.. include:: ../references.txt .. _wstat_derivation: Derivation of the WStat formula ------------------------------- you can write down the likelihood formula as .. math:: 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 :math:\mu_{\mathrm{sig}} and :math:\mu_{\mathrm{bkg}} are respectively the number of expected signal and background counts in the ON region, as defined in the :ref:stats-introduction. By taking two time the negative log likelihood and neglecting model independent and thus constant terms, we define the **WStat**. .. math:: 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 :math:\mu_{\mathrm{src}} and :math:\mu_{\mathrm{bkg}} are free the minimum of :math:W is at .. math:: \mu_{\mathrm{sig}} = n_{\mathrm{on}} - \alpha\,n_{\mathrm{off}} \\ \mu_{\mathrm{bkg}} = \alpha\,n_{\mathrm{off}} Profile Likelihood ^^^^^^^^^^^^^^^^^^ Most of the times you probably won't have a model in order to get :math:\mu_{\mathrm{bkg}}. The strategy in this case is to treat :math:\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 :math:\mu_{\mathrm{bkg}} by analytically minimizing the likelihood function. This is called 'profile likelihood'. .. math:: \frac{\mathrm d \log L}{\mathrm d \mu_{\mathrm{bkg}}} = 0 This yields a quadratic equation for :math:\mu_{\mathrm{bkg}} .. math:: \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 .. math:: \mu_{\mathrm{bkg}} = \frac{C + D}{2\alpha(\alpha + 1)} where .. math:: 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}} 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 :math:n_{\mathrm{on}} and :math:n_{\mathrm{off}} under the expectation of :math:n_{\mathrm{on}} and :math:n_{\mathrm{off}}, .. math:: 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 .. math:: 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: .. math:: -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 ^^^^^^^^^^^^ .. math:: 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 :math:n_{\mathrm{on}} or :math:n_{\mathrm{off}} are equal to zero, because of the :math:n\log{{n}} terms, that were introduced by adding the goodness of fit terms. These cases are treated as follows. If :math:n_{\mathrm{on}} = 0 the likelihood formulae read .. math:: 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}})}, and .. math:: 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 .. math:: 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 :math:n_{\mathrm{on}} \rightarrow 0. The analytical result for :math:\mu_{\mathrm{bkg}} in this case reads: .. math:: \mu_{\mathrm{bkg}} = \frac{n_{\mathrm{off}}}{\alpha + 1} When inserting this into the WStat we find the simplified expression. .. math:: W = 2\big(\mu_{\mathrm{sig}} + n_{\mathrm{off}} \log{(1 + \alpha)}\big) If :math:n_{\mathrm{off}} = 0 Wstat becomes .. math:: 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}})}}) and .. math:: \mu_{\mathrm{bkg}} = \frac{n_{\mathrm{on}}}{1+\alpha} - \frac{\mu_{\mathrm{sig}}}{\alpha} For :math:\mu_{\mathrm{sig}} > n_{\mathrm{on}} (\frac{\alpha}{1 + \alpha}), :math:\mu_{\mathrm{bkg}} becomes negative which is unphysical. Therefore we distinct two cases. The physical one where :math:\mu_{\mathrm{sig}} < n_{\mathrm{on}} (\frac{\alpha}{1 + \alpha}). is straightforward and gives .. math:: 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 :math:\mu_{\mathrm{bkg}}=0 and arrive at .. math:: W = 2\big(\mu_{\mathrm{sig}} + n_{\mathrm{on}}(\log{(n_{\mathrm{on}})} - \log{(\mu_{\mathrm{sig}})} - 1)\big)