p-values and Discovery
Louis Lyons
Oxfordl.lyons@physics.ox.ac.uk

Kyoto,May 2007
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TOPICS
Discoveries
H0 or H0 v H1p-values: For Gaussian, Poisson and multi-variate data

Goodness of Fit tests
Why 5σ?
Blind analyses

What is p good for?
Errors of 1st and 2nd kind
What a p-value is notP(theory|data) ≠ P(data|theory)
THE paradox
Optimising for discovery and exclusion

Incorporating nuisance parameters

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DISCOVERIES“Recent” history:Charm
SLAC, BNL 1974
Tau lepton
SLAC1977
BottomFNAL
1977
W,ZCERN
1983
TopFNAL
1995{Pentaquarks ~Everywhere 2002 }?
FNAL/CERN 2008?
? = Higgs, SUSY, q and l substructure, extra dimensions,
free q/monopoles, technicolour, 4th generation, black holes,…..

QUESTION: How to distinguish discoveries from fluctuations or goofs?
4Penta-quarks?
Hypothesis testing: New particle or statistical fluctuation?

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H0 or H0 versus H1 ?
H0 = null hypothesis
e.g. Standard Model, with nothing new
H1 = specific New Physics e.g. Higgs with MH = 120 GeV
H0: “Goodness of Fit” e.g. χ2 ,p-values
H0 v H1: “Hypothesis Testing” e.g. L-ratio
Measures how much data favours one hypothesis wrt other
H0 v H1 likely to be more sensitiveor6

Testing H0:
Do we have an alternative in mind?
1) Data is number (of observed events)“H1” usually gives larger number(smaller number of events if looking for oscillations)
2) Data = distribution. Calculate χ2.
Agreement between data and theory gives χ2 ~ndf
Any deviations give large χ2
So test is independent of alternative?
Counter-example: Cheating undergraduate
3) Data = number or distribution
Use L-ratio as test statistic for calculating p-value4) H0 = Standard Model7

p-values
Concept of pdf
Example: Gaussian

yμx0x

y = probability density for measurement xy = 1/(√(2π)σ) exp{-0.5*(x-μ)2/σ2}
p-value: probablity that x ≥ x0
Gives probability of “extreme” values of data ( in interesting direction)(x0-μ)/σp116%i.e. Small p = unexpected

22.3%

30.13%

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0. 003%50.3*10-68p-values, contd
Assumes:
Gaussian pdf (no long tails)
Data is unbiassed
σ is correct
If so, Gaussian xuniform p-distribution(Events at large x give small p)

0p1

9p-values for non-Gaussian distributions
e.g. Poisson counting experiment, bgd = b

P(n) = e-b * bn/n!
{P = probability, not prob density}
b=2.9
P0n

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For n=7, p = Prob( at least 7 events) = P(7) + P(8) + P(9) +…….. = 0.03
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Poisson p-values
n = integer, so p has discrete values
So p distribution cannot be uniformReplace Prob{p≤p0} = p0, for continuous pby Prob{p≤p0} ≤ p0, for discrete p(equality for possible p0)
p-values often converted into equivalent Gaussian σe.g. 3*10-7 is “5σ” (one-sided Gaussian tail)11SignificanceSignificance =S/ B ?

Potential Problems:•Uncertainty in B
•Non-Gaussian behaviour of Poisson, especially in tail
•Number of bins in histogram, no. of other histograms [FDR]•Choice of cuts

(Blind analyses)•Choice of bins(……………….)

For future experiments:• Optimising

S/B could give S =0.1, B = 10-612

Goodness of Fit Tests
Data = individual points, histogram, multi-dimensional,multi-channel
χ2 and number of degrees of freedomΔχ2 (or lnL-ratio): Looking for a peak
Unbinned Lmax?
Kolmogorov-Smirnov
Zech energy test
Combining p-values
Lots of different methods. Software available from:http://www.ge.infn.it/statisticaltoolkit

13χ2 with ν degrees of freedom?
1) ν = data – free parameters ?
Why asymptotic (apart from Poisson  Gaussian) ?
a) Fit flatish histogram withy = N {1 + 10-6 cos(x-x0)} x0 = free paramb) Neutrino oscillations: almost degenerate parametersy ~ 1 – A sin2(1.27 Δm2 L/E)
2 parameters
1 – A (1.27 Δm2 L/E)21 parameter
Small Δm214

χ2 with ν degrees of freedom?
2) Is difference in χ2 distributed as χ2 ?
H0 is true.
Also fit with H1 with k extra params
e. g. Look for Gaussian peak on top of smooth backgroundy = C(x) + A exp{-0.5 ((x-x0)/σ)2}Is χ2H0 - χ2H1 distributed as χ2 with ν = k = 3 ?

