# Testing for Lack of Fit

Size: px
Start display at page:

Transcription

1 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 the data or simply takes the wrong form, then ˆσ 2 will overestimate σ 2. An example can be seen in Figure 6.1. Alternatively, if our model is too complex and overfits the data, then ˆσ 2 will be an underestimate. y Figure 6.1: True quadratic fit shown with the solid line and incorrect linear fit shown with the dotted line. Estimate of σ 2 will be unbiased for the quadratic model but far too large for the linear model x This suggests a possible testing procedure we should compare ˆσ 2 to σ 2. There are two cases one where σ 2 is known and one where it is not. 65

2 6.1. σ 2 KNOWN σ 2 known σ 2 known may be known from past experience, knowledge of the measurement error inherent in an instrument or by definition. Recall (from Section 3.4) that ˆσ 2 χn 2 p σ n p 2 n p ˆσ 2 which leads to the test: Conclude there is a lack of fit if σ 2 χ 2 n 1 α p If a lack of fit is found, then a new model is needed. Continuing with the same data as in the weighted least squares example we test to see if a linear model is adequate. In this example, we know the variance almost exactly because each response value is the average of a large number of observations. Because of the way the weights are defined, w i 1 var y i, the known variance is implicitly equal to one. There is nothing special about one - we could define w i 99 var y i and the variance would be implicitly 99. However, we would get essentially the same result as the following analysis. > data(strongx) > g <- lm(crossx energy, weights=sdˆ-2, strongx) > summary(g) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-08 energy e-06 Residual standard error: 1.66 on 8 degrees of freedom Multiple R-Squared: 0.94, Adjusted R-squared: F-statistic: 125 on 1 and 8 degrees of freedom, p-value: 3.71e-06 Examine the R 2 - do you think the model is a good fit? Now plot the data and the fitted regression line (shown as a solid line on Figure 6.2. > plot(strongx\$energy,strongx\$crossx,xlab="energy",ylab="crossection") > abline(g\$coef) Compute the test statistic and the p-value: > 1.66ˆ2*8 [1] > 1-pchisq(22.045,8) [1] We conclude that there is a lack of fit. Just because R 2 is large does not mean that you can not do better. Add a quadratic term to the model and test again:

3 6.2. σ 2 UNKNOWN 67 Crossection Energy Figure 6.2: Linear and quadratic fits to the physics data > g2 <- lm(crossx energy + I(energyˆ2), weights=sdˆ-2, strongx) > summary(g2) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-08 energy I(energyˆ2) Residual standard error: on 7 degrees of freedom Multiple R-Squared: 0.991, Adjusted R-squared: F-statistic: 391 on 2 and 7 degrees of freedom, p-value: 6.55e-08 > 0.679ˆ2*7 [1] > 1-pchisq( ,7) [1] This time we cannot detect a lack of fit. Plot the fit: > x <- seq(0.05,0.35,by=0.01) > lines(x,g2\$coef[1]+g2\$coef[2]*x+g2\$coef[3]*xˆ2,lty=2) The curve is shown as a dotted line on the plot (thanks to lty=2). This seems clearly more appropriate than the linear model. 6.2 σ 2 unknown The ˆσ 2 that is based in the chosen regression model needs to be compared to some model-free estimate of σ 2. We can do this if we have repeated y for one or more fixed x. These replicates do need to be

4 6.2. σ 2 UNKNOWN 68 truly independent. They cannot just be repeated measurements on the same subject or unit. Such repeated measures would only reveal the within subject variability or the measurement error. We need to know the between subject variability this reflects the σ 2 described in the model. The pure error estimate of σ 2 is given by SS pe d f pe where SS pe 2 y i ȳ distinct x givenx Degrees of freedom d f pe distinctx #replicates 1 If you fit a model that assigns one parameter to each group of observations with fixed x then the ˆσ 2 from this model will be the pure error ˆσ 2. This model is just the one-way anova model if you are familiar with that. Comparing this model to the regression model amounts to the lack of fit test. This is usually the most convenient way to compute the test but if you like we can then partition the RSS into that due to lack of fit and that due to the pure error as in Table 6.1. df SS MS F Residual n-p Lack of Fit n p d f pe RSS SS pe RSS SS n p pe d f pe Pure Error d f pe SS pe SS pe d f pe Table 6.1: ANOVA for lack of fit Ratio of MS Compute the F-statistic and compare to F n p d f pe d f pe and reject if the statistic is too large. Another way of looking at this is a comparison between the model of interest and a saturated model that assigns a parameter to each unique combination of the predictors. Because the model of interest represents a special case of the saturated model where the saturated parameters satisfy the constraints of the model of interest, we can use the standard F-testing methodology. The data for this example consist of thirteen specimens of 90/10 Cu-Ni alloys with varying iron content in percent. The specimens were submerged in sea water for 60 days and the weight loss due to corrosion was recorded in units of milligrams per square decimeter per day. The data come from Draper and Smith (1998). We load in and print the data > data(corrosion) > corrosion Fe loss

