Hypothesis Tests, Significance

Transcription

1 Hypothesis Tests, Significance Significance as hypothesis test Pitfalls Avoiding pitfalls Goodness-of-fit Context is frequency statistics. 1 Frank Porter, BaBar analysis school Statistics, February 2008

2 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 H 0 : There is no new effect; against Alternative hypothesis H 1 : 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, February 2008

3 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 > 4σ, evidence for > 3σ 3 Frank Porter, BaBar analysis school Statistics, February 2008

4 Significance (more semantics...) From Physics Today, (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 Now the word measurement has been replaced by observation in the titles of two new back-to-back reports by these same groups in the 27 August Physical Review Letters.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 Japanhaveatlastproducedthefirst 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 mightbemore 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. Butitisn ta measurement? 4 Frank Porter, BaBar analysis school Statistics, February 2008

5 Computation of significance: Example Flat normal background (avg 100/bin) + gaussian signal (100 events, fixed) withmean 0, σ = Events/ Generated signal = 100 events Cut&Count signal = 194 +_ 39 events Significance = 5.0 Expected significance = x Distribution is normal here, so P = (two-tail). Note: H 0 is flat distribution with no signal. H 1 is N sig 0. 5 Frank Porter, BaBar analysis school Statistics, February 2008

6 Note: The significance is not obtained by dividing the signal estimate (194) by the uncertainty in the signal (39), 194/39 = 5.0. Thatwould 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, February 2008

7 What about Systematic Uncertainties? B(e + e Nobel prize) = 10 ± 1 ± 5. They may be important! Maybethe±5is a systematic uncertainty in the estimate of the background expectation. A 10σ statistical significance is really only a 2σ effect. They may be irrelevant Maybethe±5is 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, February 2008

8 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, February 2008

9 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. Factoris 1+ m 2 i,wherem i 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, ŷ. y 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 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. x 2 / Frank Porter, BaBar analysis school Statistics, February 2008

10 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%! x 10 Frank Porter, BaBar analysis school Statistics, February 2008

11 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., (two-tailed probability). P = P ( x > 5), for x N (0, 1) = 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. [ From: L 0 (θ = 0;x) = 1 exp 1 2πσ 2 giving 2Δ ln L = Δχ 2 = x/σ = n. ( x/σ ) 2 ], L max ( θ = x; x) = 1 2πσ, 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, February 2008

12 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, February 2008

13 The Improbable Tails Our earlier example was known to be normal sampling. Events/ Generated signal = 100 events Cut&Count signal = 194 +_ 39 events Significance = 5.0 Expected significance = x 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, February 2008

14 Nuisance parameters Systematic Unknowns 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, February 2008

15 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, February 2008

16 Stopping Problem: Normal likelihood function example Consider an experiment in which a measurement of a parameter θ corresponds to sampling from a Gaussian distribution of standard deviation one: N (x; θ, 1)dx = 1 2π e (x θ)2 /2 dx. Frequency θ 2 θ θ+2 x Suppose the experimenter has a prejudice that θ is greater than one. Subconsciously, s/he makes measurements until the sample mean, m = 1 n ni=1 x i, is greater than one, or until s/he becomes convinced (or tired) after a maximum of N measurements. The experimenter then uses the sample mean, m, toestimateθ. 16 Frank Porter, BaBar analysis school Statistics, February 2008

17 Stopping Problem: Normal likelihood function example For illustration, assume that N =2. In terms of the random variables m and n, thepdfis: f(m, n; θ) = 1 2π e 1 2 (m θ)2, n =1,m>1 0, n =1,m<1 1 1 π e (m θ)2 e (x m)2 dx n =2 Number of experiments Histogram of sampling distribution for m, withpdf given by above equation, for θ = 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, som is the maximum likelihood estimator for θ. 17 Frank Porter, BaBar analysis school Statistics, February 2008

18 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 ) Straight line is probability for a 4 σ fluctuation of a normal distrinbution θ Frank Porter, BaBar analysis school Statistics, February 2008

19 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, som 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: 0.75 Probability (θ - <θ<θ + ) θ 19 Frank Porter, BaBar analysis school Statistics, February 2008

