We need to define some concepts that are used in experiments.
|
|
- Kerry Nash
- 5 years ago
- Views:
Transcription
1 Chapter 0 Analysis of Variance (a.k.a. Designing and Analysing Experiments) Section 0. Introduction In Chapter we mentioned some different ways in which we could get data: Surveys, Observational Studies, Experiments and Published Data. We have up to now analysed datasets in many ways, but we haven't studied how to collect the data. In this chapter we will look at the design of experiments and then we will see some new techniques for examining data collected in experiments. In an experiment the experimenter usually is interested in measuring the response of an individual or object to something applied to that individual or object. For example a pharmaceutical company may want to see the effect a drug has on patients with a particular disease. We need to define some concepts that are used in experiments.
2 Section 0. Definitions The Response Variable is the variable being measured in the experiment. Factors are variables applied to the object or individual in the experiment. We are interested in m asuring the effect of the Factors on the Response Variable. Quantitative Factors are measured on a numerical scale whereas Qualitative Factors are not measured on a numerical scale. Factor-Levels are the values of the factor used in the experiment. Treatments are the Factor-Level combinations used. An Experimental Unit is the object on which the response and factors are measured. A Designed Experiment is one for which the researcher controls the specifications of the treatments and the method of assigning the experimental units to each treatment. An Observational Experiment (Study) is one in which the researcher simply observes the treatments and the response on a sample of experimental units.
3 Section 0. Single Factor Anovas 0.. Introduction Suppose we have designed and conducted our experiment and it involved applying p different treatments to p different samples chosen in some manner. We are now interested in determining whether the responses of the experimental units differs according to treatment they received. As is usual in statistics we are interested in making an inference about a population using sample data. Here we have p populations each one represents the entire population who have in the past or will in the future receive each treatment. So here we have the following notation: P = Number of treatments being compared Population or Treatment 3 p Population or Treatment Mean μ μ μ 3 μ p Population or Treatment Variance σ σ σ 3 σ p Sample Size n n n 3 n p Sample Mean x x x 3 x p Sample Variance s s s 3 s p Also x is the mean of all the measurements
4 0.. So what are we at here? What are we trying to test here anyway? Well we're interested in seeing if there is any difference in the effect of each treatment on the experimental units. That means we are interested in testing if there is a difference between μ, μ, μ 3 etc. So in the single factor ANOVA we test the following hypotheses: H 0 : μ = μ = μ 3 =. = μ p H A : At least two of the μ's are different. Before we can do anything we must make an assumption to simplify life ASSUMPTION: We assume that σ = σ = σ 3 =.. = σ p and since they're all the same we just use the symbol σ. We also assume that we are dealing with a Completely Randomised Design Definition The Completely Randomised Design is an experimental design in which independent random samples of experimental units are selected for each treatment.
5 0..5 Now how will this test work? We will calculate sample means corresponding to each of the population means. We will check whether the difference between these sample means is small enough so that it would be explained by natural variability in the data. If the difference is instead larger than could be explained by natural variability then we will conclude that the population means actually are different. We could measure how far apart the individual sample means are by calculating SST the Sum of Squares for Treatments Definition: SST = [ n ( x - x ) + n ( x - x ) +. + n p ( x p - x ) ] A better measure however is MST - The Mean Square for Treatments. Definition: MST = SST/(p-) If all the individual x 's were the same and so also equal to the total x, then MST would be zero. The further the individual x 's are from each other then the bigger MST will be. If it is really big we will reject the null hypothesis and so conclude that at least two of the population means are different.
6 But we are left with the usual question, how big is big enough to reject? To begin to answer that question we first must notice that when H 0 is true the mean value of MST is σ. But when H 0 is false the mean of MST is not σ but instead is larger than σ. There is another statistic which estimates σ accurately and unbiased whether or not H 0 is true or not, it is called the Mean Square for Error, MSE. Definition: MSE = SSE/(N-p) where SSE = (n -) s + (n -) s +(n 3 -) s (n p -) s p and N is the total number of observations in the dataset.
