Categorical Data Analysis Chapter 3
|
|
- Margaret Hood
- 6 years ago
- Views:
Transcription
1 Categorical Data Analysis Chapter 3
2 The actual coverage probability is usually a bit higher than the nominal level. Confidence intervals for association parameteres Consider the odds ratio in the 2x2 table, ˆθ = (n 11 n 22 )/(n 12 n 21 ) Unless n is vary large, its sampling distribution is highly skewed. The log transform converges more reapidly to normality. An estimated standard error for log ˆθ is ˆσ(log ˆθ) 1 = n n n n 2 2 By the large-sample normaility of log ˆθ, the Wald confidence interval for log θ is log ˆθ ± z α/2ˆσ(log ˆθ) The Wald CI for θ is exp ( ) log ˆθ ± z α/2ˆσ(log ˆθ)
3 Wald CI for difference of proportions The estimated difference of proportions ˆπ 1 ˆπ 2 is unbiased for the true difference π 1 π 2 and has the standard error π 1 (1 π 1 ) σ(ˆπ 1 ˆπ 2 ) = + π 1(1 π 1 ) n 1 n 2 The estimate ˆσ(ˆπ 1 ˆπ 2 ) replaces π i by ˆπ i. Then ˆπ 1 ˆπ 2 ± z α/2 σ(ˆπ 1 ˆπ 2 ) is a Wald CI for π 1 π 2. Like the Wald interval for a single proportion, it usually has true coverage probability less than the nominal confidence level, especially when π 1 and π 2 are near 0 or 1.
4 Wald CI for relative risk Like the odds ratio, the log relative risk log r converges to normality faster on the log scale. An estimated standard error for log r is ˆσ(log r) = 1 ˆπ1 y ˆπ 2 y 2 The Wald interval for r is ( ) exp log r ± z α/2ˆσ(log r). It tends to be somewhat conservative.
5 Example: Aspirin and Heart attacks revisited The proportions having fatal hear attacks were 18/11,034= for placebo group and 5/11,037= for aspirin group. The 95% CI for the log relative risk is log( / ) ± 1.96(0.505) = log(1.34, 9.70) Despite the very large sample sizes, due to the very low rate of heart attach deaths, the estimated effect is imprecise. The Wald 95% CI for π 1 π 2 is ± 1.96( ) = (0.0003, 0.002) The Wald 95% CI for odds ratio is log(3.62) ± 1.96(0.51) = log(1.33, 9.84)
6 Deriving standard errors with the delta method If n(t n θ) d N(0, σ 2 ) then n(g(tn ) g(θ) d N(0, [g (θ)] 2 σ 2 )
7 Score confidence interval for difference in proportions Consider testing H 0 : π 1 π 2 = 0 Let ˆπ 1 ( 0 ) and ˆπ 2 ( 0 ) denote the ML estimates of π 1 and π 2 subject to the constraint π 1 π 2 = 0. The score test statistic is z( 0 ) = (ˆπ 1 ˆπ 2 ) 0 ˆπ1 ( 0 )[1 ˆπ 1 ( 0 )] n 1 + ˆπ 2( 0 )[1 ˆπ 2 ( 0 )] n 2 The score CI is the set of 0 such that z( 0 ) < z α/2.
8 Score confidence interval for odds ratio Consider testing on the odds ratio H 0 : θ = θ 0 Let ˆµ ij (θ 0 ) be the unique expected frequeny estimates that have the same row and column margins as {n i j} and satisfy The set of θ 0 satisfying ˆµ 11 (θ 0 )ˆµ 22 (θ 0 ) ˆµ 12 (θ 0 )ˆµ 21 (θ 0 ) = θ 0 X 2 (θ 0 ) = (n ij ˆµ ij (θ 0 )) 2 /ˆµ ij (θ 0 ) < χ 2 1(α) form a 100(1 α)% score-test-based confidence interval.
9 Profile likelihood CI Consider testing on the odds ratio H 0 : θ = θ 0 The set of θ 0 satisfying G 2 (θ 0 ) = 2 i n ij log[n ij /ˆµ ij (θ 0 )] < χ 2 1(α) j form a 100(1 α)% likeli-ratio test-based CI.
