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Program for evaluation of the significance, confidence intervals and limits by direct probabilities calculations

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Program for evaluation of the significance, confidence intervals and limits by direct probabilities calculations

S.Bityukov (IHEP,Protvino), S.Erofeeva(MSA IECS,Moscow), N.Krasnikov(INR RAS, Moscow), A.Nikitenko(IC, London)

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Introduction

Duringplanning or processing of experiment we often consider

a statistical hypothesisH0: new physics is present in Nature

against hypothesisH1: new physics is absent in Nature.

The value of uncertainty in our conclusion is defined by the probabilities

= P(reject H0 | H0is true) - Type I error

and b= P(accept H0 | H0is false) - Type II error

There are many definitions of significance as a measure of excess of signal events above background. Many approaches exist also to methods

of construction of intervals and limits: confidence, tolerant, fiducial and so on.

During one of the CMS meetings Gunter Quast formulated the problem of practicians “the only remaining problem: make a choice … chosen method should be “as simple as possible, but not wrong!”

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Motivation of the work(significance)

- Gaussian limit gives the wrong answer for low value of β (tail of Poisson distribution is heavier than tail of Gaussian)
- 2. The statistics like SL (a likelihood-ratio-based test statistic) have poor statistical properties as estimator of significance
- (SL = √2•(ln L1-ln L2)= √2•(ln Q), where Q is the ratio of binned/unbinned likelihood fits for hypotheses H0and H1
- H0: signal present and H1: no signal present)

The simplest significance is the significance S_cP described at the next slide. The S_cP is quite natural significance , which allows to take into account any uncertainties by direct calculation of probabilities.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Definition of the significance S_cP

S_cP - probability from Poisson distribution with mean μ_b to

observe equal or greater than Nobs events, converted

to equivalent number of sigmas of a Gaussian distribution

(see report (page 8) by G. Quast in CMS Physics

analysis days, May 9-12, 2005, CERN

http://cmsdoc.cern.ch/~bityukov/talks/talks.html

also, see I.Narsky, NIM A450(2000)444).

The presented program ScPallows to calculate this significance with taking into account experimental systematics with statistical properties (Gaussian approximation) and theoretical systematics without any statistical properties. Also, the program calculates (if option is on) the combining significance of several channels. As is assumed all channels are independent.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Conception

Conception. The probability of making a Type II error (β) in

hypotheses test about presence of signal in experiment (H0) is

used for determination number of sigmas (of background

distribution) between expected background and observed

number of events Nobs (formula 8 in CMS CR 2002/05).

This probability is used for determination of signal significance,

i.e. the significance S_cP will be found under resolving

of equations

, where

It can be used in combining of results. Let us consider two

possible approaches.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Combining of observed results: Approach 1

Approach 1. Suppose that observed value is greater than

expected background

Let β_1 be Type II error for channel 1

(event A = background Nobs_1 with P(A)= β_1)

and β_2 be Type II error for channel 2

(event B = background Nobs_2 with P(B)= β_2).

Because event A is independent from event B then probability of simultaneous appearance of A and B equals

β_12 = P(AB) = P(A)*P(B) = β_1* β_2.

After determination of β_12 one can calculate the S_cP.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Combining of expected signals & backgrounds: Approach 2

Approach 2. Suppose that Nobs is expected sum of expected

signal (μ_s) and expected background (μ_b),

i.e. Nobs = μ_s+μ_b (the case of planned experiment).

Then the sums of expected numbers of signal (μ_s_i ) and

background (μ_b_i ) events in each channel are used as

summary μ_s and μ_b for calculation of combined significance.

Note that we take into account in this case as fluctuation of

expected background and fluctuation of expected signal.