7 0..6 The Test So MST should in general be bigger than MSE if H 0 is false. The technique we will use therefore is to compute the ratio MST/MSE and if it is big enough we will reject H 0. This ratio will be our test statistic. So far when we have encountered test statistics they have been Z's or T's, this ratio is neither, it is from a new distribution the F. Definition: The test statistic for a single-factor ANOVA is MST F = MSE with p- numerator degrees of freedom and N-p denominator degrees of freedom. These tests are all upper tailed, we want to reject when the ratio is really big. So we look up F tables and reject H 0 if F calc is bigger than some critical value of F. All of what we have seen in this chapter so far can be summarised in the following table:
8 Single Factor Anova Table - Completely Randomised Design Source df SS MS F Treatments p- SST MST= SST/(p-) MST/MSE Error N-p SSE MSE= SSE/(N-p) Total N- SStotal 0..7 Anova Computations We have seen formulae for SSE and SST however in practice these formulae are very cumbersome for performing calculations. The following formulae are easier to use: SST = [n x + n x +. + n p x p ] - N x SSTotal = N x Nx SSE = SSTotal - SST
9 0..8 Example Detergent makers always claim that their product washes whiter. We are going to test whether there is any difference between three detergents, BOLD, DAZ and PERSIL. 9 students are asked to wear white shirts and to go out for a night's drinking and whatever. The next morning we randomly assign 3 of the 9 white shirts to be washed using BOLD, 3 using DAZ and 3 using PERSIL. After the wash is over we bring the 9 shirts over to a Biology lab and examine them under a microscope for stains. The surface area that remains dirty is determined and the following results are obtained: BOLD:,, 3 DAZ: 3, 4, 4 PERSIL:, 3, Test whether there is a difference between the population means use significance level ANSWER: Analysis of Variance Source DF SS MS F P Treatment Error Total
10 Section 0.3 Randomised Block Design Sometimes randomisation can be improved upon. In the other half of this course you have seen paired T-tests. This idea of pairing can be extended to ANOVAs. Suppose we are conducting an experiment to measure the performance of a certain drug. The effect of this drug may be different on people depending on their ages, sex, blood pressure, weight etc. It is possible that the randomisation procedure does not evenly spread these characteristics among the different treatments. Sometimes it is better to force the people receiving each treatment to be the same. For example in this drug trial, we can pick groups of people with similar characteristics to receive the different treatments that way any difference we observe will be because of the treatments not because of different characteristics of the people Definition: A Randomised Block Design is an experimental procedure consisting of two steps:. The experimental units are divided into b blocks, the units chosen for a Block will be as similar as possible. There are p units in each Block where p is the number of Treatments being compared.. One experimental unit from each Block is randomly assigned to each treatment so in total there will be n = bp responses.
11 0.3. Assumptions. The observations corresponding to all blocktreatment combinations are Normally distributed.. The variances of all the Normal distributions are the same, σ. But the mean of all the Normal distributions may depend on the treatment applied and also on the block Notation and Formulae p = Number of Treatments b = Number of Blocks The average of all observations on the i th treatment is x Ti The average of all observations on the i th block is x Bi SST = b( x x) p = b x Nx p T i T i
12 b SSB = p( x x) b = p x Nx B i SSTotal = ( x x) N = x N Nx And SSE = SSTotal -SST - SSB The Test B i The purpose of the Randomised Block Design is the same as the Completely Randomised Design ie to test: H 0 : μ = μ = μ 3 =. = μ p H A : At least two of the μ's are different.
13 The procedure for the test is to complete an ANOVA table in the following format: Source df SS MS F Treatments p- SST MST= SST/(p-) MST/MSE Blocks b- SSB MSB= MSB/MSE SSB/(b-) Error N-p-b+ SSE MSE= SSE/(N-p) Total N- SStotal The last column in this table includes two F statistics, the first one F=MST/MSE is the statistic used to test H 0 : The treatment means are all the same vs H A : At least two of the treatment means differ. The second test statistic F=MSB/MSE tests whether there is a difference in the Block means. Rejection of this test indicates that the Block means differ and that the approach of using a Randomised Block Design instead of a Completely Randomised Design was a good choice.
