Lecture 2: Statistical Estimation and Testing


 Bennett Barnett
 2 years ago
 Views:
Transcription
1 Bioinformatics: Indepth PROBABILITY AND STATISTICS Spring Semester 2012 Lecture 2: Statistical Estimation and Testing Stefanie Muff 1
2 Problems in statistics 2
3 The three main questions in statistics are Estimation: estimate the unknown value of θ, given observations of X. Question: what is the most likely value for θ? Testing: test a hypothesis about the unknown value of θ. Base acceptance/rejection upon observation of X. Question: is my hypothesis compatible with the observed data? Confidence intervals: give an interval of parameter values that explain the data reasonably well. Question: which parameters would be compatible with my data? We will concentrate on the first two questions. 3
4 4
5 Given: a probability model X P θ For example: X Bin(100,p) but the probability p is unknown. How to obtain a guess of p? => Estimation! The collection x1, x2,..., xn is called (observed) sample of X1, X2,..., Xn. 5
6 Estimator, estimate 6
7 Examples of Estimators 7
8 Desirable properties of estimators 8
9 9
10 Likelihood function for discrete RVs 10
11 Likelihood function for continuous RVs The likelihood function for continuous random variables can be set equal to the density function L(x 1,x 2,..., x n ; ˆθ) =f X (x 1,x 2,...,x n ; ˆθ), whereas f X is the joint density of (X 1,X 2,..., X n ). If X 1,X 2,..., X n are independent L(x 1,x 2,..., x n ; ˆθ) =f X1 (x 1 ; ˆθ) f X2 (x 2 ; ˆθ)...f Xn (x n ; ˆθ). 11
12 Maximum likelihood estimator 12
13 Maximum likelihood estimate 13
14 Properties of MLEs 14
15 Example: ML for the binomial distribution 15
16 16
17 Compare this to the estimators on Slide 7: the ML estimator! is 17
18 The Log Likelihood 18
19 Likelihoods are not just for independent observations! 19
20 Example: Log likelihood for the binomial distribution Instead of optimizing The log likelihood x 1 log(θ) + (100 x 1 ) log(1 θ) x n log(θ) + (100 x n ) log(1 θ) = log(θ) x i + log(1 θ) (100 x i ) i i has to be optimized to obtain the ML estimator. The result is exactly the same as in the nonlog case (check as an exercise). 20
21 Example: MLE for a normal distribution Remember: f(x, µ, σ 2 )= 1 (x µ)2 e 2σ 2 2πσ 2 Given a set of n independent observations x1, x2,...,xn.the log likelihood then is log(f(x 1,...,x n ; µ, σ 2 )) = n 2 log(2π) n 2 log(σ2 ) 1 2σ 2 n (x i µ) 2 i=1 This expression has to be derived with respect to σ 2 and µ separately and be set to 0. => Obtain two equations to estimate two parameters. See example in the exercises. 21
22 MLE in practice Analytical formulas for the ML estimator can be found only in relatively simple models. In other cases, approximate ML estimators can be found by iterative numerical optimization (ExpectationMaximization algorithm, Newton Raphson algorithm) secondorder Taylor approximations. These calculations are left to the computer (R). 22
23 Statistical Testing 23
24 24
25 Introductory example revisited g a g g a t t a c g g t a c t a g a t t c a t a a a c a c t g a c a c a t c a c t g c a c t c g c t a a Two DNA sequences of length 26. Matches at 11 of 26 positions. Is this sufficient to conclude that the two sequences are evolutionarily related? In order to answer this question, we have to find out how unlikely it would be to see 11 out of 26 matches by chance. Need to know the probability distribution of the random variable describing this experiment. Can then calculate the probability of the event. This is the essence of statistical testing. 25
26 Steps in a statistical test 1. Formulate null and alternative hypotheses H0 and H1. 2. Determine a test statistic T. 3. Determine the distribution of T under H0. 4. Choose the significance level α. 5. Calculate the critical value C. 6. Obtain the data and decide. For illustration, we now go through steps 16 for the binomal test. 26
27 1. Formulate the hypotheses A hypothesis typically specifies a value in a distribution. Here: X Bin(26, p), but p is not known. The null hypothesis H0 is the default hypothesis: H 0 : X Bin(26,p), p=0.25 The alternative hypothesis H1 is the controversial hypothesis. Strong evidence is needed to accept it in favour of H0: H 1 : X Bin(26,p), p > 0.25 Aim of a test: to find evidence against H0 in order to reject it. 27
