Derivation of the WStat formula

you can write down the likelihood formula as

\[L (n_{\mathrm{on}}, n_{\mathrm{off}}, \alpha; \mu_{\mathrm{sig}}, \mu_{\mathrm{bkg}}) = \frac{(\mu_{\mathrm{sig}}+ \mu_{\mathrm{bkg}})^{n_{\mathrm{on}}}}{n_{\mathrm{on}} !} \exp{(-(\mu_{\mathrm{sig}}+ \mu_{\mathrm{bkg}}))}\times \frac{(\mu_{\mathrm{bkg}}/\alpha)^{n_{\mathrm{off}}}}{n_{\mathrm{off}} !}\exp{(-\mu_{\mathrm{bkg}}/\alpha)},\]

where \(\mu_{\mathrm{sig}}\) and \(\mu_{\mathrm{bkg}}\) are respectively the number of expected signal and background counts in the ON region, as defined in the Introduction. By taking two time the negative log likelihood and neglecting model independent and thus constant terms, we define the WStat.

\[W = 2 \big(\mu_{\mathrm{sig}} + (1 + 1/\alpha)\mu_{\mathrm{bkg}} - n_{\mathrm{on}} \log{(\mu_{\mathrm{sig}} + \mu_{\mathrm{bkg}})} - n_{\mathrm{off}} \log{(\mu_{\mathrm{bkg}}/alpha)}\big)\]

In the most general case, where \(\mu_{\mathrm{src}}\) and \(\mu_{\mathrm{bkg}}\) are free the minimum of \(W\) is at

\[\begin{split}\mu_{\mathrm{sig}} = n_{\mathrm{on}} - \alpha\,n_{\mathrm{off}} \\ \mu_{\mathrm{bkg}} = n_{\mathrm{off}}\end{split}\]

Profile Likelihood

Most of the times you probably won’t have a model in order to get \(\mu_{\mathrm{bkg}}\). The strategy in this case is to treat \(\mu_{\mathrm{bkg}}\) as so-called nuisance parameter, i.e. a free parameter that is of no physical interest. Of course you don’t want an additional free parameter for each bin during a fit. Therefore one calculates an estimator for \(\mu_{\mathrm{bkg}}\) by analytically minimizing the likelihood function. This is called ‘profile likelihood’.

\[\frac{\mathrm d \log L}{\mathrm d \mu_{\mathrm{bkg}}} = 0\]

This yields a quadratic equation for \(\mu_{\mathrm{bkg}}\)

\[\frac{\alpha\,n_{\mathrm{on}}}{\mu_{\mathrm{sig}}+\alpha \mu_{\mathrm{bkg}}} + \frac{n_{\mathrm{off}}}{\mu_{\mathrm{bkg}}} - (\alpha + 1) = 0\]

with the solution

\[\mu_{\mathrm{bkg}} = \frac{C + D}{2\alpha(\alpha + 1)}\]


\[\begin{split}C = \alpha(n_{\mathrm{on}} + n_{\mathrm{off}}) - (\alpha+1)\mu_{\mathrm{sig}} \\ D^2 = C^2 + 4 (\alpha+1)\alpha n_{\mathrm{off}} \mu_{\mathrm{sig}}\end{split}\]

Goodness of fit

The best-fit value of the WStat as defined now contains no information about the goodness of the fit. We consider the likelihood of the data \(n_{\mathrm{on}}\) and \(n_{\mathrm{off}}\) under the expectation of \(n_{\mathrm{on}}\) and \(n_{\mathrm{off}}\),

\[L (n_{\mathrm{on}}, n_{\mathrm{off}}; n_{\mathrm{on}}, n_{\mathrm{off}}) = \frac{n_{\mathrm{on}}^{n_{\mathrm{on}}}}{n_{\mathrm{on}} !} \exp{(-n_{\mathrm{on}})}\times \frac{n_{\mathrm{off}}^{n_{\mathrm{off}}}}{n_{\mathrm{off}} !} \exp{(-n_{\mathrm{off}})}\]

and add twice the log likelihood

\[2 \log L (n_{\mathrm{on}}, n_{\mathrm{off}}; n_{\mathrm{on}}, n_{\mathrm{off}}) = 2 (n_{\mathrm{on}} ( \log{(n_{\mathrm{on}})} - 1 ) + n_{\mathrm{off}} ( \log{(n_{\mathrm{off}})} - 1))\]

to WStat. In doing so, we are computing the likelihood ratio:

\[-2 \log \frac{L(n_{\mathrm{on}},n_{\mathrm{off}},\alpha; \mu_{\mathrm{sig}},\mu_{\mathrm{bkg}})} {L(n_{\mathrm{on}},n_{\mathrm{off}};n_{\mathrm{on}},n_{\mathrm{off}})}\]

Intuitively, this log-likelihood ratio should asymptotically behave like a chi-square with m-n degrees of freedom, where m is the number of measurements and n the number of model parameters.