After determination of corresponding β one can calculate

the S_cP.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Uncertainties

- The program takes into account two types of uncertainties:
- experimental systematics with statistical properties
- (we assume that this systematics has Gaussian distribution
- with known variance σ_b**2 in according with formula
- μ_b = expected background + N(0,σ _b)).
- Appr.2: In the case of the combining of channels the summary
- variance σ_b**2 is the sum of partial variance σ_b_i**2.
- b) theoretical systematics (δ_b) without any statistical properties (we assume: the worst case takes place when the background is maximal, i.e. μ_b*(1+ δ_b), but we take the signal plus the background as Nobs; more information can be found in S.Bityukov, N.Krasnikov, CMS CR 2002/05 or
- S.Bityukov, N.Krasnikov, Mod.Phys.Lett.A 13 (1998)3235)
- Appr.2: The combining δ_b isthe sum of partial δ_b_i.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Main input and output parameters

Main input parameters:

1. expected background – μ_b

2. signal = observed value (Nobs) -

expected background (μ_b) – μ_s

3. experimental uncertainty (r.m.s.) of

background with statistical properties – σ_b

4. systematics

of theoretical origin in background – δ_b

Output parameters:

1. significance S_cP, calculated by formula - dsgnf

2. significance S_cP_MC, calculated by Monte Carlo - dsgnfm

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Auxiliary input parameters

1. switch for choosing of type calculations - iflag

iflag = 1 calculations by formula (quick calculations)

iflag = 2 Monte Carlo calculations

iflag = 12 calculations by formula and by Monte Carlo

2. number of channels for calculations - nchan

(from 1 up to 10)

3. number of channels for combined S_cP - ncombi

(from 1 up to nchan)

4. parameter for Monte Carlo calculation - over

over - parameter for Monte Carlo calculations. It is a

number of Monte Carlo trials which will give value of

number events over or equal Nobs. This

parameter (and internal value dbeta) determines the

number of trials for given μ_s, μ_b, σ_b and δ_b in

routine SCPMC.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

The structure of program

Language: Fortran 77

iflag

Three different types 1. SCPFOR - calculations by formula

of calculations: 2. SCPMC - Monte Carlo calculations

12. SCPFOR + SCPMC

Main program processes the user requirements

(defined in operators DATA) and calls routines SCPFOR and/or SCPMC.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Problem and approximation

The problem which takes place during calculations is the restricted range of applicability of standard procedure DGAUSN in CERNLIB. For values of S_cP>6.2-7 the procedure gives non correct result. In this case we use as a good approximation the significance (MPL A13 (1998)3235)

S_c12 = 2 ((μ_s+μ_b) - μ_b) .

The account of the uncertainties is very simple:

theoretical systematics (δ_b)

S_c12t = 2 ((μ_s+μ_b) - (μ_b(1+δ_b)) .

experimental systematics (σ_b**2 )

μ_b

S_c12e = 2 ((μ_s+μ_b) - μ_b) ------------------- .

(μ_b+σ_b**2)

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Simplest example of program S_cP output

Example of G.Quast: bkg=2 sig=5.4. Here S1=3.8 S12=2.6 SL=2.7

Significance S_cP and/or S_cP_MC:

NN of channels = 1, Combining channels from 1 up to 1

calculation type = 12

types: (1) S_cP by formula and (2) S_cP_MC by Monte Carlo

σ_b-experimental uncertainty, i.e. μ_b = background + N(0, σ_b)

δ_b - systematics of theoretical origin without statistical properties

#ch backgr. signal σ_b δ_b S_cP S_cP_MC S_c12

1 2.00 5.40 0.0000 0.0000 2.6095 2.5923 2.612

1.4142 0.0000 1.8581 1.8759 1.847

1.4142 .50000E-01 1.8373 1.8677 1.811

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

NN of channels = 2, Combining channels from 1 up to 2

#ch backgr. signal σ_b δ_b S_cP S_cP_MC S_c12

1 1.00 5.00 0.0000 0.0000 3.2417 3.2363 2.899

1.0000 0.0000 2.2798 2.3116 2.050

1.0000 .50000E-01 2.2547 2.3177 1.990

2 5.00 1.00 0.0000 0.0000 .29489 .30434 .4248

2.2361 0.0000 .24597 .25775 .3018

2.2361 .25000 .17006 .17654 .2210

COMBINING of OBSERVED RESULTS

Combined channels(1-2) without errors 3.5051 3.5026

Combined channels(1-2) with stat. errors 2.6078 2.6402

Combined channels(1-2) both types of err. 2.5607 2.6198

COMBINING for EXPECTED SIGNAL and BACKGROUND

Sum 6.00 6.00 0.000 0.000 2.052 2.045 2.029

2.449 0.000 1.486 1.500 1.435

2.449 0.300 1.406 1.398 1.333

Program output with combining of channelsSeptember, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Range of applicability of the program ScP