28 2. Determine a test statistic A test statistic T is a numerical value that can be determined from the outcome of a chance experiment. Note that, by definition, T is a random variable as well! Here, T = number of matches between the two sequences (= X) (There is only one realization) Usually there is more than one realization in a random sample, and the test statistic depends on all realizations: Other examples: T (X 1,...,X n )= X X n n T (X 1,..., X n )= (X µ 0) ˆσ/ n = X (mean) (Tstatistic) 28
29 3. Distribution of T under H0 In case of H0 (pure chance alignment), the distribution of T is T Bin(26, 0.25) (Note that in reality Bin(26,p) is not the right distribution for this problem, we only use it to illustrate the idea of statistical testing.) 29
30 4. Choose the significance level α In our example we reject H0 if the number of matches is too high, so that it is unlikely to happen by chance. α determines what unlikely means. Let us choose α=0.05. The significance level α fixes the probability with which H0 is rejected, although it is true. Interpretation: In 5% of the cases (1 out of 20) we will find a value of T so high that we do not believe it has happened by chance  although it did! α = probability to reject a true null hypothesis = probability to make a type I error. 30
31 5. Calculate the critical value We now calculate a value C for the test statistic T, above which we consider it unlikely that H0 is true: P(T C H 0 )=α In our example with H0: T=X Bin(26,0.25) P(X 7 H 0 ) = P(X 8 H 0 ) = P(X 9 H 0 ) = P(X 10 H 0 ) = P(X 11 H 0 ) = => C = 11! 26 ( ) 26 (which is calculated as P(X k H 0 )= 0.25 i i ) i i=k 31
32 6. Decide Only now is it finally allowed to calculate the value of T. Here, we already know that T=11, since X=T. From step 5 we have the following rule: Reject H0 if T 11 and do not reject H0 if T < 11 Decision: we reject H0. Thus we do not believe that 11 out of 26 matches can happen by chance. We say: There is statistical evidence that the two sequences are related due to evolution. 32
33 Statistical significance Note: The decision to reject H0 on the previous slide depends on the significance level α. We would not have rejected H0 if α < 0.04! Whether the outcome of an experiment is statistically significant or not depends crucially on α! For α=1 any result is significant... (but meaningless). Scientific results that claim statistical significance without giving α should at least be doubted... 33
34 pvalues The pvalue is the probability to see something at least as extreme as just observed under H0. It depends on the data. In our example: P(X 11 H 0 ) = Thus the pvalue of our experiment is p= Many statistics programmes (R, SPSS,...) compute directly this. Your results are then significant if p < α. Interpretation: The pvalue tells you for which α your data would be significant. 34
35 Type I and type II errors The type I error depends on the significance level α. It is the probability to reject the null hypothesis, although it is true. The probability for a type I error is The type II error is the other kind of false decisions: it is the probability that the null hypothesis is not rejected, although it is wrong: 35
36 The power of a test The power is typically more complicated to compute, especially if H1 is unknown. 36
37 Example 37
38 BUT if we would have chosen α=0.01, the power (1β) would be lower! E.g. 1 β = P(X 11 p =0.26) = β = P(X 11 p =0.3) =
39 Fact: The decrease of the type I error comes at the expense of an increased type II error  and vice versa. There is a compromise between a low significance level α and high power 1β. 39
40 Bin(20,0.25) and Bin(20,0.3) distribution f(x) x Power if H0 : p =0.25, H1 : p =0.3 α =
41 Bin(20,0.25) and Bin(20,0.6) distribution f(x) Power if H0 : p =0.25, H1 : p =0.6 x α =
42 Bayesian Hypothesis Testing Remember: P(A j B) = P(B A j ) P(A j ) n i=1 P(B A i) P(A i ) Bayes theorem Example (from Ewans/Grant): A bag contains 10 coins, where only 3 of them are fair. The other 7 have a chance to show heads with ph=0.6. Take one coin at random and flip it five times. All five flips give heads (event D). Then: P(H)=0.3 (prior probability that coin is fair) P(H c )=0.7 (prior probability that coin is unfair) P(D H)=0.5 5 P(D H c )=
