Idea Transcript
Hypothesis Tests, Significance Significance as hypothesis test Pitfalls Avoiding pitfalls Goodness-of-fit Context is frequency statistics.
1
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Significance as Hypothesis Test When asking for the “significance” of an observation (of, perhaps a new effect), you ask for a test of the hypotheses: Null hypothesis H0 : There is no new effect; against Alternative hypothesis H1 : There is a new effect. Reject the null hypothesis (that is, claim a new effect) if the observation falls in a region that is “unlikely” if the null hypothesis is correct. “Significance” (as typically used in HEP, e.g., “a significance of 5σ”) is the probability that we erroneously reject the null hypothesis. Also called “confidence level”, or “P -value”, or the probability of a “Type I error”.
2
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Significance A 68% confidence interval does not always tell you much about significance. The tails may be non-normal. A separate analysis is generally required, which models the tails appropriately. No recommendation on when to label result as “significant”. Label implies interpretation. – No uniform prescription seems to make sense, involves judgement. Eg, bizarre new particle vs. expected branching fraction. – Not our most essential experimental role; up to consumer ultimately to decide what they want to believe. Nevertheless, people insist on making qualitative statements (“observation of”, “evidence for”, “discovery of”, “not significant”, “consistent with”) Code: “observation of” ≡
3
> 4σ, “evidence for” ≡
> 3σ
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Significance (more semantics. . . ) From Physics Today, http://www.physicstoday.org/pt/vol-54/iss-9/p19.html (coloring mine, references deleted) nb: Just an observation on human nature, no criticism should be inferred.
“In March, back-to-back papers in Physical Review Letters reported the measurement of CP symmetry violation in the decay of neutral B mesons by groups in Japan and California. Now the word “measurement” has been replaced by “observation” in the titles of two new back-to-back reports by these same groups in the 27 August Physical Review Letters. That is to say, with a lot more data and improved event reconstruction, the BaBar collaboration at SLAC and the Belle collaboration at KEK in Japan have at last produced the first compelling evidence of CP violation in any system other than the neutral K mesons.” Some people think a measurement should not be called a “measurement” unless the result is significantly different from zero. A senior Assistant Editor at a prominent journal suggested that “bounds on” might be more appropriate than “measurement” in reference to a CP asymmetry angle which was observed as consistent with zero. Finding sin 2β = 0.00 ± 0.01 would be pretty exciting. But it isn’t a “measurement”? 4
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Computation of significance: Example Flat normal background (avg 100/bin) + gaussian signal (100 events, fixed) with mean 0, σ = 1 160 140
Events/0.5
120 100 80
Generated signal = 100 events _ 39 events Cut&Count signal = 194 + Significance = 5.0 Expected significance = 2.7
60 40 20 0 -10
-8
-6
-4
-2
0
x
2
4
6
8
10
Distribution is normal here, so P = 5.7 × 10−7 (two-tail). Note: H0 is flat distribution with no signal. H1 is Nsig = 0. 5
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Note: The significance is not obtained by dividing the signal estimate (194) by the uncertainty in the signal (39), 194/39 = 5.0. That would be akin to asking how likely a signal of the estimated size would be to fluctuate to zero. It is a good approximation in this example, however, because B/S is large. The significance was estimated here with a simple cut-and-count method: – Level of background estimated from region |x| > 3. – Counts in “signal region” |x| < 3 added up. – The excess in the signal region over the estimated background is divided by the square root of the estimated background in the signal region. In this example, we assumed we knew the mean and width of the signal we were looking for, as well as the background shape. The uncertainty in the background estimate is negligible in this example. A more sophisticated fit may yield a more powerful test by incorporating the known shape of the signal. 6
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
What about Systematic Uncertainties? B(e+e− → Nobel prize) = 10 ± 1 ± 5. They may be important! – Maybe the ±5 is a systematic uncertainty in the estimate of the background expectation. A “10σ” statistical significance is really only a “2σ” effect. They may be irrelevant – Maybe the ±5 is a systematic uncertainty on the efficiency, entering as a multiplicative factor. It makes no difference to the significance whether the result is 10 ± 1 or 5 ± 0.5. They may be “fuzzy” – E.g., how is the background expectation estimated? What is the sampling distribution? – E.g., Are “theoretical” uncertainties present? 7
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Systematic Uncertainties “Blind checks” and “educated checks”: – Blind check: testing for mistakes; no correction is expected. If pass test, no contribution to systematic error. Eg, divide data into chronological subsets and compare results. – Educated check: measuring biases, corrections. May affect quoted result. Always contributes to systematic error. Eg, dependence of efficiency on model. Quote systematic uncertainty separately from statistical – Systematic uncertainty may contain statistical components, eg, MC statistics in evaluation of efficiency.
