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I am trying to double check whether beta values calculated from odds ratios and beta values calculated from the reciprocal of the same odds ratios have the same p-values and standard errors (I am calculating them from same population and using logistic regression). Regarding standard errors I have already found a question on here (What is the standard error of the inverse of a known odds ratio?) but I am not sure of the reliability of the answer. I have tried to verify this using statistical software plink and it looks that p-values and SEs stay the same (or very similar, differing only after a few decimal points) for both original betas and betas calculated from reciprocal of odds ratios. If anyone has any mathematical/statistical explanation it would be great.

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  • $\begingroup$ Those answers look ok to me. What worries you about those answers? The standard error is different, but the p-values remain the same? That is to be expected. $\endgroup$ Commented May 26, 2017 at 12:38

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I trust we agree that beta = ln(OR). If not then I have misunderstood the question.

Then -beta = -ln(OR) = ln(1/OR). But Var(-beta) = Var(beta) = Var(ln(1/OR)) so the variances of the two values of beta are the same so their standard errors are the same.

The test statistics just have different signs, z1 = beta/se(beta) and z2 = -beta/se(-beta) = -beta/se(beta) = -z1. For a two-sided test their p-values are identical.

No computation needed, just some algebra.

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