43 Now, the posterior probability that the coin was fair, given the outcome, can be calculated: P(H D) = = P(D H) P (H) P(D H) P (H)+P(D H c ) P (H c ) =0.147 This is lower than the prior distribution of H, so evidence against it. Moreover: P(H c D) = So there is a much higher posterior probability (given the outcome and the prior) that the coin I picked was unfair. The same setup works mit multiple hypotheses H1, H2,..., Hn. Identical calculations as above lead to posterior probabilities and the hypothesis with the highest posterior is chosen. 43
44 Other statistical tests There is a large variety of statistical tests. The choice of the correct test depends on the type and qualitiy of the data, the assumptions and the question to be answered. Examples: ztest ttest signtest Wilcoxontest MannWhitney / Utest χ 2 goodnessoffit test / χ 2 test for independence... 44
45 The ztest The simplest version of a ztest: Onesample problem Situation: Given n independent measurements Xi, 1 i n. Question: Can the expected value E[X]=µ be equal to, larger or lower than some theoretical value µo? Paired twosample problem Situation: Given n independent measurements Yi and Zi, 1 i n of the same feature in two different states. E.g., the blood pressure of each person is measured before and after the intake of a special drug. Question: Is there a significant difference between the two states? I.e., is the difference Xi = Yi  Zi 0 (or < 0, >0) or, equivalently: is E[X] 0? 45
46 Assumptions In the ztest it is assumed that X i N(µ X, σ 2 ) Thus the measurements should follow a normal distribution. Moreover, the variance σ 2 of Xi is known. 46
47 1. Hypotheses H 0 : X i N(µ 0, σ0 2 ), 1 i n, independent, with known variance σ2 0 H 1 : X i N(µ 1, σ 0 2 ), 1 i n, independent, with known variance σ2 0 with either µ 1 >µ 0, µ 1 <µ 0 or µ 0 µ 1 2. Test statistic Z = X µ 0 σ 0 / n 3. Distribution of Z under H0 Z N(0, 1) 47
48 4. Choose the significance level α E.g., α=5% (or a lower level, is stronger signifiance is needed). 5. Calculate the critical value The values can be looked up in a table. The most important ones (for the α=5% level) are given here: µ 1 >µ 0 : c =1.64 with R: > qnorm(0.95) => Ho is rejected, if Z > 1.64 µ 1 <µ 0 : c = 1.64 => Ho is rejected, if Z < with R: > qnorm(0.05) µ 0 µ 1 : c =1.96 => Ho is rejected, if Z > 1.96 with R: > qnorm(0.975) where do these values come from...? 48
49 Onesided test µ 1 >µ 0 : µ 1 <µ 0 : N(0,1) distribution N(0,1) distribution f(x) f(x) x Rejection range 5% x 49
50 Twosided test µ 0 µ 1 N(0,1) distribution f(x) % 2.5% x 50
Null Hypothesis Significance Testing Signifcance Level, Power, ttests Spring 2014 Jeremy Orloff and Jonathan Bloom
Null Hypothesis Significance Testing Signifcance Level, Power, ttests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is
More informationHypothesis 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
More informationStatistiek (WISB361)
Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17
More informationSufficient Statistics and Exponential Family. 1 Statistics and Sufficient Statistics. Math 541: Statistical Theory II. Lecturer: Songfeng Zheng
Math 541: Statistical Theory II Lecturer: Songfeng Zheng Sufficient Statistics and Exponential Family 1 Statistics and Sufficient Statistics Suppose we have a random sample X 1,, X n taken from a distribution
More informationStatistiek I. ttests. John Nerbonne. CLCG, Rijksuniversiteit Groningen. John Nerbonne 1/35
Statistiek I ttests John Nerbonne CLCG, Rijksuniversiteit Groningen http://wwwletrugnl/nerbonne/teach/statistieki/ John Nerbonne 1/35 ttests To test an average or pair of averages when σ is known, we
More informationPower and Sample Size Determination
Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 Power 1 / 31 Experimental Design To this point in the semester,
More informationMultiple Hypothesis Testing: The Ftest
Multiple Hypothesis Testing: The Ftest 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
More informationChapter 14: 16, 9, 12; Chapter 15: 8 Solutions When is it appropriate to use the normal approximation to the binomial distribution?
