Data Set 8: Laysan Finch Beak Widths
|
|
- Teresa Freeman
- 6 years ago
- Views:
Transcription
1 Data Set 8: Finch Beak Widths Statistical Setting This handout describes an analysis of covariance (ANCOVA) involving one categorical independent variable (with only two levels) and one quantitative covariate. Background and Data The data are from Sheila Conant and Marie Morin s study of Finch beak morphology, described in handouts in ZOOL 631. In this handout of adult female birds will be compared between two populations ( and Islands), controlling for overall size. The data are for adult female Finches captured and measured in There were 62 on Island and 10 on Island. The two variables used in the following analysis are (the width of the upper mandible, in cm) and, as a measure of body size, the (in cm). Preliminary Data Exploration Scatter Plot The most useful preliminary description of the data is a scatter plot of the response variable () plotted against the covariate (), with the different levels of the categorical variable () shown with different symbols. This plot shows that there is a slight relationship between and, and that Island birds tend both to have wider beaks and to have longer sterna
2 Descriptive Statistics These statistics support the impressions drawn from the scatter plot: both beak widths and s tend to be greater in Island birds, and there is at least some relationship between and. The standard deviations of both variables are quite similar between s. Box Plots : N MEAN MEDIAN STDEV MIN MAX Q both : N MEAN MEDIAN STDEV MIN MAX Q both correlations between and : These boxplots again show the greater s and (to a lesser extent) sternum lengths of the Island birds. They also show that all four distributions are fairly symmetrical, with no major outliers. (A number of observations in the population exceed Minitab s 1.5xIQR rule for identifying possible outliers, but none really seem exceptional, given the fairly large size of this sample.) Data Set 8: Finch Beak Widths 2
3 Analysis of Covariance For the following analyses the covariate,, was centered by subtracting off the overall mean value (1.6747). Test of Parallelism Prior to conducting the ANCOVA it is necessary to determine whether it is reasonable to fit parallel regressions to the two samples. This is done by testing for an x sternum interaction, in a general linear model: Source DF Seq SS Adj SS Adj MS F P *sternum Error Total This model fits separate regressions to the two groups, as in the following scatterplot. The slopes of these regressions are not very different, and the analysis above indicates that this difference is not at all significant statistically There is no evidence from this analysis that the parallelism assumption is not reasonable. We therefore can proceed with the ANCOVA. Data Set 8: Finch Beak Widths 3
4 ANCOVA The ANCOVA can be conducted as a general linear model as above, but without the interaction term: Source DF Seq SS Adj SS Adj MS F P Error Total These results show that there is a highly significant difference between beak widths on the two s, after adjusting for. The relationship of beak width to also is highly significant, though this is not really interesting to us (what matters is the effect on the ANOVA conclusions of including the covariate). Analysis of Effect Estimation of Adjusted Means To determine the magnitude of the (adjusted) difference between the s we need the parameter estimates. (Notice that the t-value for the - coefficient is the square root of the F for the effect: these are equivalent tests.) Term Coef SE Coef T P Constant sternum Minitab s GLM uses an indicator-variable coding for the variable in which is +1 and Island is 1. Using the preceding coefficients, the fitted regression relationships are : Ŷ i = ( ) x i = x i : Ŷ i = ( ) x i = x i With this coding, as these equations show, the coefficient for the variable is half the difference between the s. Thus mean s, for a given sternum length, are 2 x = cm larger on Island. The adjusted means for the s (means at x i = x ) are the LS means since the covariate was centered for these analyses. They are: Least Squares Means for Mean SE Mean Data Set 8: Finch Beak Widths 4
5 To obtain a confidence interval for the difference in adjusted means, note that this difference is twice the coefficient for, given above. Using the standard error of this coefficient ( ), a 99% CI for the difference would be Aptness of Covariance Model It was noted in the preliminary exploration that the variances are similar in the two populations and there are no severe outliers. The very non-significant test for a difference in regression slopes justifies the parallelism assumption. The remaining assumptions to be considered are normality and linearity. Normality: 2τˆ1 ± t 0.995, 69 2se = 2( ) ± ( ) = ± = ( , ) Normal Score Residual This plot is very slightly sigmoid rather than perfectly straight: the tails of the distribution are slightly longer than those of a normal distribution. The correlation between residuals and normal scores, however, is 0.995, which is more than good enough given the sample size. Linearity Linearity is best assessed by the usual plot of residuals vs. fitted values, with different levels of the categorical variable indicated by different symbols. This plot (next page) shows the few large positive and negative residuals, but there is no clear indication of nonlinearity in either population. Data Set 8: Finch Beak Widths 5