14 0.3.5 Example A single factor ANOVA was conducted using a Randomised Block Design. It yielded the following results: Block Treatments Do the data provide sufficient evidence to suggest that the treatment means differ? Do the data provide sufficient evidence top indicate that blocking was effective in reducing the experimental error? Source df SS MS F Treatments Blocks Error Total
15 Section 0.4 Multiple Comparisons of Means 0.4. Introduction In the previous section we have seen how to determine if a set of n means are equal or if there is a difference between at least two of the means. In rejecting the F test of the single factor Anova all that we determined was that there was a difference between at least two of the means. We did not identify which of the means were different and which the same. In an experiment with 3 treatments there are three pairs of means which may be different (μ - μ ), (μ - μ 3 ) & (μ - μ 3 ). Having established that one of these pairs differs we now must test each pair seperately. In an experiment with P treatments there will be p ie: "p choose " possible combinations of pairs of means to be tested. Each pair of means will be tested essentially using a T-test. However there are two minor differences to the way we will perform these tests. Firstly we will actually use compute Confidence Intervals instead of performing Hypothesis Tests. Secondly we must use a different set of T-tables called Bonferroni -T tables because we are carrying out multiple tests.
16 0.4. Why Confidence Intervals are just like Hypothesis Tests Suppose we compute a 90% Confidence Interval the probability that the interval contains the actual value of the Population Characteristic is 0.9. That means the probability that the interval does not contain the actual value of the test statistic is 0.. The region outside the 90% Confidence Interval is therefore just like the Rejection Region of a Two Tailed Hypothesis Test with α = 0.. This is evident from the picture. So if we find that a 90% Confidence Interval for μ - μ does not contain the value 0 that is equivalent to rejecting the Null Hypothesis of the following Two Tailed Test with significance level α = 0.: H 0 : μ - μ = 0 Vs H A : μ - μ 0
17 0.4.3 Multiple Tests and Type errors When a Hypothesis test is performed at significance level α = 0., that means that there is a 0. probability of rejecting the Null Hypothesis when in fact it is true. When we perform two tests each with α = 0., the probability of making a Type error on the first test or on the second test is now larger than 0.. Remember P(A or B) = P(A) + P(B) - P(A and B) In an experiment with p treatments we will be performing p(p-)/ tests and the combined probability of making a Type error will be large. For this reason certain new versions of the T-Tables were invented called Bonferroni T-Tables. These tables take into account that you may want to perform several linked tests and they give critical values to be used in each test so that the OVERALL probability of a Type error for the Experiment as a whole is below a certain threshold.
18 0.4.4 The Bonferroni Multiple Comparisons Procedure When comparing p treatments or populations, we decide whether two population means differ by. Computing the p(p-)/ confidence intervals below. Checking if any intervals do not contain Intervals which do not contain 0 indicate a significant difference in the population means (μ's). μ μ :. x x ± ( Bonferroni Tcritical) MSE n + MSE n.. μ p μ p : x p x p ± ( Bonferroni Tcritical) MSE n p + MSE n p The Bonferroni T critical values are computed using the Error degrees of freedom.
ANOVA: Comparing More Than Two Means
1 ANOVA: Comparing More Than Two Means 10.1 ANOVA: The Completely Randomized Design Elements of a Designed Experiment Before we begin any calculations, we need to discuss some terminology. To make this
More informationSTAT Chapter 10: Analysis of Variance
STAT 515 -- Chapter 10: Analysis of Variance Designed Experiment A study in which the researcher controls the levels of one or more variables to determine their effect on the variable of interest (called
More information1 The Randomized Block Design
1 The Randomized Block Design When introducing ANOVA, we mentioned that this model will allow us to include more than one categorical factor(explanatory) or confounding variables in the model. In a first
More informationChapter 15: Analysis of Variance
Chapter 5: Analysis of Variance 5. Introduction In this chapter, we introduced the analysis of variance technique, which deals with problems whose objective is to compare two or more populations of quantitative
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 informationSection 9.4. Notation. Requirements. Definition. Inferences About Two Means (Matched Pairs) Examples
Objective Section 9.4 Inferences About Two Means (Matched Pairs) Compare of two matched-paired means using two samples from each population. Hypothesis Tests and Confidence Intervals of two dependent means
More informationAnalysis of Variance
Analysis of Variance Math 36b May 7, 2009 Contents 2 ANOVA: Analysis of Variance 16 2.1 Basic ANOVA........................... 16 2.1.1 the model......................... 17 2.1.2 treatment sum of squares.................
More informationSTAT 115:Experimental Designs
STAT 115:Experimental Designs Josefina V. Almeda 2013 Multisample inference: Analysis of Variance 1 Learning Objectives 1. Describe Analysis of Variance (ANOVA) 2. Explain the Rationale of ANOVA 3. Compare
More informationIn ANOVA the response variable is numerical and the explanatory variables are categorical.
