (i.e. their sum)? Then it should still give the same p-value right? but the assumptions of two errors instead of one make it seem like these models definitely have to be different and not equivalent. thank you! $\endgroup$ Commented Mar 14, 2024 at 19:28
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    $\begingroup$ So compare to the OLS test of $H_0: \beta = 1$. $\endgroup$ Commented Mar 14, 2024 at 19:49