ttests and Ftests in regression


 Moses Fitzgerald
 1 years ago
 Views:
Transcription
1 ttests and Ftests in regression Johan A. Elkink University College Dublin 5 April 2012 Johan A. Elkink (UCD) t and Ftests 5 April / 25
2 Outline 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
3 Outline Simple linear regression 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
4 Outline Simple linear regression Model 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
5 Linear equations Simple linear regression Model y = Intercept +Slope x Johan A. Elkink (UCD) t and Ftests 5 April / 25
6 Linear equations Simple linear regression Model Intercept y = Intercept +Slope x Johan A. Elkink (UCD) t and Ftests 5 April / 25
7 Linear equations Simple linear regression Model x y = Intercept +Slope x Johan A. Elkink (UCD) t and Ftests 5 April / 25
8 Linear equations Simple linear regression Model Slope x x y = Intercept +Slope x Johan A. Elkink (UCD) t and Ftests 5 April / 25
9 Simple linear regression Simple regression model Model y x Johan A. Elkink (UCD) t and Ftests 5 April / 25
10 Simple linear regression Simple regression model Model y x y = Intercept +Slope x Johan A. Elkink (UCD) t and Ftests 5 April / 25
11 Simple linear regression Simple regression model Model y x y = β 0 +β 1 x Johan A. Elkink (UCD) t and Ftests 5 April / 25
12 Simple linear regression Simple regression model Model y x y i = β 0 +β 1 x i +ε i Johan A. Elkink (UCD) t and Ftests 5 April / 25
13 Notation Simple linear regression Model y i x i x ε i β k ˆβ k ŷ i Value on the dependent variable for case i Value on the independent variable for case i Mean value on the independent variable for case i The error for case i: ε i = y i ŷ i True coefficient for variable k Estimated coefficient for variable k Predicted value on the dependent variable for case i Johan A. Elkink (UCD) t and Ftests 5 April / 25
14 Ordinary Least Squares Simple linear regression Model Quickly put, the regression line is chosen to minimize the RSS; it has slope ˆβ 1, intercept ˆβ 0, and goes through the point ( x,ȳ). Furthermore, the estimate for σ 2 is ˆσ 2 = RSS/(n 2) (Verzani 2005: 280). Johan A. Elkink (UCD) t and Ftests 5 April / 25
15 Outline Simple linear regression Variance and R 2 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
16 Breakdown of variance Simple linear regression Variance and R 2 Total Sum of Squares (TSS): Explained Sum of Squares (ESS): Residual Sum of Squares (RSS): N i=1 (y i ȳ) 2 N i=1 (ŷ i ȳ) 2 N i=1 (y i ŷ i ) 2 = N i=1 ε2 i TSS = ESS +RSS Johan A. Elkink (UCD) t and Ftests 5 April / 25
17 Breakdown of variance Simple linear regression Variance and R 2 Total Sum of Squares (TSS): Explained Sum of Squares (ESS): Residual Sum of Squares (RSS): N i=1 (y i ȳ) 2 N i=1 (ŷ i ȳ) 2 N i=1 (y i ŷ i ) 2 = N i=1 ε2 i TSS = ESS +RSS Sometimes the second is called regression sum of squares (RSS) and the third errors sum of squares (ESS), which might in fact be more accurate, since ε really represents errors, not residuals, in this specification. Beware the confusion! Johan A. Elkink (UCD) t and Ftests 5 April / 25
18 Simple linear regression Variance and R 2 R 2 How much of the variance did we explain? Johan A. Elkink (UCD) t and Ftests 5 April / 25
19 Simple linear regression Variance and R 2 R 2 How much of the variance did we explain? R 2 = 1 RSS N TSS = 1 i=1 (y i ŷ i ) 2 N N i=1 (y i ȳ) = i=1 (ŷ i ȳ) 2 2 N i=1 (y i ȳ) 2 Can be interpreted as the proportion of total variance explained by the model. Johan A. Elkink (UCD) t and Ftests 5 April / 25
20 Simple linear regression Variance and R 2 R 2 How much of the variance did we explain? R 2 = 1 RSS N TSS = 1 i=1 (y i ŷ i ) 2 N N i=1 (y i ȳ) = i=1 (ŷ i ȳ) 2 2 N i=1 (y i ȳ) 2 Can be interpreted as the proportion of total variance explained by the model. For this interpretation, the model must include an intercept. Generally, one should not attach too much value to having a high R 2  it is usually more important to understand whether x affects y and by how much, rather than to understand how much of y has not yet been explained. Johan A. Elkink (UCD) t and Ftests 5 April / 25