20 Stopping Problem: Normal likelihood function example The experimenter in this scenario thinks s/he is taking n samples from a normal distribution, and makes probability statements (e.g., about significance) according to a normal distribution. S/he gets an erroneous result because of the mistake in the distribution. If the experimenter realizes that sampling was actually from a nonnormal distribution, s/he can do a more careful analysis to obtain more valid results. Related effect: First observations tend to be biased high. Example? BES early f D = 371 vs f D 220 MeV more recently. Nothing wrong with this, but best estimates should average in earlier null data. Importance of reporting null results and quoting combinable results, e.g., two-sided confidence intervals. 20 Frank Porter, BaBar analysis school Statistics, February 2008

21 The Exploratory Bump Hunt Context is the typical bump-hunting approach in HEP That is, analysis is developed concurrent with looking at the data We see a bump, how significant is it? Events / 20 MeV/c BaBar e + e γ ISR π + π J/ψ P (H 0 ) m(π + π - J/ψ) (GeV/c ) If the analysis is our typical bump-hunt approach, it is impossible to compute the significance! We don t know the sampling distribution (under null hypothesis). Hence, we cannot compute probabilities (under null hypothesis, in particular). 21 Frank Porter, BaBar analysis school Statistics, February 2008

22 Recall our earlier example: Bump Hunting Example I Flat normal background (avg 100/bin) + gaussian signal (100 events) with mean 0, σ = Events/ Generated signal = 100 events Cut&Count signal = 194 +_ 39 events Significance = 5.0 Expected significance = x Distribution is normal here, so P = (two-tail). Note: H 0 is flat distribution with no signal. H 1 is N sig 0. As long as we knew at the outset that we were looking for an effect with mean 0 and σ =1, we make correct inferences. 22 Frank Porter, BaBar analysis school Statistics, February 2008

23 Looking for a mass bump. Bump Hunting Example II 160 Evidence for a threshold enhancement! 140 Events/10 MeV Background level = 100 events/bin Signal = 40 +_ 12 events Statistical significance = 4.0σ One-tailed probability (P-value) = _ 5 X _ m(pp) - 2m(p) (MeV) 23 Frank Porter, BaBar analysis school Statistics, February 2008

24 So, what does it mean? Even though we cannot estimate probabilities, we still quote numbers intended to relay an idea of significance, usually quoted as some number of sigma. What we (usually) mean by this is: If I had done the analysis in a controlled manner, and had been interested in the observed value for the mean and width, then the null hypothesis would require a fluctuation of this number of standard deviations of a Gaussian distribution to produce a bump as large as I see 24 Frank Porter, BaBar analysis school Statistics, February 2008

25 Is it useful? With this understanding of the meaning, it is perhaps not completely useless, since we can interpret it in the context of our experience. However, our experience is that we are sometimes fooled! pentaquarks example Life is short, we have also made great discoveries with this approach. Just remember that quoted significances are highly misleading. Note: The threshold enhancement in Example II is a statistical fluctuation! The sampling distribution was the same N (100, 10) for all bins. 160 Evidence for a threshold enhancement! 140 Events/10 MeV Background level = 100 events/bin Signal = 40 +_ 12 events Statistical significance = 4.0σ One-tailed probability (P-value) = _ 5 X _ m(pp) - 2m(p) (MeV) 25 Frank Porter, BaBar analysis school Statistics, February 2008

26 Avoiding pitfalls Model those tails! Conservatism Based on our experience with the failings of our methodology, we don t claim new discoveries lightly. Do it better: Be Blind 26 Frank Porter, BaBar analysis school Statistics, February 2008

27 Itcanbedone better Better means with more meaningful significance estimates, and avoidance of bias. Do a blind analysis Blind means you design the experiment (analysis) before you look at the results. Goal is to know the sampling distribution. BaBar routinely blinds it s analyses, when there is a well-defined quantity (e.g., CP asymmetry or branching fraction) being measured. 27 Frank Porter, BaBar analysis school Statistics, February 2008

28 Blinding Methodologies There are several basic approaches to blinding an analysis, which may be chosen according to the problem. Many variations on these themes! Don t look inside the box You can look, but keep the answer hidden Obscure the real data (e.g., adding simulated signal to data) Design on a dataset that will be thrown away Divide and conquer (e.g., BNL muon g 2 result, ω p (magnetic field) and ω a (muon precession) were analyzed independently, before combining to obtain g 2; arxiv:hep v3) [Reference: Klein & Roodman, Ann.Rev.Nucl.Part.Sci. 55 (2005) 141.] 28 Frank Porter, BaBar analysis school Statistics, February 2008