6 residuals fits Conclusion Island birds have wider beaks than do Island birds. Since Island birds also are larger (as measured by ), and larger birds tend to have wider beaks, some of the difference in s between the s could be attributed to the difference in overall size. The analysis of covariance, however, shows that there is a statistically significant difference in s even after adjusting for the difference in s (P < 0.001); the 99% confidence interval for the adjusted difference in mean s is (0.0156, ) (in cm, ). The analysis also indicates that the relationship between and beak width is roughly linear, is statistically significant (P = 0.013), and does not differ significantly between the two populations (P = 0.577). These results are shown in the plot on the next page, with the fitted regression relationships superimposed on the data, and the adjusted mean scores shown by the (green) diamond and (blue) triangle at x = Data Set 8: Finch Beak Widths 6
7 Comparison with ANOVA To understand the effect (if any) of including the covariate in this analysis, it is interesting to examine the results of a simple ANOVA (or equivalently, a two-sample t-test with pooled standard error). Source DF Seq SS Adj SS Adj MS F P Error Total Term Coeff Stdev t-value P Constant The unadjusted difference in mean s is somewhat (about 9%) larger than the adjusted difference: since Island birds had larger s as well as s, and the two variables were positively related, the adjustment to a standard reduced the difference between s. On the other hand, removing the proportion of within- variability explainable by increased the precision of the analysis: the ANCOVA MSE was smaller than that for the ANOVA ( vs , a 7% decrease). As a result, the standard error of the adjusted difference is slightly (about 2%) smaller than that of the unadjusted mean. The net result of these somewhat offsetting differences is that the effect is more significant in the ANOVA (the F is larger), though it is still highly significant in the ANCOVA. Data Set 8: Finch Beak Widths 7
Multiple Regression Examples
Multiple Regression Examples Example: Tree data. we have seen that a simple linear regression of usable volume on diameter at chest height is not suitable, but that a quadratic model y = β 0 + β 1 x +
More informationStatistics 512: Solution to Homework#11. Problems 1-3 refer to the soybean sausage dataset of Problem 20.8 (ch21pr08.dat).
Statistics 512: Solution to Homework#11 Problems 1-3 refer to the soybean sausage dataset of Problem 20.8 (ch21pr08.dat). 1. Perform the two-way ANOVA without interaction for this model. Use the results
More informationAnalysis of Covariance. The following example illustrates a case where the covariate is affected by the treatments.
Analysis of Covariance In some experiments, the experimental units (subjects) are nonhomogeneous or there is variation in the experimental conditions that are not due to the treatments. For example, a
More informationModel Building Chap 5 p251
Model Building Chap 5 p251 Models with one qualitative variable, 5.7 p277 Example 4 Colours : Blue, Green, Lemon Yellow and white Row Blue Green Lemon Insects trapped 1 0 0 1 45 2 0 0 1 59 3 0 0 1 48 4
More informationData Set 1A: Algal Photosynthesis vs. Salinity and Temperature
Data Set A: Algal Photosynthesis vs. Salinity and Temperature Statistical setting These data are from a controlled experiment in which two quantitative variables were manipulated, to determine their effects
More informationChapter 12: Multiple Regression
Chapter 12: Multiple Regression 12.1 a. A scatterplot of the data is given here: Plot of Drug Potency versus Dose Level Potency 0 5 10 15 20 25 30 0 5 10 15 20 25 30 35 Dose Level b. ŷ = 8.667 + 0.575x
More informationRegression Analysis. Regression: Methodology for studying the relationship among two or more variables
Regression Analysis Regression: Methodology for studying the relationship among two or more variables Two major aims: Determine an appropriate model for the relationship between the variables Predict the
More informationANOVA: Analysis of Variation
ANOVA: Analysis of Variation The basic ANOVA situation Two variables: 1 Categorical, 1 Quantitative Main Question: Do the (means of) the quantitative variables depend on which group (given by categorical
More informationConfidence Interval for the mean response
Week 3: Prediction and Confidence Intervals at specified x. Testing lack of fit with replicates at some x's. Inference for the correlation. Introduction to regression with several explanatory variables.