1 ANOVA ANOVA means ANalysis Of VAriance. The ANOVA is a tool for studying the influence of one or more qualitative variables on the mean of a numerical variable in a population. In ANOVA the response
More informationOne-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
More informationSummary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)
Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ
More informationwith the usual assumptions about the error term. The two values of X 1 X 2 0 1
Sample questions 1. A researcher is investigating the effects of two factors, X 1 and X 2, each at 2 levels, on a response variable Y. A balanced two-factor factorial design is used with 1 replicate. The
More informationUnit 27 One-Way Analysis of Variance
Unit 27 One-Way Analysis of Variance Objectives: To perform the hypothesis test in a one-way analysis of variance for comparing more than two population means Recall that a two sample t test is applied
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 informationChapter 10. Design of Experiments and Analysis of Variance
Chapter 10 Design of Experiments and Analysis of Variance Elements of a Designed Experiment Response variable Also called the dependent variable Factors (quantitative and qualitative) Also called the independent
More information16.3 One-Way ANOVA: The Procedure
16.3 One-Way ANOVA: The Procedure Tom Lewis Fall Term 2009 Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term 2009 1 / 10 Outline 1 The background 2 Computing formulas 3 The ANOVA Identity 4 Tom
More informationLAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2
LAB 2. HYPOTHESIS TESTING IN THE BIOLOGICAL SCIENCES- Part 2 Data Analysis: The mean egg masses (g) of the two different types of eggs may be exactly the same, in which case you may be tempted to accept
More informationStatistics For Economics & Business
Statistics For Economics & Business Analysis of Variance In this chapter, you learn: Learning Objectives The basic concepts of experimental design How to use one-way analysis of variance to test for differences
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 information1 DV is normally distributed in the population for each level of the within-subjects factor 2 The population variances of the difference scores
One-way Prepared by: Prof. Dr Bahaman Abu Samah Department of Professional Development and Continuing Education Faculty of Educational Studies Universiti Putra Malaysia Serdang The purpose is to test the
More informationSIMPLE REGRESSION ANALYSIS. Business Statistics
SIMPLE REGRESSION ANALYSIS Business Statistics CONTENTS Ordinary least squares (recap for some) Statistical formulation of the regression model Assessing the regression model Testing the regression coefficients
More informationIn a one-way ANOVA, the total sums of squares among observations is partitioned into two components: Sums of squares represent:
Activity #10: AxS ANOVA (Repeated subjects design) Resources: optimism.sav So far in MATH 300 and 301, we have studied the following hypothesis testing procedures: 1) Binomial test, sign-test, Fisher s
More informationIntroduction to Business Statistics QM 220 Chapter 12
Department of Quantitative Methods & Information Systems Introduction to Business Statistics QM 220 Chapter 12 Dr. Mohammad Zainal 12.1 The F distribution We already covered this topic in Ch. 10 QM-220,
More informationNotes for Week 13 Analysis of Variance (ANOVA) continued WEEK 13 page 1
Notes for Wee 13 Analysis of Variance (ANOVA) continued WEEK 13 page 1 Exam 3 is on Friday May 1. A part of one of the exam problems is on Predictiontervals : When randomly sampling from a normal population
More informationMultiple Regression Analysis
Multiple Regression Analysis y = β 0 + β 1 x 1 + β 2 x 2 +... β k x k + u 2. Inference 0 Assumptions of the Classical Linear Model (CLM)! So far, we know: 1. The mean and variance of the OLS estimators
More informationOne-factor analysis of variance (ANOVA)
One-factor analysis of variance (ANOVA) March 1, 2017 psych10.stanford.edu Announcements / Action Items Schedule update: final R lab moved to Week 10 Optional Survey 5 coming soon, due on Saturday Last
More informationEcon 3790: Business and Economic Statistics. Instructor: Yogesh Uppal
Econ 3790: Business and Economic Statistics Instructor: Yogesh Uppal Email: yuppal@ysu.edu Chapter 13, Part A: Analysis of Variance and Experimental Design Introduction to Analysis of Variance Analysis
More informationComparing the means of more than two groups
Comparing the means of more than two groups Chapter 15 Analysis of variance (ANOVA) Like a t-test, but can compare more than two groups Asks whether any of two or more means is different from any other.
