Interaction between quantitative predictors


 Ophelia Lyons
 1 years ago
 Views:
Transcription
1 Interaction between quantitative predictors In a firstorder model like the ones we have discussed, the association between E(y) and a predictor x j does not depend on the value of the other predictors in the model. See Fig. 4.1: relation between E(y) and x 1 is the same regardless of the value of x 2 : all the prediction lines are parallel. If, however, the association between response and one of the predictors depends on the value of other predictors, then a firstorder model is no longer appropriate. We say that there is an interaction among predictors. Stat Fall
2 Interaction (cont d) Example: a company wishes to estimate the association between sales of a beauty product (y) and two potential predictors of sales in each of n markets: $ spent on daytime TV ads in ith market (x 1 ) and average number of years of education of females in ith market. Intuitively, this is what we would expect: Advertisement expenses will tend to increase sales (up to a point). In cities where women are highly educated (on the average), less of them will be watching TV during the day. The effect of $ in ads on sales may then also depend on education of potential consumers. Stat Fall
3 Interaction (cont d) A figure to represent the association between ads and sales for different levels of education will be drawn in class. How do we include an interaction term in the model? With k = 2 predictors: y i = β 0 + β 1 x 1i + β 2 x 2i + β 3 x 1i x 2i + ɛ i, where the assumptions about the model are the same as before. An interaction between two predictors is a secondorder term in the model. Stat Fall
4 Interaction (cont d) In sales example, we would expect that β 3 < 0: as education increases (and more women are out working), the strength of the association between daytime TV ads on sales decreases. In other words, daytime ads are expected to be more effective in markets where more women are at home watching TV during the day than in markets where most women are not watching TV. In general, with k predictors, we can include pairwise interactions between any two, as appropriate. Higher order interactions (e.g. x j x l x t denoting the threeway interaction between the jth, lth and tth predictors) can also be included in the model, but are much harder to interpret from a subject matter point of view. Stat Fall
5 Interaction (cont d) When predictors interact, the interpretation of all the β s changes. If the model is y i = β 0 + β 1 x 1i + β 2 x 2i + β 3 x 1i x 2i + ɛ i, β 0 is still interpreted as before. (β 1 + β 3 x 2 ) is change in E(y) when x 1 increases by one unit and x 2 is held fixed. (β 2 + β 3 x 1 ) is change in E(y) when x 2 increases by one unit and x 1 is held fixed. Association between E(y) and x 1 depends on level of x 2, unless β 3 = 0, in which case interaction does not exist. Stat Fall
6 Interaction (cont d) In sales example, suppose we find that: b 0 = 5, b 1 = 3, b 2 = 0.5, b 3 = 0.2. Interpretation? Number of units sold can be expected to change by 3 0.2x 2 when ad expenses increase by $1 given education. Number of units sold can be expected to change by x 1 when education of potential customers increases by one year, given ad expenditures. In a market with 12 years of average education, we expect that sales will increase by 30.2(12) = 0.6 units if ad expenditures increase by $1. In a market with average education equal to 8 years, an additional $1 spent on daytime ads would be associated to an increase of about 1.4 units in expected sales. Stat Fall
7 Interaction (cont d) How do we draw inferences in models with interaction terms? Steps would be the same as in any multiple regression model: 1. Do a global F test of the utility of the model. The null hypothesis in this case is H 0 : β 1 = β 2 =... = β k = 0, tested against the alternative that says that at least one of the β s is different from If F test leads to rejection of H 0, then do a t test on each of the β s associated to interaction terms. 3. If interaction between x j and x k is significant, do not test hypothesis for β j and β k ; if the interaction is important, the individual x s must be important too (some statisticians would argue different here). Stat Fall
8 Second order model with quadratic predictors Sometimes, the association between E(y) and x j quadratic. is not linear but A second order model with one predictor is: y i = β 0 + β 1 x 1i + β 2 x 2 1i + ɛ i. If β 2 > 0: association is concave upwards (bowl shape). If β 2 < 0: concave downwards (mound shape). β 2 is known as a rate of curvature parameter. Stat Fall
