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NumberCountingUtils.h
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1// @(#)root/roostats:$Id$
2// Author: Kyle Cranmer 28/07/2008
3
4/*************************************************************************
5 * Copyright (C) 1995-2008, Rene Brun and Fons Rademakers. *
6 * All rights reserved. *
7 * *
8 * For the licensing terms see $ROOTSYS/LICENSE. *
9 * For the list of contributors see $ROOTSYS/README/CREDITS. *
10 *************************************************************************/
11
12#ifndef RooStats_NumberCountingUtils
13#define RooStats_NumberCountingUtils
14
15//_________________________________________________
16/*
17BEGIN_HTML
18<h2>NumberCountingUtils</h2>
19<p>
20These are RooStats standalone utilities
21that calculate the p-value or Z value (eg. significance in
221-sided Gaussian standard deviations) for a number counting experiment.
23This is a hypothesis test between background only and signal-plus-background.
24The background estimate has uncertainty derived from an auxiliary or sideband
25measurement.
26</p>
27<p>
28This is based on code and comments from Bob Cousins
29and on the following papers:
30<p>
31<ul>
32<li>Evaluation of three methods for calculating statistical significance when incorporating a
33systematic uncertainty into a test of the background-only hypothesis for a Poisson process<br />
34Authors: Robert D. Cousins, James T. Linnemann, Jordan Tucker<br />
35http://arxiv.org/abs/physics/0702156<br />
36NIM A 595 (2008) 480--501</li>
37
38<li>
39Statistical Challenges for Searches for New Physics at the LHC<br />
40Authors: Kyle Cranmer<br />
41http://arxiv.org/abs/physics/0511028
42</li>
43<li>
44 Measures of Significance in HEP and Astrophysics<br />
45 Authors: J. T. Linnemann<br />
46 http://arxiv.org/abs/physics/0312059
47</li>
48</ul>
49<p>
50The problem is treated in a fully frequentist fashion by
51interpreting the relative background uncertainty as
52being due to an auxiliary or sideband observation
53that is also Poisson distributed with only background.
54Finally, one considers the test as a ratio of Poisson means
55where an interval is well known based on the conditioning on the total
56number of events and the binomial distribution.
57</p>
58
59<p>
60In short, this is an exact frequentist solution to the problem of
61a main measurement x distributed as a Poisson around s+b and a sideband or
62auxiliary measurement y distributed as a Poisson around tau*b. Eg.
63</p>
64END_HTML
65BEGIN_LATEX
66L(x,y|s,b,#tau) = Pois(x|s+b) Pois(y|#tau b)
67END_LATEX
68BEGIN_HTML
69<pre>
70Naming conventions:
71Exp = Expected
72Obs = Observed
73P = p-value
74Z = Z-value or significance in sigma (one-sided convention)
75</pre>
76END_HTML
77*/
78//
79
80#include "RtypesCore.h"
81
82
83namespace RooStats{
84
85 namespace NumberCountingUtils {
86
87
88 // Expected P-value for s=0 in a ratio of Poisson means.
89 // Here the background and its uncertainty are provided directly and
90 // assumed to be from the double Poisson counting setup described in the
91 // BinomialWithTau functions.
92 // Normally one would know tau directly, but here it is determiend from
93 // the background uncertainty. This is not strictly correct, but a useful
94 // approximation.
95 Double_t BinomialExpZ(Double_t sExp, Double_t bExp, Double_t fractionalBUncertainty);
96
97 // See BinomialWithTauExpP
99
100 // See BinomialObsP
101 Double_t BinomialObsZ(Double_t nObs, Double_t bExp, Double_t fractionalBUncertainty);
102
103 // See BinomialWithTauObsP
105
106 // See BinomialExpP
107 Double_t BinomialExpP(Double_t sExp, Double_t bExp, Double_t fractionalBUncertainty);
108
109 // Expected P-value for s=0 in a ratio of Poisson means.
110 // Based on two expectations, a main measurement that might have signal
111 // and an auxiliarly measurement for the background that is signal free.
112 // The expected background in the auxiliary measurement is a factor
113 // tau larger than in the main measurement.
115
116 // P-value for s=0 in a ratio of Poisson means.
117 // Here the background and its uncertainty are provided directly and
118 // assumed to be from the double Poisson counting setup.
119 // Normally one would know tau directly, but here it is determiend from
120 // the background uncertainty. This is not strictly correct, but a useful
121 // approximation.
122 Double_t BinomialObsP(Double_t nObs, Double_t, Double_t fractionalBUncertainty);
123
124 // P-value for s=0 in a ratio of Poisson means.
125 // Based on two observations, a main measurement that might have signal
126 // and an auxiliarly measurement for the background that is signal free.
127 // The expected background in the auxiliary measurement is a factor
128 // tau larger than in the main measurement.
130
131
132 }
133}
134
135#endif
double Double_t
Definition RtypesCore.h:59
Double_t BinomialWithTauExpZ(Double_t sExp, Double_t bExp, Double_t tau)
Double_t BinomialObsZ(Double_t nObs, Double_t bExp, Double_t fractionalBUncertainty)
Double_t BinomialWithTauObsZ(Double_t nObs, Double_t bExp, Double_t tau)
Double_t BinomialWithTauObsP(Double_t nObs, Double_t bExp, Double_t tau)
Double_t BinomialObsP(Double_t nObs, Double_t, Double_t fractionalBUncertainty)
Double_t BinomialWithTauExpP(Double_t sExp, Double_t bExp, Double_t tau)
Double_t BinomialExpZ(Double_t sExp, Double_t bExp, Double_t fractionalBUncertainty)
Double_t BinomialExpP(Double_t sExp, Double_t bExp, Double_t fractionalBUncertainty)
Namespace for the RooStats classes.
Definition Asimov.h:19