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 informationAMS7: WEEK 7. CLASS 1. More on Hypothesis Testing Monday May 11th, 2015
AMS7: WEEK 7. CLASS 1 More on Hypothesis Testing Monday May 11th, 2015 Testing a Claim about a Standard Deviation or a Variance We want to test claims about or 2 Example: Newborn babies from mothers taking
More information2 Hand-out 2. Dr. M. P. M. M. M c Loughlin Revised 2018
Math 403 - P. & S. III - Dr. McLoughlin - 1 2018 2 Hand-out 2 Dr. M. P. M. M. M c Loughlin Revised 2018 3. Fundamentals 3.1. Preliminaries. Suppose we can produce a random sample of weights of 10 year-olds
More informationCh. 1: Data and Distributions
Ch. 1: Data and Distributions Populations vs. Samples How to graphically display data Histograms, dot plots, stem plots, etc Helps to show how samples are distributed Distributions of both continuous and
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 informationChapter 9 Inferences from Two Samples
Chapter 9 Inferences from Two Samples 9-1 Review and Preview 9-2 Two Proportions 9-3 Two Means: Independent Samples 9-4 Two Dependent Samples (Matched Pairs) 9-5 Two Variances or Standard Deviations Review
More informationSampling Distributions: Central Limit Theorem
Review for Exam 2 Sampling Distributions: Central Limit Theorem Conceptually, we can break up the theorem into three parts: 1. The mean (µ M ) of a population of sample means (M) is equal to the mean (µ)
More informationHypothesis testing: Steps
Review for Exam 2 Hypothesis testing: Steps Repeated-Measures ANOVA 1. Determine appropriate test and hypotheses 2. Use distribution table to find critical statistic value(s) representing rejection region
More informationHypothesis T e T sting w ith with O ne O One-Way - ANOV ANO A V Statistics Arlo Clark Foos -
Hypothesis Testing with One-Way ANOVA Statistics Arlo Clark-Foos Conceptual Refresher 1. Standardized z distribution of scores and of means can be represented as percentile rankings. 2. t distribution
More informationWhile you wait: Enter the following in your calculator. Find the mean and sample variation of each group. Bluman, Chapter 12 1
While you wait: Enter the following in your calculator. Find the mean and sample variation of each group. Bluman, Chapter 12 1 Chapter 12 Analysis of Variance McGraw-Hill, Bluman, 7th ed., Chapter 12 2
More informationStatistical methods for comparing multiple groups. Lecture 7: ANOVA. ANOVA: Definition. ANOVA: Concepts
Statistical methods for comparing multiple groups Lecture 7: ANOVA Sandy Eckel seckel@jhsph.edu 30 April 2008 Continuous data: comparing multiple means Analysis of variance Binary data: comparing multiple
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 informationChapter 12 - Lecture 2 Inferences about regression coefficient
Chapter 12 - Lecture 2 Inferences about regression coefficient April 19th, 2010 Facts about slope Test Statistic Confidence interval Hypothesis testing Test using ANOVA Table Facts about slope In previous
More informationCHAPTER 9: HYPOTHESIS TESTING
CHAPTER 9: HYPOTHESIS TESTING THE SECOND LAST EXAMPLE CLEARLY ILLUSTRATES THAT THERE IS ONE IMPORTANT ISSUE WE NEED TO EXPLORE: IS THERE (IN OUR TWO SAMPLES) SUFFICIENT STATISTICAL EVIDENCE TO CONCLUDE
More informationAnalysis of Variance (ANOVA)
Analysis of Variance ANOVA) Compare several means Radu Trîmbiţaş 1 Analysis of Variance for a One-Way Layout 1.1 One-way ANOVA Analysis of Variance for a One-Way Layout procedure for one-way layout Suppose
More informationHypothesis testing: Steps
Review for Exam 2 Hypothesis testing: Steps Exam 2 Review 1. Determine appropriate test and hypotheses 2. Use distribution table to find critical statistic value(s) representing rejection region 3. Compute
More informationAnalysis of variance (ANOVA) ANOVA. Null hypothesis for simple ANOVA. H 0 : Variance among groups = 0
Analysis of variance (ANOVA) ANOVA Comparing the means of more than two groups Like a t-test, but can compare more than two groups Asks whether any of two or more means is different from any other. In
More informationModule 03 Lecture 14 Inferential Statistics ANOVA and TOI
Introduction of Data Analytics Prof. Nandan Sudarsanam and Prof. B Ravindran Department of Management Studies and Department of Computer Science and Engineering Indian Institute of Technology, Madras Module
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 informationFormal Statement of Simple Linear Regression Model
Formal Statement of Simple Linear Regression Model Y i = β 0 + β 1 X i + ɛ i Y i value of the response variable in the i th trial β 0 and β 1 are parameters X i is a known constant, the value of the predictor
More informationHYPOTHESIS TESTING. Hypothesis Testing
MBA 605 Business Analytics Don Conant, PhD. HYPOTHESIS TESTING Hypothesis testing involves making inferences about the nature of the population on the basis of observations of a sample drawn from the population.