10 Example: Aspirin and heart attacks profile LRT CI for odds ratio: (1.44, 2.34) score CI for difference in proportion: (0.0047, ) score CI for relative risk (1.43, 2.30) score CI for odds ratio: (1.44, 2.33)
11 Testing independence in two-way contingency tables Consider the hypothesis of statistical independence H 0 : π ij = π i+ π +j for all i and j The Pearson X 2 test statistic is X 2 = (n ij ˆµ ij ) 2 ˆµ i j ij where ˆµ ij = nˆπ i+ˆπ +j = n i+ n +j /n which is the expected cell count under independence. Under H 0, X 2 follows an asymptotic chi-square distribution with. df = (IJ 1) (I 1) (J 1) = (I 1)(J 1) The likelihood-ratio test produces a different statistic: G 2 = 2 n ij log(n ij /ˆµ ij ) i j which follows the same asymptotic distribution as X 2
12 Adequacy of Chi-Squared approximations When there are independent multinomial samples, independence between the row and column corresponds to homogeneity of each outcome probability among the rows or columns with fixed margin. The limiting chi-squared results still hold. The convergence of the actual sampling distribution of X 2 or G 2 to the chi-squared distribution applies as n grows for a fixed number of cells. The adequacy of the approximation depends on both n and the number of cells. The size of n/ij that produces adequate approximations for X 2 tends to decrease as IJ increases.
13 Adequacy of Chi-Squared approximations Research has shown that X 2 performs adequately with smaller n and more sparse tables than G 2. The distribution of G 2 is usually poorly approximated by chi-squared when n/ij < 5. Chi-squared approximations for both tend to be poor for tables containing both very small and moderately large µ ij. Small-sample methods are available whenever it is doubtful.
14 Example: Education and belief in God X 2 = 76.1, G 2 = 73.2 with df = (3 1)(6 1) = 10. The P-values are < These statistics provide extremely strong evidence of an association.
15 Following-up Chi-squared tests When a test of independence has a small p-value, what does it say about the strength of the association? Not much, the smaller the p-value, the stronger the evidence that AN association exists It does not tell you that the association is very strong To understand more about assoication, do 1) a residual analysis 2) Consider partitioning the Chi-square statistics into independent pieces to examine association in subtables
16 Residuals Pearson residuals e ij = n ij ˆµ ij ˆµij Pearson residuals have asymptotic variances less than 1, averaging [(I 1)(J 1)]/IJ Standardized residuals r ij = n ij ˆµ ij ˆµ ij (1 p i+ )(1 p +j ) In 2x2 tables, df = 1 and r 11 = r 12 = r 21 = r 22, and any r 2 ij = X 2 A standardized residual that exceeds about 2 or 3 in absolute value indicates lack of fit of H 0.
17 Example: Education and Belief in God revisited n 36 = 293, ˆµ 36 = 358.8, p 3+ = 581/2000 = , p +6 = 1235/2000 = r 36 = ( )/ 358.8( )( ) = 6.7 We can infer that in the population in 2008, fewer people at the highest level of education would have responded know God exists than if the variables were truly independent.
18 Example: Mosaic plot
19 Partitioning Chi-Squared After rejecting independence, the next question could be Are there individual comparisons more significant than others? Partitioning may show the association is largly dependent on certain categories or groupings of categories For IXJ tables, one way to partition G 2 to the G 2 of the (I 1)(J 1) separate 2x2 tables is The G 2 s of the (I 1)(J 1) tables are independent.
20 Example: Origin of Schizophrenia Here G 2 = with df = 4. To understand the association better, we partition G 2 into 4 independent components.
21 Example the order of the above tables, G 2 = 0.29, 1.36, 12.95, 8.43 respectively. In The psychoanalytic school seems more likely than the other schools to ascribe the origins of schizophrenia as being a combination Of those who chose either the biogenic or environmental origin, members of the psychoanalytic school where somewhat more likely than the other schools to choose the environmental origin.
22 Partitioning For G 2, exact partitioning occurs Pearson X 2 does not have this property, but since X 2 and G 2 are asymptotically equivalent, X 2 can be used for subtables too The selection of subtables is not unique. To initiate the process, you can use your residual analysis to identify the most extreme cells and begin there Association measures such as odds ratio, relative risk, difference of proportions, and association factors and their CI can also be used describe strength of association in subtables.
23 Rules for partitioning The df for the subtables must sum to the df for the full table Each cell count in the full table must be a cell count in one and only one subtable Each marginal total of the full table must be a marginal total for one and only one subtable
n y π y (1 π) n y +ylogπ +(n y)log(1 π).
Tests for a binomial probability π Let Y bin(n,π). The likelihood is L(π) = n y π y (1 π) n y and the log-likelihood is L(π) = log n y +ylogπ +(n y)log(1 π). So L (π) = y π n y 1 π. 1 Solving for π gives
More informationSTAT 705: Analysis of Contingency Tables
STAT 705: Analysis of Contingency Tables Timothy Hanson Department of Statistics, University of South Carolina Stat 705: Analysis of Contingency Tables 1 / 45 Outline of Part I: models and parameters Basic
More informationCategorical Variables and Contingency Tables: Description and Inference
Categorical Variables and Contingency Tables: Description and Inference STAT 526 Professor Olga Vitek March 3, 2011 Reading: Agresti Ch. 1, 2 and 3 Faraway Ch. 4 3 Univariate Binomial and Multinomial Measurements
More informationSTA 4504/5503 Sample Exam 1 Spring 2011 Categorical Data Analysis. 1. Indicate whether each of the following is true (T) or false (F).
