Don t forget the degrees of freedom: evaluating uncertainty from small numbers of repeated measurements

Save this PDF as:

Size: px
Start display at page:

Download "Don t forget the degrees of freedom: evaluating uncertainty from small numbers of repeated measurements"

Transcription

1 Don t forget the degrees of freedom: evaluating uncertainty from small numbers of repeated measurements Blair Hall Talk given via internet to the 35 th ANAMET Meeting, October 20, c Measurement Standards Laboratory of New Zealand / 11

2 Is there a need for accurate uncertainty statements about complex quantities when dealing with small numbers of repeated measurements? The GUM introduced a calculation of effective degrees of freedom to deal with the case of small numbers of repeat measurements. Effective degrees of freedom: extends a classical statistical concept needed to calculate coverage factors currently out of favour with metrology theorists GUM method does not apply to complex quantities There is an effective degrees of freedom calculation for complex problems too. No one seems to be using it. The notion of effective degrees of freedom (DoF) was introduced in the GUM. The basic tool for uncertainty propagation in the GUM is the Law of Propagation of Uncertainty (LPU), which evaluates a standard uncertainty for the measurement result. The LPU takes into account the standard uncertainties of all measurement influence quantities and the sensitivity of the measurement to each influence. In classical statistics, the degrees of freedom is a measure of how reliable the sample standard deviation is as an estimate of the population standard deviation. The GUM adopted this notion, and expanded it to help quantify how good a combined standard uncertainty is as an estimate of the standard deviation of the distribution of measurement errors. So, in addition to the LPU, there is a calculation in the GUM that propagates information about finite sample sizes. It is called the Welch-Satterthwaite formula and obtains an effective degrees of freedom for the result. That number, ν eff, is used to calculate a coverage factor for the expanded uncertainty. Without an accurate value for ν eff, the coverage probability of the uncertainty statement will be incorrect. c Measurement Standards Laboratory of New Zealand / 11

3 The GUM extended classical methods Suppose you wish to measure the physical quantity θ = µ 1 +µ 2 You can observe x 1 i = µ 1 +ε 1 i, x 2 i = µ 2 +ε 2 i where ε 1 i and ε 2 i are measurement errors in each observation. ε 1 i N(0,σ 1 ), ε 2 i N(0,σ 2 ) Samples of sizes n 1 and n 2 are taken. The sample means x 1, x 2 provide an estimate of θ y = x 1 +x 2 The standard error in the sample means estimates σ 1 n1 u(x 1 ), σ 2 n2 u(x 2 ) To understand the thinking behind effective degrees of freedom, we need to consider classical statistics. In this case, we imagine two independent noise sources that perturb observations of µ 1 and µ 2. These random errors have Gaussian distributions with zero mean. We do not know the respective variances, σ1 2 and σ2 2 : they are estimated from our observations. We collect n 1 observations for µ 1, and n 2 for µ 2. The sample statistics are s(x 1 ) 2 = 1 n 1 (n 1 1) x 1 = 1 n 1 x 1 i, x 2 = 1 n 2 n 1 n 2 n 1 i=1 (x 1 i x 1 ) 2, s(x 2 ) 2 = i=1 i=1 x 2 i 1 n 2 (n 2 1) n 2 (x 2 i x 2 ) 2 In the conventional GUM notation, we would write x 1, instead of x 1, and u(x 1 ) 2, instead of s(x 1 ) 2, etc. i=1 c Measurement Standards Laboratory of New Zealand / 11

4 ... extended GUM methods: Welch-Satterthwaite The standard uncertainty associated with our estimate of θ is u(y) = u(x 1 ) 2 +u(x 2 ) 2 The effective degrees of freedom ν eff are found (approximately) using the Welch- Satterthwaite formula u(y) 4 ν eff = u(x 1) 4 n u(x 2) 4 n 2 1, A number of degrees-of-freedom ν eff is an effective measure of the sample size. It is used to find a coverage factor k p, for a given level of confidence p. The expanded uncertainty is then U = k p u(y) y y U y +U The last step in which the degrees of freedom are evaluated is approximate. In this (textbook) problem the distribution of error in y is also Gaussian, but it is difficult to obtain an expression for the associated sampling distribution of u(y). If we had more than two influence quantities, then there is no known solution. The Welch-Satterthwaite formula is an approximate method that works well in most cases. The coverage factor is found from tables of the Student t-distribution k p = t α,νeff, with α = (1+p)/2 t α,ν is the 100α th percentile of the t-distribution with ν degrees of freedom c Measurement Standards Laboratory of New Zealand / 11