More informationThe t-statistic. Student s t Test
The t-statistic 1 Student s t Test When the population standard deviation is not known, you cannot use a z score hypothesis test Use Student s t test instead Student s t, or t test is, conceptually, very
More 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 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 information1. The (dependent variable) is the variable of interest to be measured in the experiment.
Chapter 10 Analysis of variance (ANOVA) 10.1 Elements of a designed experiment 1. The (dependent variable) is the variable of interest to be measured in the experiment. 2. are those variables whose effect
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 information10/31/2012. One-Way ANOVA F-test
PSY 511: Advanced Statistics for Psychological and Behavioral Research 1 1. Situation/hypotheses 2. Test statistic 3.Distribution 4. Assumptions One-Way ANOVA F-test One factor J>2 independent samples
More informationAssignment #7. Chapter 12: 18, 24 Chapter 13: 28. Due next Friday Nov. 20 th by 2pm in your TA s homework box
Assignment #7 Chapter 12: 18, 24 Chapter 13: 28 Due next Friday Nov. 20 th by 2pm in your TA s homework box Lab Report Posted on web-site Dates Rough draft due to TAs homework box on Monday Nov. 16 th
More informationOne-way Analysis of Variance. Major Points. T-test. Ψ320 Ainsworth
One-way Analysis of Variance Ψ30 Ainsworth Major Points Problem with t-tests and multiple groups The logic behind ANOVA Calculations Multiple comparisons Assumptions of analysis of variance Effect Size
More informationWeek 14 Comparing k(> 2) Populations
Week 14 Comparing k(> 2) Populations Week 14 Objectives Methods associated with testing for the equality of k(> 2) means or proportions are presented. Post-testing concepts and analysis are introduced.
More informationOHSU OGI Class ECE-580-DOE :Design of Experiments Steve Brainerd
Why We Use Analysis of Variance to Compare Group Means and How it Works The question of how to compare the population means of more than two groups is an important one to researchers. Let us suppose that
More informationReview. One-way ANOVA, I. What s coming up. Multiple comparisons
Review One-way ANOVA, I 9.07 /15/00 Earlier in this class, we talked about twosample z- and t-tests for the difference between two conditions of an independent variable Does a trial drug work better than
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Motivations for the ANOVA We defined the F-distribution, this is mainly used in
More informationChapter 10: Analysis of variance (ANOVA)
Chapter 10: Analysis of variance (ANOVA) ANOVA (Analysis of variance) is a collection of techniques for dealing with more general experiments than the previous one-sample or two-sample tests. We first
More informationThe legacy of Sir Ronald A. Fisher. Fisher s three fundamental principles: local control, replication, and randomization.
1 Chapter 1: Research Design Principles The legacy of Sir Ronald A. Fisher. Fisher s three fundamental principles: local control, replication, and randomization. 2 Chapter 2: Completely Randomized Design
More informationTwo-Sample Inferential Statistics
The t Test for Two Independent Samples 1 Two-Sample Inferential Statistics In an experiment there are two or more conditions One condition is often called the control condition in which the treatment is
More informationCorrelation Analysis
Simple Regression Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the
More informationMultiple Linear Regression
Multiple Linear Regression Simple linear regression tries to fit a simple line between two variables Y and X. If X is linearly related to Y this explains some of the variability in Y. In most cases, there
More informationQuestion. Hypothesis testing. Example. Answer: hypothesis. Test: true or not? Question. Average is not the mean! μ average. Random deviation or not?
Hypothesis testing Question Very frequently: what is the possible value of μ? Sample: we know only the average! μ average. Random deviation or not? Standard error: the measure of the random deviation.