STA 4504/5503 Sample Exam 1 Spring 2011 Categorical Data Analysis 1. Indicate whether each of the following is true (T) or false (F). (a) T In 2 2 tables, statistical independence is equivalent to a population
More informationSTA 4504/5503 Sample Exam 1 Spring 2011 Categorical Data Analysis. 1. Indicate whether each of the following is true (T) or false (F).
STA 4504/5503 Sample Exam 1 Spring 2011 Categorical Data Analysis 1. Indicate whether each of the following is true (T) or false (F). (a) (b) (c) (d) (e) In 2 2 tables, statistical independence is equivalent
More informationST3241 Categorical Data Analysis I Two-way Contingency Tables. Odds Ratio and Tests of Independence
ST3241 Categorical Data Analysis I Two-way Contingency Tables Odds Ratio and Tests of Independence 1 Inference For Odds Ratio (p. 24) For small to moderate sample size, the distribution of sample odds
More informationST3241 Categorical Data Analysis I Two-way Contingency Tables. 2 2 Tables, Relative Risks and Odds Ratios
ST3241 Categorical Data Analysis I Two-way Contingency Tables 2 2 Tables, Relative Risks and Odds Ratios 1 What Is A Contingency Table (p.16) Suppose X and Y are two categorical variables X has I categories
More informationSome comments on Partitioning
Some comments on Partitioning Dipankar Bandyopadhyay Department of Biostatistics, Virginia Commonwealth University BIOS 625: Categorical Data & GLM 1/30 Partitioning Chi-Squares We have developed tests
More informationTesting Independence
Testing Independence Dipankar Bandyopadhyay Department of Biostatistics, Virginia Commonwealth University BIOS 625: Categorical Data & GLM 1/50 Testing Independence Previously, we looked at RR = OR = 1
More informationGood Confidence Intervals for Categorical Data Analyses. Alan Agresti
Good Confidence Intervals for Categorical Data Analyses Alan Agresti Department of Statistics, University of Florida visiting Statistics Department, Harvard University LSHTM, July 22, 2011 p. 1/36 Outline
More informationConfidence Intervals, Testing and ANOVA Summary
Confidence Intervals, Testing and ANOVA Summary 1 One Sample Tests 1.1 One Sample z test: Mean (σ known) Let X 1,, X n a r.s. from N(µ, σ) or n > 30. Let The test statistic is H 0 : µ = µ 0. z = x µ 0
More informationSolutions for Examination Categorical Data Analysis, March 21, 2013
STOCKHOLMS UNIVERSITET MATEMATISKA INSTITUTIONEN Avd. Matematisk statistik, Frank Miller MT 5006 LÖSNINGAR 21 mars 2013 Solutions for Examination Categorical Data Analysis, March 21, 2013 Problem 1 a.
More informationSections 3.4, 3.5. Timothy Hanson. Department of Statistics, University of South Carolina. Stat 770: Categorical Data Analysis
Sections 3.4, 3.5 Timothy Hanson Department of Statistics, University of South Carolina Stat 770: Categorical Data Analysis 1 / 22 3.4 I J tables with ordinal outcomes Tests that take advantage of ordinal
More informationUnit 9: Inferences for Proportions and Count Data
Unit 9: Inferences for Proportions and Count Data Statistics 571: Statistical Methods Ramón V. León 12/15/2008 Unit 9 - Stat 571 - Ramón V. León 1 Large Sample Confidence Interval for Proportion ( pˆ p)
More informationModeling and inference for an ordinal effect size measure
STATISTICS IN MEDICINE Statist Med 2007; 00:1 15 Modeling and inference for an ordinal effect size measure Euijung Ryu, and Alan Agresti Department of Statistics, University of Florida, Gainesville, FL
More informationLecture 25. Ingo Ruczinski. November 24, Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University
Lecture 25 Department of Biostatistics Johns Hopkins Bloomberg School of Public Health Johns Hopkins University November 24, 2015 1 2 3 4 5 6 7 8 9 10 11 1 Hypothesis s of homgeneity 2 Estimating risk
More informationReview of Statistics 101
Review of Statistics 101 We review some important themes from the course 1. Introduction Statistics- Set of methods for collecting/analyzing data (the art and science of learning from data). Provides methods
More informationLog-linear Models for Contingency Tables
Log-linear Models for Contingency Tables Statistics 149 Spring 2006 Copyright 2006 by Mark E. Irwin Log-linear Models for Two-way Contingency Tables Example: Business Administration Majors and Gender A
More informationStatistics 3858 : Contingency Tables
Statistics 3858 : Contingency Tables 1 Introduction Before proceeding with this topic the student should review generalized likelihood ratios ΛX) for multinomial distributions, its relation to Pearson
More informationMultinomial Logistic Regression Models
Stat 544, Lecture 19 1 Multinomial Logistic Regression Models Polytomous responses. Logistic regression can be extended to handle responses that are polytomous, i.e. taking r>2 categories. (Note: The word
More informationUnit 9: Inferences for Proportions and Count Data
Unit 9: Inferences for Proportions and Count Data Statistics 571: Statistical Methods Ramón V. León 1/15/008 Unit 9 - Stat 571 - Ramón V. León 1 Large Sample Confidence Interval for Proportion ( pˆ p)
