Practical Statistics for Particle Physics
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Estimating Confidence Regions of Common Measures of (Baseline, Treatment Effect) On
Estimating confidence regions of common measures of (baseline, treatment effect) on dichotomous outcome of a population Li Yin1 and Xiaoqin Wang2* 1Department of Medical Epidemiology and Biostatistics, Karolinska Institute, Box 281, SE- 171 77, Stockholm, Sweden 2Department of Electronics, Mathematics and Natural Sciences, University of Gävle, SE-801 76, Gävle, Sweden (Email: [email protected]). *Corresponding author Abstract In this article we estimate confidence regions of the common measures of (baseline, treatment effect) in observational studies, where the measure of baseline is baseline risk or baseline odds while the measure of treatment effect is odds ratio, risk difference, risk ratio or attributable fraction, and where confounding is controlled in estimation of both baseline and treatment effect. To avoid high complexity of the normal approximation method and the parametric or non-parametric bootstrap method, we obtain confidence regions for measures of (baseline, treatment effect) by generating approximate distributions of the ML estimates of these measures based on one logistic model. Keywords: baseline measure; effect measure; confidence region; logistic model 1 Introduction Suppose that one conducts a randomized trial to investigate the effect of a dichotomous treatment z on a dichotomous outcome y of certain population, where = 0, 1 indicate the 푧 1 active respective control treatments while = 0, 1 indicate positive respective negative outcomes. With a sufficiently large sample,푦 covariates are essentially unassociated with treatments z and thus are not confounders. Let R = pr( = 1 | ) be the risk of = 1 given 푧 z. Then R is marginal with respect to covariates and thus푦 conditional푧 on treatment푦 z only, so 푧 R is also called marginal risk. -
Theory Pest.Pdf
Theory for PEST Users Zhulu Lin Dept. of Crop and Soil Sciences University of Georgia, Athens, GA 30602 [email protected] October 19, 2005 Contents 1 Linear Model Theory and Terminology 2 1.1 Amotivationexample ...................... 2 1.2 General linear regression model . 3 1.3 Parameterestimation. 6 1.3.1 Ordinary least squares estimator . 6 1.3.2 Weighted least squares estimator . 7 1.4 Uncertaintyanalysis ....................... 9 1.4.1 Variance-covariance matrix of βˆ and estimation of σ2 . 9 1.4.2 Confidence interval for βj ................ 9 1.4.3 Confidence region for β ................. 10 1.4.4 Confidence interval for E(y0) .............. 10 1.4.5 Prediction interval for a future observation y0 ..... 11 2 Nonlinear regression model 13 2.1 Linearapproximation. 13 2.2 Nonlinear least squares estimator . 14 2.3 Numericalmethods ........................ 14 2.3.1 Steepest Descent algorithm . 16 2.3.2 Gauss-Newton algorithm . 16 2.3.3 Levenberg-Marquardt algorithm . 18 2.3.4 Newton’smethods . 19 1 2.4 Uncertainty analysis . 22 2.4.1 Confidence intervals for parameter and model prediction 22 2.4.2 Nonlinear calibration-constrained method . 23 3 Miscellaneous 27 3.1 Convergence criteria . 27 3.2 Derivatives computation . 28 3.3 Parameter estimation of compartmental models . 28 3.4 Initial values and prior information . 29 3.5 Parametertransformation . 30 1 Linear Model Theory and Terminology Before discussing parameter estimation and uncertainty analysis for nonlin- ear models, we need to review linear model theory as many of the ideas and methods of estimation and analysis (inference) in nonlinear models are essentially linear methods applied to a linear approximate of the nonlinear models. -
Random Vectors
Random Vectors x is a p×1 random vector with a pdf probability density function f(x): Rp→R. Many books write X for the random vector and X=x for the realization of its value. E[X]= ∫ x f.(x) dx = µ Theorem: E[Ax+b]= AE[x]+b Covariance Matrix E[(x-µ)(x-µ)’]=var(x)=Σ (note the location of transpose) Theorem: Σ=E[xx’]-µµ’ If y is a random variable: covariance C(x,y)= E[(x-µ)(y-ν)’] Theorem: For constants a, A, var (a’x)=a’Σa, var(Ax+b)=AΣA’, C(x,x)=Σ, C(x,y)=C(y,x)’ Theorem: If x, y are independent RVs, then C(x,y)=0, but not conversely. Theorem: Let x,y have same dimension, then var(x+y)=var(x)+var(y)+C(x,y)+C(y,x) Normal Random Vectors The Central Limit Theorem says that if a focal random variable x consists of the sum of many other independent random variables, then the focal random variable will asymptotically have a 2 distribution that is basically of the form e−x , which we call “normal” because it is so common. 2 ⎛ x−µ ⎞ 1 − / 2 −(x−µ) (x−µ) / 2 1 ⎜ ⎟ 1 2 Normal random variable has pdf f (x) = e ⎝ σ ⎠ = e σ 2πσ2 2πσ2 Denote x p×1 normal random variable with pdf 1 −1 f (x) = e−(x−µ)'Σ (x−µ) (2π)p / 2 Σ 1/ 2 where µ is the mean vector and Σ is the covariance matrix: x~Np(µ,Σ). -
The Likelihood Principle