8
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
y’
Systematic Uncertainties Example: D mixing and DCSD revisited – Want simple procedure. Willing to accept approximation. – Scale statistical-only contour uniformly along ray from best-fit value. – Factor is 1 + m2i , where mi is an estimate of the effect of systematic uncertainty i measured in units of the statistical uncertainty. This estimate is obtained by determining the effect of the systematic uncertainty on x 2, y.
0.04 0.02 -0 -0.02 2
Physical (x’ , y’) 2 Central (x’ , y’) No CPV 95% CL CPV allowed 95% CL CPV allowed, stat only 95% CL CP conserved 95% CL CP conserved, stat only
-0.04 -0.06 -0.5
0
0.5
1
1.5
2
2
x’ / 10
2.5
-3
Method conservative (≡ lazy) in sense that scaling for a given systematic in one direction is applied uniformly in all directions. On the other hand, a linear approximation is being made.
9
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Aside: Significance as “nσ” HEP parlance is to say an effect has, e.g., “5σ” significance. At face value, this means the observation is “5 standard deviations” away from the mean: σ ≡ (x − x¯ )2.
But we often don’t really mean this. Note that a 5σ effect of this sort may not be improbable: P(x)
0.8
P (|x − x¯ | = 5σ ) = 20% ! 0.1
0.1 -1
10
0
1
x
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Aside: Significance as “nσ” (continued) Instead, we often mean that the probability (P -value) for the effect is given by the probability of a fluctuation in a normal distribution 5σ from the mean, i.e., P = P (|x| > 5), for x ∈ N (0, 1) = 5.7 × 10−7
(two-tailed probability). But sometimes we really do mean 5σ, usually presuming that the sampling distribution is approximately normal. [This may not be an accurate presumption when far out in the tails!] √ Also now popular to call −2Δ ln L the “n” in “nσ”. 2 From: L0(θ = 0; x) = √ 1 exp − 12 x/σ , Lmax( θ = x; x) = √ 1 , 2πσ 2πσ √ giving −2Δ ln L = Δχ2 = x/σ = n. Desirable to be more concise by quoting probabilities, or “P -values” as is common in the statistics world. At least say what you mean! 11
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Estimating significance: Pitfalls What are the dangers? In a nutshell: Unknown or unknowable sampling distributions Ways to not know the distribution: The Improbable Tails Systematic Unknowns The Stopping Problem The exploratory Bump Hunt
12
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
The Improbable Tails 160 140
Our earlier example was known to be normal sampling.
Events/0.5
120 100 80
Generated signal = 100 events _ 39 events Cut&Count signal = 194 + Significance = 5.0 Expected significance = 2.7
60 40 20 0 -10
-8
-6
-4
-2
0
x
2
4
6
8
10
Often, this is true approximately (central limit theorem). But for significance, often interested in distribution far into the tails. The normal approximation may be very bad here! If there is any doubt, need to compute the actual distribution. Typically this is done with a “toy Monte Carlo” to simulate the distribution of the significance statistic. To get to the tails, this may require a fair amount of computing time. Still need to be wary of pushing calculation beyond its validity as a model of the actual distribution. BaBar (and others) now routinely performs these calculations.
13
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Systematic Unknowns Nuisance parameters – Unknown, but relevant parameters. Estimated somehow, but with some uncertainty. Even if sampling distribution is known, cannot in general derive exact P -values in lower dimensional parameter space. – Central Limit Theorem is our friend. – Can try other values besides best estimate of nuisance parameter. – See our discussion of confidence intervals. “Theoretical” systematic uncertainties. Guesses, no sampling distribution. – Use worst case values when evaluating significance. requires understanding what is meant by the theory “errors”. – Or, give the dependence, e.g., as a range.