Relevant for assessing whether enhancement in data is just astatistical fluctuation, or something more interesting
N.B. Under H0 (y = C(x)) :A=0 (boundary of physical region)
x0 and σ undefined15

Is difference in χ2 distributed as χ2 ?
Demortier:
H0 = quadratic bgd
H1 = ……………… +Gaussian of fixed width,variable location & ampl

Protassov, van Dyk, Connors, ….
H0 = continuum(a) H1 = narrow emission line(b) H1 = wider emission line(c) H1 = absorption line

Nominal significance level = 5%16Is difference in χ2 distributed as χ2 ?, contd.

So need to determine the Δχ2 distribution by Monte CarloN.B.
1) Determining Δχ2 for hypothesis H1 when data is generated
according to H0 is not trivial, because there will be lots oflocal minima
2) If we are interested in 5σ significance level, needs lots ofMC simulations (or intelligent MC generation)

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Unbinned Lmax and Goodness of Fit?
Find params by maximising L
So larger L better than smaller L
So Lmax gives Goodness of Fit ??

BadGood?

Great?

Monte Carlo distributionof unbinned Lmax

FrequencyLmax

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Not necessarily:
L(data,params)

pdf

fixed vary
Contrast pdf(data,params)Lparamvary fixeddatae.g. p(t,λ) = λ *exp(- λt)

Max at λ=1/t

Max at t = 0Lpt

λ19

Example 1: Exponential distribution
Fit exponential λ to times t1, t2 ,t3 …….

[Joel Heinrich, CDF 5639]

L =   e  ti = -N(1 + ln t )lnLmaxavi.e. lnLmax depends only on AVERAGE t, but is
INDEPENDENT OF DISTRIBUTION OF t(except for……..)(Average t is a sufficient statistic)
Variation of Lmax in Monte Carlo is due to variations in samples‟ average t , but

NOT TO BETTER OR WORSE FIT

pdf
Same average tsame Lmaxt20

Example 21  cos 2 d cos 1  / 3dN

L=i

1   cos 2 i
1  / 3cos θpdf (and likelihood) depends only on cos2θiInsensitive to sign of cosθi
So data can be in very bad agreement with expected distribution
e.g. all data with cosθ < 0 , but Lmax does not know about it.

Example of general principle
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Example 3
Fit to Gaussian with variable μ, fixed σ21  x  pdf exp{  }2   2
1lnLmax = N(-0.5 ln2π – lnσ) – 0.5 Σ(xi – xav)2 /σ2constant~variance(x)i.e. Lmax depends only on variance(x),which is not relevant for fitting μ(μest = xav)

Smaller than expected variance(x) results in larger Lmaxx

Worse fit, larger Lmaxx

Better fit, lower Lmax

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Lmax and Goodness of Fit?
Conclusion:
L has sensible properties with respect to parameters
NOT with respect to dataLmax within Monte Carlo peak is NECESSARY
not SUFFICIENT(‘Necessary’ doesn’t mean that you have to do it!)23

Goodness of Fit:
Kolmogorov-Smirnov
Compares data and model cumulative plots
Uses largest discrepancy between dists.
Model can be analytic or MC sample
Uses individual data points
Not so sensitive to deviations in tails(so variants of K-S exist)
Not readily extendible to more dimensions
Distribution-free conversion to p; depends on n
(but not when free parameters involved – needs MC)

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Goodness of fit: „Energy‟ test
Assign +ve charge to data; -ve charge to M.C.

Calculate „electrostatic energy E‟ of charges
If distributions agree, E ~ 0
If distributions don‟t overlap, E is positivev2

Assess significance of magnitude of E by MC

N.B.

v1

1) Works in many dimensions2) Needs metric for each variable (make variances similar?)3) E ~ Σ qiqj f(Δr = |ri – rj|) ,f = 1/(Δr + ε) or –ln(Δr + ε)

Performance insensitive to choice of small ε
See Aslan and Zech‟s paper at:
http://www.ippp.dur.ac.uk/Workshops/02/statistics/program.shtml
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Combining different p-values
Several results quote p-values for same effect: p1, p2, p3…..
e.g. 0.9, 0.001, 0.3 ……..
What is combined significance?
Not just p1*p2*p3…..
If 10 expts each have p ~ 0.5, product ~ 0.001 and is clearly
NOT correct combined pn 1

S = z *  (-ln z)j /j! ,j 0z = p1p2p3…….