Chapter 14: 16, 9, 1; Chapter 15: 8 Solutions 141 When is it appropriate to use the normal approximation to the binomial distribution? The usual recommendation is that the approximation is good if np
More informationStatistics  Written Examination MEC Students  BOVISA
Statistics  Written Examination MEC Students  BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.
More informationComparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Comparison of frequentist and Bayesian inference. Class 20, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Be able to explain the difference between the pvalue and a posterior
More informationStatistical Inference and ttests
1 Statistical Inference and ttests Objectives Evaluate the difference between a sample mean and a target value using a onesample ttest. Evaluate the difference between a sample mean and a target value
More informationQuantitative Biology Lecture 5 (Hypothesis Testing)
15 th Oct 2015 Quantitative Biology Lecture 5 (Hypothesis Testing) Gurinder Singh Mickey Atwal Center for Quantitative Biology Summary Classification Errors Statistical significance Ttests Qvalues (Traditional)
More information15.0 More Hypothesis Testing
15.0 More Hypothesis Testing 1 Answer Questions Type I and Type II Error Power Calculation Bayesian Hypothesis Testing 15.1 Type I and Type II Error In the philosophy of hypothesis testing, the null hypothesis
More informationStatistical Significance and Bivariate Tests
Statistical Significance and Bivariate Tests BUS 735: Business Decision Making and Research 1 1.1 Goals Goals Specific goals: Refamiliarize ourselves with basic statistics ideas: sampling distributions,
More informationHypothesis Testing COMP 245 STATISTICS. Dr N A Heard. 1 Hypothesis Testing 2 1.1 Introduction... 2 1.2 Error Rates and Power of a Test...
Hypothesis Testing COMP 45 STATISTICS Dr N A Heard Contents 1 Hypothesis Testing 1.1 Introduction........................................ 1. Error Rates and Power of a Test.............................
More informationModule 7: Hypothesis Testing I Statistics (OA3102)
Module 7: Hypothesis Testing I Statistics (OA3102) Professor Ron Fricker Naval Postgraduate School Monterey, California Reading assignment: WM&S chapter 10.110.5 Revision: 212 1 Goals for this Module
More informationSECOND PART, LECTURE 4: CONFIDENCE INTERVALS
Massimo Guidolin Massimo.Guidolin@unibocconi.it Dept. of Finance STATISTICS/ECONOMETRICS PREP COURSE PROF. MASSIMO GUIDOLIN SECOND PART, LECTURE 4: CONFIDENCE INTERVALS Lecture 4: Confidence Intervals
More informationLAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING
LAB 4 INSTRUCTIONS CONFIDENCE INTERVALS AND HYPOTHESIS TESTING In this lab you will explore the concept of a confidence interval and hypothesis testing through a simulation problem in engineering setting.