21 Simple linear regression Variance and R 2 R 2 How much of the variance did we explain? R 2 = 1 RSS N TSS = 1 i=1 (y i ŷ i ) 2 N N i=1 (y i ȳ) = i=1 (ŷ i ȳ) 2 2 N i=1 (y i ȳ) 2 Can be interpreted as the proportion of total variance explained by the model. For this interpretation, the model must include an intercept. Generally, one should not attach too much value to having a high R 2  it is usually more important to understand whether x affects y and by how much, rather than to understand how much of y has not yet been explained. For simple linear regression (i.e. one independent variable), R 2 is the same as the correlation coefficient, Pearson s r, squared. Johan A. Elkink (UCD) t and Ftests 5 April / 25
22 Outline Inference 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
23 Outline Inference ttest 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
24 Inference from regression Inference ttest In linear regression, the sampling distribution of the coefficient estimates form a normal distribution, which is approximated by a t distribution due to approximating σ by s. Johan A. Elkink (UCD) t and Ftests 5 April / 25
25 Inference from regression Inference ttest In linear regression, the sampling distribution of the coefficient estimates form a normal distribution, which is approximated by a t distribution due to approximating σ by s. Thus we can calculate a confidence interval for each estimated coefficient. Johan A. Elkink (UCD) t and Ftests 5 April / 25
26 Inference ttest Inference from regression In linear regression, the sampling distribution of the coefficient estimates form a normal distribution, which is approximated by a t distribution due to approximating σ by s. Thus we can calculate a confidence interval for each estimated coefficient. Or perform a hypothesis test along the lines of: H 0 :β 1 = 0 H 1 :β 1 0 Johan A. Elkink (UCD) t and Ftests 5 April / 25
27 Inference from regression Inference ttest To calculate the confidence interval, we need to calculate the standard error of the coefficient. Johan A. Elkink (UCD) t and Ftests 5 April / 25
28 Inference from regression Inference ttest To calculate the confidence interval, we need to calculate the standard error of the coefficient. Rule of thumb to get the 95% confidence interval: β 2SE < β < β +2SE Johan A. Elkink (UCD) t and Ftests 5 April / 25
29 Inference from regression Inference ttest To calculate the confidence interval, we need to calculate the standard error of the coefficient. Rule of thumb to get the 95% confidence interval: β 2SE < β < β +2SE Thus if β is positive, we are 95% certain it is different from zero when β 2SE > 0. Johan A. Elkink (UCD) t and Ftests 5 April / 25
30 Inference from regression Inference ttest To calculate the confidence interval, we need to calculate the standard error of the coefficient. Rule of thumb to get the 95% confidence interval: β 2SE < β < β +2SE Thus if β is positive, we are 95% certain it is different from zero when β 2SE > 0. (Or when the t value is greater than 2 or less than 2.) Johan A. Elkink (UCD) t and Ftests 5 April / 25
31 Outline Inference Ftest 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
32 Breakdown of variance Inference Ftest Total Sum of Squares (TSS): Explained Sum of Squares (ESS): Residual Sum of Squares (RSS): N i=1 (y i ȳ) 2 N i=1 (ŷ i ȳ) 2 N i=1 (y i ŷ i ) 2 = N i=1 ε2 i TSS = ESS +RSS Johan A. Elkink (UCD) t and Ftests 5 April / 25
33 Inference Ftest Ftest In simple linear regression, we can do an Ftest: H 0 :β 1 = 0 H 1 :β 1 0 Johan A. Elkink (UCD) t and Ftests 5 April / 25
34 Inference Ftest Ftest In simple linear regression, we can do an Ftest: H 0 :β 1 = 0 H 1 :β 1 0 F = with 1 and n 2 degrees of freedom. ESS/1 RSS/(n 2) = ESS ˆσ 2 F 1,n 2 Johan A. Elkink (UCD) t and Ftests 5 April / 25
35 Inference Ftest Ftest In simple linear regression, we can do an Ftest: H 0 :β 1 = 0 H 1 :β 1 0 F = with 1 and n 2 degrees of freedom. ESS/1 RSS/(n 2) = ESS ˆσ 2 F 1,n 2 For multiple regression, this would generalize to: F = ESS/(k 1) RSS/(n k) F k 1,n k Johan A. Elkink (UCD) t and Ftests 5 April / 25