(e.g. For 2 measurements, S = z * (1 - lnz) ≥ z )Slight problem: Formula is not associative
Combining {{p1 and p2}, and then p3} gives different answer
from {{p3 and p2}, and then p1} , or all together
Due to different options for “more extreme than x1, x2, x3”.
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Combining different p-values
Conventional:
Are set of p-values consistent with H0?
SLEUTH:How significant is smallest p?

p2

1-S = (1-psmallest)np1

Combined S
Conventional
SLEUTH

p1 = 0.01
p2 = 0.01
p2 = 1p1 = 10-4
p2 = 10-4
p2 = 1

1.0 10-3
2.0 10-2

1.9 10-7
2.0 10-4

5.6 10-2
2.0 10-2

1.0 10-3
2.0 10-4
27Why 5σ?
• Past experience with 3σ, 4σ,… signals• Look elsewhere effect:
Different cuts to produce data
Different bins (and binning) of this histogram
Different distributions Collaboration did/could look atDefined in SLEUTH• Bayesian priors:
P(H0|data)
P(data|H0) * P(H0)P(H1|data)
P(data|H1) * P(H1)
Bayes posteriors

Likelihoods

PriorsPrior for {H0 = S.M.} >>> Prior for {H1 = New Physics}
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Sleutha quasi-model-independent search strategy for newphysics
Assumptions:
1. Exclusive final state
2. Large ∑pT
3. An excess

0608025(prediction) d(hep-ph)0001001

Rigorously
compute the trials
factor associatedwith looking
everywhere 29

pseudo discovery

Sleuth- < 8e-08
PWbbjj~P < 4e-0530

BLIND ANALYSES
Why blind analysis?
Methods of blinding

Selections, corrections, method

Add random number to result *Study procedure with simulation only
Look at only first fraction of data
Keep the signal box closed
Keep MC parameters hidden
Keep unknown fraction visible for each bin

After analysis is unblinded, ……..
* Luis Alvarez suggestion re “discovery” of free quarks
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What is p good for?
Used to test whether data is consistent with H0
Reject H0 if p is small : p≤α (How small?)Sometimes make wrong decision:
Reject H0 when H0 is true: Error of 1st kind
Should happen at rate αOR
Fail to reject H0 when something else
(H1,H2,…) is true:
Error of 2nd kind
Rate at which this happens depends on……….
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Errors of 2nd kind: How often?
e.g.1. Does data line on straight line?

Calculate χ2
Reject if χ2 ≥ 20yxError of 1st kind: χ2 ≥ 20 Reject H0 when true
Error of 2nd kind: χ2 ≤ 20 Accept H0 when in fact quadratic or..
How often depends on:
Size of quadratic term
Magnitude of errors on data, spread in x-values,…….
How frequently quadratic term is present33

Errors of 2nd kind: How often?
e.g. 2. Particle identification (TOF, dE/dx, Čerenkov,…….)
Particles are π or μ
Extract p-value for H0 = π from PID information
π and μ have similar masses

p01Of particles that have p ~ 1% („reject H0‟), fraction that are π isa) ~ half,for equal mixture of π and μb) almost all, for “pure” π beam34c) very few, for “pure” μ beam

What is p good for?
Selecting sample of wanted eventse.g. kinematic fit to select t t eventstbW, bjj, Wμν tbW, bjj, Wjj
Convert χ2 from kinematic fit to p-value
Choose cut on χ2 to select t t events
Error of 1st kind: Loss of efficiency for t t events
Error of 2nd kind: Background from other processes
Loose cut (large χ2max , small pmin): Good efficiency, larger bgd
Tight cut (small χ2max , larger pmin): Lower efficiency, small bgd
Choose cut to optimise analysis:
More signal events: Reduced statistical error
More background: Larger systematic error
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p-value is not ……..
Does NOT measure Prob(H0 is true)i.e. It is NOT P(H0|data)
It is P(data|H0)
N.B. P(H0|data)≠ P(data|H0)
P(theory|data) ≠ P(data|theory)“Of all results with p ≤ 5%, half will turn out to be wrong”
N.B. Nothing wrong with this statement
e.g. 1000 tests of energy conservation~50 should have p ≤ 5%, and so reject H0 = energy
conservation
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Of these 50 results, all are likely to be “wrong”

P (Data;Theory) P (Theory;Data)

Theory = male or female
Data = pregnant or not pregnant

P (pregnant ; female) ~ 3%

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P (Data;Theory) P (Theory;Data)