More informationChapter 7 Part 2. Hypothesis testing Power
Chapter 7 Part 2 Hypothesis testing Power November 6, 2008 All of the normal curves in this handout are sampling distributions Goal: To understand the process of hypothesis testing and the relationship
More information1 Maximum likelihood estimation
COS 424: Interacting with Data Lecturer: David Blei Lecture #4 Scribes: Wei Ho, Michael Ye February 14, 2008 1 Maximum likelihood estimation 1.1 MLE of a Bernoulli random variable (coin flips) Given N
More informationThe 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
More informationConfindence Intervals and Probability Testing
Confindence Intervals and Probability Testing PO7001: Quantitative Methods I Kenneth Benoit 3 November 2010 Using probability distributions to assess sample likelihoods Recall that using the µ and σ from
More informationChapter 1112 1 Review
Chapter 1112 Review Name 1. In formulating hypotheses for a statistical test of significance, the null hypothesis is often a statement of no effect or no difference. the probability of observing the data
More informationNonparametric Test Procedures
Nonparametric Test Procedures 1 Introduction to Nonparametrics Nonparametric tests do not require that samples come from populations with normal distributions or any other specific distribution. Hence
More informationTwoSample TTests Assuming Equal Variance (Enter Means)
Chapter 4 TwoSample TTests Assuming Equal Variance (Enter Means) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when the variances of
More informationNull Hypothesis Significance Testing Signifcance Level, Power, ttests. 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom
Null Hypothesis Significance Testing Signifcance Level, Power, ttests 18.05 Spring 2014 Jeremy Orloff and Jonathan Bloom Simple and composite hypotheses Simple hypothesis: the sampling distribution is
More informationm (t) = e nt m Y ( t) = e nt (pe t + q) n = (pe t e t + qe t ) n = (qe t + p) n
1. For a discrete random variable Y, prove that E[aY + b] = ae[y] + b and V(aY + b) = a 2 V(Y). Solution: E[aY + b] = E[aY] + E[b] = ae[y] + b where each step follows from a theorem on expected value from
More informationA crash course in probability and Naïve Bayes classification
Probability theory A crash course in probability and Naïve Bayes classification Chapter 9 Random variable: a variable whose possible values are numerical outcomes of a random phenomenon. s: A person s
More informationTwoSample TTests Allowing Unequal Variance (Enter Difference)
Chapter 45 TwoSample TTests Allowing Unequal Variance (Enter Difference) Introduction This procedure provides sample size and power calculations for one or twosided twosample ttests when no assumption
More informationChapter 8 Introduction to Hypothesis Testing
Chapter 8 Student Lecture Notes 81 Chapter 8 Introduction to Hypothesis Testing Fall 26 Fundamentals of Business Statistics 1 Chapter Goals After completing this chapter, you should be able to: Formulate
More informationBayesian Updating with Discrete Priors Class 11, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
1 Learning Goals Bayesian Updating with Discrete Priors Class 11, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1. Be able to apply Bayes theorem to compute probabilities. 2. Be able to identify
More informationInferential Statistics
Inferential Statistics Sampling and the normal distribution Zscores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are
More informationChapter 3: Nonparametric Tests
B. Weaver (15Feb00) Nonparametric Tests... 1 Chapter 3: Nonparametric Tests 3.1 Introduction Nonparametric, or distribution free tests are socalled because the assumptions underlying their use are fewer
More information7 Hypothesis testing  one sample tests
7 Hypothesis testing  one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X
More information14.0 Hypothesis Testing
14.0 Hypothesis Testing 1 Answer Questions Hypothesis Tests Examples 14.1 Hypothesis Tests A hypothesis test (significance test) is a way to decide whether the data strongly support one point of view or
More informationTHE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.
THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM
More informationLesson 1: Comparison of Population Means Part c: Comparison of Two Means
Lesson : Comparison of Population Means Part c: Comparison of Two Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis
More informationExamination 110 Probability and Statistics Examination
Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiplechoice test questions. The test is a threehour examination
More informationMONT 107N Understanding Randomness Solutions For Final Examination May 11, 2010
MONT 07N Understanding Randomness Solutions For Final Examination May, 00 Short Answer (a) (0) How are the EV and SE for the sum of n draws with replacement from a box computed? Solution: The EV is n times
More informationPeople have thought about, and defined, probability in different ways. important to note the consequences of the definition:
PROBABILITY AND LIKELIHOOD, A BRIEF INTRODUCTION IN SUPPORT OF A COURSE ON MOLECULAR EVOLUTION (BIOL 3046) Probability The subject of PROBABILITY is a branch of mathematics dedicated to building models
More informationHypothesis Test for Mean Using Given Data (Standard Deviation Knownztest)
Hypothesis Test for Mean Using Given Data (Standard Deviation Knownztest) A hypothesis test is conducted when trying to find out if a claim is true or not. And if the claim is true, is it significant.
