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Goodness of fit, confidence intervals and limits Jorge Andre Swieca School Campos do Jordão, January,2003 fourth lecture

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References Statistical Data Analysis, G. Cowan, Oxford, 1998 Statistics, A guide to the Use of Statistical Methods in the Physical Sciences, R. Barlow, J. Wiley & Sons, 1989; Particle Data Group (PDG) Review of Particle Physics, 2002 electronic edition. Data Analysis, Statistical and Computational Methods for Scientists and Engineers, S. Brandt, Third Edition, Springer, 1999

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Limits Tens, como Hamlet, o pavor do desconhecido? Mas o que é conhecido? O que é que tu conheces, Para que chames desconhecido a qualquer coisa em especial? Álvaro de Campos (Fernando Pessoa) Se têm a verdade, guardem-na! Lisbon Revisited, Álvaro de Campos

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Statistical tests How well the data stand in agreement with given predicted probabilities – hypothesis. null hypothesis H 0 alternative function of measured variables: test statistics error first kind significance level power = error second kind power to discriminate against H 1

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Neyman-Pearson lemma Where to place t cut ? H 0 signal H 1 background 1-D: efficiency (and purity) m-D: def. of acceptance region is not obvious Neyman-Pearson lemma: highest power (highest signal purity) for a given significance level α region of t-space such that determined by the desired efficiency

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Goodness of fit how well a given null hypothesis H 0 is compatible with the observed data (no reference to other alternative hypothesis) coins: N tosses, n h, n t = N - n h coin fair? H and T equal? test statistic: n h binomial distribution, p=0.5 N=20, n h =17 E[n h ]=Np=10 0 1 2 3 17 18 19 20 10

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Goodness of fit P=0.0026 P-value: probability P, under H 0, obtain a result as compatible of less with H 0 than the one actually observed. P-value is a random variable, α is a constant specified before carrying out the test Bayesian statistics: use the Bayes theorem to assign a probability to H 0 (specify the prior probability) P value is often interpreted incorrectly as a prob. to H 0 P-value: fraction of times on would obtain data as compatible with H 0 or less so if the experiment (20 coin tosses) were repeated under similar circunstances

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Goodness of fit Easy to identify the region of values of t with equal or less degree of compatibility with the hypothesis than the observed value (alternate hypothesis: p 0.5) optional stopping problem

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Significance of an observed signal Whether a discrepancy between data and expectation is sufficiently significant to merit a claim for a new discovery signal event n s, Poisson variable ν S background event n b, Poisson variable ν b prob. to observe n events: experiment: n obs events, quantify our degree of confidence in the discovery of a new effect (ν S 0) How likely is to find n obs events or more from background alone?

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Significance of an observed signal Ex: expect ν b =0.5, n obs = 5 P(n>n obs )=1.7x10 -4 this is not the prob. of the hypothesis ν S =0 ! this is the prob., under the hypothesis ν S =0, of obtaining as many events as observed or more.

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Significance of an observed signal How to report the measurement? estimate of ν : misleading: only two std. deviations from zero impression that ν S is not very incompatible with zero yes:prob. that a Poisson variable of mean ν b will fluctuate up to n obs or higher no:prob. that a variable with mean n obs will fluctuate down to ν b or lower

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Pearsons test histogram of x with N bins niνiniνi construct a statistic which reflects the level of agreement between observed and expected histograms data aprox. gaussian, Poisson distributed with follow a distribution for N degrees of freedom regardless of the distribution of x distribution free larger larger discrepancy between data and the hypothesis

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Pearsons test (rule of thumb for a good fit)

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Pearsons test

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Before Poisson variable with Set n tot = fixed n i dist. as multinomial with prob. Not testing the total number of expected and observed Events, but only the distribution of x. large number on entries in each bin p i known Follows a distribution for N-1 degrees of freedom In general, if m parameters estimated from data, n d = N - m

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ML: estimator for θ Standard deviation as stat. error n observations of x, hypothesis p.d.f f(x;θ) analytic method RCF bound Monte Carlo graphical standard deviation measurement repeated estimates each based on n obs.: estimator dist. centered around true value θ and with true estimated by and Most practical estimators: becomes approx. Gaussian in the large sample limit.

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Classical confidence intervals n obs. of x, evaluate an estimator for a param. θ obtained and its p.d.f. (for a given θ unknown) prob. α prob. β

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Classical confidence intervals prob. for estimator to be inside the belt regardless of θ monotonic incresing functions of θ

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Classical confidence intervals Usually: central confidence interval a: hypothetical value of for which a fraction of the repeated estimt. would be higher than the obtain.

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Classical confidence intervals Relationship between a conf. interval and a test of goodness of fit: test the hypothesys using having equal or less agreement than the result obtained P-value = α (random variable) and θ = a is specified Confidence interval: α is specified first, a is a random quantity depending on the data

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Classical confidence intervals Many experiments: the interval would include the true value in It does not mean that the probability that the true value of is in the fixed interval is Frequency interpretation: is not a random variable, but the interval fluctuates since it is constructed from data.

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Gaussian distributed Simple and very important application Central limit theorem: any estimator linear function of sum of random variables becomes Gaussian in the large sample limit. known, experiment resulted in

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Gaussian distributed

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Choose quantile 10.682710.8413 20.954420.9772 30.997330.9987 Choose confidence level 0.901.6450.901.282 0.951.9600.951.645 0.992.5760.992.326.

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