More information1 Introduction to Minitab
1 Introduction to Minitab Minitab is a statistical analysis software package. The software is freely available to all students and is downloadable through the Technology Tab at my.calpoly.edu. When you
More informationANOVA Situation The F Statistic Multiple Comparisons. 1-Way ANOVA MATH 143. Department of Mathematics and Statistics Calvin College
1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College An example ANOVA situation Example (Treating Blisters) Subjects: 25 patients with blisters Treatments: Treatment A, Treatment
More information9 Correlation and Regression
9 Correlation and Regression SW, Chapter 12. Suppose we select n = 10 persons from the population of college seniors who plan to take the MCAT exam. Each takes the test, is coached, and then retakes the
More informationGeneral Linear Model (Chapter 4)
General Linear Model (Chapter 4) Outcome variable is considered continuous Simple linear regression Scatterplots OLS is BLUE under basic assumptions MSE estimates residual variance testing regression coefficients
More informationIntroduction to Linear regression analysis. Part 2. Model comparisons
Introduction to Linear regression analysis Part Model comparisons 1 ANOVA for regression Total variation in Y SS Total = Variation explained by regression with X SS Regression + Residual variation SS Residual
More informationCorrelation and Simple Linear Regression
Correlation and Simple Linear Regression Sasivimol Rattanasiri, Ph.D Section for Clinical Epidemiology and Biostatistics Ramathibodi Hospital, Mahidol University E-mail: sasivimol.rat@mahidol.ac.th 1 Outline
More informationSTA 108 Applied Linear Models: Regression Analysis Spring Solution for Homework #6
STA 8 Applied Linear Models: Regression Analysis Spring 011 Solution for Homework #6 6. a) = 11 1 31 41 51 1 3 4 5 11 1 31 41 51 β = β1 β β 3 b) = 1 1 1 1 1 11 1 31 41 51 1 3 4 5 β = β 0 β1 β 6.15 a) Stem-and-leaf
More informationInference for the Regression Coefficient
Inference for the Regression Coefficient Recall, b 0 and b 1 are the estimates of the slope β 1 and intercept β 0 of population regression line. We can shows that b 0 and b 1 are the unbiased estimates
More information1 A Review of Correlation and Regression
1 A Review of Correlation and Regression SW, Chapter 12 Suppose we select n = 10 persons from the population of college seniors who plan to take the MCAT exam. Each takes the test, is coached, and then
More informationVIII. ANCOVA. A. Introduction
VIII. ANCOVA A. Introduction In most experiments and observational studies, additional information on each experimental unit is available, information besides the factors under direct control or of interest.
More information1-Way ANOVA MATH 143. Spring Department of Mathematics and Statistics Calvin College
1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College Spring 2010 The basic ANOVA situation Two variables: 1 Categorical, 1 Quantitative Main Question: Do the (means of) the quantitative
More information23. Inference for regression
23. Inference for regression The Practice of Statistics in the Life Sciences Third Edition 2014 W. H. Freeman and Company Objectives (PSLS Chapter 23) Inference for regression The regression model Confidence
More informationCorrelation & Simple Regression
Chapter 11 Correlation & Simple Regression The previous chapter dealt with inference for two categorical variables. In this chapter, we would like to examine the relationship between two quantitative variables.
More informationInference for Regression Inference about the Regression Model and Using the Regression Line
Inference for Regression Inference about the Regression Model and Using the Regression Line PBS Chapter 10.1 and 10.2 2009 W.H. Freeman and Company Objectives (PBS Chapter 10.1 and 10.2) Inference about
More informationSTA 302 H1F / 1001 HF Fall 2007 Test 1 October 24, 2007
STA 302 H1F / 1001 HF Fall 2007 Test 1 October 24, 2007 LAST NAME: SOLUTIONS FIRST NAME: STUDENT NUMBER: ENROLLED IN: (circle one) STA 302 STA 1001 INSTRUCTIONS: Time: 90 minutes Aids allowed: calculator.
