1. 8. Hypotheses 8.2 The Likelihood ratio at work K. Desch – Statistical methods of data analysis SS10

2. 8. Hypotheses 8.2 The Likelihood ratio at work K. Desch – Statistical methods of data analysis SS10 in general, use MC to caluclate the p.d.f. for Q under the s+b and b hypotheses

3. 8. Hypotheses 8.2 The Likelihood ratio at work K. Desch – Statistical methods of data analysis SS10 Alex Read, "Modified Frequentist Analysis of Search Results (The CLs Method)" CERN 2000-205 (30 May 2000) http://cdsweb.cern.ch/record/451614/files/open-2000-205.pdf

4. 8. Hypotheses 8.2 The Likelihood ratio at work K. Desch – Statistical methods of data analysis SS10 Combination of several channels is easy: For Poissonian p.d.f.s a channel contributes according to ln(1+s/b) to the log LHR!

5. 8. Hypotheses 8.2 The Likelihood ratio at work K. Desch – Statistical methods of data analysis SS10

6. 8. Hypotheses 8.2 The Likelihood ratio at work K. Desch – Statistical methods of data analysis SS10 Example: „Channel“ can mean: - different experiments - different search topologies (event types) in a single experiment - different bins (e.g. reconstructed mass, discr. variable) in the same topology with different s/b

7. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 In fall 2000 the 4 LEP experiments released their (perliminary) results in the search of the Higgs boson. This was a search of 4 different experiments (detectors) (ALEPH, DELPHI, L3, OPAL) each in 4 different search topologies with small numbers of background and small numbers of expected signal events (in case there was a Higgs boson) No single experiment had enough sensitivity to discover a Higgs at m H >~ 110 GeV Combination was done using the likelihood ratio as a test statistic

8. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 Production:

9. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 Decay:

10. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 Search topologies:

11. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 For each selected candidate event two main discriminating variables were used: b-quark content of the jets (Higgs decays to b-bbar) reconstructed mass of the two jets which are assigned to the Higgs candidate (signal peaks at Higgs mass, background peaks at Z mass)

12. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 For each bin in (m rec, b-tag) the expected s/b was calculated from MC simulation (for a range of hypothetical true Higgs masses) For each hypothetical true Higgs mass („test mass“) a LHR test was performed

13. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 simplest combination throw all eggs in one basket then LHR-Test for 1 channel  sensitivity not optimal

14. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 Full LHR test statistic for each experiment (for a single test mass m=114 GeV) full black line: observed -2 lnQ

15. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 Full LHR test statistic for each search topology (for a single test mass m=114 GeV)

16. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 Discovery ? (1- CL b ) < Prob(n *  ) „100*(1-CL b )“ percent of experiments would yield a test statistic -2lnQ smaller than the observed one, if no signal would be present“

17. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 Exclusion ? CL s (m test ) < 5% „95% CL exclusion“ „Only 100*CL s percent of experiments would yield a -2lnQ larger (i.e. more BG-like) than the observed one if a signal (at the test mass) would be present

18. 8. Hypotheses 8.3 A real example: Higgs search at LEP K. Desch – Statistical methods of data analysis SS10 In the end data were reanalyzed (and some problems were found in the s/b in some bins). Final result less significant (for a signal):