36 Outline Exercises 1 Simple linear regression Model Variance and R 2 2 Inference ttest Ftest 3 Exercises Johan A. Elkink (UCD) t and Ftests 5 April / 25
37 Exercises Exercise Open oecd 1960.sav. Repeat for both industry (IND) and services (AGR): 1 Regress percentage in sector on income per capita. 2 Interpret the regression results. 3 Evaluate the model fit. 4 Interpret the t and Ftests. Johan A. Elkink (UCD) t and Ftests 5 April / 25
38 Exercises Exercise Open bes class data.sav and investigate: whether leftright selfplacement (lr) and explains trust in politics (trustpol) whether leftright selfplacement influences attitude towards EU membership (eumember) Johan A. Elkink (UCD) t and Ftests 5 April / 25
Simple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationCHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression
Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the
More informationMultiple Linear Regression
Multiple Linear Regression A regression with two or more explanatory variables is called a multiple regression. Rather than modeling the mean response as a straight line, as in simple regression, it is
More informationSimple Linear Regression, Scatterplots, and Bivariate Correlation
1 Simple Linear Regression, Scatterplots, and Bivariate Correlation This section covers procedures for testing the association between two continuous variables using the SPSS Regression and Correlate analyses.
More informationUnit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression
Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a
More informationFactors affecting online sales
Factors affecting online sales Table of contents Summary... 1 Research questions... 1 The dataset... 2 Descriptive statistics: The exploratory stage... 3 Confidence intervals... 4 Hypothesis tests... 4
More informationRegression Analysis: A Complete Example
Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty
More informationChapter 7: Simple linear regression Learning Objectives
Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) 
More information" Y. Notation and Equations for Regression Lecture 11/4. Notation:
Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through
More information2. What is the general linear model to be used to model linear trend? (Write out the model) = + + + or
Simple and Multiple Regression Analysis Example: Explore the relationships among Month, Adv.$ and Sales $: 1. Prepare a scatter plot of these data. The scatter plots for Adv.$ versus Sales, and Month versus
More informationSPSS Guide: Regression Analysis
SPSS Guide: Regression Analysis I put this together to give you a stepbystep guide for replicating what we did in the computer lab. It should help you run the tests we covered. The best way to get familiar
More informationUsing R for Linear Regression
Using R for Linear Regression In the following handout words and symbols in bold are R functions and words and symbols in italics are entries supplied by the user; underlined words and symbols are optional
More informationHYPOTHESIS TESTING: CONFIDENCE INTERVALS, TTESTS, ANOVAS, AND REGRESSION
HYPOTHESIS TESTING: CONFIDENCE INTERVALS, TTESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate
More information1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96
1 Final Review 2 Review 2.1 CI 1propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years
More informationModule 5: Statistical Analysis
Module 5: Statistical Analysis To answer more complex questions using your data, or in statistical terms, to test your hypothesis, you need to use more advanced statistical tests. This module reviews the
More informationStat 412/512 CASE INFLUENCE STATISTICS. Charlotte Wickham. stat512.cwick.co.nz. Feb 2 2015
Stat 412/512 CASE INFLUENCE STATISTICS Feb 2 2015 Charlotte Wickham stat512.cwick.co.nz Regression in your field See website. You may complete this assignment in pairs. Find a journal article in your field
More information2. Simple Linear Regression
Research methods  II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according
More information17. SIMPLE LINEAR REGRESSION II
17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.