5 6.2. σ 2 UNKNOWN 69 We fit a straight line model: > g <- lm(loss Fe, data=corrosion) > summary(g) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 Fe e-09 Residual standard error: 3.06 on 11 degrees of freedom Multiple R-Squared: 0.97, Adjusted R-squared: F-statistic: 352 on 1 and 11 degrees of freedom, p-value: 1.06e-09 Check the fit graphically see Figure 6.3. > plot(corrosion\$fe,corrosion\$loss,xlab="iron content",ylab="weight loss") > abline(g\$coef) Weight loss Iron content Figure 6.3: Linear fit to the Cu-Ni corrosion data. Group means denoted by black diamonds We have an R 2 of 97% and an apparently good fit to the data. We now fit a model that reserves a parameter for each group of data with the same value of x. This is accomplished by declaring the predictor to be a factor. We will describe this in more detail in a later chapter > ga <- lm(loss factor(fe), data=corrosion) The fitted values are the means in each group - put these on the plot: > points(corrosion\$fe,ga\$fit,pch=18)

6 6.2. σ 2 UNKNOWN 70 We can now compare the two models in the usual way: > anova(g,ga) Analysis of Variance Table Model 1: loss Fe Model 2: loss factor(fe) Res.Df Res.Sum Sq Df Sum Sq F value Pr(>F) The low p-value indicates that we must conclude that there is a lack of fit. The reason is that the pure error sd is substantially less than the regression standard error of We might investigate models other than a straight line although no obvious alternative is suggested by the plot. Before considering other models, I would first find out whether the replicates are genuine perhaps the low pure error SD can be explained by some correlation in the measurements. Another possible explanation is unmeasured third variable is causing the lack of fit. When there are replicates, it is impossible to get a perfect fit. Even when there is parameter assigned to each group of x-values, the residual sum of squares will not be zero. For the factor model above, the R 2 is 99.7%. So even this saturated model does not attain a 100% value for R 2. For these data, it s a small difference but in other cases, the difference can be substantial. In these cases, one should realize that the maximum R 2 that may be attained might be substantially less than 100% and so perceptions about what a good value for R 2 should be downgraded appropriately. These methods are good for detecting lack of fit, but if the null hypothesis is accepted, we cannot conclude that we have the true model. After all, it may be that we just did not have enough data to detect the inadequacies of the model. All we can say is that the model is not contradicted by the data. When there are no replicates, it may be possible to group the responses for similar x but this is not straightforward. It is also possible to detect lack of fit by less formal, graphical methods. A more general question is how good a fit do you really want? By increasing the complexity of the model, it is possible to fit the data more closely. By using as many parameters as data points, we can fit the data exactly. Very little is achieved by doing this since we learn nothing beyond the data itself and any predictions made using such a model will tend to have very high variance. The question of how complex a model to fit is difficult and fundamental. For example, we can fit the mean responses for the example above exactly using a sixth order polynomial: > gp <- lm(loss Fe+I(Feˆ2)+I(Feˆ3)+I(Feˆ4)+I(Feˆ5)+I(Feˆ6),corrosion) Now look at this fit: > plot(loss Fe, data=corrosion,ylim=c(60,130)) > points(corrosion\$fe,ga\$fit,pch=18) > grid <- seq(0,2,len=50) > lines(grid,predict(gp,data.frame(fe=grid))) as shown in Figure 6.4. The fit of this model is excellent for example: > summary(gp)\$r.squared [1] but it is clearly riduculous. There is no plausible reason corrosion loss should suddenly drop at 1.7 and thereafter increase rapidly. This is a consequence of overfitting the data. This illustrates the need not to become too focused on measures of fit like R 2.