More informationCHAPTER 17 CHI-SQUARE AND OTHER NONPARAMETRIC TESTS FROM: PAGANO, R. R. (2007)
FROM: PAGANO, R. R. (007) I. INTRODUCTION: DISTINCTION BETWEEN PARAMETRIC AND NON-PARAMETRIC TESTS Statistical inference tests are often classified as to whether they are parametric or nonparametric Parameter
More informationLecture 8: Summary Measures
Lecture 8: Summary Measures Dipankar Bandyopadhyay, Ph.D. BMTRY 711: Analysis of Categorical Data Spring 2011 Division of Biostatistics and Epidemiology Medical University of South Carolina Lecture 8:
More informationUNIVERSITY OF TORONTO. Faculty of Arts and Science APRIL 2010 EXAMINATIONS STA 303 H1S / STA 1002 HS. Duration - 3 hours. Aids Allowed: Calculator
UNIVERSITY OF TORONTO Faculty of Arts and Science APRIL 2010 EXAMINATIONS STA 303 H1S / STA 1002 HS Duration - 3 hours Aids Allowed: Calculator LAST NAME: FIRST NAME: STUDENT NUMBER: There are 27 pages
More informationDiscrete Multivariate Statistics
Discrete Multivariate Statistics Univariate Discrete Random variables Let X be a discrete random variable which, in this module, will be assumed to take a finite number of t different values which are
More informationCategorical data analysis Chapter 5
Categorical data analysis Chapter 5 Interpreting parameters in logistic regression The sign of β determines whether π(x) is increasing or decreasing as x increases. The rate of climb or descent increases
More informationTopic 21 Goodness of Fit
Topic 21 Goodness of Fit Contingency Tables 1 / 11 Introduction Two-way Table Smoking Habits The Hypothesis The Test Statistic Degrees of Freedom Outline 2 / 11 Introduction Contingency tables, also known
More informationChi-Square. Heibatollah Baghi, and Mastee Badii
1 Chi-Square Heibatollah Baghi, and Mastee Badii Different Scales, Different Measures of Association Scale of Both Variables Nominal Scale Measures of Association Pearson Chi-Square: χ 2 Ordinal Scale
More informationParametric versus Nonparametric Statistics-when to use them and which is more powerful? Dr Mahmoud Alhussami
Parametric versus Nonparametric Statistics-when to use them and which is more powerful? Dr Mahmoud Alhussami Parametric Assumptions The observations must be independent. Dependent variable should be continuous
More informationHomework 1 Solutions
36-720 Homework 1 Solutions Problem 3.4 (a) X 2 79.43 and G 2 90.33. We should compare each to a χ 2 distribution with (2 1)(3 1) 2 degrees of freedom. For each, the p-value is so small that S-plus reports
More informationBIOS 625 Fall 2015 Homework Set 3 Solutions
BIOS 65 Fall 015 Homework Set 3 Solutions 1. Agresti.0 Table.1 is from an early study on the death penalty in Florida. Analyze these data and show that Simpson s Paradox occurs. Death Penalty Victim's
More informationSTAC51: Categorical data Analysis
STAC51: Categorical data Analysis Mahinda Samarakoon January 26, 2016 Mahinda Samarakoon STAC51: Categorical data Analysis 1 / 32 Table of contents Contingency Tables 1 Contingency Tables Mahinda Samarakoon
More informationCentral Limit Theorem ( 5.3)
Central Limit Theorem ( 5.3) Let X 1, X 2,... be a sequence of independent random variables, each having n mean µ and variance σ 2. Then the distribution of the partial sum S n = X i i=1 becomes approximately
More information8 Nominal and Ordinal Logistic Regression
8 Nominal and Ordinal Logistic Regression 8.1 Introduction If the response variable is categorical, with more then two categories, then there are two options for generalized linear models. One relies on
More informationSession 3 The proportional odds model and the Mann-Whitney test
Session 3 The proportional odds model and the Mann-Whitney test 3.1 A unified approach to inference 3.2 Analysis via dichotomisation 3.3 Proportional odds 3.4 Relationship with the Mann-Whitney test Session
More information" M A #M B. Standard deviation of the population (Greek lowercase letter sigma) σ 2
Notation and Equations for Final Exam Symbol Definition X The variable we measure in a scientific study n The size of the sample N The size of the population M The mean of the sample µ The mean of the
More informationReview of One-way Tables and SAS
Stat 504, Lecture 7 1 Review of One-way Tables and SAS In-class exercises: Ex1, Ex2, and Ex3 from http://v8doc.sas.com/sashtml/proc/z0146708.htm To calculate p-value for a X 2 or G 2 in SAS: http://v8doc.sas.com/sashtml/lgref/z0245929.htmz0845409
More informationAnalysis of data in square contingency tables
Analysis of data in square contingency tables Iva Pecáková Let s suppose two dependent samples: the response of the nth subject in the second sample relates to the response of the nth subject in the first
More information13.1 Categorical Data and the Multinomial Experiment
Chapter 13 Categorical Data Analysis 13.1 Categorical Data and the Multinomial Experiment Recall Variable: (numerical) variable (i.e. # of students, temperature, height,). (non-numerical, categorical)
More informationThe goodness-of-fit test Having discussed how to make comparisons between two proportions, we now consider comparisons of multiple proportions.