14
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
The Stopping Problem There is a strong tendency to work on an analysis until we are convinced that we got it “right”, then we stop. Simple example: “Keep sampling” until we are satisfied. Motivate our example: • Ample historical evidence that experimental measurements are sometimes biased by some preconception of what the answer “should be”. For example, a preconception could be based on the result of another experiment, or on some theoretical prejudice. • A model for such a biased experiment is that the experimenter works “hard” until s/he gets the expected result, and then quits. Let’s Consider a simple example of a distribution which could result from such a scenario.
15
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Stopping Problem: Normal likelihood function example
1
N (x; θ, 1)dx = √
2π
2/2 −(x−θ) e dx.
Frequency
• Consider an experiment in which a measurement of a parameter θ corresponds to sampling from a Gaussian distribution of standard deviation one:
θ− 2
θ x
θ+2
• Suppose the experimenter has a prejudice that θ is greater than one. • Subconsciously, s/he makes measurements until the sample mean, m = 1 n x , is greater than one, or until s/he becomes convinced (or i=1 i n tired) after a maximum of N measurements. • The experimenter then uses the sample mean, m, to estimate θ.
16
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Stopping Problem: Normal likelihood function example For illustration, assume that N = 2. In terms of the random variables m and n, the pdf is: ⎧ 1 2 ⎪ n = 1, m > 1 ⎨ √12π e− 2 (m−θ) , f (m, n; θ) = 0, n = 1, m < 1
⎪ 2 2 1 ⎩ 1 e−(m−θ) −(x−m) dx n = 2 −∞ e π
4000
Histogram of sampling distribution for m, with pdf given by above equation, for θ = 0.
Number of experiments
3000
2000
1000
0 -4
-2
0
2
4
Sample mean
The likelihood function, as a function of θ, has the shape of a normal distribution, given any experimental result. The peak is at θ = m, so m is the maximum likelihood estimator for θ. 17
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Stopping Problem: Normal likelihood function example In spite of the normal form of the likelihood function, the sample mean is not sampled from a normal distribution. The “4σ” tail is more probable (for some θ) than the experimenter thinks.
Probability( θ < m - 4s )
0.00007 0.00006 0.00005 0.00004 0.00003 0.00002
Straight line is probability for a “4σ” fluctuation of a normal distrinbution.
0.00001 0 -8
-6
-4
-2
0
2
4
θ
18
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Stopping Problem: Normal likelihood function example The likelihood function, as a function of θ, has the shape of a normal distribution, given any experimental result. The peak is at θ = m, so m is the maximum likelihood estimator for θ. In spite of the normal form of the likelihood function, the sample mean is not sampled from a normal distribution. The interval defined by where the likelihood function falls by e−1/2 does not correspond to a 68% CI:
Probability (θ-=0 Fix mean 40
80 x
Any signal Float mean 0
40
80 x
Fits to data with no signal.
37
500
H1 fit
Counts/bin 800 500 800
800 500
Counts/bin 500 800
Any signal Fix mean
0
H0 fit
H1 fit
H1 fit
Any signal Fix mean H1 fit
H1 fit
Any signal Float mean
Signal>=0 Fix mean 0
40
80
0
x
40
80 x
Fits to data with a signal.
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Significance – Conclusions Evaluation of significance is a “hypothesis test”. It is essentially the same problem as evaluating confidence intervals. – Except for the more obvious role played by improbable tails. Pitfalls amount to (not) knowing the sampling distribution. Techniques exist to avoid pitfalls: – Simulating the sampling distribution – Vary the nuisance/theory parameters – Blind your experiment Doing it properly requires patience and discipline; the benefit is a more meaningful, convincing result to yourself and to others.
38
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Goodness-of-Fit No perfect general goodness-of-fit test: – Given a dataset generated under null hypothesis, can usually find a test which rejects the null hypothesis (ie, choosing the test after you see the data is dangerous). – Given a dataset generated under alternative hypothesis, can usually find a test for which the null passes (ie, should think about what you want to test for). Nominal recommendation: – If you have a specific question, test for that. – χ2 test when valid. – Consider more general likelihood ratio test, Kolmogorov-Smirnov, etc., otherwise. – Monte Carlo evaluation of distribution of test statistic.