9 Quadratic predictors  Example Example 4.6, page 198. Data: y is immunoglobin in blood (indicator of immunity, in mgrs) and x is maximum oxygen uptake (indicator of fitness, in ml/kg) measured on 30 individuals. Range: x (32, 70). See scatter plot of data. Model: with usual assumptions. y i = β 0 + β 1 x i + β 2 x 2 i + ɛ i, Stat Fall
10 Quadratic predictors  Example Results: b 0 = 1, 464, b 1 = 88.3 and b 2 = 0.54, so that the prediction equation is ŷ = 1, x 0.54x 2. R 2 a = 0.93 so about 93% of the variability observed in immunoglobin can be associated to fitness. Interpretation of coefficients: The intercept is meaningless. Cannot have negative immunoglobin. b 1 no longer has a simple interpretation. It is NOT the expected change in y when x increases by one. The quadratic term b 2 is negative: response curves downwards as x increases. Stat Fall
11 Quadratic predictors  Example Be cautious with extrapolations! See Fig Concavity of response implies that for large enough x the E(y) will begin to decrease. This makes no sense from a physiology point of view. Nonsensical predictions may occur if the model is used outside of the range of the data! Stat Fall
12 Quadratic predictors  Example First test of hypotheses is F test for entire model. We test: H 0 : β 1 = β 2 = 0, against H a : at least one of the two 0. In this example, F = which we know will be larger than the critical value even without looking at the table. We reject H 0 : maximal oxygen uptake contributes information about immunoglobin levels in the blood. Next step is to decide whether curvature is important or not. Stat Fall
13 Quadratic predictors  Example We now test for significance of the quadratic effect: H 0 : β 2 = 0 against H a : β 2 0 (or we can do a onetailed test too). t statistic is t = b 2 /ˆσ b2 = 0.536/0.158 = 3.39 which we compare to a table value with α/2 = and n 3 degrees of freedom. We reject H 0. Interpretation: There is strong evidence that immunoglobin levels increase more slowly per unit increase in maximal oxygen uptake in individuals with high aerobic fitness than in those with low aerobic fitness. If we had failed to reject H 0 : β 2 = 0, we would conclude that the association between y and x is linear. Stat Fall
14 Estimation and prediction Same concepts as before. With the model we might wish to: 1. Estimate the expected mean value of the response at a certain value of the predictor(s). 2. Predict a single response for some value of the predictor. In both cases, the point estimator (predictor) is ŷ = b 0 + b 1 x p + b 2 x 2 p for x = x p. The standard error of ŷ depends on whether we predict a mean or a single value. As before, ˆσ (y ŷ) > ˆσŷ. Calculations are complex, so we use the computer to get these standard errors and CIs. Stat Fall
15 Estimation and prediction In example, suppose we wish to obtain 1. The expected mean immunoglobin levels for people with oxygen uptake of x p = 40 ml/kg. 2. The expected immunoglobin level for a person with x p = 40 ml/kg. In both cases, point estimator is ŷ = 1, (40) 0.536(40) 2 = 1, JMP and SAS will give the (1 α)% CI for the mean or for a single prediction. Stat Fall
16 Estimation and prediction From CI we can derive ˆσŷ or ˆσ (y ŷ) recalling that (1 α)% Lower bound of CI = ŷ t α/2,n k 1 std error. Then Std error = ŷ Lower bound t α/2,n k 1. We can also derive the std errors using the upper bound of the CI as follows: Upper bound ŷ Std error =. t α/2,n k 1 Stat Fall
17 Estimation and prediction In example, the 95% CI for the mean immunoglobin at x = 40 ml/kg is (1, 156.2, 1, 263.6). Then: ˆσŷ = 1, , = Also, since the 95% CI for a single response is (985, 1, 434.8): ˆσ (y ŷ) = 1, = Stat Fall
18 More complex models: interaction + curvature Consider the following complete secondorder model with two predictors: See Fig y = β 0 + β 1 x 1 + β 2 x 2 + β 3 x 1 x 2 + β 4 x β 5 x ɛ. A complete second order model with three predictors includes 3 firstorder terms, 3 squared terms, 3 twoway interactions, and 1 threeway interaction. The number of terms in complete models gets out of hand fast. Samples often not large enough to fit all possible terms. Use subjectmatter knowledge to decide which terms to include. Stat Fall