More informationSTAT 3900/4950 MIDTERM TWO Name: Spring, 2015 (print: first last ) Covered topics: Two-way ANOVA, ANCOVA, SLR, MLR and correlation analysis
STAT 3900/4950 MIDTERM TWO Name: Spring, 205 (print: first last ) Covered topics: Two-way ANOVA, ANCOVA, SLR, MLR and correlation analysis Instructions: You may use your books, notes, and SPSS/SAS. NO
More informationInferences for Regression
Inferences for Regression An Example: Body Fat and Waist Size Looking at the relationship between % body fat and waist size (in inches). Here is a scatterplot of our data set: Remembering Regression In
More informationAnalysis of Covariance
Analysis of Covariance (ANCOVA) Bruce A Craig Department of Statistics Purdue University STAT 514 Topic 10 1 When to Use ANCOVA In experiment, there is a nuisance factor x that is 1 Correlated with y 2
More informationTABLES AND FORMULAS FOR MOORE Basic Practice of Statistics
TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Exploring Data: Distributions Look for overall pattern (shape, center, spread) and deviations (outliers). Mean (use a calculator): x = x 1 + x
More informationMultiple Regression an Introduction. Stat 511 Chap 9
Multiple Regression an Introduction Stat 511 Chap 9 1 case studies meadowfoam flowers brain size of mammals 2 case study 1: meadowfoam flowering designed experiment carried out in a growth chamber general
More informationMultiple Regression. Inference for Multiple Regression and A Case Study. IPS Chapters 11.1 and W.H. Freeman and Company
Multiple Regression Inference for Multiple Regression and A Case Study IPS Chapters 11.1 and 11.2 2009 W.H. Freeman and Company Objectives (IPS Chapters 11.1 and 11.2) Multiple regression Data for multiple
More informationSoil Phosphorus Discussion
Solution: Soil Phosphorus Discussion Summary This analysis is ambiguous: there are two reasonable approaches which yield different results. Both lead to the conclusion that there is not an independent
More informationAnswer Keys to Homework#10
Answer Keys to Homework#10 Problem 1 Use either restricted or unrestricted mixed models. Problem 2 (a) First, the respective means for the 8 level combinations are listed in the following table A B C Mean
More informationStatistical Modelling in Stata 5: Linear Models
Statistical Modelling in Stata 5: Linear Models Mark Lunt Arthritis Research UK Epidemiology Unit University of Manchester 07/11/2017 Structure This Week What is a linear model? How good is my model? Does
More informationINFERENCE FOR REGRESSION
CHAPTER 3 INFERENCE FOR REGRESSION OVERVIEW In Chapter 5 of the textbook, we first encountered regression. The assumptions that describe the regression model we use in this chapter are the following. We
More informationAssignment 9 Answer Keys
Assignment 9 Answer Keys Problem 1 (a) First, the respective means for the 8 level combinations are listed in the following table A B C Mean 26.00 + 34.67 + 39.67 + + 49.33 + 42.33 + + 37.67 + + 54.67
More informationFormula for the t-test
Formula for the t-test: How the t-test Relates to the Distribution of the Data for the Groups Formula for the t-test: Formula for the Standard Error of the Difference Between the Means Formula for the
More informationStat 501, F. Chiaromonte. Lecture #8
Stat 501, F. Chiaromonte Lecture #8 Data set: BEARS.MTW In the minitab example data sets (for description, get into the help option and search for "Data Set Description"). Wild bears were anesthetized,
More informationSTATISTICS 110/201 PRACTICE FINAL EXAM
STATISTICS 110/201 PRACTICE FINAL EXAM Questions 1 to 5: There is a downloadable Stata package that produces sequential sums of squares for regression. In other words, the SS is built up as each variable
More information28. SIMPLE LINEAR REGRESSION III
28. SIMPLE LINEAR REGRESSION III Fitted Values and Residuals To each observed x i, there corresponds a y-value on the fitted line, y = βˆ + βˆ x. The are called fitted values. ŷ i They are the values of
More informationRegression and Models with Multiple Factors. Ch. 17, 18
Regression and Models with Multiple Factors Ch. 17, 18 Mass 15 20 25 Scatter Plot 70 75 80 Snout-Vent Length Mass 15 20 25 Linear Regression 70 75 80 Snout-Vent Length Least-squares The method of least
More informationW&M CSCI 688: Design of Experiments Homework 2. Megan Rose Bryant
W&M CSCI 688: Design of Experiments Homework 2 Megan Rose Bryant September 25, 201 3.5 The tensile strength of Portland cement is being studied. Four different mixing techniques can be used economically.