More informationCHAPTER 9, 10. Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities:
CHAPTER 9, 10 Hypothesis Testing Similar to a courtroom trial. In trying a person for a crime, the jury needs to decide between one of two possibilities: The person is guilty. The person is innocent. To
More informationAnalysis Of Variance Compiled by T.O. Antwi-Asare, U.G
Analysis Of Variance Compiled by T.O. Antwi-Asare, U.G 1 ANOVA Analysis of variance compares two or more population means of interval data. Specifically, we are interested in determining whether differences
More informationSTAT Chapter 8: Hypothesis Tests
STAT 515 -- Chapter 8: Hypothesis Tests CIs are possibly the most useful forms of inference because they give a range of reasonable values for a parameter. But sometimes we want to know whether one particular
More informationECO220Y Simple Regression: Testing the Slope
ECO220Y Simple Regression: Testing the Slope Readings: Chapter 18 (Sections 18.3-18.5) Winter 2012 Lecture 19 (Winter 2012) Simple Regression Lecture 19 1 / 32 Simple Regression Model y i = β 0 + β 1 x
More informationChapter 8 Student Lecture Notes 8-1. Department of Economics. Business Statistics. Chapter 12 Chi-square test of independence & Analysis of Variance
Chapter 8 Student Lecture Notes 8-1 Department of Economics Business Statistics Chapter 1 Chi-square test of independence & Analysis of Variance ECON 509 Dr. Mohammad Zainal Chapter Goals After completing
More informationStatistical Inference. Hypothesis Testing
Statistical Inference Hypothesis Testing Previously, we introduced the point and interval estimation of an unknown parameter(s), say µ and σ 2. However, in practice, the problem confronting the scientist
More informationWhat If There Are More Than. Two Factor Levels?
What If There Are More Than Chapter 3 Two Factor Levels? Comparing more that two factor levels the analysis of variance ANOVA decomposition of total variability Statistical testing & analysis Checking
More informationLecture notes 13: ANOVA (a.k.a. Analysis of Variance)
Lecture notes 13: ANOVA (a.k.a. Analysis of Variance) Outline: Testing for a difference in means Notation Sums of squares Mean squares The F distribution The ANOVA table Part II: multiple comparisons Worked
More informationINTRODUCTION TO ANALYSIS OF VARIANCE
CHAPTER 22 INTRODUCTION TO ANALYSIS OF VARIANCE Chapter 18 on inferences about population means illustrated two hypothesis testing situations: for one population mean and for the difference between two
More informationa Sample By:Dr.Hoseyn Falahzadeh 1
In the name of God Determining ee the esize eof a Sample By:Dr.Hoseyn Falahzadeh 1 Sample Accuracy Sample accuracy: refers to how close a random sample s statistic is to the true population s value it
More informationEcon 3790: Business and Economics Statistics. Instructor: Yogesh Uppal
Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal yuppal@ysu.edu Sampling Distribution of b 1 Expected value of b 1 : Variance of b 1 : E(b 1 ) = 1 Var(b 1 ) = σ 2 /SS x Estimate of
More informationChapter 23: Inferences About Means
Chapter 3: Inferences About Means Sample of Means: number of observations in one sample the population mean (theoretical mean) sample mean (observed mean) is the theoretical standard deviation of the population
More informationIntroduction to Statistical Inference Lecture 10: ANOVA, Kruskal-Wallis Test
Introduction to Statistical Inference Lecture 10: ANOVA, Kruskal-Wallis Test la Contents The two sample t-test generalizes into Analysis of Variance. In analysis of variance ANOVA the population consists
More informationMultiple t Tests. Introduction to Analysis of Variance. Experiments with More than 2 Conditions
Introduction to Analysis of Variance 1 Experiments with More than 2 Conditions Often the research that psychologists perform has more conditions than just the control and experimental conditions You might
More informationChapter 3 Multiple Regression Complete Example
Department of Quantitative Methods & Information Systems ECON 504 Chapter 3 Multiple Regression Complete Example Spring 2013 Dr. Mohammad Zainal Review Goals After completing this lecture, you should be
More informationPLSC PRACTICE TEST ONE
PLSC 724 - PRACTICE TEST ONE 1. Discuss briefly the relationship between the shape of the normal curve and the variance. 2. What is the relationship between a statistic and a parameter? 3. How is the α
More informationLecture 9 Two-Sample Test. Fall 2013 Prof. Yao Xie, H. Milton Stewart School of Industrial Systems & Engineering Georgia Tech
