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Discussion on significance

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ATLAS Statistics Forum

CERN/Phone, 2 December, 2009

Glen Cowan

Physics Department

Royal Holloway, University of London

www.pp.rhul.ac.uk/~cowan

Discussion on significance

The standard way to quantify the significance of a discovery

is to give the p-value of the background-only hypothesis H0:

p = Prob( data equally or more incompatible with H0 | H0 )

Requires a definition of what data values constitute a lesser

level of compatibility with H0 relative to the level found with

the observed data.

Define this to get high probability to reject H0 if a

particular signal model (or class of models) is true.

Note that actual confidence in whether a real discovery is made

depends also on other factors, e.g., plausibility of signal, degree

to which it describes the data, reliability of the model used to

find the p-value. p-value is really only first step!

Discussion on significance

Often define significance Z as the number of standard deviations

that a Gaussian variable would fluctuate in one direction

to give the same p-value.

TMath::Prob

TMath::NormQuantile

Z = 5 corresponds to p = 2.87 × 10-7

Discussion on significance

The significance with which one rejects the SM depends on

the particular data set obtained.

To characterize the sensitivity of a planned analysis, give the

expected (e.g., mean or median) significance assuming a

given signal model.

To determine accurately could in principle require an MC study.

Often sufficient to evaluate with representative (e.g. “Asimov”)

data.

Discussion on significance

Suppose we measure n events, expect s signal, b background.

n ~ Poisson(s+b)

Find p-value of s = 0 hypothesis.

data values with n ≥ nobs constitute lesser compatibility

Discussion on significance

Equivalently can write expectation value of n as

where m is a strength parameter (background-only is m = 0).

To test a value of m, construct likelihood ratio

where muhat is the Maximum Likelihood Estimator (MLE),

which we constrain to be positive:

Discussion on significance

Also define

High values correspond to increasing incompatibility with m.

For discovery we are testing m = 0. We find

The p-value is

Discussion on significance

Here we will have:

and so the p-value is same as before. But for large enough n,

we can regard qm as continuous, and find

Furthermore, for large enough n, the distribution of qm approaches

a form related to the chi-square distribution for 1 d.o.f.

Complications arise from requirement that m be positive, but

end result simple. For test of m = 0 (discovery), significance is

Discussion on significance

Find median significance from median n, which is approximately

s + b when this is sufficiently large.

Or, if using the approximate formula based on chi-square,

approximate median by substituting s + b for n (“Asimov” data)

For s << b, expanding logarithm and keeping terms to O(s2),

Discussion on significance

Suppose b consists of several components, and that these are

not precisely known but estimated from subsidiary measurements:

n ~ Poisson,

mi ~ Poisson,

Likelihood function for full set of measurements is:

Discussion on significance

To account for the nuisance parameters (systematics), test m

with the profile likelihood ratio:

Double hat: maximize

L for the given m

Single hats: maximize

L wrt m and b.

Important point is that qm = -2 ln l(m) still related to chi-square

distribution even with nuisance parameters (for sufficiently large

sample), so retain the simple formula for significance:

Discussion on significance

From recent hypernews (Tetiana Hrynova, Xavier Prudent),

Consider s = 20.4, b = 2.5 ± 1.5. What is “correct” sensitivity?

First suppose b = 2.5 exactly, then:

1) Use MC to find median, assuming s = 20.4, of

Best

2) Use formula based on chi-square approx. for likelihood ratio:

Good for s+b > dozen?

3) Use

Here OK for s << b, b > dozen?

Discussion on significance

To take into account the uncertainty in the background, need to

understand the origin of the 2.5 ± 1.5.

Is this e.g. an estimate based on a Poisson measurement?

Use profile likelihood for nuisance parameter b.

Or is it a Gaussian prior (truncated at zero) with mean 2.5, s = 1.5?

Use “Cousins-Highland”

Discussion on significance

Key is to view p-value as the basic quantity of interest; Z is

equivalent, and all “magic formulae” are various approximations

for Z.

Also other considerations for discovery (and limits) beyond

p-value, e.g., level to which signal described by data, plausibility

of signal model, reliability of model for p-value, …

Also consider e.g. Bayes factors for complementary info.

StatForum should move towards firm recommendations on

what formulae to use where possible, but cannot investigate

every approximation – analysts must take some responsibility here.

Draft note (INT) attached to agenda on discovery significance;

will also have partner note on limits.

Discussion on significance