More informationBill Burton Albert Einstein College of Medicine william.burton@einstein.yu.edu April 28, 2014 EERS: Managing the Tension Between Rigor and Resources 1
Bill Burton Albert Einstein College of Medicine william.burton@einstein.yu.edu April 28, 2014 EERS: Managing the Tension Between Rigor and Resources 1 Calculate counts, means, and standard deviations Produce
More informationPremaster Statistics Tutorial 4 Full solutions
Premaster Statistics Tutorial 4 Full solutions Regression analysis Q1 (based on Doane & Seward, 4/E, 12.7) a. Interpret the slope of the fitted regression = 125,000 + 150. b. What is the prediction for
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 informationChapter 13 Introduction to Linear Regression and Correlation Analysis
Chapter 3 Student Lecture Notes 3 Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing
More informationNCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )
Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates
More informationMultiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear.
Multiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear. In the main dialog box, input the dependent variable and several predictors.
More informationThe importance of graphing the data: Anscombe s regression examples
The importance of graphing the data: Anscombe s regression examples Bruce Weaver Northern Health Research Conference Nipissing University, North Bay May 3031, 2008 B. Weaver, NHRC 2008 1 The Objective
More information4. Simple regression. QBUS6840 Predictive Analytics. https://www.otexts.org/fpp/4
4. Simple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/4 Outline The simple linear model Least squares estimation Forecasting with regression Nonlinear functional forms Regression
More informationIllustration (and the use of HLM)
Illustration (and the use of HLM) Chapter 4 1 Measurement Incorporated HLM Workshop The Illustration Data Now we cover the example. In doing so we does the use of the software HLM. In addition, we will
More informationCourse Objective This course is designed to give you a basic understanding of how to run regressions in SPSS.
SPSS Regressions Social Science Research Lab American University, Washington, D.C. Web. www.american.edu/provost/ctrl/pclabs.cfm Tel. x3862 Email. SSRL@American.edu Course Objective This course is designed
More informationPenalized regression: Introduction
Penalized regression: Introduction Patrick Breheny August 30 Patrick Breheny BST 764: Applied Statistical Modeling 1/19 Maximum likelihood Much of 20thcentury statistics dealt with maximum likelihood
More information5. Linear Regression
5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4
More informationCoefficient of Determination
Coefficient of Determination The coefficient of determination R 2 (or sometimes r 2 ) is another measure of how well the least squares equation ŷ = b 0 + b 1 x performs as a predictor of y. R 2 is computed
More informationLinear Models in STATA and ANOVA
Session 4 Linear Models in STATA and ANOVA Page Strengths of Linear Relationships 42 A Note on NonLinear Relationships 44 Multiple Linear Regression 45 Removal of Variables 48 Independent Samples
More informationAnswer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade
Statistics Quiz Correlation and Regression  ANSWERS 1. Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Boston and Montreal. Boston makes their measurements
More informationTeaching Business Statistics through Problem Solving
Teaching Business Statistics through Problem Solving David M. Levine, Baruch College, CUNY with David F. Stephan, Two Bridges Instructional Technology CONTACT: davidlevine@davidlevinestatistics.com Typical
More information2013 MBA Jump Start Program. Statistics Module Part 3
2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just
More informationMULTIPLE LINEAR REGRESSION ANALYSIS USING MICROSOFT EXCEL. by Michael L. Orlov Chemistry Department, Oregon State University (1996)
MULTIPLE LINEAR REGRESSION ANALYSIS USING MICROSOFT EXCEL by Michael L. Orlov Chemistry Department, Oregon State University (1996) INTRODUCTION In modern science, regression analysis is a necessary part
More informationLecture 11: Confidence intervals and model comparison for linear regression; analysis of variance
Lecture 11: Confidence intervals and model comparison for linear regression; analysis of variance 14 November 2007 1 Confidence intervals and hypothesis testing for linear regression Just as there was
More informationSlopeIntercept Equation. Example
1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the yintercept. Determine
More informationtable to see that the probability is 0.8413. (b) What is the probability that x is between 16 and 60? The zscores for 16 and 60 are: 60 38 = 1.