Lecture 9 Two-Sample Test Fall 2013 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milton Stewart School of Industrial Systems & Engineering Georgia Tech Computer exam 1 18 Histogram 14 Frequency 9 5 0 75 83.33333333
More informationTopic 6. Two-way designs: Randomized Complete Block Design [ST&D Chapter 9 sections 9.1 to 9.7 (except 9.6) and section 15.8]
Topic 6. Two-way designs: Randomized Complete Block Design [ST&D Chapter 9 sections 9.1 to 9.7 (except 9.6) and section 15.8] The completely randomized design Treatments are randomly assigned to e.u. such
More informationAnalysis of Variance. ภาว น ศ ร ประภาน ก ล คณะเศรษฐศาสตร มหาว ทยาล ยธรรมศาสตร
Analysis of Variance ภาว น ศ ร ประภาน ก ล คณะเศรษฐศาสตร มหาว ทยาล ยธรรมศาสตร pawin@econ.tu.ac.th Outline Introduction One Factor Analysis of Variance Two Factor Analysis of Variance ANCOVA MANOVA Introduction
More informationSleep data, two drugs Ch13.xls
Model Based Statistics in Biology. Part IV. The General Linear Mixed Model.. Chapter 13.3 Fixed*Random Effects (Paired t-test) ReCap. Part I (Chapters 1,2,3,4), Part II (Ch 5, 6, 7) ReCap Part III (Ch
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 informationAn Analysis of College Algebra Exam Scores December 14, James D Jones Math Section 01
An Analysis of College Algebra Exam s December, 000 James D Jones Math - Section 0 An Analysis of College Algebra Exam s Introduction Students often complain about a test being too difficult. Are there
More informationBALANCED INCOMPLETE BLOCK DESIGNS
BALANCED INCOMPLETE BLOCK DESIGNS V.K. Sharma I.A.S.R.I., Library Avenue, New Delhi -110012. 1. Introduction In Incomplete block designs, as their name implies, the block size is less than the number of
More informationSTAT Chapter 11: Regression
STAT 515 -- Chapter 11: Regression Mostly we have studied the behavior of a single random variable. Often, however, we gather data on two random variables. We wish to determine: Is there a relationship
More informationOne sided tests. An example of a two sided alternative is what we ve been using for our two sample tests:
One sided tests So far all of our tests have been two sided. While this may be a bit easier to understand, this is often not the best way to do a hypothesis test. One simple thing that we can do to get
More informationDepartment of Economics. Business Statistics. Chapter 12 Chi-square test of independence & Analysis of Variance ECON 509. Dr.
Department of Economics Business Statistics Chapter 1 Chi-square test of independence & Analysis of Variance ECON 509 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should be able
More informationCS 5014: Research Methods in Computer Science
Computer Science Clifford A. Shaffer Department of Computer Science Virginia Tech Blacksburg, Virginia Fall 2010 Copyright c 2010 by Clifford A. Shaffer Computer Science Fall 2010 1 / 207 Correlation and
More informationChapter 14 Student Lecture Notes Department of Quantitative Methods & Information Systems. Business Statistics. Chapter 14 Multiple Regression
Chapter 14 Student Lecture Notes 14-1 Department of Quantitative Methods & Information Systems Business Statistics Chapter 14 Multiple Regression QMIS 0 Dr. Mohammad Zainal Chapter Goals After completing
More informationTwo sided, two sample t-tests. a) IQ = 100 b) Average height for men = c) Average number of white blood cells per cubic millimeter is 7,000.
Two sided, two sample t-tests. I. Brief review: 1) We are interested in how a sample compares to some pre-conceived notion. For example: a) IQ = 100 b) Average height for men = 5 10. c) Average number
More informationThis gives us an upper and lower bound that capture our population mean.
Confidence Intervals Critical Values Practice Problems 1 Estimation 1.1 Confidence Intervals Definition 1.1 Margin of error. The margin of error of a distribution is the amount of error we predict when
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 informationOne-Way Analysis of Variance. With regression, we related two quantitative, typically continuous variables.
One-Way Analysis of Variance With regression, we related two quantitative, typically continuous variables. Often we wish to relate a quantitative response variable with a qualitative (or simply discrete)
More informationSign test. Josemari Sarasola - Gizapedia. Statistics for Business. Josemari Sarasola - Gizapedia Sign test 1 / 13
Josemari Sarasola - Gizapedia Statistics for Business Josemari Sarasola - Gizapedia 1 / 13 Definition is a non-parametric test, a special case for the binomial test with p = 1/2, with these applications:
More information