5 Errors and sampling The sample mean and standard deviation are not equal to the parameters of the population (actual distribution) sample population Y sample population The figures on the left illustrate the effect of sample size. The figures show two independent sets of 25 measurements of a quantity X. A random error with a standard deviation of 0.1 has been added to X to simulate measurement errors. In the top figure, the sample mean = and the sd = In the bottom figure, the sample mean = and the sd = Note that we can neither measure the value of the quantity, nor the amplitude of the error, exactly. The figure on the right shows how the variability of sampling statistics leads to occasional uncertainty statements (expanded uncertainties) that do not cover the measurand. The coverage probability indicates the frequency, over many independent measurements, with which uncertainty statements will cover the measurand. c Measurement Standards Laboratory of New Zealand / 11

6 Yes, Welch-Satterthwaite does work! In the GUM, a number of effective degrees of freedom is needed to calculate the expanded uncertainty for a given coverage probability. An alternative, propagation of distribution (PoD) method does not use DoF. This is currently favoured by many metrology theoreticians and is described in the BIPM GUM supplements. Proponents of PoD assert its superiority. However, the view that PoD is correct by definition, which is held by many, leads to confusion among practitioners. The confusion stems from the lack of a clear, commonly held, understanding of coverage probability. Studies [1,2] show WS performing better than PoD (always) when coverage probability is interpreted in terms of many independent experiments. [1] H Tanaka and K Ehara, Validity of expanded uncertainties evaluated using the Monte Carlo Method, in Advanced Mathematical and Computational Tools in Metrology and Testing (World Scientific, 2009) pp [2] B D Hall and R Willink, Does Welch-Satterthwaite make a good uncertainty estimate?, Metrologia 38 (2001) pp The PoD method is often referred to as the Monte Carlo Method. There are, however, many ways of using Monte Carlo methods for numerical computations. In fact, the implementation of PoD, as described in the GUM supplements, uses a Monte Carlo method. The GUM approach to uncertainty calculation assumes that the underlying distribution of error in a measurement result is Gaussian. Then, it uses the effective DoF and the Student t-distribution to obtain a coverage factor k, which scales the extent of the expanded uncertainty. Clearly, the Gaussian assumption will not always hold. However, in most well-designed measurement scenarios it is reasonable. That is why the GUM approach is so useful. Some practitioners have expressed concern that the expanded uncertainty calculated by GUM methods was incorrect. References in [2] are an early example of this. However, such concerns do not seem to be well-considered. Coverage probability has to be thought of as the long-run-average rate for uncertainty intervals to cover the measurand in repeated and independent experiments. While this criterion cannot readily be verified in practical measurements, it is the only reasonable interpretation of coverage probability that has been given to date. c Measurement Standards Laboratory of New Zealand / 11

7 Complex uncertainty regions The uncertainty region is an ellipse. It s orientation and shape depend on the standard uncertainties in the real and imaginary components u(x re ), u(x im ) and on the correlation r between x re and x im The size of the ellipse is set by a two-dimensional coverage factor k 2,p, which is a function of the degrees of freedom. imag k 2,p u(x im ) x im φ k 2,p u(x re ) real x re The standard uncertainties in the real and imaginary components are equal to the square root of the diagonal terms in the covariance matrix. The degrees of freedom associated with the standard uncertainties in the real and imaginary components will be the same. The principal axis lengths are not, in general, proportional to u(x re ) and u(x im ). The length of the axes is however proportional to the eigenvalues of the covariance matrix. c Measurement Standards Laboratory of New Zealand / 11