Final result

\[W = 2 \big(\mu_{\mathrm{sig}} + (1 + \alpha)\mu_{\mathrm{bkg}} - n_{\mathrm{on}} - n_{\mathrm{off}} - n_{\mathrm{on}} (\log{(\mu_{\mathrm{sig}} + \alpha \mu_{\mathrm{bkg}}) - \log{(n_{\mathrm{on}})}}) - n_{\mathrm{off}} (\log{(\mu_{\mathrm{bkg}})} - \log{(n_{\mathrm{off}})})\big)\]

Special cases

The above formula is undefined if \(n_{\mathrm{on}}\) or \(n_{\mathrm{off}}\) are equal to zero, because of the \(n\log{{n}}\) terms, that were introduced by adding the goodness of fit terms. These cases are treated as follows.

If \(n_{\mathrm{on}} = 0\) the likelihood formulae read

\[L (0, n_{\mathrm{off}}, \alpha; \mu_{\mathrm{sig}}, \mu_{\mathrm{bkg}}) = \exp{(-(\mu_{\mathrm{sig}}+\alpha \mu_{\mathrm{bkg}}))}\times \frac{(\mu_{\mathrm{bkg}})^{n_{\mathrm{off}}}}{n_{\mathrm{off}} !}\exp{(-\mu_{\mathrm{bkg}})},\]


\[L (0, n_{\mathrm{off}}; 0, n_{\mathrm{off}}) = \frac{n_{\mathrm{off}}^{n_{\mathrm{off}}}}{n_{\mathrm{off}} !} \exp{(-n_{\mathrm{off}})}\]

WStat is derived by taking 2 times the negative log likelihood and adding the goodness of fit term as ever

\[W = 2 \big(\mu_{\mathrm{sig}} + (1 + \alpha)\mu_{\mathrm{bkg}} - n_{\mathrm{off}} - n_{\mathrm{off}} (\log{(\mu_{\mathrm{bkg}})} - \log{(n_{\mathrm{off}})})\big)\]

Note that this is the limit of the original Wstat formula for \(n_{\mathrm{on}} \rightarrow 0\).

The analytical result for \(\mu_{\mathrm{bkg}}\) in this case reads:

\[\mu_{\mathrm{bkg}} = \frac{n_{\mathrm{off}}}{\alpha + 1}\]

When inserting this into the WStat we find the simplified expression.

\[W = 2\big(\mu_{\mathrm{sig}} + n_{\mathrm{off}} \log{(1 + \alpha)}\big)\]

If \(n_{\mathrm{off}} = 0\) Wstat becomes

\[W = 2 \big(\mu_{\mathrm{sig}} + (1 + \alpha)\mu_{\mathrm{bkg}} - n_{\mathrm{on}} - n_{\mathrm{on}} (\log{(\mu_{\mathrm{sig}} + \alpha \mu_{\mathrm{bkg}}) - \log{(n_{\mathrm{on}})}})\]


\[\mu_{\mathrm{bkg}} = \frac{n_{\mathrm{on}}}{1+\alpha} - \frac{\mu_{\mathrm{sig}}}{\alpha}\]

For \(\mu_{\mathrm{sig}} > n_{\mathrm{on}} (\frac{\alpha}{1 + \alpha})\), \(\mu_{\mathrm{bkg}}\) becomes negative which is unphysical.

Therefore we distinct two cases. The physical one where

\(\mu_{\mathrm{sig}} < n_{\mathrm{on}} (\frac{\alpha}{1 + \alpha})\).

is straightforward and gives

\[W = -2\big(\mu_{\mathrm{sig}} \left(\frac{1}{\alpha}\right) + n_{\mathrm{on}} \log{\left(\frac{\alpha}{1 + \alpha}\right)\big)}\]

For the unphysical case, we set \(\mu_{\mathrm{bkg}}=0\) and arrive at

\[W = 2\big(\mu_{\mathrm{sig}} + n_{\mathrm{on}}(\log{(n_{\mathrm{on}})} - \log{(\mu_{\mathrm{sig}})} - 1)\big)\]