More informationVariance. Standard deviation VAR = = value. Unbiased SD = SD = 10/23/2011. Functional Connectivity Correlation and Regression.
10/3/011 Functional Connectivity Correlation and Regression Variance VAR = Standard deviation Standard deviation SD = Unbiased SD = 1 10/3/011 Standard error Confidence interval SE = CI = = t value for
More informationBasic Business Statistics, 10/e
Chapter 4 4- Basic Business Statistics th Edition Chapter 4 Introduction to Multiple Regression Basic Business Statistics, e 9 Prentice-Hall, Inc. Chap 4- Learning Objectives In this chapter, you learn:
More informationInference for Regression Simple Linear Regression
Inference for Regression Simple Linear Regression IPS Chapter 10.1 2009 W.H. Freeman and Company Objectives (IPS Chapter 10.1) Simple linear regression p Statistical model for linear regression p Estimating
More informationConfidence Intervals, Testing and ANOVA Summary
Confidence Intervals, Testing and ANOVA Summary 1 One Sample Tests 1.1 One Sample z test: Mean (σ known) Let X 1,, X n a r.s. from N(µ, σ) or n > 30. Let The test statistic is H 0 : µ = µ 0. z = x µ 0
More informationMathematical Notation Math Introduction to Applied Statistics
Mathematical Notation Math 113 - Introduction to Applied Statistics Name : Use Word or WordPerfect to recreate the following documents. Each article is worth 10 points and should be emailed to the instructor
More informationAnalysis of Bivariate Data
Analysis of Bivariate Data Data Two Quantitative variables GPA and GAES Interest rates and indices Tax and fund allocation Population size and prison population Bivariate data (x,y) Case corr® 2 Independent
More informationAcknowledgements. Outline. Marie Diener-West. ICTR Leadership / Team INTRODUCTION TO CLINICAL RESEARCH. Introduction to Linear Regression
INTRODUCTION TO CLINICAL RESEARCH Introduction to Linear Regression Karen Bandeen-Roche, Ph.D. July 17, 2012 Acknowledgements Marie Diener-West Rick Thompson ICTR Leadership / Team JHU Intro to Clinical
More informationSTAT 350. Assignment 4
STAT 350 Assignment 4 1. For the Mileage data in assignment 3 conduct a residual analysis and report your findings. I used the full model for this since my answers to assignment 3 suggested we needed the
More informationPre-Calculus Multiple Choice Questions - Chapter S8
1 If every man married a women who was exactly 3 years younger than he, what would be the correlation between the ages of married men and women? a Somewhat negative b 0 c Somewhat positive d Nearly 1 e
More informationHomework 2: Simple Linear Regression
STAT 4385 Applied Regression Analysis Homework : Simple Linear Regression (Simple Linear Regression) Thirty (n = 30) College graduates who have recently entered the job market. For each student, the CGPA
More informationy = a + bx 12.1: Inference for Linear Regression Review: General Form of Linear Regression Equation Review: Interpreting Computer Regression Output
12.1: Inference for Linear Regression Review: General Form of Linear Regression Equation y = a + bx y = dependent variable a = intercept b = slope x = independent variable Section 12.1 Inference for Linear
More informationChapter 8. Linear Regression. Copyright 2010 Pearson Education, Inc.