29 Blind Analysis Many BaBar results are obtained in blind analyses. avoid introduction of bias. Purpose is to Not all: Exploratory analyses not blind, eg, discovery of the DsJ (2317)+. Example of a blind analysis: B ± K ± e + e After event selection, look for signal in distribution of two kinematic variables ΔE and m ES. Monte Carlo, control sample (usually a type of data resembling signal) studies of entire sideband and fit region. Data only looked at in large sideband region prior to unblinding. Δ E (GeV) B + K + e + e - Signal MC signal region large sideband fit region mes (GeV/c 2 ) Data after unblinding 29 Frank Porter, BaBar analysis school Statistics, February 2008

30 Blind Analysis (continued) Issue: Updating a blind analysis with additional data. Simply add data, no change in analysis. May be impractical, undesirable, eg, re-reconstruction of entire dataset; improvements in tools such as particle identification. Sometimes would like to work harder to optimize analysis, or optimize on different criterion (eg, for most precision instead of most sensitivity). Notion of reblinding and re-optimization. Considered safe to use variables which have not been inspected too carefully in blind region. Not called a blind analysis. Pure approach: Do a blind analysis only on the new dateset. OK to use whatever was learned from the old dataset. Can combine the results from the old and the new data. 30 Frank Porter, BaBar analysis school Statistics, February 2008

31 Blind Analysis Hiding the answer In this approach, no data is hidden, just the answer (typically, a number). For example, a hidden, possibly random, offset may be applied to the real answer to prevent it from being seen before the analysis is designed. Δt(Hidden) = ( ) 1 1 (TAG)Δt + Offset, Top: Not hidden; Bottom: Hidden BaBar, from Klein& Roodman, op. cit. where TAG = ±1 according to the B flavor tag. This algorithm permits viewing the decay time distributions without revealing the asymmetry. 31 Frank Porter, BaBar analysis school Statistics, February 2008

32 Blinding a bump hunt No different in principle than our other blind analyses. The more you know (about the signal you are interested in), the better you can do. Yes, it is harder. It takes discipline, and it might even cost sensitivity. But it is at least worth thinking about. 32 Frank Porter, BaBar analysis school Statistics, February 2008

33 What cuts do I make? Again, the more we know, the better off we are! If we have good simulation of background processes, or background data, that s great! We ll use it. The less we know, the bigger the risk that we won t be optimal. That s life. Being non-optimal doesn t mean wrong. One approach: Sacrifice a portion of data for cut optimization. Ignore any signals. Or tune on them... If signal is real, it should show up in remaining blinded data. If it is a fluctuation, it will disappear. Events/10 MeV Evidence for a threshold enhancement! Background level = 100 events/bin Signal = 40 +_ 12 events Statistical significance = 4.0σ One-tailed probability (P-value) = _ 5 X _ m(pp) - 2m(p) (MeV) Systematic effects will show up both places. But that s another matter. 33 Frank Porter, BaBar analysis school Statistics, February 2008

34 Significance/GOF: Counting Degrees of Freedom The following situation arises with some frequency (with variations): I do two fits to the same dataset (say a histogram with N bins): Fit A has n A parameters, with χ 2 A [or perhaps 2lnL A]. Fit B has a subset n B of the parameters in fit A, withχ 2 B,wherethe n A n B other parameters (call them θ) are fixed at zero. What is the distribution of χ 2 B χ2 A? See SWiG thread: Luc Demortier in: [this is recommended reading] 34 Frank Porter, BaBar analysis school Statistics, February 2008

35 Counting Degrees of Freedom (continued) In the asymptotic limit (that is, as long as the normal sampling distribution is a valid approximation), Δχ 2 χ 2 B χ2 A is the same as a likelihood ratio ( 2lnλ) statistic for the test: H 0 : θ =0 against H 1 : some θ = 0 Δχ 2 is distributed according to a χ 2 (NDOF = n A n B ) distribution as long as: Parameter estimates in λ are consistent (converge to the correct values) under H 0. Parameter values in the null hypothesis are interior points of the maintained hypothesis (union of H 0 and H 1 ). There are no additional nuisance parameters under the alternative hypothesis. 35 Frank Porter, BaBar analysis school Statistics, February 2008

