rs_numbercountingutils.py File Reference

Detailed Description

'Number Counting Utils' RooStats tutorial

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

• http://arxiv.org/abs/0905.3831
• http://arxiv.org/abs/physics/physics/0702156
• http://arxiv.org/abs/physics/0511028

expected p-value = 0.0009416504675382001  Z value (Gaussian sigma) = 3.108043895747122
observed p-value = 0.0009416504675382001  Z value (Gaussian sigma) = 3.108043895747122
observed p-value = 0.0009416504675382269  Z value (Gaussian sigma) = 3.1080438957471137
observed p-value = 0.0009416504675382269  Z value (Gaussian sigma) = 3.1080438957471137

 
import ROOT
 
# From the root prompt, you can see the full list of functions by using tab-completion
# ~~~{.bash}
# root [0] 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 [1] 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
sExpected = 50
bExpected = 100
relativeBkgUncert = 0.1
 
pExp = ROOT.RooStats.NumberCountingUtils.BinomialExpP(sExpected, bExpected, relativeBkgUncert)
zExp = ROOT.RooStats.NumberCountingUtils.BinomialExpZ(sExpected, bExpected, relativeBkgUncert)
print("expected p-value = {}  Z value (Gaussian sigma) = {}".format(pExp, zExp))
 
# -------------------------------------------------
# Expected p-values and significance with background uncertainty
observed = 150
pObs = ROOT.RooStats.NumberCountingUtils.BinomialObsP(observed, bExpected, relativeBkgUncert)
zObs = ROOT.RooStats.NumberCountingUtils.BinomialObsZ(observed, bExpected, relativeBkgUncert)
print("observed p-value = {}  Z value (Gaussian sigma) = {}".format(pObs, zObs))
 
# ---------------------------------------------------------
# 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 [0] RooStats::NumberCountingUtils::BinomialWithTauExpP(
# Double_t BinomialWithTauExpP(Double_t sExp, Double_t bExp, Double_t tau)
# ~~~
 
# --------------------------------------------------------------
# Expected p-values and significance with background uncertainty
tau = 1
 
pExpWithTau = ROOT.RooStats.NumberCountingUtils.BinomialWithTauExpP(sExpected, bExpected, tau)
zExpWithTau = ROOT.RooStats.NumberCountingUtils.BinomialWithTauExpZ(sExpected, bExpected, tau)
print("observed p-value = {}  Z value (Gaussian sigma) = {}".format(pExpWithTau, zExpWithTau))
 
# ---------------------------------------------------------------
# Expected p-values and significance with background uncertainty
pObsWithTau = ROOT.RooStats.NumberCountingUtils.BinomialWithTauObsP(observed, bExpected, tau)
zObsWithTau = ROOT.RooStats.NumberCountingUtils.BinomialWithTauObsZ(observed, bExpected, tau)
print("observed p-value = {}  Z value (Gaussian sigma) = {}".format(pObsWithTau, zObsWithTau))

Date

July 2022

Authors

Artem Busorgin, Kyle Cranmer (C++ version)

Definition in file rs_numbercountingutils.py.