Theory = male or female
Data = pregnant or not pregnant

P (pregnant ; female) ~ 3%but
P (female ; pregnant) >>>3%
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Aside: Bayes‟ Theorem
P(A and B) = P(A|B) * P(B) = P(B|A) * P(A)N(A and B)/Ntot = N(A and B)/NB * NB/Ntot
If A and B are independent, P(A|B) = P(A)Then P(A and B) = P(A) * P(B), but not otherwise
e.g. P(Rainy and Sunday) = P(Rainy)
But P(Rainy and Dec) = P(Rainy|Dec) * P(Dec)
25/365=25/31* 31/365
Bayes‟ Th:

P(A|B) = P(B|A) * P(A) / P(B)39More and more data
1) Eventually p(data|H0) will be small, even if data and H0
are very similar.
p-value does not tell you how different they are.
2) Also, beware of multiple (yearly?) looks at data.
“Repeated tests eventually sure
to reject H0, independent ofvalue of α”
Probably not too serious –< ~10 times per experiment.

40More “More and more data”

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PARADOX
Histogram with 100 bins
Fit 1 parameter
Smin: χ2 with NDF = 99 (Expected χ2 = 99 ± 14)
For our data, Smin(p0) = 90
Is p1 acceptable if S(p1) = 115?
1) YES.
2) NO.

Very acceptable χ2 probability
σp from S(p0 +σp) = Smin +1 = 91
But S(p1) – S(p0) = 25
So p1 is 5σ away from best value
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Comparing data with different hypotheses

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Choosing between 2 hypotheses
Possible methods:Δχ2lnL–ratio
Bayesian evidence
Minimise “cost”

48Optimisation for Discovery and Exclusion
Giovanni Punzi, PHYSTAT2003:“Sensitivity for searches for new signals and its optimisation”
http://www.slac.stanford.edu/econf/C030908/proceedings.html
Simplest situation: Poisson counting experiment,
Bgd = b, Possible signal = s, nobs counts(More complex: Multivariate data,lnL-ratio)
Traditional sensitivity:
Median limit when s=0
Median σ when s ≠ 0 (averaged over s?)
Punzi criticism: Not most useful criteria
Separate optimisations
491) No sensitivity
H02) Maybe3) Easy separation

H1nβncrit αProcedure: Choose α (e.g. 95%, 3σ, 5σ ?) and CL for β (e.g. 95%)
Given b, α determines ncrits defines β. For s > smin, separation of curves  discovery or excln
smin = Punzi measure of sensitivity For s ≥ smin, 95% chance of 5σ discovery
Optimise cuts for smallest smin

Now data:

If nobs ≥ ncrit, discovery at level α

If nobs < ncrit, no discovery. If βobs < 1 – CL, exclude H1501) No sensitivity
Data almost always falls in peakβ as large as 5%, so 5% chance of H1 exclusion even when no sensitivity. (CLs)2) Maybe
If data fall above ncrit, discovery
Otherwise, and nobs  βobs small, exclude H1(95% exclusion is easier than 5σ discovery)
But these may not happen  no decision

3) Easy separation
Always gives discovery or exclusion (or both!)

Disc Excl 1)

2)

3)

NoNoNoYes

Yes

No()

Yes

Yes

!
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Incorporating systematics in p-values
Simplest version:
Observe n events
Poisson expectation for background only is b ± σbσb may come from:acceptance problems
jet energy scaledetector alignment
limited MC or data statistics for backgrounds
theoretical uncertainties
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Luc Demortier,“p-values: What they are andhow we use them”, CDF memo June 2006
http://www-cdfd.fnal.gov/~luc/statistics/cdf0000.ps

Includes discussion of several ways ofincorporating nuisance parameters
Desiderata:
Uniformity of p-value (averaged over ν, orfor each ν?)
p-value increases as σν increases
GeneralityMaintains power for discovery
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Ways to incorporate nuisance params in p-values• Supremum• Conditioning• Prior Predictive

Maximise p over all ν. Very conservative
Good, if applicableBox. Most common in HEP
p = ∫p(ν) π(ν) dν••••

Posterior predictive Averages p over posterior
Plug-in
Uses best estimate of ν, without error
L-ratio
Confidence interval Berger and Boos.
p = Sup{p(ν)} + β, where 1-β Conf Int for ν• Generalised frequentist Generalised test statistic
Performances compared by Demortier54

Summary• P(H0|data) ≠ P(data|H0)• p-value is NOT probability of hypothesis, givendata• Many different Goodness of Fit tests – most needMC for statistic  p-value• For comparing hypotheses, Δχ2 is better than χ21
and χ22• Blind analysis avoids personal choice issues• Worry about systematics
PHYSTAT Workshop at CERN, June 27  29 2007
“Statistical issues for LHC Physics Analyses”
55Final messageSend interesting statistical issues tol.lyons@physics.ox.ac.uk

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