Review Problems for Exam 3 Math 1040 1 1. Find the probability that a standard normal random variable is less than 2.37. Looking up 2.37 on the normal table, we see that the probability is 0.9911. 2. Find
More informationRegression stepbystep using Microsoft Excel
Step 1: Regression stepbystep using Microsoft Excel Notes prepared by Pamela Peterson Drake, James Madison University Type the data into the spreadsheet The example used throughout this How to is a regression
More informationElements of statistics (MATH04871)
Elements of statistics (MATH04871) Prof. Dr. Dr. K. Van Steen University of Liège, Belgium December 10, 2012 Introduction to Statistics Basic Probability Revisited Sampling Exploratory Data Analysis 
More informationRegression Analysis (Spring, 2000)
Regression Analysis (Spring, 2000) By Wonjae Purposes: a. Explaining the relationship between Y and X variables with a model (Explain a variable Y in terms of Xs) b. Estimating and testing the intensity
More informationStatistics courses often teach the twosample ttest, linear regression, and analysis of variance
2 Making Connections: The TwoSample ttest, Regression, and ANOVA In theory, there s no difference between theory and practice. In practice, there is. Yogi Berra 1 Statistics courses often teach the twosample
More informationCorrelation and Simple Linear Regression
Correlation and Simple Linear Regression We are often interested in studying the relationship among variables to determine whether they are associated with one another. When we think that changes in a
More informationPredictor Coef StDev T P Constant 970667056 616256122 1.58 0.154 X 0.00293 0.06163 0.05 0.963. S = 0.5597 RSq = 0.0% RSq(adj) = 0.
Statistical analysis using Microsoft Excel Microsoft Excel spreadsheets have become somewhat of a standard for data storage, at least for smaller data sets. This, along with the program often being packaged
More informationPITFALLS IN TIME SERIES ANALYSIS. Cliff Hurvich Stern School, NYU
PITFALLS IN TIME SERIES ANALYSIS Cliff Hurvich Stern School, NYU The t Test If x 1,..., x n are independent and identically distributed with mean 0, and n is not too small, then t = x 0 s n has a standard
More informationAugust 2012 EXAMINATIONS Solution Part I
August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,
More informationSimple Linear Regression
Chapter Nine Simple Linear Regression Consider the following three scenarios: 1. The CEO of the local Tourism Authority would like to know whether a family s annual expenditure on recreation is related
More informationMISSING DATA TECHNIQUES WITH SAS. IDRE Statistical Consulting Group
MISSING DATA TECHNIQUES WITH SAS IDRE Statistical Consulting Group ROAD MAP FOR TODAY To discuss: 1. Commonly used techniques for handling missing data, focusing on multiple imputation 2. Issues that could
More information11. Analysis of Casecontrol Studies Logistic Regression
Research methods II 113 11. Analysis of Casecontrol Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:
More informationSTATISTICA Formula Guide: Logistic Regression. Table of Contents
: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 SigmaRestricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary
More informationLinear Regression. Chapter 5. Prediction via Regression Line Number of new birds and Percent returning. Least Squares
Linear Regression Chapter 5 Regression Objective: To quantify the linear relationship between an explanatory variable (x) and response variable (y). We can then predict the average response for all subjects
More information5. Multiple regression
5. Multiple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/5 QBUS6840 Predictive Analytics 5. Multiple regression 2/39 Outline Introduction to multiple linear regression Some useful
More information1 Simple Linear Regression I Least Squares Estimation
Simple Linear Regression I Least Squares Estimation Textbook Sections: 8. 8.3 Previously, we have worked with a random variable x that comes from a population that is normally distributed with mean µ and
More informationMultiple Linear Regression in Data Mining
Multiple Linear Regression in Data Mining Contents 2.1. A Review of Multiple Linear Regression 2.2. Illustration of the Regression Process 2.3. Subset Selection in Linear Regression 1 2 Chap. 2 Multiple
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationWe extended the additive model in two variables to the interaction model by adding a third term to the equation.