The goodness-of-fit test Having discussed how to make comparisons between two proportions, we now consider comparisons of multiple proportions. A common problem of this type is concerned with determining
More informationij i j m ij n ij m ij n i j Suppose we denote the row variable by X and the column variable by Y ; We can then re-write the above expression as
page1 Loglinear Models Loglinear models are a way to describe association and interaction patterns among categorical variables. They are commonly used to model cell counts in contingency tables. These
More informationNATIONAL UNIVERSITY OF SINGAPORE EXAMINATION (SOLUTIONS) ST3241 Categorical Data Analysis. (Semester II: )
NATIONAL UNIVERSITY OF SINGAPORE EXAMINATION (SOLUTIONS) Categorical Data Analysis (Semester II: 2010 2011) April/May, 2011 Time Allowed : 2 Hours Matriculation No: Seat No: Grade Table Question 1 2 3
More informationSTAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and Q-Q plots. March 8, 2015
STAT 135 Lab 6 Duality of Hypothesis Testing and Confidence Intervals, GLRT, Pearson χ 2 Tests and Q-Q plots March 8, 2015 The duality between CI and hypothesis testing The duality between CI and hypothesis
More informationST3241 Categorical Data Analysis I Multicategory Logit Models. Logit Models For Nominal Responses
ST3241 Categorical Data Analysis I Multicategory Logit Models Logit Models For Nominal Responses 1 Models For Nominal Responses Y is nominal with J categories. Let {π 1,, π J } denote the response probabilities
More informationSolution to Tutorial 7
1. (a) We first fit the independence model ST3241 Categorical Data Analysis I Semester II, 2012-2013 Solution to Tutorial 7 log µ ij = λ + λ X i + λ Y j, i = 1, 2, j = 1, 2. The parameter estimates are
More informationSTAT 7030: Categorical Data Analysis
STAT 7030: Categorical Data Analysis 5. Logistic Regression Peng Zeng Department of Mathematics and Statistics Auburn University Fall 2012 Peng Zeng (Auburn University) STAT 7030 Lecture Notes Fall 2012
More informationOne-sample categorical data: approximate inference
One-sample categorical data: approximate inference Patrick Breheny October 6 Patrick Breheny Biostatistical Methods I (BIOS 5710) 1/25 Introduction It is relatively easy to think about the distribution
More informationCDA Chapter 3 part II
CDA Chapter 3 part II Two-way tables with ordered classfications Let u 1 u 2... u I denote scores for the row variable X, and let ν 1 ν 2... ν J denote column Y scores. Consider the hypothesis H 0 : X
More informationThe material for categorical data follows Agresti closely.
