## Bias correction unveiled [BE/BA News]

Hi Achievwin,

❝ ❝ Furthermore, we need the standard error $$\small{s_\textrm{d}}$$ of the point estimate

❝ I did not find this one in the guidance can you point out to me what line number this is in the new guidance?

That’s a little bit tricky. I’ll try again.

We have the regulatory constant $$\small{\theta_\textrm{s}=\frac{\log_{e}1.25}{\sigma_\textrm{w0}}=0.89257\ldots}\tag{1}$$
In the guidance line 998 the scaled average BE limit is given by $$\small{\theta\equiv\left(\frac{\log_{e}1.25}{\sigma_\textrm{w0}}\right)^2\tag{2}}$$which is simply what I used in my previous post by $$\small{\theta=\theta_\textrm{s}^2}\tag{3}$$ Of course, $$\small{\left(\widehat{Y}_T-\widehat{Y}_R\right)^2=PE^2}\tag{4}$$ In the guidance line 1003 we have $$\small{crit=\left(\widehat{Y}_T-\widehat{Y}_R\right)^2-\theta\cdot s_\textrm{wR}^2}\tag{5}$$ That’s the same as $$\small{crit=PE^2-\theta_\textrm{s}^2\cdot s_\textrm{wR}^2}\tag{6}$$Here the confusion starts. You find my $${\color{Red}{s_\textrm{d}}}$$ in lines 1083 and 1219 as the stderr of the difference.

In the respective next lines you find

x=estimate**2–stderr**2;

where it acts as a bias correction of the point estimate.

Hence, actually not $$(5)$$ or $$(6)$$ has to be used but $$\small{\begin{matrix} crit=PE^2-{\color{Red}{s_\textrm{d}^2}}-\theta_\textrm{s}^2\cdot s_\textrm{wR}^2 =\\ \left(\widehat{Y}_T-\widehat{Y}_R\right)^2-{\color{Red}{se^2}}-\theta \cdot s_\textrm{wR}^2 \end{matrix}}\tag{7}$$ The bias correction is not mentioned anywhere in the guidance and therefore, $$(5)$$ without one is extremely misleading.1,2 It is evident only if you inspect the SAS code. See Detlew’s post about the background given by Donald Schuirmann in 2016.

1. Troubles likely if one naïvely implements the formulas in another software…
2. This formula is used in all papers of Endrényi and Tóthfalusi.

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