Quadratic Models We extended the additive model in two variables to the interaction model by adding a third term to the equation. Similarly, we can extend the linear model in one variable to the quadratic
More informationTesting for Lack of Fit
Chapter 6 Testing for Lack of Fit How can we tell if a model fits the data? If the model is correct then ˆσ 2 should be an unbiased estimate of σ 2. If we have a model which is not complex enough to fit
More informationCantonMassillon PM2.5 Nonattainment Area Monitor Missing Data Analysis
CantonMassillon PM2.5 Nonattainment Area Monitor Missing Data Analysis The current CantonMassillon nonattainment area is located in norast Ohio and includes Stark County. The area has two monitors measuring
More informationSimple Linear Regression
STAT 101 Dr. Kari Lock Morgan Simple Linear Regression SECTIONS 9.3 Confidence and prediction intervals (9.3) Conditions for inference (9.1) Want More Stats??? If you have enjoyed learning how to analyze
More informationFinancial Risk Management Exam Sample Questions/Answers
Financial Risk Management Exam Sample Questions/Answers Prepared by Daniel HERLEMONT 1 2 3 4 5 6 Chapter 3 Fundamentals of Statistics FRM99, Question 4 Random walk assumes that returns from one time period
More informationLAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK DEPARTMENT OF MATHEMATICS, ENGINEERING, AND COMPUTER SCIENCE
LAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK DEPARTMENT OF MATHEMATICS, ENGINEERING, AND COMPUTER SCIENCE MAT 119 STATISTICS AND ELEMENTARY ALGEBRA 5 Lecture Hours, 2 Lab Hours, 3 Credits Pre
More informationUsing Excel for Statistical Analysis
Using Excel for Statistical Analysis You don t have to have a fancy pants statistics package to do many statistical functions. Excel can perform several statistical tests and analyses. First, make sure
More informationHow To Run Statistical Tests in Excel
How To Run Statistical Tests in Excel Microsoft Excel is your best tool for storing and manipulating data, calculating basic descriptive statistics such as means and standard deviations, and conducting
More informationOverview Classes. 123 Logistic regression (5) 193 Building and applying logistic regression (6) 263 Generalizations of logistic regression (7)
Overview Classes 123 Logistic regression (5) 193 Building and applying logistic regression (6) 263 Generalizations of logistic regression (7) 24 Loglinear models (8) 54 1517 hrs; 5B02 Building and
More informationStatistics 305: Introduction to Biostatistical Methods for Health Sciences
Statistics 305: Introduction to Biostatistical Methods for Health Sciences Modelling the Log Odds Logistic Regression (Chap 20) Instructor: Liangliang Wang Statistics and Actuarial Science, Simon Fraser
More informationCategorical Data Analysis
Richard L. Scheaffer University of Florida The reference material and many examples for this section are based on Chapter 8, Analyzing Association Between Categorical Variables, from Statistical Methods
More informationChapter 23. Inferences for Regression
Chapter 23. Inferences for Regression Topics covered in this chapter: Simple Linear Regression Simple Linear Regression Example 23.1: Crying and IQ The Problem: Infants who cry easily may be more easily
More informationCase Study in Data Analysis Does a drug prevent cardiomegaly in heart failure?
Case Study in Data Analysis Does a drug prevent cardiomegaly in heart failure? Harvey Motulsky hmotulsky@graphpad.com This is the first case in what I expect will be a series of case studies. While I mention
More informationA Primer on Forecasting Business Performance
A Primer on Forecasting Business Performance There are two common approaches to forecasting: qualitative and quantitative. Qualitative forecasting methods are important when historical data is not available.
More informationApplying Statistics Recommended by Regulatory Documents
Applying Statistics Recommended by Regulatory Documents Steven Walfish President, Statistical Outsourcing Services steven@statisticaloutsourcingservices.com 301325 32531293129 About the Speaker Mr. Steven
More informationCurve Fitting. Before You Begin
Curve Fitting Chapter 16: Curve Fitting Before You Begin Selecting the Active Data Plot When performing linear or nonlinear fitting when the graph window is active, you must make the desired data plot
More informationSAS Software to Fit the Generalized Linear Model
SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling
More informationChapter 5 Estimating Demand Functions
Chapter 5 Estimating Demand Functions 1 Why do you need statistics and regression analysis? Ability to read market research papers Analyze your own data in a simple way Assist you in pricing and marketing
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 informationData Analysis Tools. Tools for Summarizing Data
Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool
More informationCurriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 20092010
Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 20092010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different
More informationAnalyzing Intervention Effects: Multilevel & Other Approaches. Simplest Intervention Design. Better Design: Have Pretest
Analyzing Intervention Effects: Multilevel & Other Approaches Joop Hox Methodology & Statistics, Utrecht Simplest Intervention Design R X Y E Random assignment Experimental + Control group Analysis: t
More informationT O P I C 1 2 Techniques and tools for data analysis Preview Introduction In chapter 3 of Statistics In A Day different combinations of numbers and types of variables are presented. We go through these
More informationANALYSING LIKERT SCALE/TYPE DATA, ORDINAL LOGISTIC REGRESSION EXAMPLE IN R.