8 Level of Confidence A coverage factor can be calculated from ν and the coverage probability p k 2 2,p(ν) = 2ν ν 1 F 2,ν 1(p), F 2,ν 1 (p) is the upper 100p th percentile of the F-distribution ν is a measure of the sample size (c.f. ν = N 1) when ν is infinite, k 2 2,p( ) = χ 2 2,p Complex coverage factors k 2,p are not the same as the k p used in real uncertainty statements: ν k 2,0.95 k 0.95 ν k 2,0.95 k A useful relation in the infinite dof case is χ 2 2,p = 2log e (1 p) It is worth emphasizing the fact that the area of the uncertainty region is scaled by k 2,p squared. So the coverage factor is important when ν is small. What is the effect of ignoring the degrees of freedom completely? If an uncertainty statement is evaluated using the coverage factor for infinite degrees of freedom, when in fact there are finite degrees of freedom, then the actual coverage probability will be less then nominal. For example, this table shows how actual coverage probability falls off if the coverage factors k p = 1.96 and k 2,p = 2.45 are used to evaluate 95% expanded uncertainty instead of correct values for three sample sizes (ν = 3,5,10). ν coverage (real) coverage (complex) 3 85% 66% 5 89% 78% 10 92% 88% Clearly, it is important to take the degrees of freedom into account when calculating a coverage factor. This is even more important for uncertainty statements about complex quantities. c Measurement Standards Laboratory of New Zealand / 11

9 Effective degrees of freedom There is a method for calculating ν eff in the complex case [1]. As in the GUM (WS) method, there are some restrictions: Correlation is allowed between real and imaginary components of individual influence quantities Correlation is not allowed between real and imaginary components of different influence quantities Tests show that the method provides good overall performance in terms of coverage [1] R Willink and B D Hall, A classical method for uncertainty analysis with multidimensional data, Metrologia 39 (2002) pp Reference 1 describes three methods for evaluating an effective degrees of freedom. The Total Variance method is preferred for propagation of uncertainty. An appendix to the paper shows how to apply the total variance method to the complex (bivariate) problem. There are some cases for which the Total Variance method did not perform well, but this is not unexpected. The Welch-Satterthwaite formula is also known to have problems in some cases. Coverage can fall well below nominal when there is a very small number of samples for one quantity and a very large number of samples for the other (e.g., n 1 = 3 and n 2 = ). In such cases, the mean coverage, over a wide range of population parameters, drops slightly below 95% (e.g., 93% for n 1 = 3 and n 2 =, which was the worst case in the study). However, the range of coverage observed becomes wide (for n 1 = 3 and n 2 =, the minimum coverage was 81% and the maximum 99%). In other words, there are a few populations with rather extreme covariance structures that are not handled well by the total variance calculation. c Measurement Standards Laboratory of New Zealand / 11

10 The PoD alternative The PoD method uses a form of bivariate t-distribution to model a quantity estimated from a small sample of measurements. There are problems with this approach: There is not a unique choice of t-distribution in the bivariate case One t-distribution does not provide 95% coverage in the simplest case [1] (one quantity). E.g, for n = 3, the mean coverage is approximately 77%, rising to approximately 92% for n = 8 Another type is correct for single quantities, but becomes very conservative when applied to the sum of two quantities [1] The analytic method gives significantly better performance in all cases [1] [1] B D Hall, Monte Carlo uncertainty calculations with small-sample estimates of complex quantities, Metrologia 43 (2006) pp ; B D Hall, Complex-value Monte Carlo uncertainty calculations with few observations, CPEM Digest (9-14 July 2006, Torino, Italy). Wikipedia may be consulted for references to multivariate t-distribution literature: Student distribution c Measurement Standards Laboratory of New Zealand / 11