7 6.2. σ 2 UNKNOWN 71 loss Fe Figure 6.4: Polynomial fit to the corrosion data

### Multiple 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

### " 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

### DEPARTMENT 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,

### Statistical Models in R

Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Linear Models in R Regression Regression analysis is the appropriate

### Lecture 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

### Stat 5303 (Oehlert): Tukey One Degree of Freedom 1

Stat 5303 (Oehlert): Tukey One Degree of Freedom 1 > catch

### SPSS Guide: Regression Analysis

SPSS Guide: Regression Analysis I put this together to give you a step-by-step 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

### 5. Linear Regression

5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4

### 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

### Univariate Regression

Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is

### Outline. 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

### NCSS 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

### POLYNOMIAL 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

### Basic 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

### 1. 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 1-propZint 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

### Comparing 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

### Using 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

### 2. 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

### Multiple 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.

### 17. 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.

### E(y i ) = x T i β. yield of the refined product as a percentage of crude specific gravity vapour pressure ASTM 10% point ASTM end point in degrees F

Random and Mixed Effects Models (Ch. 10) Random effects models are very useful when the observations are sampled in a highly structured way. The basic idea is that the error associated with any linear,

### 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

### Regression step-by-step using Microsoft Excel

Step 1: Regression step-by-step 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

### One-Way Analysis of Variance: A Guide to Testing Differences Between Multiple Groups

One-Way Analysis of Variance: A Guide to Testing Differences Between Multiple Groups In analysis of variance, the main research question is whether the sample means are from different populations. The

### EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION

EDUCATION AND VOCABULARY MULTIPLE REGRESSION IN ACTION EDUCATION AND VOCABULARY 5-10 hours of input weekly is enough to pick up a new language (Schiff & Myers, 1988). Dutch children spend 5.5 hours/day

### Week 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

### Unit 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

### Week 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

### Part 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

### Section 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

### Final 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

### Correlation 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

### Chapter 7. One-way ANOVA

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

### MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS

MULTIPLE REGRESSION AND ISSUES IN REGRESSION ANALYSIS MSR = Mean Regression Sum of Squares MSE = Mean Squared Error RSS = Regression Sum of Squares SSE = Sum of Squared Errors/Residuals α = Level of Significance

### ORTHOGONAL POLYNOMIAL CONTRASTS INDIVIDUAL DF COMPARISONS: EQUALLY SPACED TREATMENTS

ORTHOGONAL POLYNOMIAL CONTRASTS INDIVIDUAL DF COMPARISONS: EQUALLY SPACED TREATMENTS Many treatments are equally spaced (incremented). This provides us with the opportunity to look at the response curve

### Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software

STATA Tutorial Professor Erdinç Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software 1.Wald Test Wald Test is used

### Section 14 Simple Linear Regression: Introduction to Least Squares Regression

Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship

### Regression 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

### Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this

### 2013 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

### Chapter 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

### Module 5: Multiple Regression Analysis

Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College

### Factors 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

### 1.5 Oneway Analysis of Variance

Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

### 1. 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

### General Regression Formulae ) (N-2) (1 - r 2 YX

General Regression Formulae Single Predictor Standardized Parameter Model: Z Yi = β Z Xi + ε i Single Predictor Standardized Statistical Model: Z Yi = β Z Xi Estimate of Beta (Beta-hat: β = r YX (1 Standard

### MULTIPLE 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

### Normality Testing in Excel

Normality Testing in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com

### Data 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

### N-Way Analysis of Variance

N-Way Analysis of Variance 1 Introduction A good example when to use a n-way ANOVA is for a factorial design. A factorial design is an efficient way to conduct an experiment. Each observation has data

### t Tests in Excel The Excel Statistical Master By Mark Harmon Copyright 2011 Mark Harmon

t-tests in Excel By Mark Harmon Copyright 2011 Mark Harmon No part of this publication may be reproduced or distributed without the express permission of the author. mark@excelmasterseries.com www.excelmasterseries.com

### The 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 30-31, 2008 B. Weaver, NHRC 2008 1 The Objective

### DATA 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

### 1 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

### You have data! What s next?

You have data! What s next? Data Analysis, Your Research Questions, and Proposal Writing Zoo 511 Spring 2014 Part 1:! Research Questions Part 1:! Research Questions Write down > 2 things you thought were

### INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of

### IAPRI Quantitative Analysis Capacity Building Series. Multiple regression analysis & interpreting results

IAPRI Quantitative Analysis Capacity Building Series Multiple regression analysis & interpreting results How important is R-squared? R-squared Published in Agricultural Economics 0.45 Best article of the

### Section 1: Simple Linear Regression

Section 1: Simple Linear Regression Carlos M. Carvalho The University of Texas McCombs School of Business http://faculty.mccombs.utexas.edu/carlos.carvalho/teaching/ 1 Regression: General Introduction

### Outline. Dispersion Bush lupine survival Quasi-Binomial family

Outline 1 Three-way interactions 2 Overdispersion in logistic regression Dispersion Bush lupine survival Quasi-Binomial family 3 Simulation for inference Why simulations Testing model fit: simulating the

### Curve 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

### Interaction between quantitative predictors

Interaction between quantitative predictors In a first-order 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

### SIMPLE LINEAR CORRELATION. r can range from -1 to 1, and is independent of units of measurement. Correlation can be done on two dependent variables.

