 ROOT   Reference Guide rs_numbercountingutils.C File Reference

## Detailed Description

This tutorial shows an example of the RooStats standalone utilities that calculate the p-value or Z value (eg. significance in 1-sided Gaussian standard deviations) for a number counting experiment. This is a hypothesis test between background only and signal-plus-background. The background estimate has uncertainty derived from an auxiliary or sideband measurement.

Documentation for these utilities can be found here: http://root.cern.ch/root/html/RooStats__NumberCountingUtils.html

This problem is often called a proto-type problem for high energy physics. In some references it is referred to as the on/off problem.

The problem is treated in a fully frequentist fashion by interpreting the relative background uncertainty as being due to an auxiliary or sideband observation that is also Poisson distributed with only background. Finally, one considers the test as a ratio of Poisson means where an interval is well known based on the conditioning on the total number of events and the binomial distribution. For more on this, see

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expected p-value =0.00094165 Z value (Gaussian sigma) = 3.10804
observed p-value =0.00094165 Z value (Gaussian sigma) = 3.10804
expected p-value =0.00094165 Z value (Gaussian sigma) = 3.10804
observed p-value =0.00094165 Z value (Gaussian sigma) = 3.10804
#include <iostream>
using namespace RooFit;
using namespace RooStats; // the utilities are in the RooStats namespace
using namespace std;
void rs_numbercountingutils()
{
// From the root prompt, you can see the full list of functions by using tab-completion
// ~~~{.bash}
// root  RooStats::NumberCountingUtils:: <tab>
// BinomialExpZ
// BinomialWithTauExpZ
// BinomialObsZ
// BinomialWithTauObsZ
// BinomialExpP
// BinomialWithTauExpP
// BinomialObsP
// BinomialWithTauObsP
// ~~~
// For each of the utilities you can inspect the arguments by tab completion
// ~~~{.bash}
// root  NumberCountingUtils::BinomialExpZ( <tab>
// Double_t BinomialExpZ(Double_t sExp, Double_t bExp, Double_t fractionalBUncertainty)
// ~~~
// -------------------------------------------------
// Here we see common usages where the experimenter
// has a relative background uncertainty, without
// explicit reference to the auxiliary or sideband
// measurement
// -------------------------------------------------------------
// Expected p-values and significance with background uncertainty
double sExpected = 50;
double bExpected = 100;
double relativeBkgUncert = 0.1;
double pExp = NumberCountingUtils::BinomialExpP(sExpected, bExpected, relativeBkgUncert);
double zExp = NumberCountingUtils::BinomialExpZ(sExpected, bExpected, relativeBkgUncert);
cout << "expected p-value =" << pExp << " Z value (Gaussian sigma) = " << zExp << endl;
// -------------------------------------------------
// Expected p-values and significance with background uncertainty
double observed = 150;
double pObs = NumberCountingUtils::BinomialObsP(observed, bExpected, relativeBkgUncert);
double zObs = NumberCountingUtils::BinomialObsZ(observed, bExpected, relativeBkgUncert);
cout << "observed p-value =" << pObs << " Z value (Gaussian sigma) = " << zObs << endl;
// ---------------------------------------------------------
// Here we see usages where the experimenter has knowledge
// about the properties of the auxiliary or sideband
// measurement. In particular, the ratio tau of background
// in the auxiliary measurement to the main measurement.
// Large values of tau mean small background uncertainty
// because the sideband is very constraining.
// Usage:
// ~~~{.bash}
// root  RooStats::NumberCountingUtils::BinomialWithTauExpP(
// Double_t BinomialWithTauExpP(Double_t sExp, Double_t bExp, Double_t tau)
// ~~~
// --------------------------------------------------------------
// Expected p-values and significance with background uncertainty
double tau = 1;
double pExpWithTau = NumberCountingUtils::BinomialWithTauExpP(sExpected, bExpected, tau);
double zExpWithTau = NumberCountingUtils::BinomialWithTauExpZ(sExpected, bExpected, tau);
cout << "expected p-value =" << pExpWithTau << " Z value (Gaussian sigma) = " << zExpWithTau << endl;
// ---------------------------------------------------------------
// Expected p-values and significance with background uncertainty
double pObsWithTau = NumberCountingUtils::BinomialWithTauObsP(observed, bExpected, tau);
double zObsWithTau = NumberCountingUtils::BinomialWithTauObsZ(observed, bExpected, tau);
cout << "observed p-value =" << pObsWithTau << " Z value (Gaussian sigma) = " << zObsWithTau << endl;
}

Definition in file rs_numbercountingutils.C.

RooStats::NumberCountingUtils::BinomialWithTauExpP
Double_t BinomialWithTauExpP(Double_t sExp, Double_t bExp, Double_t tau)
Definition: NumberCountingUtils.cxx:70
RooStats::NumberCountingUtils::BinomialExpZ
Double_t BinomialExpZ(Double_t sExp, Double_t bExp, Double_t fractionalBUncertainty)
Definition: NumberCountingUtils.cxx:122
RooStats::NumberCountingUtils::BinomialObsP
Double_t BinomialObsP(Double_t nObs, Double_t, Double_t fractionalBUncertainty)
Definition: NumberCountingUtils.cxx:86
RooStats::NumberCountingUtils::BinomialObsZ
Double_t BinomialObsZ(Double_t nObs, Double_t bExp, Double_t fractionalBUncertainty)
Definition: NumberCountingUtils.cxx:133
RooStats::NumberCountingUtils::BinomialWithTauObsZ
Double_t BinomialWithTauObsZ(Double_t nObs, Double_t bExp, Double_t tau)
Definition: NumberCountingUtils.cxx:138
RooStats::NumberCountingUtils::BinomialWithTauExpZ
Double_t BinomialWithTauExpZ(Double_t sExp, Double_t bExp, Double_t tau)
Definition: NumberCountingUtils.cxx:127
RooFit
The namespace RooFit contains mostly switches that change the behaviour of functions of PDFs (or othe...
Definition: RooCFunction1Binding.h:29
RooStats::NumberCountingUtils::BinomialWithTauObsP
Double_t BinomialWithTauObsP(Double_t nObs, Double_t bExp, Double_t tau)
Definition: NumberCountingUtils.cxx:107
RooStatsUtils.h
RooStats
Namespace for the RooStats classes.
Definition: Asimov.h:19
RooStats::NumberCountingUtils::BinomialExpP
Double_t BinomialExpP(Double_t sExp, Double_t bExp, Double_t fractionalBUncertainty)
Definition: NumberCountingUtils.cxx:25