11 Any discussion? Is there a need to evaluate uncertainty for small numbers of repeat measurements? Effective degrees of freedom will be required if calculations of uncertainty for complex quantities use the extended GUM (LPU) method. Is there any foreseeable need for this? If so, how will degrees of freedom be handled? Is it well-known that the 1D (WS) method does not apply (e.g., phase and magnitude uncertainies)? The GUM Supplements favour the PoD method, which does not always generate results compatible with the conventional notion of confidence level. Is this acceptable? Some advocates of PoD suggest that the actual coverage is irrelevant. Is that OK with you? Some colleagues feel it best to follow a common methodology, regardless of correctness (i.e., for consistency) As far as I know, few laboratories report the uncertainty of a complex quantity as a region of the complex plane (NPL, being an exception). For real quantities, many national labs report report uncertainties with a coverage factor of k = 2, suggesting that degrees of freedom is not considered to be a concern. Perhaps the situation for complex quantities will be similar. Is it understood that separate, independent, univariate statements of uncertainty, for instance in the magnitude and phase of a complex quantity, do not provide the same coverage as a bivariate uncertainty region? RF uncertainty calculations can become pretty complicated (e.g., VNA). So software will be used in future. If that software is designed according to current guidelines, the small-sample problem will not be addressed. Both software development and producing BIPM uncertainty guides are slow processes. If this issue is not addressed within current activities, it is likely to remain a problem for a long time. c Measurement Standards Laboratory of New Zealand / 11

ISO GUM, Uncertainty Quantification, and Philosophy of Statistics

ISO GUM, Uncertainty Quantification, and Philosophy of Statistics Gunnar Taraldsen NTNU and SINTEF January 21th 2015 1 2 Abstract and history In 1978, recognizing the lack of international consensus on

COMPARATIVE EVALUATION OF UNCERTAINTY TR- ANSFORMATION FOR MEASURING COMPLEX-VALU- ED QUANTITIES

Progress In Electromagnetics Research Letters, Vol. 39, 141 149, 2013 COMPARATIVE EVALUATION OF UNCERTAINTY TR- ANSFORMATION FOR MEASURING COMPLEX-VALU- ED QUANTITIES Yu Song Meng * and Yueyan Shan National

Chi-Square Distribution. is distributed according to the chi-square distribution. This is usually written

Chi-Square Distribution If X i are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable is distributed according to the chi-square distribution. This

4. Introduction to Statistics

Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

Uncertainty of Measurement Results

Quality Assurance and Quality Control Uncertainty of Measurement Results A measurement result is an information. Without knowledge of its uncertainty it is just a rumour. Albert Weckenmann Overview Uncertainty

Statistical Intervals. Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage

7 Statistical Intervals Chapter 7 Stat 4570/5570 Material from Devore s book (Ed 8), and Cengage Confidence Intervals The CLT tells us that as the sample size n increases, the sample mean X is close to

Hypothesis Testing. Chapter Introduction

Contents 9 Hypothesis Testing 553 9.1 Introduction............................ 553 9.2 Hypothesis Test for a Mean................... 557 9.2.1 Steps in Hypothesis Testing............... 557 9.2.2 Diagrammatic

On Small Sample Properties of Permutation Tests: A Significance Test for Regression Models

On Small Sample Properties of Permutation Tests: A Significance Test for Regression Models Hisashi Tanizaki Graduate School of Economics Kobe University (tanizaki@kobe-u.ac.p) ABSTRACT In this paper we

JCGM 101:2008. First edition 2008

Evaluation of measurement data Supplement 1 to the Guide to the expression of uncertainty in measurement Propagation of distributions using a Monte Carlo method Évaluation des données de mesure Supplément

E205 Final: Version B

Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random

Multivariate Normal Distribution

Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues

G104 - Guide for Estimation of Measurement Uncertainty In Testing. December 2014

Page 1 of 31 G104 - Guide for Estimation of Measurement Uncertainty In Testing December 2014 2014 by A2LA All rights reserved. No part of this document may be reproduced in any form or by any means without

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for

Market Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk

Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day

Estimation; Sampling; The T distribution

Estimation; Sampling; The T distribution I. Estimation A. In most statistical studies, the population parameters are unknown and must be estimated. Therefore, developing methods for estimating as accurately

Regression analysis in the Assistant fits a model with one continuous predictor and one continuous response and can fit two types of models:

This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. The simple regression procedure in the

Chapter 11: Two Variable Regression Analysis

Department of Mathematics Izmir University of Economics Week 14-15 2014-2015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions

Summary of Probability

Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible

Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

Regression in SPSS. Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology

Regression in SPSS Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology John P. Bentley Department of Pharmacy Administration University of

Calculating Interval Forecasts

Calculating Chapter 7 (Chatfield) Monika Turyna & Thomas Hrdina Department of Economics, University of Vienna Summer Term 2009 Terminology An interval forecast consists of an upper and a lower limit between

Testing: is my coin fair?