Chapter 8 Linear Regression Copyright 2010 Pearson Education, Inc. Fat Versus Protein: An Example The following is a scatterplot of total fat versus protein for 30 items on the Burger King menu: Copyright
More informationExamination paper for TMA4255 Applied statistics
Department of Mathematical Sciences Examination paper for TMA4255 Applied statistics Academic contact during examination: Anna Marie Holand Phone: 951 38 038 Examination date: 16 May 2015 Examination time
More informationAnalysis of Variance
Statistical Techniques II EXST7015 Analysis of Variance 15a_ANOVA_Introduction 1 Design The simplest model for Analysis of Variance (ANOVA) is the CRD, the Completely Randomized Design This model is also
More informationIntro to Linear Regression
Intro to Linear Regression Introduction to Regression Regression is a statistical procedure for modeling the relationship among variables to predict the value of a dependent variable from one or more predictor
More informationCategorical Predictor Variables
Categorical Predictor Variables We often wish to use categorical (or qualitative) variables as covariates in a regression model. For binary variables (taking on only 2 values, e.g. sex), it is relatively
More informationLecture 18: Simple Linear Regression
Lecture 18: Simple Linear Regression BIOS 553 Department of Biostatistics University of Michigan Fall 2004 The Correlation Coefficient: r The correlation coefficient (r) is a number that measures the strength
More information[4+3+3] Q 1. (a) Describe the normal regression model through origin. Show that the least square estimator of the regression parameter is given by
Concordia University Department of Mathematics and Statistics Course Number Section Statistics 360/1 40 Examination Date Time Pages Final June 2004 3 hours 7 Instructors Course Examiner Marks Y.P. Chaubey
More informationIntro to Linear Regression
Intro to Linear Regression Introduction to Regression Regression is a statistical procedure for modeling the relationship among variables to predict the value of a dependent variable from one or more predictor
More informationEstimating σ 2. We can do simple prediction of Y and estimation of the mean of Y at any value of X.
Estimating σ 2 We can do simple prediction of Y and estimation of the mean of Y at any value of X. To perform inferences about our regression line, we must estimate σ 2, the variance of the error term.
More informationCh Inference for Linear Regression
Ch. 12-1 Inference for Linear Regression ACT = 6.71 + 5.17(GPA) For every increase of 1 in GPA, we predict the ACT score to increase by 5.17. population regression line β (true slope) μ y = α + βx mean
More informationBasic Business Statistics 6 th Edition
Basic Business Statistics 6 th Edition Chapter 12 Simple Linear Regression Learning Objectives In this chapter, you learn: How to use regression analysis to predict the value of a dependent variable based
More informationTable of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z).
Table of z values and probabilities for the standard normal distribution. z is the first column plus the top row. Each cell shows P(X z). For example P(X 1.04) =.8508. For z < 0 subtract the value from
More informationGROUPED DATA E.G. FOR SAMPLE OF RAW DATA (E.G. 4, 12, 7, 5, MEAN G x / n STANDARD DEVIATION MEDIAN AND QUARTILES STANDARD DEVIATION
FOR SAMPLE OF RAW DATA (E.G. 4, 1, 7, 5, 11, 6, 9, 7, 11, 5, 4, 7) BE ABLE TO COMPUTE MEAN G / STANDARD DEVIATION MEDIAN AND QUARTILES Σ ( Σ) / 1 GROUPED DATA E.G. AGE FREQ. 0-9 53 10-19 4...... 80-89
More information1. An article on peanut butter in Consumer reports reported the following scores for various brands
SMAM 314 Review Exam 1 1. An article on peanut butter in Consumer reports reported the following scores for various brands Creamy 56 44 62 36 39 53 50 65 45 40 56 68 41 30 40 50 50 56 65 56 45 40 Crunchy
More informationThis document contains 3 sets of practice problems.
P RACTICE PROBLEMS This document contains 3 sets of practice problems. Correlation: 3 problems Regression: 4 problems ANOVA: 8 problems You should print a copy of these practice problems and bring them
More informationChapter 14. Multiple Regression Models. Multiple Regression Models. Multiple Regression Models
Chapter 14 Multiple Regression Models 1 Multiple Regression Models A general additive multiple regression model, which relates a dependent variable y to k predictor variables,,, is given by the model equation
More informationThe entire data set consists of n = 32 widgets, 8 of which were made from each of q = 4 different materials.