39
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Goodness-of-Fit (continued) But recognize when test may not answer desired question, eg, in sin 2β analysis, a likelihood ratio (or a χ2) test on the time distribution may
Entries / 0.6 ps
have little sensitivity to testing goodness-of-fit of the asymmetry. 150
0
B tags
Background
−0
100
B tags
Raw Asymmetry
50
0 0.5
0
-0.5 -5
40
0
5
Δt (ps)
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Consistency of two correlated results BaBar has encountered several times the question of whether a new analysis is consistent with an old analysis. Often, new analysis is a combination of additional data plus changed (improved. . . ) analysis of original data. The stickiest issue is handling the correlation in testing for consistency in the overlapping data. People sometimes have difficulty understanding that statistical differences can arise even comparing results based on the same events. Given a sampling θ1 , θ2 from a bivariate normal distribution N (θ, σ1, σ2, ρ), with θ1 = θ2 = θ, the difference Δθ ≡ θ2 − θ1 is N (0, σ)-distributed with σ 2 = σ12 + σ22 − 2ρσ1σ2 . If the correlation is unknown, all we can say is that the variance of the difference is in the range (σ1 − σ2)2 . . . (σ1 + σ2)2. If we at least believe ρ ≥ 0 then the maximum variance of the difference is σ12 + σ22. 41
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Consistency – Simple example of two analyses on same events Suppose we measure a neutrino mass, m, in a sample of n = 10 independent events. The measurements are xi, i = 1, . . . , 10. Assume the sampling distribution for xi is N (m, σi). We may form unbiased estimator, m 1, for m: m 1 =
1 n x i=1 i n
±
1 n2
n
2 i=1 σi .
The result (from a MC) is m 1 = 0.058 ± 0.039. Then we notice that we have some further information which might be useful: we know the experimental resolutions, σi for each measurement. We form another unbiased estimator, m 2, for m: n
m 2 =
xi i=1 σ 2 n 1i i=1 σ 2 i
± n1
1 i=1 σ 2 i
.
The result (from the same MC) is m 1 = 0.000 ± 0.016. 42
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Example continued The results are certainly correlated, so question of consistency arises (we know the error on the difference is between 0.023 and 0.055). In this example, the difference between the results is 0.058 ± 0.036, where the 0.036 error includes the correlation (ρ = 0.41).
43
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Consistency – Evaluating the Correlation Art Snyder developed an approximate formula for evaluating the correlation in a comparison of maximum likelihood analyses (eg, in one-dimensional case). Suppose we perform two maximum likelihood analysis, with event likelihoods L1, L2, on the same set of events [nb, may use different information in each analysis]. The results are estimators θ1, θ2 for parameter θ. The correlation coefficient ρ may be estimated according to: N d ln L2i d ln L1i i=1 Ri dθ |θ=θ1 dθ |θ=θ2 ρ ≈ , N d2 ln L1i N d2 ln L2i | θ=θ 2 i=1 dθ i=1 dθ2 |θ=θ0 0 where (θ0 is an expansion reference point) d ln L d ln L 2 2 ln L ln L d d 1i 1i 2i 2i Ri = 1 − ( 1 − ( θ θ1 − θ0) | − θ ) | | | . θ=θ 2 0 θ=θ 0 0 dθ2 dθ θ=θ0 dθ2 dθ θ=θ0 If θ0 ≈ θ1 ≈ θ2 , then ρ≈σ ˜θ1 σ ˜θ2
N d ln L1i i=1
where
σ ˜θ2k
≡ 1/ 44
N dLki i=1
dθ
|θ=θ0
2
dθ
|θ=θ0
d ln L2i | , dθ θ=θ0
.
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008
Consistency – Example: sin 2β ¯ pairs – PRL, vol 87, 27 August 2001: 32 × 106 B B sin 2β = 0.59 ± 0.14(stat) ± 0.05(syst) ¯ pairs – SLAC-PUB-9153, March 2002: 62 × 106 B B sin 2β = 0.75 ± 0.09(stat) ± 0.04(syst) Second result includes the earlier data, re-reconstructed. Analysis involves multivariate maximum likelihood fits; reprocessing changes, eg, relative likelihood for an event to be signal or background. Not simply counting events. Question: are the two results statistically consistent? If these were independent data sets, a difference of 0.16 ± 0.17 would not be a worry. The issue is the correlation. A specialized analysis deriving from the previous formula is performed on the events in common between the two analyses. A correlation of ρ = 0.87 is deduced, yielding a difference of ∼ 2.2σ. 45
Frank Porter, BaBar analysis school – Statistics, 12-15 February 2008