Testing: is my coin fair? Formally: we want to make some inference about P(head) Try it: toss coin several times (say 7 times) Assume that it is fair ( P(head)= ), and see if this assumption is compatible

Sample size and power calculations using the noncentral t-distribution

The Stata Journal (004) 4, Number, pp. 14 153 Sample size and power calculations using the noncentral t-distribution David A. Harrison ICNARC, London, UK david@icnarc.org Anthony R. Brady ICNARC, London,

Multiple Hypothesis Testing: The F-test

Multiple Hypothesis Testing: The F-test Matt Blackwell December 3, 2008 1 A bit of review When moving into the matrix version of linear regression, it is easy to lose sight of the big picture and get lost

Boutique AFNOR pour : JEAN-LUC BERTRAND-KRAJEWSKI le 16/10/2009 14:35. GUIDE 98-3/Suppl.1

Boutique AFNOR pour : JEAN-LUC BERTRAND-KRAJEWSKI le 16/10/009 14:35 ISO/CEI GUIDE 98-3/S1:008:008-1 GUIDE 98-3/Suppl.1 Uncertainty of measurement Part 3: Guide to the expression of uncertainty in measurement

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

CONFIDENCE INTERVALS ON µ WHEN σ IS UNKNOWN

CONFIDENCE INTERVALS ON µ WHEN σ IS UNKNOWN A. Introduction 1. this situation, where we do not know anything about the population but the sample characteristics, is far and away the most common circumstance

UNCERTAINTY AND CONFIDENCE IN MEASUREMENT

Ελληνικό Στατιστικό Ινστιτούτο Πρακτικά 8 ου Πανελληνίου Συνεδρίου Στατιστικής (2005) σελ.44-449 UNCERTAINTY AND CONFIDENCE IN MEASUREMENT Ioannis Kechagioglou Dept. of Applied Informatics, University

Quiz 1(1): Experimental Physics Lab-I Solution

Quiz 1(1): Experimental Physics Lab-I Solution 1. Suppose you measure three independent variables as, x = 20.2 ± 1.5, y = 8.5 ± 0.5, θ = (30 ± 2). Compute the following quantity, q = x(y sin θ). Find the

Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0

Probability & Statistics Primer Gregory J. Hakim University of Washington 2 January 2009 v2.0 This primer provides an overview of basic concepts and definitions in probability and statistics. We shall

Rutherford scattering: a simple model

Rutherford scattering: a simple model Scott D. Morrison MIT Department of Physics (Dated: December 10, 2008) We performed a Rutherford scattering experiment with gold foil and alpha particles emitted by

Fixed vs. Random Effects

Statistics 203: Introduction to Regression and Analysis of Variance Fixed vs. Random Effects Jonathan Taylor - p. 1/19 Today s class Implications for Random effects. One-way random effects ANOVA. Two-way

Two-Sample T-Test from Means and SD s

Chapter 07 Two-Sample T-Test from Means and SD s Introduction This procedure computes the two-sample t-test and several other two-sample tests directly from the mean, standard deviation, and sample size.

Meta-Analysis Fixed effect vs. random effects

Meta-Analysis Fixed effect vs. random effects Michael Borenstein Larry Hedges Hannah Rothstein www.meta-analysis.com 2007 Borenstein, Hedges, Rothstein 1 Section: Fixed effect vs. random effects models...4

Determining the Sample Size: One Sample

Sample Sie Laboratory Determining the Sample Sie: One Sample OVERVIEW In this lab, you will be determining the sample sie for population means and population proportions. This lab is intended to help you

Confidence Intervals for the Difference Between Two Means

Chapter 47 Confidence Intervals for the Difference Between Two Means Introduction This procedure calculates the sample size necessary to achieve a specified distance from the difference in sample means

Section 7.2 Confidence Intervals for Population Proportions

Section 7.2 Confidence Intervals for Population Proportions 2012 Pearson Education, Inc. All rights reserved. 1 of 83 Section 7.2 Objectives Find a point estimate for the population proportion Construct