19 More complex models: Example Example 4.7, page 213: Study to determine whether weight of package (x 1 ) and distance delivered (x 2 ) are associated to shipping costs (y) in a small regional express delivery service. See scatter plots. Complete secondorder model fitted with JMP. Data Express on class web site. Results: See output. Stat Fall
20 More complex models: Example Interpretation of results: Since RMSE = 0.44, about 95% of shipping costs will fall within $0.89 of their predicted values. R 2 a = 0.99: almost all of the variability in shipping costs can be explained by the model. F statistic = on 5 and 14 df. Highly significant, model is useful. Weight is associated to cost both linearly and quadratically. Distance only linearly. Interaction between weight and cost is positive: effect of weight on cost is not independent of distance. Stat Fall
We 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 informationEstimation of σ 2, the variance of ɛ
Estimation of σ 2, the variance of ɛ The variance of the errors σ 2 indicates how much observations deviate from the fitted surface. If σ 2 is small, parameters β 0, β 1,..., β k will be reliably estimated
More informationHypothesis testing  Steps
Hypothesis testing  Steps Steps to do a twotailed test of the hypothesis that β 1 0: 1. Set up the hypotheses: H 0 : β 1 = 0 H a : β 1 0. 2. Compute the test statistic: t = b 1 0 Std. error of b 1 =
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 informationPart 2: Analysis of Relationship Between Two Variables
Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable
More informatione = random error, assumed to be normally distributed with mean 0 and standard deviation σ
1 Linear Regression 1.1 Simple Linear Regression Model The linear regression model is applied if we want to model a numeric response variable and its dependency on at least one numeric factor variable.
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 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 informationHYPOTHESIS TESTING AND TYPE I AND TYPE II ERROR
HYPOTHESIS TESTING AND TYPE I AND TYPE II ERROR Hypothesis is a conjecture (an inferring) about one or more population parameters. Null Hypothesis (H 0 ) is a statement of no difference or no relationship
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 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 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 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 informationSimple 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 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 informationRegression. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.
Class: Date: Regression Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Given the least squares regression line y8 = 5 2x: a. the relationship between
More informationStatistics Review PSY379
Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses
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 informationChapter 9, Part A Hypothesis Tests. Learning objectives
Chapter 9, Part A Hypothesis Tests Slide 1 Learning objectives 1. Understand how to develop Null and Alternative Hypotheses 2. Understand Type I and Type II Errors 3. Able to do hypothesis test about population
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 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 informationINTERPRETING THE ONEWAY ANALYSIS OF VARIANCE (ANOVA)
INTERPRETING THE ONEWAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the oneway ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of
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 information1. The parameters to be estimated in the simple linear regression model Y=α+βx+ε ε~n(0,σ) are: a) α, β, σ b) α, β, ε c) a, b, s d) ε, 0, σ
STA 3024 Practice Problems Exam 2 NOTE: These are just Practice Problems. This is NOT meant to look just like the test, and it is NOT the only thing that you should study. Make sure you know all the material
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 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 informationPearson's Correlation Tests
Chapter 800 Pearson's Correlation Tests Introduction The correlation coefficient, ρ (rho), is a popular statistic for describing the strength of the relationship between two variables. The correlation
More informationA Short Tour of the Predictive Modeling Process
Chapter 2 A Short Tour of the Predictive Modeling Process Before diving in to the formal components of model building, we present a simple example that illustrates the broad concepts of model building.