1 01/28/99 ãMarc Nerlove 1999 Chapter 1: The Likelihood Principle "What has now appeared is that the mathematical concept of probability is ... inadequate to express our mental confidence or diffidence in making ... inferences, and that the mathematical quantity which usually appears to be appropriate for measuring our order of preference among different possible populations does not in fact obey the laws of probability. To distinguish it from probability, I have used the term 'Likelihood' to designate this quantity; since both the words 'likelihood' and 'probability' are loosely used in common speech to cover both kinds of relationship." R. A. Fisher, Statistical Methods for Research Workers, 1925. "What we can find from a sample is the likelihood of any particular value of r [a parameter], if we define the likelihood as a quantity proportional to the probability that, from a particular population having that particular value of r, a sample having the observed value r [a statistic] should be obtained. So defined, probability and likelihood are quantities of an entirely different nature." R. A. Fisher, "On the 'Probable Error' of a Coefficient of Correlation Deduced from a Small Sample," Metron, 1:3-32, 1921. Introduction The likelihood principle as stated by Edwards (1972, p. 30) is that Within the framework of a statistical model, all the information which the data provide concerning the relative merits of two hypotheses is contained in the likelihood ratio of those hypotheses on the data. ...For a continuum of hypotheses, this principle -
Statistical Theory
Statistical Theory Prof. Gesine Reinert November 23, 2009 Aim: To review and extend the main ideas in Statistical Inference, both from a frequentist viewpoint and from a Bayesian viewpoint. This course serves not only as background to other courses, but also it will provide a basis for developing novel inference methods when faced with a new situation which includes uncertainty. Inference here includes estimating parameters and testing hypotheses. Overview • Part 1: Frequentist Statistics { Chapter 1: Likelihood, sufficiency and ancillarity. The Factoriza- tion Theorem. Exponential family models. { Chapter 2: Point estimation. When is an estimator a good estima- tor? Covering bias and variance, information, efficiency. Methods of estimation: Maximum likelihood estimation, nuisance parame- ters and profile likelihood; method of moments estimation. Bias and variance approximations via the delta method. { Chapter 3: Hypothesis testing. Pure significance tests, signifi- cance level. Simple hypotheses, Neyman-Pearson Lemma. Tests for composite hypotheses. Sample size calculation. Uniformly most powerful tests, Wald tests, score tests, generalised likelihood ratio tests. Multiple tests, combining independent tests. { Chapter 4: Interval estimation. Confidence sets and their con- nection with hypothesis tests. Approximate confidence intervals. Prediction sets. { Chapter 5: Asymptotic theory. Consistency. Asymptotic nor- mality of maximum likelihood estimates, score tests. Chi-square approximation for generalised likelihood ratio tests. Likelihood confidence regions. Pseudo-likelihood tests. • Part 2: Bayesian Statistics { Chapter 6: Background. Interpretations of probability; the Bayesian paradigm: prior distribution, posterior distribution, predictive distribution, credible intervals. Nuisance parameters are easy. 1 { Chapter 7: Bayesian models. Sufficiency, exchangeability. De Finetti's Theorem and its intepretation in Bayesian statistics. { Chapter 8: Prior distributions. Conjugate priors. -
Statistical Data Analysis Stat 4: Confidence Intervals, Limits, Discovery
Statistical Data Analysis Stat 4: confidence intervals, limits, discovery London Postgraduate Lectures on Particle Physics; University of London MSci course PH4515 Glen Cowan Physics Department Royal Holloway, University of London [email protected] www.pp.rhul.ac.uk/~cowan Course web page: www.pp.rhul.ac.uk/~cowan/stat_course.html G. Cowan Statistical Data Analysis / Stat 4 1 Interval estimation — introduction In addition to a ‘point estimate’ of a parameter we should report an interval reflecting its statistical uncertainty. Desirable properties of such an interval may include: communicate objectively the result of the experiment; have a given probability of containing the true parameter; provide information needed to draw conclusions about the parameter possibly incorporating stated prior beliefs. Often use +/- the estimated standard deviation of the estimator. In some cases, however, this is not adequate: estimate near a physical boundary, e.g., an observed event rate consistent with zero. We will look briefly at Frequentist and Bayesian intervals. G. Cowan Statistical Data Analysis / Stat 4 2 Frequentist confidence intervals Consider an estimator for a parameter θ and an estimate We also need for all possible θ its sampling distribution Specify upper and lower tail probabilities, e.g., α = 0.05, β = 0.05, then find functions uα(θ) and vβ(θ) such that: G. Cowan Statistical Data Analysis / Stat 4 3 Confidence interval from the confidence belt The region between uα(θ) and vβ(θ) is called the confidence belt. Find points where observed estimate intersects the confidence belt. This gives the confidence interval [a, b] Confidence level = 1 - α - β = probability for the interval to cover true value of the parameter (holds for any possible true θ). -