The Method of Least Squares

33 The Method of Least Squares KEY WORDS confidence interval, critical sum of squares, dependent variable, empirical model, experimental error, independent variable, joint confidence region, least squares,

The Delta Method and Applications

Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and

Applications of the Central Limit Theorem

Applications of the Central Limit Theorem October 23, 2008 Take home message. I expect you to know all the material in this note. We will get to the Maximum Liklihood Estimate material very soon! 1 Introduction

MAT X Hypothesis Testing - Part I

MAT 2379 3X Hypothesis Testing - Part I Definition : A hypothesis is a conjecture concerning a value of a population parameter (or the shape of the population). The hypothesis will be tested by evaluating

AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

POL 571: Expectation and Functions of Random Variables

POL 571: Expectation and Functions of Random Variables Kosuke Imai Department of Politics, Princeton University March 10, 2006 1 Expectation and Independence To gain further insights about the behavior

The purpose of this section is to derive the asymptotic distribution of the Pearson chi-square statistic. k (n j np j ) 2. np j.

Chapter 7 Pearson s chi-square test 7. Null hypothesis asymptotics Let X, X 2, be independent from a multinomial(, p) distribution, where p is a k-vector with nonnegative entries that sum to one. That

15.1 Least Squares as a Maximum Likelihood Estimator

15.1 Least Squares as a Maximum Likelihood Estimator 651 should provide (i) parameters, (ii) error estimates on the parameters, and (iii) a statistical measure of goodness-of-fit. When the third item suggests

Calculating VaR. Capital Market Risk Advisors CMRA

Calculating VaR Capital Market Risk Advisors How is VAR Calculated? Sensitivity Estimate Models - use sensitivity factors such as duration to estimate the change in value of the portfolio to changes in

Hypothesis Testing in the Classical Regression Model

LECTURE 5 Hypothesis Testing in the Classical Regression Model The Normal Distribution and the Sampling Distributions It is often appropriate to assume that the elements of the disturbance vector ε within

Continuous Random Variables and Probability Distributions. Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage

4 Continuous Random Variables and Probability Distributions Stat 4570/5570 Material from Devore s book (Ed 8) Chapter 4 - and Cengage Continuous r.v. A random variable X is continuous if possible values

Why do we measure central tendency? Basic Concepts in Statistical Analysis

Why do we measure central tendency? Basic Concepts in Statistical Analysis Chapter 4 Too many numbers Simplification of data Descriptive purposes What is central tendency? Measure of central tendency A

Week 4: Standard Error and Confidence Intervals

Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

Joint IMEKO TC-1 & XXXIV MKM Conference 2002 Wrocław, 8-12 September 2002 CALCULATION OF EXPANDED UNCERTAINTY

Joint IMEKO TC-1 & XXXIV MKM Conference 00 Wrocław, 8-1 September 00 Jerzy M. KORCZYŃSKI * ndrzej HETMN CLCULTION OF EXPNDED UNCERTINTY The main purpose of the paper is to present a method of calculation

Standard Deviation Calculator

CSS.com Chapter 35 Standard Deviation Calculator Introduction The is a tool to calculate the standard deviation from the data, the standard error, the range, percentiles, the COV, confidence limits, or

Ch. 12 The Analysis of Variance. can be modelled as normal random variables. come from populations having the same variance.

Ch. 12 The Analysis of Variance Residual Analysis and Model Assessment Model Assumptions: The measurements X i,j can be modelled as normal random variables. are statistically independent. come from populations

Hypothesis Testing hypothesis testing approach formulation of the test statistic

Hypothesis Testing For the next few lectures, we re going to look at various test statistics that are formulated to allow us to test hypotheses in a variety of contexts: In all cases, the hypothesis testing

Meta-Analysis Fixed effect vs. random effects

Meta-Analysis Fixed effect vs. random effects Michael Borenstein Larry Hedges Hannah Rothstein www.meta-analysis.com 2007 Borenstein, Hedges, Rothstein 1 Section: Fixed effect vs. random effects models...