36 Counting Degrees of Freedom (continued) A commonly-encountered situation violates these requirements: We compare fits with and without a signal component to estimate significance of the signal. If the signal fit has, e.g., a parameter for mass, this constitutes an additional nuisance parameter under the alternative hypothesis. If the fit for signal constrains the yield to be non-negative, this violates the interior point requirement. Frequency H0: Fit with no signal H1: Float signal yield Curve: χ 2 (1 dof) Frequency H0: Fit with no signal H1: Float signal yield, constrained _> 0 Curve: χ 2 (1 dof) Frequency H0: Fit with no signal H1: Float signal yield and location Blue: χ 2 (1 dof) Red: χ 2 (2 dof) Δχ (H0-H1) Δχ (H0-H1) Δχ (H0-H1) 36 Frank Porter, BaBar analysis school Statistics, February 2008

37 Counting Degrees of Freedom - Sample fits H0 fit H1 fit H0 fit H1 fit Counts/bin H1 fit Any signal Fix mean H1 fit Counts/bin H1 fit Any signal Fix mean H1 fit Signal>=0 Fix mean x Any signal Float mean x Signal>=0 Fix mean x Any signal Float mean x Fits to data with no signal. Fits to data with a signal. 37 Frank Porter, BaBar analysis school Statistics, February 2008

38 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, February 2008

39 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. χ 2 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, February 2008

40 Goodness-of-Fit (continued) But recognize when test may not answer desired question, eg, in sin 2β analysis, a likelihood ratio (or a χ 2 ) test on the time distribution may have little sensitivity to testing goodness-of-fit of the asymmetry. Entries / 0.6 ps B 0 tags B 0 tags Background 50 Raw Asymmetry Δt (ps) 40 Frank Porter, BaBar analysis school Statistics, February 2008

41 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 θ 1, θ 2 from a bivariate normal distribution N(θ, σ 1,σ 2,ρ), with θ 1 = θ 2 = θ, the difference Δθ θ 2 θ 1 is N(0,σ)-distributed with σ 2 = σ σ2 2 2ρσ 1σ 2. If the correlation is unknown, all we can say is that the variance of the difference is in the range (σ 1 σ 2 ) 2...(σ 1 + σ 2 ) 2. If we at least believe ρ 0 then the maximum variance of the difference is σ1 2 + σ Frank Porter, BaBar analysis school Statistics, February 2008

42 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 x i,i =1,...,10. Assume the sampling distribution for x i is N(m, σ i ). We may form unbiased estimator, m 1,form: m 1 = 1 n ni=1 x i ± 1 n 2 ni=1 σ 2 i. The result (from a MC) is m 1 =0.058 ± Then we notice that we have some further information which might be useful: we know the experimental resolutions, σ i for each measurement. We form another unbiased estimator, m 2,form: m 2 = ni=1 x i σ 2 i ni=1 1 σ 2 i ± 1 ni=1 1 σ 2 i. The result (from the same MC) is m 1 =0.000 ± Frank Porter, BaBar analysis school Statistics, February 2008

43 Example continued The results are certainly correlated, so question of consistency arises (we know the error on the difference is between and 0.055). In this example, the difference between the results is ± 0.036, wherethe error includes the correlation (ρ =0.41). 43 Frank Porter, BaBar analysis school Statistics, February 2008

44 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 L 1, L 2,on the same set of events [nb, may use different information in each analysis]. The results are estimators θ 1, θ 2 for parameter θ. The correlation coefficient ρ may be estimated according to: ρ ( N N i=1 R i d ln L 1i i=1 dθ d ln L 2i θ= θ1 dθ θ= θ2 d 2 ln L 1i dθ 2 θ=θ0 )( N i=1 d 2 ln L 2i dθ 2 θ=θ0 ), where (θ 0 is an expansion reference point) R i = [ 1 ( θ 1 θ 0 ) d2 ln L 1i dθ 2 If θ 0 θ 1 θ 2,then / ][ d ln L1i θ=θ0 dθ θ= θ0 1 ( θ 2 θ 0 ) d2 ln L 2i dθ 2 / ] d ln L2i θ=θ0 dθ θ= θ0. ρ σ θ1 σ θ2 N i=1 d ln L 1i dθ d ln L 2i θ= θ0 dθ θ= θ0, where σ θ 2 k 1/ ( ) 2 N dlki i=1 dθ θ=θ Frank Porter, BaBar analysis school Statistics, February 2008

45 Consistency Example: sin 2β B B pairs PRL, vol 87, 27 August 2001: sin 2β =0.59 ± 0.14(stat) ± 0.05(syst) B B pairs SLAC-PUB-9153, March 2002: 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, February 2008