Confidence Intervals for Functions of Variance Components Kok-Leong Chiang Iowa State University
Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 2000 Confidence intervals for functions of variance components Kok-Leong Chiang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Chiang, Kok-Leong, "Confidence intervals for functions of variance components " (2000). Retrospective Theses and Dissertations. 13890. https://lib.dr.iastate.edu/rtd/13890 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while ottiers may be f^ any type of computer printer. The quality of this reproduction is dependent upon the quality of ttw copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author dkl not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g.. maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overiaps. -
On the Relation Between Frequency Inference and Likelihood
On the Relation between Frequency Inference and Likelihood Donald A. Pierce Radiation Effects Research Foundation 5-2 Hijiyama Park Hiroshima 732, Japan [email protected] 1. Introduction Modern higher-order asymptotic theory for frequency inference, as largely described in Barn- dorff-Nielsen & Cox (1994), is fundamentally likelihood-oriented. This has led to various indica- tions of more unity in frequency and Bayesian inferences than previously recognised, e.g. Sweeting (1987, 1992). A main distinction, the stronger dependence of frequency inference on the sample space—the outcomes not observed—is isolated in a single term of the asymptotic formula for P- values. We capitalise on that to consider from an asymptotic perspective the relation of frequency inferences to the likelihood principle, with results towards a quantification of that as follows. Whereas generally the conformance of frequency inference to the likelihood principle is only to first order, i.e. O(n–1/2), there is at least one major class of applications, likely more, where the confor- mance is to second order, i.e. O(n–1). There the issue is the effect of censoring mechanisms. On a more subtle point, it is often overlooked that there are few, if any, practical settings where exact frequency inferences exist broadly enough to allow precise comparison to the likelihood principle. To the extent that ideal frequency inference may only be defined to second order, there is some- times—perhaps usually—no conflict at all. The strong likelihood principle is that in even totally unrelated experiments, inferences from data yielding the same likelihood function should be the same (the weak one being essentially the sufficiency principle). -
Monte Carlo Methods for Confidence Bands in Nonlinear Regression Shantonu Mazumdar University of North Florida
UNF Digital Commons UNF Graduate Theses and Dissertations Student Scholarship 1995 Monte Carlo Methods for Confidence Bands in Nonlinear Regression Shantonu Mazumdar University of North Florida Suggested Citation Mazumdar, Shantonu, "Monte Carlo Methods for Confidence Bands in Nonlinear Regression" (1995). UNF Graduate Theses and Dissertations. 185. https://digitalcommons.unf.edu/etd/185 This Master's Thesis is brought to you for free and open access by the Student Scholarship at UNF Digital Commons. It has been accepted for inclusion in UNF Graduate Theses and Dissertations by an authorized administrator of UNF Digital Commons. For more information, please contact Digital Projects. © 1995 All Rights Reserved Monte Carlo Methods For Confidence Bands in Nonlinear Regression by Shantonu Mazumdar A thesis submitted to the Department of Mathematics and Statistics in partial fulfillment of the requirements for the degree of Master of Science in Mathematical Sciences UNIVERSITY OF NORTH FLORIDA COLLEGE OF ARTS AND SCIENCES May 1995 11 Certificate of Approval The thesis of Shantonu Mazumdar is approved: Signature deleted (Date) ¢)qS- Signature deleted 1./( j ', ().-I' 14I Signature deleted (;;-(-1J' he Depa Signature deleted Chairperson Accepted for the College: Signature deleted Dean Signature deleted ~-3-9£ III Acknowledgment My sincere gratitude goes to Dr. Donna Mohr for her constant encouragement throughout my graduate program, academic assistance and concern for my graduate success. The guidance, support and knowledge that I received from Dr. Mohr while working on this thesis is greatly appreciated. My gratitude is extended to the members of my reviewing committee, Dr. Ping Sa and Dr. Peter Wludyka for their invaluable input into the writing of this thesis. -
1 the Likelihood Principle