Statistics for Management II-STAT 362-Final Review

Statistics for Management II-STAT 362-Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

2. Describing Data. We consider 1. Graphical methods 2. Numerical methods 1 / 56

2. Describing Data We consider 1. Graphical methods 2. Numerical methods 1 / 56 General Use of Graphical and Numerical Methods Graphical methods can be used to visually and qualitatively present data and

A Summary of Error Propagation

A Summary of Error Propagation Suppose you measure some quantities a, b, c,... with uncertainties δa, δb, δc,.... Now you want to calculate some other quantity Q which depends on a and b and so forth.

General and statistical principles for certification of RM ISO Guide 35 and Guide 34

General and statistical principles for certification of RM ISO Guide 35 and Guide 34 / REDELAC International Seminar on RM / PT 17 November 2010 Dan Tholen,, M.S. Topics Role of reference materials in

Probability and Statistics Lecture 9: 1 and 2-Sample Estimation

Probability and Statistics Lecture 9: 1 and -Sample Estimation to accompany Probability and Statistics for Engineers and Scientists Fatih Cavdur Introduction A statistic θ is said to be an unbiased estimator

Regression, least squares

Regression, least squares Joe Felsenstein Department of Genome Sciences and Department of Biology Regression, least squares p.1/24 Fitting a straight line X Two distinct cases: The X values are chosen

2.3 Simple Random Sampling

2.3 Simple Random Sampling Simple random sampling without replacement (srswor) of size n is the probability sampling design for which a fixed number of n units are selected from a population of N units

Introduction to Statistics for Computer Science Projects

Introduction Introduction to Statistics for Computer Science Projects Peter Coxhead Whole modules are devoted to statistics and related topics in many degree programmes, so in this short session all I

EA guidelines on the expression. in quantitative testing

Publication Reference EA-4/16 G:2003 EA guidelines on the expression of uncertainty in quantitative testing PURPOSE The purpose of this document is to harmonise the evaluation of uncertainties associated

European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

THE STANDARD ERROR OF THE LAB SCIENTIST

THE STNDRD ERROR OF THE L SCIENTIST and other common statistical misconceptions in the scientific literature. Ingo Ruczinski Department of iostatistics, Johns Hopkins University Some Quotes Instead of

Outline. Correlation & Regression, III. Review. Relationship between r and regression

Outline Correlation & Regression, III 9.07 4/6/004 Relationship between correlation and regression, along with notes on the correlation coefficient Effect size, and the meaning of r Other kinds of correlation

Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

Ratchet Equity Indexed Annuities

Ratchet Equity Indexed Annuities Mary R Hardy University of Waterloo Waterloo Ontario N2L 3G1 Canada Tel: 1-519-888-4567 Fax: 1-519-746-1875 email: mrhardy@uwaterloo.ca Abstract The Equity Indexed Annuity

STATISTICS FOR PSYCH MATH REVIEW GUIDE

STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.

Software Support for Metrology Best Practice Guide No. 6. Uncertainty and Statistical Modelling M G Cox, M P Dainton and P M Harris

Software Support for Metrology Best Practice Guide No. 6 Uncertainty and Statistical Modelling M G Cox, M P Dainton and P M Harris March 2001 Software Support for Metrology Best Practice Guide No. 6 Uncertainty

Inference in Regression Analysis. Dr. Frank Wood

Inference in Regression Analysis Dr. Frank Wood Inference in the Normal Error Regression Model Y i = β 0 + β 1 X i + ɛ i Y i value of the response variable in the i th trial β 0 and β 1 are parameters

An Introduction to Bayesian Statistics

An Introduction to Bayesian Statistics Robert Weiss Department of Biostatistics UCLA School of Public Health robweiss@ucla.edu April 2011 Robert Weiss (UCLA) An Introduction to Bayesian Statistics UCLA

Bayesian Classification

CS 650: Computer Vision Bryan S. Morse BYU Computer Science Statistical Basis Training: Class-Conditional Probabilities Suppose that we measure features for a large training set taken from class ω i. Each

LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12,

LINEAR ALGEBRA: NUMERICAL METHODS. Version: August 12, 2000 45 4 Iterative methods 4.1 What a two year old child can do Suppose we want to find a number x such that cos x = x (in radians). This is a nonlinear

