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I am working with summary statistics from a genome-wide association study (p-values, odds ratio, standard error).

The test statistic in the dataset was inflated, so I have had to correct for this inflation (using genomic correction). I now have p-values adjusted for the inflation, and would like to provide an estimated effect size in terms of odds ratio and standard error.

Is there a way to estimate the odds ratio and corresponding standard error, given only the p-value?

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No. It's like trying to find two unknowns using a single equation. Using an example of a t-test, the p-value $p$ is a function of the observed t-statistic $t_0$, and the latter is equal to the point estimate, $b$, divided by its standard error, $se(b)$. Knowing $b$ is equivalent to knowing the odds ratio.

If you know only $p$, you can back out $t_0$ but not $b$ and $se(b)$ separately.

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  • $\begingroup$ That makes sense. Is there any way to estimate se(b) and b, if you know p and you know the sample size? $\endgroup$
    – user3745089
    Commented Aug 22, 2014 at 13:49
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    $\begingroup$ No. Instead, I think that it's reasonable to say that the "inflated" odds ratio is the same as the original one, and then you'll be able to back out the "inflated" s.e. from the p-value. As an example, suppose I fit a Poisson regression to the data that is actually Negative Binomial. The point estimate of $b$ and the odds ratio will be the same in both cases, but if I stick to Poisson I will get an incorrect s.e. and the p-value. If your genomic correction does a similar thing, i.e. corrects the s.e. rather than the point estimate, then you'll be fine. $\endgroup$
    – James
    Commented Aug 22, 2014 at 14:30

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