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 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 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 informationDEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9
DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,
More informationLogs Transformation in a Regression Equation
Fall, 2001 1 Logs as the Predictor Logs Transformation in a Regression Equation The interpretation of the slope and intercept in a regression change when the predictor (X) is put on a log scale. In this
More informationEPS 625 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM
EPS 6 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM ANCOVA One Continuous Dependent Variable (DVD Rating) Interest Rating in DVD One Categorical/Discrete Independent Variable
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 informationChapter 3 Quantitative Demand Analysis
Managerial Economics & Business Strategy Chapter 3 uantitative Demand Analysis McGrawHill/Irwin Copyright 2010 by the McGrawHill Companies, Inc. All rights reserved. Overview I. The Elasticity Concept
More informationInternational Statistical Institute, 56th Session, 2007: Phil Everson
Teaching Regression using American Football Scores Everson, Phil Swarthmore College Department of Mathematics and Statistics 5 College Avenue Swarthmore, PA198, USA Email: peverso1@swarthmore.edu 1. Introduction
More informationSampling and Hypothesis Testing
Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus
More informationWeek 5: Multiple Linear Regression
BUS41100 Applied Regression Analysis Week 5: Multiple Linear Regression Parameter estimation and inference, forecasting, diagnostics, dummy variables Robert B. Gramacy The University of Chicago Booth School
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 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 informationSimple linear regression
Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between
More information6. Statistical Inference: Significance Tests
6. Statistical Inference: Significance Tests Goal: Use statistical methods to check hypotheses such as Women's participation rates in elections in France is higher than in Germany. (an effect) Ethnic divisions
More informationFinal Exam Practice Problem Answers
Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal
More informationWeek TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500 6 8480
1) The S & P/TSX Composite Index is based on common stock prices of a group of Canadian stocks. The weekly close level of the TSX for 6 weeks are shown: Week TSX Index 1 8480 2 8470 3 8475 4 8510 5 8500
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 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 informationChapter 7 Section 7.1: Inference for the Mean of a Population
Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used
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 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 informationChapter 8. Hypothesis Testing
Chapter 8 Hypothesis Testing Hypothesis In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing
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 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 informationOutline. Topic 4  Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares
Topic 4  Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test  Fall 2013 R 2 and the coefficient of correlation
More informationLecture Notes Module 1
Lecture Notes Module 1 Study Populations A study population is a clearly defined collection of people, animals, plants, or objects. In psychological research, a study population usually consists of a specific
More informationSimple Regression Theory II 2010 Samuel L. Baker
SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the
More informationANOVA  Analysis of Variance
ANOVA  Analysis of Variance ANOVA  Analysis of Variance Extends independentsamples t test Compares the means of groups of independent observations Don t be fooled by the name. ANOVA does not compare
More informationData Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression
Data Mining and Data Warehousing Henryk Maciejewski Data Mining Predictive modelling: regression Algorithms for Predictive Modelling Contents Regression Classification Auxiliary topics: Estimation of prediction
More informationMultinomial and Ordinal Logistic Regression
Multinomial and Ordinal Logistic Regression ME104: Linear Regression Analysis Kenneth Benoit August 22, 2012 Regression with categorical dependent variables When the dependent variable is categorical,
More informationComparing Nested Models
Comparing Nested Models ST 430/514 Two models are nested if one model contains all the terms of the other, and at least one additional term. The larger model is the complete (or full) model, and the smaller
More informationSTAT 350 Practice Final Exam Solution (Spring 2015)
PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects
More informationCorrelational Research
Correlational Research Chapter Fifteen Correlational Research Chapter Fifteen Bring folder of readings The Nature of Correlational Research Correlational Research is also known as Associational Research.
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 informationMind on Statistics. Chapter 13
Mind on Statistics Chapter 13 Sections 13.113.2 1. Which statement is not true about hypothesis tests? A. Hypothesis tests are only valid when the sample is representative of the population for the question
More informationPOLYNOMIAL AND MULTIPLE REGRESSION. Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model.
Polynomial Regression POLYNOMIAL AND MULTIPLE REGRESSION Polynomial regression used to fit nonlinear (e.g. curvilinear) data into a least squares linear regression model. It is a form of linear regression
More information12: Analysis of Variance. Introduction
1: Analysis of Variance Introduction EDA Hypothesis Test Introduction In Chapter 8 and again in Chapter 11 we compared means from two independent groups. In this chapter we extend the procedure to consider
More informationUnit 26 Estimation with Confidence Intervals
Unit 26 Estimation with Confidence Intervals Objectives: To see how confidence intervals are used to estimate a population proportion, a population mean, a difference in population proportions, or a difference
More informationMGT 267 PROJECT. Forecasting the United States Retail Sales of the Pharmacies and Drug Stores. Done by: Shunwei Wang & Mohammad Zainal
MGT 267 PROJECT Forecasting the United States Retail Sales of the Pharmacies and Drug Stores Done by: Shunwei Wang & Mohammad Zainal Dec. 2002 The retail sale (Million) ABSTRACT The present study aims
More informationANALYSIS OF TREND CHAPTER 5
ANALYSIS OF TREND CHAPTER 5 ERSH 8310 Lecture 7 September 13, 2007 Today s Class Analysis of trends Using contrasts to do something a bit more practical. Linear trends. Quadratic trends. Trends in SPSS.