Comparing Two Populations OPRE 6301

Comparing Two Populations OPRE 6301 Introduction... In many applications, we are interested in hypotheses concerning differences between the means of two populations. For example, we may wish to decide

Chapter 7. One-way ANOVA

Chapter 7 One-way ANOVA One-way ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The t-test of Chapter 6 looks

Chapter Five. Hypothesis Testing: Concepts

Chapter Five The Purpose of Hypothesis Testing... 110 An Initial Look at Hypothesis Testing... 112 Formal Hypothesis Testing... 114 Introduction... 114 Null and Alternate Hypotheses... 114 Procedure for

3.6: General Hypothesis Tests

3.6: General Hypothesis Tests The χ 2 goodness of fit tests which we introduced in the previous section were an example of a hypothesis test. In this section we now consider hypothesis tests more generally.

Module 10: Mixed model theory II: Tests and confidence intervals

St@tmaster 02429/MIXED LINEAR MODELS PREPARED BY THE STATISTICS GROUPS AT IMM, DTU AND KU-LIFE Module 10: Mixed model theory II: Tests and confidence intervals 10.1 Notes..................................

Predictability of Average Inflation over Long Time Horizons

Predictability of Average Inflation over Long Time Horizons Allan Crawford, Research Department Uncertainty about the level of future inflation adversely affects the economy because it distorts savings

Factor Analysis. Chapter 420. Introduction

Chapter 420 Introduction (FA) is an exploratory technique applied to a set of observed variables that seeks to find underlying factors (subsets of variables) from which the observed variables were generated.

Least Squares Estimation

Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

Association Between Variables

Contents 11 Association Between Variables 767 11.1 Introduction............................ 767 11.1.1 Measure of Association................. 768 11.1.2 Chapter Summary.................... 769 11.2 Chi

1 Hypotheses test about µ if σ is not known

1 Hypotheses test about µ if σ is not known In this section we will introduce how to make decisions about a population mean, µ, when the standard deviation is not known. In order to develop a confidence

Robust Design: Statistical Analysis for Taguchi Methods

Robust Design: Statistical Analysis for Taguchi Methods In the previous lecture, we saw how to set up the design of a series of experiments to test different configurations of potential designs under different

Pooling and Meta-analysis. Tony O Hagan

Pooling and Meta-analysis Tony O Hagan Pooling Synthesising prior information from several experts 2 Multiple experts The case of multiple experts is important When elicitation is used to provide expert

Intro to Parametric Statistics,

Descriptive Statistics vs. Inferential Statistics vs. Population Parameters Intro to Parametric Statistics, Assumptions & Degrees of Freedom Some terms we will need Normal Distributions Degrees of freedom

(Uncertainty) 2. How uncertain is your uncertainty budget?

(Uncertainty) 2 How uncertain is your uncertainty budget? Paper Author and Presenter: Dr. Henrik S. Nielsen HN Metrology Consulting, Inc 10219 Coral Reef Way, Indianapolis, IN 46256 Phone: (317) 849 9577,

CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS

LESSON SEVEN CONFIDENCE INTERVALS FOR MEANS AND PROPORTIONS An interval estimate for μ of the form a margin of error would provide the user with a measure of the uncertainty associated with the point estimate.

Lecture 7: Binomial Test, Chisquare

Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two

Conventional And Robust Paired And Independent-Samples t Tests: Type I Error And Power Rates

Journal of Modern Applied Statistical Methods Volume Issue Article --3 Conventional And And Independent-Samples t Tests: Type I Error And Power Rates Katherine Fradette University of Manitoba, umfradet@cc.umanitoba.ca

Finite Population Sampling

Finite Population Sampling Fernando TUSELL May 2012 Outline Introduction Sampling of independent observations Sampling without replacement Stratified sampling Taking samples Introduction Sampling of independent

Determining Consensus Values in Interlaboratory Comparisons and Proficiency Testing

Determining Consensus Values in Interlaboratory Comparisons and Proficiency Testing 1. Abstract Speaker/Author: Dr. Henrik S. Nielsen HN Metrology Consulting, Inc. HN Proficiency Testing, Inc. Indianapolis,

Inferences About Differences Between Means Edpsy 580

Inferences About Differences Between Means Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Inferences About Differences Between Means Slide