ISyE8843A, Brani Vidakovic Handout 2 1 The Likelihood Principle Likelihood principle concerns foundations of statistical inference and it is often invoked in arguments about correct statistical reasoning. Let f(xjθ) be a conditional distribution for X given the unknown parameter θ. For the observed data, X = x, the function `(θ) = f(xjθ), considered as a function of θ, is called the likelihood function. The name likelihood implies that, given x, the value of θ is more likely to be the true parameter than θ0 if f(xjθ) > f(xjθ0): Likelihood Principle. In the inference about θ, after x is observed, all relevant experimental information is contained in the likelihood function for the observed x. Furthermore, two likelihood functions contain the same information about θ if they are proportional to each other. Remark. The maximum-likelihood estimation does satisfy the likelihood principle. Figure 1: Leonard Jimmie Savage; Born: November 20, 1917, Detroit, Michigan; Died: November 1, 1971, New Haven, Connecticut The following example quoted by Lindley and Phillips (1976) is an argument of Leonard Savage dis- cussed at Purdue Symposium 1962. It shows that the inference can critically depend on the likelihood principle. Example 1: Testing fairness. Suppose we are interested in testing θ, the unknown probability of heads for possibly biased coin. Suppose, H0 : θ = 1=2 v:s: H1 : θ > 1=2: 1 An experiment is conducted and 9 heads and 3 tails are observed. This information is not sufficient to fully specify the model f(xjθ): A rashomonian analysis follows: ² Scenario 1: Number of flips, n = 12 is predetermined. -
Model Selection by Normalized Maximum Likelihood
Myung, J. I., Navarro, D. J. and Pitt, M. A. (2006). Model selection by normalized maximum likelihood. Journal of Mathematical Psychology, 50, 167-179 http://dx.doi.org/10.1016/j.jmp.2005.06.008 Model selection by normalized maximum likelihood Jay I. Myunga;c, Danielle J. Navarrob and Mark A. Pitta aDepartment of Psychology Ohio State University bDepartment of Psychology University of Adelaide, Australia cCorresponding Author Abstract The Minimum Description Length (MDL) principle is an information theoretic approach to in- ductive inference that originated in algorithmic coding theory. In this approach, data are viewed as codes to be compressed by the model. From this perspective, models are compared on their ability to compress a data set by extracting useful information in the data apart from random noise. The goal of model selection is to identify the model, from a set of candidate models, that permits the shortest description length (code) of the data. Since Rissanen originally formalized the problem using the crude `two-part code' MDL method in the 1970s, many significant strides have been made, especially in the 1990s, with the culmination of the development of the refined `universal code' MDL method, dubbed Normalized Maximum Likelihood (NML). It represents an elegant solution to the model selection problem. The present paper provides a tutorial review on these latest developments with a special focus on NML. An application example of NML in cognitive modeling is also provided. Keywords: Minimum Description Length, Model Complexity, Inductive Inference, Cognitive Modeling. Preprint submitted to Journal of Mathematical Psychology 13 August 2019 To select among competing models of a psychological process, one must decide which criterion to use to evaluate the models, and then make the best inference as to which model is preferable. -
P Values, Hypothesis Testing, and Model Selection: It’S De´Ja` Vu All Over Again1
FORUM P values, hypothesis testing, and model selection: it’s de´ja` vu all over again1 It was six men of Indostan To learning much inclined, Who went to see the Elephant (Though all of them were blind), That each by observation Might satisfy his mind. ... And so these men of Indostan Disputed loud and long, Each in his own opinion Exceeding stiff and strong, Though each was partly in the right, And all were in the wrong! So, oft in theologic wars The disputants, I ween, Rail on in utter ignorance Of what each other mean, And prate about an Elephant Not one of them has seen! —From The Blind Men and the Elephant: A Hindoo Fable, by John Godfrey Saxe (1872) Even if you didn’t immediately skip over this page (or the entire Forum in this issue of Ecology), you may still be asking yourself, ‘‘Haven’t I seen this before? Do we really need another Forum on P values, hypothesis testing, and model selection?’’ So please bear with us; this elephant is still in the room. We thank Paul Murtaugh for the reminder and the invited commentators for their varying perspectives on the current shape of statistical testing and inference in ecology. Those of us who went through graduate school in the 1970s, 1980s, and 1990s remember attempting to coax another 0.001 out of SAS’s P ¼ 0.051 output (maybe if I just rounded to two decimal places ...), raising a toast to P ¼ 0.0499 (and the invention of floating point processors), or desperately searching the back pages of Sokal and Rohlf for a different test that would cross the finish line and satisfy our dissertation committee.