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

8. Hypotheses 8.2 The Likelihood ratio at work

K. Desch – Statistical methods of data analysis SS10

data

data

data

n(s b)

nsdatan

(b)

data

(s b)e

n ! sQ e (1 )

bbe

n !



in general, use MCto caluclate thep.d.f. for Qunder the s+band b hypotheses



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



Combination of several channels is easy:

For Poissonian p.d.f.s

tot ichan

Q Q

data

data

n(s b)

pois data datatot n

(b)chan chanpois data

data

(s b)e

P (n ;s b) n !Q

P (n ;b) be

n !

i i itot data,i i i i tot data,i

chan chani i

s b slnQ n ln( ) (s b b ) s n ln(1 )

b b

a channel contributes according to ln(1+s/b) to the log LHR!





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

8. Hypotheses 8.3 A real example: Higgs search at LEP


In fall 2000 the 4 LEP experiments released their (perliminary) resultsin the search of the Higgs boson.

This was a search of 4 different experiments (detectors) (ALEPH, DELPHI, L3, OPAL) each in 4 different search topologieswith small numbers of background and small numbers of expectedsignal events (in case there was a Higgs boson)

No single experiment had enough sensitivity to discover a Higgs atmH >~ 110 GeV

Combination was done using the likelihood ratio as a test statistic



Production:



Decay:



Search topologies:



For each selected candidate event two main discriminating variableswere used:

b-quark content of the jets(Higgs decays to b-bbar)

reconstructed mass of thetwo jets which are assignedto the Higgs candidate(signal peaks at Higgs mass, background peaks at Z mass)



For each bin in (mrec, 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 wasperformed



simplest combination

throw all eggs in one basketthen LHR-Test for 1 channel

sensitivity not optimal



Full LHR test statistic for each experiment (for a single test mass m=114 GeV)

full black line:observed -2 lnQ



Full LHR test statistic for each search topology(for a single test mass m=114 GeV)



Discovery ? (1- CLb) < Prob(n * )

„100*(1-CLb)“ percent of experiments would yield a test statistic -2lnQ smaller than the observed one, if no signal would be present“



Exclusion ? CLs (mtest) < 5% „95% CL exclusion“

„Only 100*CLs 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



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):

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