One-Way ANOVA Summary The One-Way ANOVA procedure is designed to construct a statistical model describing the impact of a single categorical factor X on a dependent variable Y. Tests are run to determine
More informationChapter 14 Student Lecture Notes 14-1
Chapter 14 Student Lecture Notes 14-1 Business Statistics: A Decision-Making Approach 6 th Edition Chapter 14 Multiple Regression Analysis and Model Building Chap 14-1 Chapter Goals After completing this
More informationInstitutionen för matematik och matematisk statistik Umeå universitet November 7, Inlämningsuppgift 3. Mariam Shirdel
Institutionen för matematik och matematisk statistik Umeå universitet November 7, 2011 Inlämningsuppgift 3 Mariam Shirdel (mash0007@student.umu.se) Kvalitetsteknik och försöksplanering, 7.5 hp 1 Uppgift
More informationSMAM 314 Practice Final Examination Winter 2003
SMAM 314 Practice Final Examination Winter 2003 You may use your textbook, one page of notes and a calculator. Please hand in the notes with your exam. 1. Mark the following statements True T or False
More informationK. Model Diagnostics. residuals ˆɛ ij = Y ij ˆµ i N = Y ij Ȳ i semi-studentized residuals ω ij = ˆɛ ij. studentized deleted residuals ɛ ij =
K. Model Diagnostics We ve already seen how to check model assumptions prior to fitting a one-way ANOVA. Diagnostics carried out after model fitting by using residuals are more informative for assessing
More informationunadjusted model for baseline cholesterol 22:31 Monday, April 19,
unadjusted model for baseline cholesterol 22:31 Monday, April 19, 2004 1 Class Level Information Class Levels Values TRETGRP 3 3 4 5 SEX 2 0 1 Number of observations 916 unadjusted model for baseline cholesterol
More informationCh 13 & 14 - Regression Analysis
Ch 3 & 4 - Regression Analysis Simple Regression Model I. Multiple Choice:. A simple regression is a regression model that contains a. only one independent variable b. only one dependent variable c. more
More informationNature vs. nurture? Lecture 18 - Regression: Inference, Outliers, and Intervals. Regression Output. Conditions for inference.
Understanding regression output from software Nature vs. nurture? Lecture 18 - Regression: Inference, Outliers, and Intervals In 1966 Cyril Burt published a paper called The genetic determination of differences
More informationPART I. (a) Describe all the assumptions for a normal error regression model with one predictor variable,
Concordia University Department of Mathematics and Statistics Course Number Section Statistics 360/2 01 Examination Date Time Pages Final December 2002 3 hours 6 Instructors Course Examiner Marks Y.P.
More informationusing the beginning of all regression models
Estimating using the beginning of all regression models 3 examples Note about shorthand Cavendish's 29 measurements of the earth's density Heights (inches) of 14 11 year-old males from Alberta study Half-life
More informationInference for Regression Inference about the Regression Model and Using the Regression Line, with Details. Section 10.1, 2, 3
Inference for Regression Inference about the Regression Model and Using the Regression Line, with Details Section 10.1, 2, 3 Basic components of regression setup Target of inference: linear dependency
More informationMULTIPLE REGRESSION METHODS
DEPARTMENT OF POLITICAL SCIENCE AND INTERNATIONAL RELATIONS Posc/Uapp 816 MULTIPLE REGRESSION METHODS I. AGENDA: A. Residuals B. Transformations 1. A useful procedure for making transformations C. Reading:
More informationResiduals from regression on original data 1
Residuals from regression on original data 1 Obs a b n i y 1 1 1 3 1 1 2 1 1 3 2 2 3 1 1 3 3 3 4 1 2 3 1 4 5 1 2 3 2 5 6 1 2 3 3 6 7 1 3 3 1 7 8 1 3 3 2 8 9 1 3 3 3 9 10 2 1 3 1 10 11 2 1 3 2 11 12 2 1
More informationIs economic freedom related to economic growth?
Is economic freedom related to economic growth? It is an article of faith among supporters of capitalism: economic freedom leads to economic growth. The publication Economic Freedom of the World: 2003
More informationRegression Analysis: Basic Concepts
The simple linear model Regression Analysis: Basic Concepts Allin Cottrell Represents the dependent variable, y i, as a linear function of one independent variable, x i, subject to a random disturbance
More informationMATH 1150 Chapter 2 Notation and Terminology
MATH 1150 Chapter 2 Notation and Terminology Categorical Data The following is a dataset for 30 randomly selected adults in the U.S., showing the values of two categorical variables: whether or not the
More informationSix Sigma Black Belt Study Guides
Six Sigma Black Belt Study Guides 1 www.pmtutor.org Powered by POeT Solvers Limited. Analyze Correlation and Regression Analysis 2 www.pmtutor.org Powered by POeT Solvers Limited. Variables and relationships
More informationDraft Proof - Do not copy, post, or distribute. Chapter Learning Objectives REGRESSION AND CORRELATION THE SCATTER DIAGRAM
1 REGRESSION AND CORRELATION As we learned in Chapter 9 ( Bivariate Tables ), the differential access to the Internet is real and persistent. Celeste Campos-Castillo s (015) research confirmed the impact
More informationPLS205!! Lab 9!! March 6, Topic 13: Covariance Analysis
PLS205!! Lab 9!! March 6, 2014 Topic 13: Covariance Analysis Covariable as a tool for increasing precision Carrying out a full ANCOVA Testing ANOVA assumptions Happiness! Covariable as a Tool for Increasing
More informationSMAM 314 Exam 42 Name
SMAM 314 Exam 42 Name Mark the following statements True (T) or False (F) (10 points) 1. F A. The line that best fits points whose X and Y values are negatively correlated should have a positive slope.
