Wing Sheung Chan

Statistical interpretation and results 107 each other. Other systematic uncertainties are modelled by NPs with Gaussian constraints, where bin-to-bin correlations are preserved in the parameterisation. 6.1.2. Test statistics and hypothesis tests The significance of an excess is determined in p -value using the modified frequentist CL s method. The test statistic employed is the log-likelihood ratio q µ s , which is modified into q 0 for testing the background-only (null) hypothesis. q µ s is constructed based on the likelihood ratio λ ( µ s ) : λ ( µ s ) =   L µ s , ˆˆ b ( µ s ) , ˆˆ θ ( µ s ) L ˆ µ s , ˆ b , ˆ θ if ˆ µ s ≥ 0 , L µ s , ˆˆ b ( µ s ) , ˆˆ θ ( µ s ) L 0 , ˆˆ b (0) , ˆˆ θ (0) if ˆ µ s < 0 , (6.2) where the single-hatted quantities are the best-fit values and the doubled-hatted quantities are values that maximise the likelihood while given a fixed value of µ s . For quantifying the significance of a potential discovery, the test statistic used is q 0 = ( − 2 ln λ ( µ s = 1) if ˆ µ s ≥ 0 , 0 if ˆ µ s < 0 . (6.3) When testing a specific background-plus-signal model, the test statistic is given by q µ s = ( − 2 ln λ ( µ s ) if ˆ µ s ≤ µ s , 0 if ˆ µ s > µ s . (6.4) This choice of test statistics ensures that q 0 is zero when the best-fit value of µ s is smaller than zero, i.e. the number of events is below the SM expectation. It also ensures that no exclusion limit will be set below ˆ µ s . The level of (dis)agreement between data and the null hypothesis is quantified by the p -value p 0 = Z ∞ q obs f ( q 0 | 0) d q 0 , (6.5) where q obs is the value of the test statistic observed from data and f is the probability distribution of the test statistic. For testing the alternative hypotheses and setting exclusion limits, the CL s value CL s = p µ s 1 − p 0 (6.6)