More informationL10: Probability, statistics, and estimation theory
L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian
More informationDesign and Analysis of Equivalence Clinical Trials Via the SAS System
Design and Analysis of Equivalence Clinical Trials Via the SAS System Pamela J. Atherton Skaff, Jeff A. Sloan, Mayo Clinic, Rochester, MN 55905 ABSTRACT An equivalence clinical trial typically is conducted
More informationTwosample hypothesis testing, II 9.07 3/16/2004
Twosample hypothesis testing, II 9.07 3/16/004 Small sample tests for the difference between two independent means For twosample tests of the difference in mean, things get a little confusing, here,
More informationChapter 9: Hypothesis Testing Sections
Chapter 9: Hypothesis Testing Sections 9.1 Problems of Testing Hypotheses Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 TwoSided Alternatives 9.6 Comparing the
More informationInferences About Differences Between Means Edpsy 580
Inferences About Differences Between Means Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at UrbanaChampaign Inferences About Differences Between Means Slide
More informationHYPOTHESIS TESTING WITH SPSS:
HYPOTHESIS TESTING WITH SPSS: A NONSTATISTICIAN S GUIDE & TUTORIAL by Dr. Jim Mirabella SPSS 14.0 screenshots reprinted with permission from SPSS Inc. Published June 2006 Copyright Dr. Jim Mirabella CHAPTER
More informationMeasuring the Power of a Test
Textbook Reference: Chapter 9.5 Measuring the Power of a Test An economic problem motivates the statement of a null and alternative hypothesis. For a numeric data set, a decision rule can lead to the rejection
More informationTesting Hypotheses About Proportions
Chapter 11 Testing Hypotheses About Proportions Hypothesis testing method: uses data from a sample to judge whether or not a statement about a population may be true. Steps in Any Hypothesis Test 1. Determine
More information93.4 Likelihood ratio test. NeymanPearson lemma
93.4 Likelihood ratio test NeymanPearson lemma 91 Hypothesis Testing 91.1 Statistical Hypotheses Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental
More informationData Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments  Introduction
Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments  Introduction
More informationCONTENTS OF DAY 2. II. Why Random Sampling is Important 9 A myth, an urban legend, and the real reason NOTES FOR SUMMER STATISTICS INSTITUTE COURSE
1 2 CONTENTS OF DAY 2 I. More Precise Definition of Simple Random Sample 3 Connection with independent random variables 3 Problems with small populations 8 II. Why Random Sampling is Important 9 A myth,
More informationExperimental Design. Power and Sample Size Determination. Proportions. Proportions. Confidence Interval for p. The Binomial Test
Experimental Design Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin Madison November 3 8, 2011 To this point in the semester, we have largely
More informationGeneral Method: Difference of Means. 3. Calculate df: either WelchSatterthwaite formula or simpler df = min(n 1, n 2 ) 1.
General Method: Difference of Means 1. Calculate x 1, x 2, SE 1, SE 2. 2. Combined SE = SE1 2 + SE2 2. ASSUMES INDEPENDENT SAMPLES. 3. Calculate df: either WelchSatterthwaite formula or simpler df = min(n
More informationChapter 9: Hypothesis Testing Sections
Chapter 9: Hypothesis Testing Sections 9.1 Problems of Testing Hypotheses Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 TwoSided Alternatives 9.5 The t Test 9.6
More informationOutline of Topics. Statistical Methods I. Types of Data. Descriptive Statistics
Statistical Methods I Tamekia L. Jones, Ph.D. (tjones@cog.ufl.edu) Research Assistant Professor Children s Oncology Group Statistics & Data Center Department of Biostatistics Colleges of Medicine and Public
More informationTesting a claim about a population mean
Introductory Statistics Lectures Testing a claim about a population mean One sample hypothesis test of the mean Department of Mathematics Pima Community College Redistribution of this material is prohibited
More informationLikelihood: Frequentist vs Bayesian Reasoning
"PRINCIPLES OF PHYLOGENETICS: ECOLOGY AND EVOLUTION" Integrative Biology 200B University of California, Berkeley Spring 2009 N Hallinan Likelihood: Frequentist vs Bayesian Reasoning Stochastic odels and
More informationModule 5 Hypotheses Tests: Comparing Two Groups
Module 5 Hypotheses Tests: Comparing Two Groups Objective: In medical research, we often compare the outcomes between two groups of patients, namely exposed and unexposed groups. At the completion of this
More informationChapter 7. Section Introduction to Hypothesis Testing
Section 7.1  Introduction to Hypothesis Testing Chapter 7 Objectives: State a null hypothesis and an alternative hypothesis Identify type I and type II errors and interpret the level of significance Determine
More informationChapter 9: Hypothesis Testing Sections
Chapter 9: Hypothesis Testing Sections  we are still here Skip: 9.2 Testing Simple Hypotheses Skip: 9.3 Uniformly Most Powerful Tests Skip: 9.4 TwoSided Alternatives 9.5 The t Test 9.6 Comparing the
More informationProbability and Statistics Lecture 9: 1 and 2Sample 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
More informationConfidence intervals, t tests, P values
Confidence intervals, t tests, P values Joe Felsenstein Department of Genome Sciences and Department of Biology Confidence intervals, t tests, P values p.1/31 Normality Everybody believes in the normal
More informationStructure of the Data. Paired Samples. Overview. The data from a paired design can be tabulated in this form. Individual Y 1 Y 2 d i = Y 1 Y
Structure of the Data Paired Samples Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison Statistics 371 11th November 2005 The data from a paired design can be tabulated
More informationIntroduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.
Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.
More informationINTRODUCTORY STATISTICS
INTRODUCTORY STATISTICS FIFTH EDITION Thomas H. Wonnacott University of Western Ontario Ronald J. Wonnacott University of Western Ontario WILEY JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore
More informationGood luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:
Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours
More informationStatistics Graduate Courses
Statistics Graduate Courses STAT 7002Topics in StatisticsBiological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.
More informationStatistical Inference
Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this
More informationIntroduction. Hypothesis Testing. Hypothesis Testing. Significance Testing
Introduction Hypothesis Testing Mark Lunt Arthritis Research UK Centre for Ecellence in Epidemiology University of Manchester 13/10/2015 We saw last week that we can never know the population parameters
More informationSECOND PART, LECTURE 3: HYPOTHESIS TESTING
Massimo Guidolin Massimo.Guidolin@unibocconi.it Dept. of Finance STATISTICS/ECONOMETRICS PREP COURSE PROF. MASSIMO GUIDOLIN SECOND PART, LECTURE 3: HYPOTHESIS TESTING Lecture 3: Hypothesis Testing Prof.
More informationNonparametric tests, Bootstrapping
Nonparametric tests, Bootstrapping http://www.isrec.isbsib.ch/~darlene/embnet/ Hypothesis testing review 2 competing theories regarding a population parameter: NULL hypothesis H ( straw man ) ALTERNATIVEhypothesis
More informationTerminology. 2 There is no mathematical difference between the errors, however. The bottom line is that we choose one type
Hypothesis Testing 10.2.1 Terminology The null hypothesis H 0 is a nothing hypothesis, whose interpretation could be that nothing has changed, there is no difference, there is nothing special taking place,
More informationWhat is Bayesian statistics and why everything else is wrong
What is Bayesian statistics and why everything else is wrong 1 Michael Lavine ISDS, Duke University, Durham, North Carolina Abstract We use a single example to explain (1), the Likelihood Principle, (2)
More informationHow to Conduct a Hypothesis Test
How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some
More informationHypothesis Testing I
ypothesis Testing I The testing process:. Assumption about population(s) parameter(s) is made, called null hypothesis, denoted. 2. Then the alternative is chosen (often just a negation of the null hypothesis),
More informationLecture 9: Bayesian hypothesis testing
Lecture 9: Bayesian hypothesis testing 5 November 27 In this lecture we ll learn about Bayesian hypothesis testing. 1 Introduction to Bayesian hypothesis testing Before we go into the details of Bayesian
More informationName: Date: Use the following to answer questions 34:
Name: Date: 1. Determine whether each of the following statements is true or false. A) The margin of error for a 95% confidence interval for the mean increases as the sample size increases. B) The margin
More informationClass 19: Two Way Tables, Conditional Distributions, ChiSquare (Text: Sections 2.5; 9.1)
Spring 204 Class 9: Two Way Tables, Conditional Distributions, ChiSquare (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationIntroduction to Regression and Data Analysis
Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it
More informationHypothesis Testing: General Framework 1 1
Hypothesis Testing: General Framework Lecture 2 K. Zuev February 22, 26 In previous lectures we learned how to estimate parameters in parametric and nonparametric settings. Quite often, however, researchers
More informationAnalysis of numerical data S4
Basic medical statistics for clinical and experimental research Analysis of numerical data S4 Katarzyna Jóźwiak k.jozwiak@nki.nl 3rd November 2015 1/42 Hypothesis tests: numerical and ordinal data 1 group:
More informationC. The null hypothesis is not rejected when the alternative hypothesis is true. A. population parameters.