Exam 2 is Wednesday March 8 4 sheets of notes The material for categorical data follows Agresti closely A categorical variable is one for which the measurement scale consists of a set of categories Categorical
More informationContingency Tables Part One 1
Contingency Tables Part One 1 STA 312: Fall 2012 1 See last slide for copyright information. 1 / 32 Suggested Reading: Chapter 2 Read Sections 2.1-2.4 You are not responsible for Section 2.5 2 / 32 Overview
More informationSTA 303 H1S / 1002 HS Winter 2011 Test March 7, ab 1cde 2abcde 2fghij 3
STA 303 H1S / 1002 HS Winter 2011 Test March 7, 2011 LAST NAME: FIRST NAME: STUDENT NUMBER: ENROLLED IN: (circle one) STA 303 STA 1002 INSTRUCTIONS: Time: 90 minutes Aids allowed: calculator. Some formulae
More informationReports of the Institute of Biostatistics
Reports of the Institute of Biostatistics No 02 / 2008 Leibniz University of Hannover Natural Sciences Faculty Title: Properties of confidence intervals for the comparison of small binomial proportions
More informationPoisson regression: Further topics
Poisson regression: Further topics April 21 Overdispersion One of the defining characteristics of Poisson regression is its lack of a scale parameter: E(Y ) = Var(Y ), and no parameter is available to
More informationOne-Way Tables and Goodness of Fit
Stat 504, Lecture 5 1 One-Way Tables and Goodness of Fit Key concepts: One-way Frequency Table Pearson goodness-of-fit statistic Deviance statistic Pearson residuals Objectives: Learn how to compute the
More informationLISA Short Course Series Generalized Linear Models (GLMs) & Categorical Data Analysis (CDA) in R. Liang (Sally) Shan Nov. 4, 2014
LISA Short Course Series Generalized Linear Models (GLMs) & Categorical Data Analysis (CDA) in R Liang (Sally) Shan Nov. 4, 2014 L Laboratory for Interdisciplinary Statistical Analysis LISA helps VT researchers
More informationOptimal exact tests for complex alternative hypotheses on cross tabulated data
Optimal exact tests for complex alternative hypotheses on cross tabulated data Daniel Yekutieli Statistics and OR Tel Aviv University CDA course 29 July 2017 Yekutieli (TAU) Optimal exact tests for complex
More informationI i=1 1 I(J 1) j=1 (Y ij Ȳi ) 2. j=1 (Y j Ȳ )2 ] = 2n( is the two-sample t-test statistic.
Serik Sagitov, Chalmers and GU, February, 08 Solutions chapter Matlab commands: x = data matrix boxplot(x) anova(x) anova(x) Problem.3 Consider one-way ANOVA test statistic For I = and = n, put F = MS
More informationST3241 Categorical Data Analysis I Generalized Linear Models. Introduction and Some Examples
ST3241 Categorical Data Analysis I Generalized Linear Models Introduction and Some Examples 1 Introduction We have discussed methods for analyzing associations in two-way and three-way tables. Now we will
More informationLecture 01: Introduction
Lecture 01: Introduction Dipankar Bandyopadhyay, Ph.D. BMTRY 711: Analysis of Categorical Data Spring 2011 Division of Biostatistics and Epidemiology Medical University of South Carolina Lecture 01: Introduction
More informationLecture 14: Introduction to Poisson Regression
Lecture 14: Introduction to Poisson Regression Ani Manichaikul amanicha@jhsph.edu 8 May 2007 1 / 52 Overview Modelling counts Contingency tables Poisson regression models 2 / 52 Modelling counts I Why
More informationModelling counts. Lecture 14: Introduction to Poisson Regression. Overview
Modelling counts I Lecture 14: Introduction to Poisson Regression Ani Manichaikul amanicha@jhsph.edu Why count data? Number of traffic accidents per day Mortality counts in a given neighborhood, per week
More informationSTAT Chapter 13: Categorical Data. Recall we have studied binomial data, in which each trial falls into one of 2 categories (success/failure).
STAT 515 -- Chapter 13: Categorical Data Recall we have studied binomial data, in which each trial falls into one of 2 categories (success/failure). Many studies allow for more than 2 categories. Example
More informationReview. Timothy Hanson. Department of Statistics, University of South Carolina. Stat 770: Categorical Data Analysis
Review Timothy Hanson Department of Statistics, University of South Carolina Stat 770: Categorical Data Analysis 1 / 22 Chapter 1: background Nominal, ordinal, interval data. Distributions: Poisson, binomial,
More informationHypothesis Testing, Power, Sample Size and Confidence Intervals (Part 2)
Hypothesis Testing, Power, Sample Size and Confidence Intervals (Part 2) B.H. Robbins Scholars Series June 23, 2010 1 / 29 Outline Z-test χ 2 -test Confidence Interval Sample size and power Relative effect
More informationChapter 2: Describing Contingency Tables - I
: Describing Contingency Tables - I Dipankar Bandyopadhyay Department of Biostatistics, Virginia Commonwealth University BIOS 625: Categorical Data & GLM [Acknowledgements to Tim Hanson and Haitao Chu]
More informationContingency Tables. Safety equipment in use Fatal Non-fatal Total. None 1, , ,128 Seat belt , ,878
Contingency Tables I. Definition & Examples. A) Contingency tables are tables where we are looking at two (or more - but we won t cover three or more way tables, it s way too complicated) factors, each
More informationChapter 10. Chapter 10. Multinomial Experiments and. Multinomial Experiments and Contingency Tables. Contingency Tables.