ANALYSING LIKERT SCALE/TYPE DATA, ORDINAL LOGISTIC REGRESSION EXAMPLE IN R. 1. Motivation. Likert items are used to measure respondents attitudes to a particular question or statement. One must recall
More informationSection 13, Part 1 ANOVA. Analysis Of Variance
Section 13, Part 1 ANOVA Analysis Of Variance Course Overview So far in this course we ve covered: Descriptive statistics Summary statistics Tables and Graphs Probability Probability Rules Probability
More informationINTRODUCTION TO MULTIPLE CORRELATION
CHAPTER 13 INTRODUCTION TO MULTIPLE CORRELATION Chapter 12 introduced you to the concept of partialling and how partialling could assist you in better interpreting the relationship between two primary
More informationCOMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES.
277 CHAPTER VI COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. This chapter contains a full discussion of customer loyalty comparisons between private and public insurance companies
More informationData Mining Techniques Chapter 5: The Lure of Statistics: Data Mining Using Familiar Tools
Data Mining Techniques Chapter 5: The Lure of Statistics: Data Mining Using Familiar Tools Occam s razor.......................................................... 2 A look at data I.........................................................
More informationbusiness statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar
business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel
More informationData Mining Techniques Chapter 6: Decision Trees
Data Mining Techniques Chapter 6: Decision Trees What is a classification decision tree?.......................................... 2 Visualizing decision trees...................................................
More informationAnalysing Questionnaires using Minitab (for SPSS queries contact ) Graham.Currell@uwe.ac.uk
Analysing Questionnaires using Minitab (for SPSS queries contact ) Graham.Currell@uwe.ac.uk Structure As a starting point it is useful to consider a basic questionnaire as containing three main sections:
More informationDirections for using SPSS
Directions for using SPSS Table of Contents Connecting and Working with Files 1. Accessing SPSS... 2 2. Transferring Files to N:\drive or your computer... 3 3. Importing Data from Another File Format...
More informationPoint Biserial Correlation Tests
Chapter 807 Point Biserial Correlation Tests Introduction The point biserial correlation coefficient (ρ in this chapter) is the productmoment correlation calculated between a continuous random variable
More informationSpecifications for this HLM2 run
One way ANOVA model 1. How much do U.S. high schools vary in their mean mathematics achievement? 2. What is the reliability of each school s sample mean as an estimate of its true population mean? 3. Do
More informationUNIT 1: COLLECTING DATA
Core Probability and Statistics Probability and Statistics provides a curriculum focused on understanding key data analysis and probabilistic concepts, calculations, and relevance to realworld applications.
More informationIntroduction to Linear Regression
14. Regression A. Introduction to Simple Linear Regression B. Partitioning Sums of Squares C. Standard Error of the Estimate D. Inferential Statistics for b and r E. Influential Observations F. Regression
More informationDATA INTERPRETATION AND STATISTICS
PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE
More informationThe VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.
Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium
More informationAlgebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
More informationKSTAT MINIMANUAL. Decision Sciences 434 Kellogg Graduate School of Management
KSTAT MINIMANUAL Decision Sciences 434 Kellogg Graduate School of Management Kstat is a set of macros added to Excel and it will enable you to do the statistics required for this course very easily. To
More informationNote on growth and growth accounting
CHAPTER 0 Note on growth and growth accounting 1. Growth and the growth rate In this section aspects of the mathematical concept of the rate of growth used in growth models and in the empirical analysis
More informationBowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition
Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Online Learning Centre Technology StepbyStep  Excel Microsoft Excel is a spreadsheet software application
More information430 Statistics and Financial Mathematics for Business
Prescription: 430 Statistics and Financial Mathematics for Business Elective prescription Level 4 Credit 20 Version 2 Aim Students will be able to summarise, analyse, interpret and present data, make predictions
More informationBasic Statistics and Data Analysis for Health Researchers from Foreign Countries
Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Volkert Siersma siersma@sund.ku.dk The Research Unit for General Practice in Copenhagen Dias 1 Content Quantifying association
More information