More informationExample. Multiple Regression. Review of ANOVA & Simple Regression /749 Experimental Design for Behavioral and Social Sciences
36-309/749 Experimental Design for Behavioral and Social Sciences Sep. 29, 2015 Lecture 5: Multiple Regression Review of ANOVA & Simple Regression Both Quantitative outcome Independent, Gaussian errors
More informationHistogram of Residuals. Residual Normal Probability Plot. Reg. Analysis Check Model Utility. (con t) Check Model Utility. Inference.
Steps for Regression Simple Linear Regression Make a Scatter plot Does it make sense to plot a line? Check Residual Plot (Residuals vs. X) Are there any patterns? Check Histogram of Residuals Is it Normal?
More informationSimple Linear Regression. Steps for Regression. Example. Make a Scatter plot. Check Residual Plot (Residuals vs. X)
Simple Linear Regression 1 Steps for Regression Make a Scatter plot Does it make sense to plot a line? Check Residual Plot (Residuals vs. X) Are there any patterns? Check Histogram of Residuals Is it Normal?
More information2.4.3 Estimatingσ Coefficient of Determination 2.4. ASSESSING THE MODEL 23
2.4. ASSESSING THE MODEL 23 2.4.3 Estimatingσ 2 Note that the sums of squares are functions of the conditional random variables Y i = (Y X = x i ). Hence, the sums of squares are random variables as well.
More informationINTRODUCTION TO DESIGN AND ANALYSIS OF EXPERIMENTS
GEORGE W. COBB Mount Holyoke College INTRODUCTION TO DESIGN AND ANALYSIS OF EXPERIMENTS Springer CONTENTS To the Instructor Sample Exam Questions To the Student Acknowledgments xv xxi xxvii xxix 1. INTRODUCTION
More informationExample: Four levels of herbicide strength in an experiment on dry weight of treated plants.
The idea of ANOVA Reminders: A factor is a variable that can take one of several levels used to differentiate one group from another. An experiment has a one-way, or completely randomized, design if several
More informationTables Table A Table B Table C Table D Table E 675
BMTables.indd Page 675 11/15/11 4:25:16 PM user-s163 Tables Table A Standard Normal Probabilities Table B Random Digits Table C t Distribution Critical Values Table D Chi-square Distribution Critical Values
More informationSteps for Regression. Simple Linear Regression. Data. Example. Residuals vs. X. Scatterplot. Make a Scatter plot Does it make sense to plot a line?
Steps for Regression Simple Linear Regression Make a Scatter plot Does it make sense to plot a line? Check Residual Plot (Residuals vs. X) Are there any patterns? Check Histogram of Residuals Is it Normal?
More information10 Model Checking and Regression Diagnostics
10 Model Checking and Regression Diagnostics The simple linear regression model is usually written as i = β 0 + β 1 i + ɛ i where the ɛ i s are independent normal random variables with mean 0 and variance
More informationSMAM 319 Exam 1 Name. 1.Pick the best choice for the multiple choice questions below (10 points 2 each)
SMAM 319 Exam 1 Name 1.Pick the best choice for the multiple choice questions below (10 points 2 each) A b In Metropolis there are some houses for sale. Superman and Lois Lane are interested in the average
More informationSociology 6Z03 Review II
Sociology 6Z03 Review II John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review II Fall 2016 1 / 35 Outline: Review II Probability Part I Sampling Distributions Probability
More informationLecture Slides. Elementary Statistics. by Mario F. Triola. and the Triola Statistics Series
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 13 Nonparametric Statistics 13-1 Overview 13-2 Sign Test 13-3 Wilcoxon Signed-Ranks
More information