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An Answer and a Question Limits: Combining 2 results Significance: Does 2 give 2 ?

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### An Answer and a QuestionLimits: Combining 2 resultsSignificance: Does 2 give 2?

Roger Barlow

BIRS meeting

July 2006

Revisit s+b

- Calculator (used in BaBar) based on Cousins and Highland: frequentist for s, Bayesian integration for and b
- See http://www.slac.stanford.edu/~barlow/java/statistics2.html and C.P.C. 149 (2002) 97
- 3 different priors (uniform in ,1/ , ln )

Combining Limits?

With 2 measurements

x=1.1 0.1 and x=1.2 0.1

the combination is obvious

With 2 measurements

x<1.1 @ 90% CL and x<1.2 @ 90% CL

all we can say is x<1.1 @ 90% CL

Frequentist problem

Given N1 events with effcy 1 , background b1

N2 events with effcy 2 , background b2

(Could be 2 experiments, or 2 channels in same experiment)

For significance need to calculate, given source strength s, probability of result {N1 ,N2 } or less.

Constraint

If 1 = 2 and b1=b2 then N1+N2 is sufficient. So cannot just take lower left quadrant as ‘less’.

(And the example given yesterday is trivial)

Suggestion

- Could estimate s by maximising log (Poisson) likelihood

-(i s +bi) + Ni ln (i s +bi)

Hence

Ni i /( i s +bi) - i =0

- Order results by the value of s they give from solving this
- Easier than it looks. For a given {Ni } this quantity is monotonic decreasing with s. Solve once to get sdata , explore s space generating many {Ni } : sign of Ni i /( i sdata +bi) - i tells you whether this estimated s is greater or less than sdata

Message

- This is implemented in the code – ‘Add experiment’ button (up to 10)
- Comments as to whether this is useful are welcome

Significance

Analysis looking for bumps

Pure background gives 2old of 60 for 37 dof (Prob 1%).

Not good but not totally impossible

Fit to background+bump (4 new parameters) gives better 2new of 28

Question: Is this significant?

Answer: Yes

Question: How much?

Answer:

Significance is(2 new- 2old )

= (60-28)=5.65

Schematic only!!

No reference to any experimental data, real or fictitious

Puzzle. How does a 3 sigma discrepancy become a 5 sigma discovery?

Justification?

- ‘We always do it this way’
- ‘Belle does it this way’
- ‘CLEO does it this way’

Possible Justification

Likelihood Ratio Test

a.k.a. Maximum Likelihood Ratio Test

If M1 and M2 are models with max. likelihoods L1 and L2 for the data, then 2ln(L2 / L1) is distributed as a 2 with N1 - N2 degrees of freedom

Provided that

- M2 contains M1
- Ns are large
- Errors are Gaussian
- Models are linear

Does it matter?

- Investigate with toy MC
- Generate with Uniform distribution in 100 bins, <events/bin>=100. 100 is large and Poisson is reasonably Gaussian
- Fit with
- Uniform distribution (99 dof)
- Linear distribution (98 dof)
- Cubic (96 dof):a0+a1 x + a2 x2 + a3 x3
- Flat+Gaussian (96 dof): a0+a1 exp(-0.5(x- a2)2/a32)

Cubic is linear: Gaussian is not linear in a2and a3

Calculate 2 probabilities of differences in models

Compare linear and uniform models. 1 dof. Probability flat Method OK

Compare flat+gaussian and uniform models. 3 dof. Probability very unflat

Method invalid

Peak at low P corresponds to large 2 i.e. false claims of significant signal

Compare cubic and uniform models. 3 dof. Probability flat Method OK

Not all parameters are equally useful

If 2 models have the same number of parameters and both contain the true model, one can give better results than the other.This tells us nothing about the data

Shows 2 for flat+gauss v. cubic

Same number of parameters

Flat+gauss tends to be lower

Conclude: 2 does not give 2?

But surely…

- In the large N limit, ln L

is parabolic in fitted

parameters.

- Model 2 contains Model 1

with a2=0 etc. So expect ln L

to increase by equivalent of 3

in chi squared.

Question. What is wrong with this argument? Asymptopic? Different probability? Or is it right and the previous analysis is wrong?

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