Chapter 10 Multinomial Experiments and Contingency Tables 1 Chapter 10 Multinomial Experiments and Contingency Tables 10-1 1 Overview 10-2 2 Multinomial Experiments: of-fitfit 10-3 3 Contingency Tables:
More informationNew Bayesian methods for model comparison
Back to the future New Bayesian methods for model comparison Murray Aitkin murray.aitkin@unimelb.edu.au Department of Mathematics and Statistics The University of Melbourne Australia Bayesian Model Comparison
More informationPart 1.) We know that the probability of any specific x only given p ij = p i p j is just multinomial(n, p) where p k1 k 2
Problem.) I will break this into two parts: () Proving w (m) = p( x (m) X i = x i, X j = x j, p ij = p i p j ). In other words, the probability of a specific table in T x given the row and column counts
More informationInterval estimation. October 3, Basic ideas CLT and CI CI for a population mean CI for a population proportion CI for a Normal mean
Interval estimation October 3, 2018 STAT 151 Class 7 Slide 1 Pandemic data Treatment outcome, X, from n = 100 patients in a pandemic: 1 = recovered and 0 = not recovered 1 1 1 0 0 0 1 1 1 0 0 1 0 1 0 0
More informationLikelihood-based inference with missing data under missing-at-random
Likelihood-based inference with missing data under missing-at-random Jae-kwang Kim Joint work with Shu Yang Department of Statistics, Iowa State University May 4, 014 Outline 1. Introduction. Parametric
More information2 and F Distributions. Barrow, Statistics for Economics, Accounting and Business Studies, 4 th edition Pearson Education Limited 2006
and F Distributions Lecture 9 Distribution The distribution is used to: construct confidence intervals for a variance compare a set of actual frequencies with expected frequencies test for association
More informationPartition of the Chi-Squared Statistic in a Contingency Table
Partition of the Chi-Squared Statistic in a Contingency Table by Jo Ann Colas Thesis submitted submitted to the Faculty of Graduate and Postdoctoral Studies In partial fulfillment of the requirements For
More informationSection 4.6 Simple Linear Regression
Section 4.6 Simple Linear Regression Objectives ˆ Basic philosophy of SLR and the regression assumptions ˆ Point & interval estimation of the model parameters, and how to make predictions ˆ Point and interval
More informationThe t-statistic. Student s t Test
The t-statistic 1 Student s t Test When the population standard deviation is not known, you cannot use a z score hypothesis test Use Student s t test instead Student s t, or t test is, conceptually, very
More information1/24/2008. Review of Statistical Inference. C.1 A Sample of Data. C.2 An Econometric Model. C.4 Estimating the Population Variance and Other Moments
/4/008 Review of Statistical Inference Prepared by Vera Tabakova, East Carolina University C. A Sample of Data C. An Econometric Model C.3 Estimating the Mean of a Population C.4 Estimating the Population
More informationGeneralized Linear Models. Kurt Hornik
Generalized Linear Models Kurt Hornik Motivation Assuming normality, the linear model y = Xβ + e has y = β + ε, ε N(0, σ 2 ) such that y N(μ, σ 2 ), E(y ) = μ = β. Various generalizations, including general
More informationDescribing Contingency tables
Today s topics: Describing Contingency tables 1. Probability structure for contingency tables (distributions, sensitivity/specificity, sampling schemes). 2. Comparing two proportions (relative risk, odds
More informationMISCELLANEOUS TOPICS RELATED TO LIKELIHOOD. Copyright c 2012 (Iowa State University) Statistics / 30
MISCELLANEOUS TOPICS RELATED TO LIKELIHOOD Copyright c 2012 (Iowa State University) Statistics 511 1 / 30 INFORMATION CRITERIA Akaike s Information criterion is given by AIC = 2l(ˆθ) + 2k, where l(ˆθ)
More informationSTATS 200: Introduction to Statistical Inference. Lecture 29: Course review
STATS 200: Introduction to Statistical Inference Lecture 29: Course review Course review We started in Lecture 1 with a fundamental assumption: Data is a realization of a random process. The goal throughout
More information2 Describing Contingency Tables
2 Describing Contingency Tables I. Probability structure of a 2-way contingency table I.1 Contingency Tables X, Y : cat. var. Y usually random (except in a case-control study), response; X can be random
More informationInterpret Standard Deviation. Outlier Rule. Describe the Distribution OR Compare the Distributions. Linear Transformations SOCS. Interpret a z score
Interpret Standard Deviation Outlier Rule Linear Transformations Describe the Distribution OR Compare the Distributions SOCS Using Normalcdf and Invnorm (Calculator Tips) Interpret a z score What is an
More informationLing 289 Contingency Table Statistics