NN of channels = 3, Combining channels from 1 up to 1

calculation type = 12

#ch backgr. signal σ_b δ_b S_cP S_cP_MC S_c12

1 500.00 100.00 0.0000 0.0000 4.3205 4.3144 4.268

22.361 0.0000 3.1131 3.0378 3.018

22.361 25.000 2.3040 2.3390 2.210

2 300.00 120.00 0.0000 0.0000 6.5145 0.0 6.347

17.321 0.0000 4.8231 4.6070 4.488

17.321 15.000 4.1453 4.0227 3.835

3 15000.0 1000.0 0.0000 0.0000 6.2873 0.0 8.033

122.47 0.0000 6.1213 0.0 5.680

122.47 575.00 2.4477 0.0 2.369

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Motivation of the work(confidence intervals)

Suppose f(n;m)describes the Poisson distribution of probabilities andg(m;n)is the density of Gamma-distribution G1,1+nthen

(Eq.1)

where

and n is the observed number of casual events appearing in Poisson flow for certain period of time. This identity shows that in our case the distribution of the probability of a true value of Poisson distribution parameter (the confidence density)for observed value nis the Gamma-distribution with mode nand mean value n+1,i.e. observed value n corresponds to the most probable value of parameter. The Poisson and Gamma distributions are statistically dual distributions. As shown, we for these distributions can reconstruct only single confidence density.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Program Limsb

The unique of confidence density allows to construct the confidence intervals by simplest (and correct) way:

we reconstruct for observed value n the correspondent confidence

density and by direct calculations of probabilities determine the

confidence intervals and/or confidence limits.

Now the program Limsb constructs the central confidence interval and the confidence interval of minimal length for observed value n.

Input: values EPS, CL and array DLAMB.

The testing set of observed values is given in data array DLAMB.

The value EPS determines the precision of calculations.

The value CL determines the confidence level of intervals.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Simplest example of program Limsb output

Confidence limits: eps, CL = 9.99999975E-05 0.899999976

central and shortes confidence intervals

NN ev left bound right bound left tail upper prob. lenght

.10000E-01

Central 0.05308130 3.015071 0.04998137 0.9500018 2.961990 Minimal 1.30385E-08 2.319871 1.082756E-08 0.9000000 2.319871 .10000

Central 0.07074530 3.186507 0.04999527 0.9500008 3.115762 Minimal 6.053597E-09 2.473754 8.71932E-10 0.9000000 2.473754 .50000

Central 0.17588568 3.907293 0.04998513 0.9499978 3.731407 Minimal 0.00512708 3.128773 0.00027532 0.9002753 3.123645 1.0000

Central 0.35530150 4.743777 0.04998505 0.94999635 4.3884754

Minimal 0.08397551 3.932307 0.00333463 0.90290034 3.8483307

10000.

Central 9837.07715 10166.06 0.04999995 0.94999993 328.98340 Minimal 9836.24023 10165.22 0.04913389 0.94913387 328.97656

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Conclusion

Programs ScP and Limsb can be found in Web page

http://cmsdoc.cern.ch/~bityukov

We are ready to include in program Limsb the calculation of the confidence intervals of Poisson distribution parameter for signal events in presence of background (formula O.Helene, which appears in our approach by natural way, see hep-ex/0108020).

We are grateful to Vladimir Gavrilov, Vassili Katchanov, and

Albert De Roeck for the interest and support of this work.

We would like to thank Bob Cousins, Vladimir Obraztsov and

Claudia Wulz for discussions and useful comments.

September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

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