More informationNull Hypothesis H 0. The null hypothesis (denoted by H 0
Hypothesis test In statistics, a hypothesis is a claim or statement about a property of a population. A hypothesis test (or test of significance) is a standard procedure for testing a claim about a property
More informationEconometrics Simple Linear Regression
Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight
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 informationPearson s correlation
Pearson s correlation Introduction Often several quantitative variables are measured on each member of a sample. If we consider a pair of such variables, it is frequently of interest to establish if there
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 informationMINITAB ASSISTANT WHITE PAPER
MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. OneWay
More informationRecall this chart that showed how most of our course would be organized:
Chapter 4 OneWay ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical
More informationChapter 5 Analysis of variance SPSS Analysis of variance
Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means Oneway ANOVA To test the null hypothesis that several population means are equal,
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 informationNonlinear Regression Functions. SW Ch 8 1/54/
Nonlinear Regression Functions SW Ch 8 1/54/ The TestScore STR relation looks linear (maybe) SW Ch 8 2/54/ But the TestScore Income relation looks nonlinear... SW Ch 8 3/54/ Nonlinear Regression General
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Final Exam Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A researcher for an airline interviews all of the passengers on five randomly
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 information12.5: CHISQUARE GOODNESS OF FIT TESTS
125: ChiSquare Goodness of Fit Tests CD121 125: CHISQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability
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 informationAn analysis method for a quantitative outcome and two categorical explanatory variables.
Chapter 11 TwoWay ANOVA An analysis method for a quantitative outcome and two categorical explanatory variables. If an experiment has a quantitative outcome and two categorical explanatory variables that
More informationCHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING
CHAPTER 11 SECTION 2: INTRODUCTION TO HYPOTHESIS TESTING MULTIPLE CHOICE 56. In testing the hypotheses H 0 : µ = 50 vs. H 1 : µ 50, the following information is known: n = 64, = 53.5, and σ = 10. The standardized
More information1 SAMPLE SIGN TEST. NonParametric Univariate Tests: 1 Sample Sign Test 1. A nonparametric equivalent of the 1 SAMPLE TTEST.
NonParametric Univariate Tests: 1 Sample Sign Test 1 1 SAMPLE SIGN TEST A nonparametric equivalent of the 1 SAMPLE TTEST. ASSUMPTIONS: Data is nonnormally distributed, even after log transforming.
More informationX X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1)
CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.
More informationDifference of Means and ANOVA Problems
Difference of Means and Problems Dr. Tom Ilvento FREC 408 Accounting Firm Study An accounting firm specializes in auditing the financial records of large firm It is interested in evaluating its fee structure,particularly
More informationGeneralized Linear Models
Generalized Linear Models We have previously worked with regression models where the response variable is quantitative and normally distributed. Now we turn our attention to two types of models where the
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 informationSection A. Index. Section A. Planning, Budgeting and Forecasting Section A.2 Forecasting techniques... 1. Page 1 of 11. EduPristine CMA  Part I
Index Section A. Planning, Budgeting and Forecasting Section A.2 Forecasting techniques... 1 EduPristine CMA  Part I Page 1 of 11 Section A. Planning, Budgeting and Forecasting Section A.2 Forecasting
More informationYiming Peng, Department of Statistics. February 12, 2013
Regression Analysis Using JMP Yiming Peng, Department of Statistics February 12, 2013 2 Presentation and Data http://www.lisa.stat.vt.edu Short Courses Regression Analysis Using JMP Download Data to Desktop
More informationAn Introduction to Statistical Tests for the SAS Programmer Sara Beck, Fred Hutchinson Cancer Research Center, Seattle, WA
ABSTRACT An Introduction to Statistical Tests for the SAS Programmer Sara Beck, Fred Hutchinson Cancer Research Center, Seattle, WA Often SAS Programmers find themselves in situations where performing
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 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 informationThe Multiple Regression Model: Hypothesis Tests and the Use of Nonsample Information
Chapter 8 The Multiple Regression Model: Hypothesis Tests and the Use of Nonsample Information An important new development that we encounter in this chapter is using the F distribution to simultaneously
More informationPrinciples of Hypothesis Testing for Public Health
Principles of Hypothesis Testing for Public Health Laura Lee Johnson, Ph.D. Statistician National Center for Complementary and Alternative Medicine johnslau@mail.nih.gov Fall 2011 Answers to Questions
More informationElementary Statistics Sample Exam #3
Elementary Statistics Sample Exam #3 Instructions. No books or telephones. Only the supplied calculators are allowed. The exam is worth 100 points. 1. A chi square goodness of fit test is considered to
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 information