Ling 289 Contingency Table Statistics Roger Levy and Christopher Manning This is a summary of the material that we ve covered on contingency tables. Contingency tables: introduction Odds ratios Counting,
More informationDecomposition of Parsimonious Independence Model Using Pearson, Kendall and Spearman s Correlations for Two-Way Contingency Tables
International Journal of Statistics and Probability; Vol. 7 No. 3; May 208 ISSN 927-7032 E-ISSN 927-7040 Published by Canadian Center of Science and Education Decomposition of Parsimonious Independence
More informationChapter 4: Generalized Linear Models-II
: Generalized Linear Models-II Dipankar Bandyopadhyay Department of Biostatistics, Virginia Commonwealth University BIOS 625: Categorical Data & GLM [Acknowledgements to Tim Hanson and Haitao Chu] D. Bandyopadhyay
More informationLecture 5: ANOVA and Correlation
Lecture 5: ANOVA and Correlation Ani Manichaikul amanicha@jhsph.edu 23 April 2007 1 / 62 Comparing Multiple Groups Continous data: comparing means Analysis of variance Binary data: comparing proportions
More informationMeans or "expected" counts: j = 1 j = 2 i = 1 m11 m12 i = 2 m21 m22 True proportions: The odds that a sampled unit is in category 1 for variable 1 giv
Measures of Association References: ffl ffl ffl Summarize strength of associations Quantify relative risk Types of measures odds ratio correlation Pearson statistic ediction concordance/discordance Goodman,
More informationInstitute of Actuaries of India
Institute of Actuaries of India Subject CT3 Probability & Mathematical Statistics May 2011 Examinations INDICATIVE SOLUTION Introduction The indicative solution has been written by the Examiners with the
More informationNominal Data. Parametric Statistics. Nonparametric Statistics. Parametric vs Nonparametric Tests. Greg C Elvers
Nominal Data Greg C Elvers 1 Parametric Statistics The inferential statistics that we have discussed, such as t and ANOVA, are parametric statistics A parametric statistic is a statistic that makes certain
More informationLoglikelihood and Confidence Intervals
Stat 504, Lecture 2 1 Loglikelihood and Confidence Intervals The loglikelihood function is defined to be the natural logarithm of the likelihood function, l(θ ; x) = log L(θ ; x). For a variety of reasons,
More informationOHSU OGI Class ECE-580-DOE :Design of Experiments Steve Brainerd
Why We Use Analysis of Variance to Compare Group Means and How it Works The question of how to compare the population means of more than two groups is an important one to researchers. Let us suppose that
More informationPractical Meta-Analysis -- Lipsey & Wilson
Overview of Meta-Analytic Data Analysis Transformations, Adjustments and Outliers The Inverse Variance Weight The Mean Effect Size and Associated Statistics Homogeneity Analysis Fixed Effects Analysis
More informationCorrespondence Analysis
Correspondence Analysis Q: when independence of a 2-way contingency table is rejected, how to know where the dependence is coming from? The interaction terms in a GLM contain dependence information; however,
More informationHypothesis Testing hypothesis testing approach
Hypothesis Testing In this case, we d be trying to form an inference about that neighborhood: Do people there shop more often those people who are members of the larger population To ascertain this, we
More informationCohen s s Kappa and Log-linear Models
Cohen s s Kappa and Log-linear Models HRP 261 03/03/03 10-11 11 am 1. Cohen s Kappa Actual agreement = sum of the proportions found on the diagonals. π ii Cohen: Compare the actual agreement with the chance
More informationPOLI 443 Applied Political Research
POLI 443 Applied Political Research Session 6: Tests of Hypotheses Contingency Analysis Lecturer: Prof. A. Essuman-Johnson, Dept. of Political Science Contact Information: aessuman-johnson@ug.edu.gh College
More informationHYPOTHESIS TESTING: THE CHI-SQUARE STATISTIC
1 HYPOTHESIS TESTING: THE CHI-SQUARE STATISTIC 7 steps of Hypothesis Testing 1. State the hypotheses 2. Identify level of significant 3. Identify the critical values 4. Calculate test statistics 5. Compare
More informationFrequency Distribution Cross-Tabulation
Frequency Distribution Cross-Tabulation 1) Overview 2) Frequency Distribution 3) Statistics Associated with Frequency Distribution i. Measures of Location ii. Measures of Variability iii. Measures of Shape
More informationThe 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 9 Pearson s chi-square test 9. 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
More informationAnswer Key for STAT 200B HW No. 8
Answer Key for STAT 200B HW No. 8 May 8, 2007 Problem 3.42 p. 708 The values of Ȳ for x 00, 0, 20, 30 are 5/40, 0, 20/50, and, respectively. From Corollary 3.5 it follows that MLE exists i G is identiable
More information