SIMPLE LINEAR CORRELATION Simple linear correlation is a measure of the degree to which two variables vary together, or a measure of the intensity of the association between two variables. Correlation

### 12: 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

### Calculating P-Values. Parkland College. Isela Guerra Parkland College. Recommended Citation

Parkland College A with Honors Projects Honors Program 2014 Calculating P-Values Isela Guerra Parkland College Recommended Citation Guerra, Isela, "Calculating P-Values" (2014). A with Honors Projects.

### Chapter 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) -

### 11. Analysis of Case-control Studies Logistic Regression

Research methods II 113 11. Analysis of Case-control Studies Logistic Regression This chapter builds upon and further develops the concepts and strategies described in Ch.6 of Mother and Child Health:

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way 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

### Psychology 205: Research Methods in Psychology

Psychology 205: Research Methods in Psychology Using R to analyze the data for study 2 Department of Psychology Northwestern University Evanston, Illinois USA November, 2012 1 / 38 Outline 1 Getting ready

### One-Way Analysis of Variance

One-Way Analysis of Variance Note: Much of the math here is tedious but straightforward. We ll skim over it in class but you should be sure to ask questions if you don t understand it. I. Overview A. We

Lecture 16: Generalized Additive Models Regression III: Advanced Methods Bill Jacoby Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Goals of the Lecture Introduce Additive Models

### Exercise 1.12 (Pg. 22-23)

Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

### Nonlinear 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

### Chapter 4 and 5 solutions

Chapter 4 and 5 solutions 4.4. Three different washing solutions are being compared to study their effectiveness in retarding bacteria growth in five gallon milk containers. The analysis is done in a laboratory,

### Introduction 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

### KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management

KSTAT MINI-MANUAL 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

### Elementary 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

### Chicago Insurance Redlining - a complete example

Chapter 12 Chicago Insurance Redlining - a complete example In a study of insurance availability in Chicago, the U.S. Commission on Civil Rights attempted to examine charges by several community organizations

### Doing Multiple Regression with SPSS. In this case, we are interested in the Analyze options so we choose that menu. If gives us a number of choices:

Doing Multiple Regression with SPSS Multiple Regression for Data Already in Data Editor Next we want to specify a multiple regression analysis for these data. The menu bar for SPSS offers several options:

### Answer: 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

### Analysis of Variance ANOVA

Analysis of Variance ANOVA Overview We ve used the t -test to compare the means from two independent groups. Now we ve come to the final topic of the course: how to compare means from more than two populations.

### Least Squares Estimation

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

### Penalized regression: Introduction

Penalized regression: Introduction Patrick Breheny August 30 Patrick Breheny BST 764: Applied Statistical Modeling 1/19 Maximum likelihood Much of 20th-century statistics dealt with maximum likelihood

### STAT 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

Not Your Dad s Magic Eight Ball Prepared for the NCSL Fiscal Analysts Seminar, October 21, 2014 Jim Landers, Office of Fiscal and Management Analysis, Indiana Legislative Services Agency Actual Forecast

### Stat 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

### BIOL 933 Lab 6 Fall 2015. Data Transformation

BIOL 933 Lab 6 Fall 2015 Data Transformation Transformations in R General overview Log transformation Power transformation The pitfalls of interpreting interactions in transformed data Transformations

### Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption

Two-sample t-tests. - Independent samples - Pooled standard devation - The equal variance assumption Last time, we used the mean of one sample to test against the hypothesis that the true mean was a particular

### Predictor Coef StDev T P Constant 970667056 616256122 1.58 0.154 X 0.00293 0.06163 0.05 0.963. S = 0.5597 R-Sq = 0.0% R-Sq(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

### Simple 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

### Simple Methods and Procedures Used in Forecasting

Simple Methods and Procedures Used in Forecasting The project prepared by : Sven Gingelmaier Michael Richter Under direction of the Maria Jadamus-Hacura What Is Forecasting? Prediction of future events

### COMPARISONS 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

### Bill 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

### CHAPTER 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

### Introduction to General and Generalized Linear Models

Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby

### ANOVA. February 12, 2015

ANOVA February 12, 2015 1 ANOVA models Last time, we discussed the use of categorical variables in multivariate regression. Often, these are encoded as indicator columns in the design matrix. In [1]: %%R