Sample Multiple Choice Questions for the material since Midterm 2. Sample questions from Midterms and 2 are also representative of questions that may appear on the final exam.. A randomly selected sample
More informationStat260: Bayesian Modeling and Inference Lecture Date: February 1, Lecture 3
Stat26: Bayesian Modeling and Inference Lecture Date: February 1, 21 Lecture 3 Lecturer: Michael I. Jordan Scribe: Joshua G. Schraiber 1 Decision theory Recall that decision theory provides a quantification
More informationHypothesis Testing. Bluman Chapter 8
CHAPTER 8 Learning Objectives C H A P T E R E I G H T Hypothesis Testing 1 Outline 81 Steps in Traditional Method 82 z Test for a Mean 83 t Test for a Mean 84 z Test for a Proportion 85 2 Test for
More informationSingle sample hypothesis testing, II 9.07 3/02/2004
Single sample hypothesis testing, II 9.07 3/02/2004 Outline Very brief review Onetailed vs. twotailed tests Small sample testing Significance & multiple tests II: Data snooping What do our results mean?
More informationProbability of rejecting the null hypothesis when
Sample Size The first question faced by a statistical consultant, and frequently the last, is, How many subjects (animals, units) do I need? This usually results in exploring the size of the treatment
More informationInferential Statistics. Probability. From Samples to Populations. Katie RommelEsham Education 504
Inferential Statistics Katie RommelEsham Education 504 Probability Probability is the scientific way of stating the degree of confidence we have in predicting something Tossing coins and rolling dice
More informationInvariance and optimality Linear rank statistics Permutation tests in R. Rank Tests. Patrick Breheny. October 7. STA 621: Nonparametric Statistics
Rank Tests October 7 Power Invariance and optimality Permutation testing allows great freedom to use a wide variety of test statistics, all of which lead to exact levelα tests regardless of the distribution
More informationTwosample hypothesis testing, I 9.07 3/09/2004
Twosample hypothesis testing, I 9.07 3/09/2004 But first, from last time More on the tradeoff between Type I and Type II errors The null and the alternative: Sampling distribution of the mean, m, given
More informationHYPOTHESIS TESTING: POWER OF THE TEST
HYPOTHESIS TESTING: POWER OF THE TEST The first 6 steps of the 9step test of hypothesis are called "the test". These steps are not dependent on the observed data values. When planning a research project,
More informationChapter 8 Hypothesis Testing Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing
Chapter 8 Hypothesis Testing 1 Chapter 8 Hypothesis Testing 81 Overview 82 Basics of Hypothesis Testing 83 Testing a Claim About a Proportion 85 Testing a Claim About a Mean: s Not Known 86 Testing
More informationBayesian probability: P. State of the World: X. P(X your information I)
Bayesian probability: P State of the World: X P(X your information I) 1 First example: bag of balls Every probability is conditional to your background knowledge I : P(A I) What is the (your) probability
More informationQUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NONPARAMETRIC TESTS
QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NONPARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.
More informationChapter 2. Hypothesis testing in one population
Chapter 2. Hypothesis testing in one population Contents Introduction, the null and alternative hypotheses Hypothesis testing process Type I and Type II errors, power Test statistic, level of significance
More informationChapter 8: Introduction to Hypothesis Testing
Chapter 8: Introduction to Hypothesis Testing We re now at the point where we can discuss the logic of hypothesis testing. This procedure will underlie the statistical analyses that we ll use for the remainder
More informationStudy Guide for the Final Exam
Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make
More information4. Introduction to Statistics
Statistics for Engineers 41 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
More informationBiodiversity Data Analysis: Testing Statistical Hypotheses By Joanna Weremijewicz, Simeon Yurek, Steven Green, Ph. D. and Dana Krempels, Ph. D.
Biodiversity Data Analysis: Testing Statistical Hypotheses By Joanna Weremijewicz, Simeon Yurek, Steven Green, Ph. D. and Dana Krempels, Ph. D. In biological science, investigators often collect biological
More information