# ANOVA Analysis of Variance

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 ANOVA Analysis of Variance What is ANOVA and why do we use it? Can test hypotheses about mean differences between more than 2 samples. Can also make inferences about the effects of several different IVs, each with several different levels. Uses sample data to draw inferences about populations Looks at the ratio of treatment effect (differences between groups) to error (difference between individual scores and their group means) A factor is an independent variable (IV). o It can be between-group o Within-subject (or repeated measures) Mixed designs a bit of both o Main effect o Effect of a factor averaged across all other factors Interactions o Effect of a particular combination of factors i.e. 1 factor at a specific level of another factor. ANOVA as Regression It is important to understand that regression and ANOVA are identical approaches except for the nature of the explanatory variables (IVs). For example, it is a small step from having three levels of a shade factor (say light, medium and heavy shade cloths) then carrying out a one-way analysis of variance, to measuring the light intensity in the three treatments and carrying out a regression with light intensity as the explanatory variable. Some experiments combine regression and analysis of variance by fitting a series of regression lines, one in each of several levels of a given factor (this is called analysis of covariance ANCOVA).

2 We ll come back to this later. Why can t you just use several t-tests? If each time you do a t-test it carries a 5% chance of making a Type 1 Error (i.e. rejecting H 0 when it is actually true: finding an effect when none exists) Each time you carry out a t-test on the same set of data, the Type 1 Error rate increases: Number of Tests Type 1 Error rate 1 1 (.95) = 5% 2 1 (.95) 2 = 9.75% 3 1 (.95) 3 = 14.3% 4 1 (.95) 4 = 18.5% 5 1 (.95) 5 = 22.6% 6 1 (.95) 6 = 26.5% This is the familywise error increased probability of falsely rejecting H 0 Using ANOVA allows us make multiple comparisons with the same data, without increasing the familywise error Logic: The F-Ratio ANOVA looks at the variance within data set to find out where it comes from Why do scores within a data set vary? Compares the variance due to the IV with the variance due to individual differences. The difference will be significant when the between group variability is significantly more than the within-group variability. = h = + If treatment effect = 0, then = =1 If treatment effect, then F (The following 3 diagrams have been adapted from Lecture notes on ANOVA1 by Dr Wendy Johnson)

3 If all groups are the same then any variance between groups is due to chance the mean of each group is the same as the grand mean, and you can predict any one score from the grand mean (of all your samples together) o H 0 can t be rejected; no evidence for differences between the means. If groups are different then the variance between groups is not due to chance alone, but a treatment effect (between groups variation) plus error (within group variation). You can predict one score from the mean of its individual group o H 0 is rejected in favour of H 1 there is a difference between the means of the groups

4 To decide which is the better predictor, we divide all the variance into within group variance (a measure of how much each score differs from its group mean) and between group variance (how much each score differs from the grand mean) Steps for one-way ANOVA 1. Calculate SS T Total sum of squares Based on the best guess for each participant being the grand mean Total sum of squared differences of each score from the grand mean. = = 1

5 df T = (N 1) (i.e. one less than the total sample size) 2. Calculate SS M Model sum of squares (AKA Treatment sum of squares, SS A ). This is how much variance comes from the fact that data points belong to different groups Based on the best guess for each participant being the mean of the group to which they belong The difference between each participant s predicted score (i.e. the mean of the group to which it belongs) and the grand mean a. Calculate difference between each group mean and the grand mean b. Square each of these differences c. Multiply each result by the number of participants within that group d. Add the values for each group together 3. Calculate SS R = = 1 Residual sum of squares (AKA error SS, SS E ) Where k is the number of groups How much variation in the scores cannot be explained by the model?

6 o Could be caused by individual differences basically anything other than the experimental manipulation Can be calculated by: = Or look at the differences between each score and the mean of the group it belongs to. o Distance between each data point and its group mean is squared and summed. Or, using variance (s 2 ): = = 1 i.e. The sum of the variance of each group multiplied by one less than the number of participants in each group. = = 1 1 = i.e. Total sample size (N) minus the number of groups (k) 4. Calculate MS M and MS R SS M and SS R deal with totals (sums), and are therefore influenced by how many in each sample o Need to divide by df before calculating the F ratio = 5. Calculate F =, = 1-way ANOVA table Source SS df MS F Critical F Factor SS M k-1 MS M= MS M qf(0.95,df M,df R ) SS M /k-1 MS R Error SS R k(n-1) MS R= SS R /k(n-1) Total SS T kn-1

7 6. For higher-way designs Factorial ANOVA looks at effects of 2 or more IVs on a DV, as well as the effect of the interaction between them. e.g. 2-way unrelated design = = Calculate MS ANOVA Assumptions when can you use an ANOVA? When parametric assumptions are met: 1. DV is interval or ratio 2. Data is usually in row x column format each row represents a case or participant and each column a group / category 3. Data comes from a random sample 4. Observations are independent of each other Shouldn t be able to predict an observation from either other observations in its group, or from corresponding cases in other groups. 5. Homogeneity of Variance Variability is equal across all treatment conditions/groups o Most common violation: floor/ceiling effects; As mean floor/ceiling, variability 0 o E.g. Likert scales, Counts, Time/length/volume measurements (>0 by definition) o Apply Welch correction for violations default in 1-way ANOVA in R. Test using Levene s test check for R

8 o Levene test measures heteroscedasticity violations of homogeneity of variance o If significant, then there is some reason to be worried o p-value will only be approximately correct o Brown-Forsythe/Welch tests robust tests ANOVA is robust against this if cell sizes are equal 6. DV is normally distributed Look at frequency distribution charts o Check for skew (long tails) floor/ceiling effects, Multimodal distributions o Fields: P-P plots of data versus expected o 1-sample Kolmogorov-Smirnoff test o Compares empirical cumulative distributions to hypothetical distribution with same mean and standard deviation BUT ANOVA is robust if this is violated, if sample size is large, and there is Homogeneity of Variance 7. Sphericity all the covariances across pairs of levels of the IV are similar Mauchly's test for sphericity: if significant, the usual F-tables don t apply Use corrections in the corrected F tables o Greenhouse-Geisser or Huynh-Feldt probabilities. Types of ANOVA 1. One-way ANOVA One independent Variable (IV), explanatory variable or factor, with 3 or more levels (otherwise you d use a t-test which would give the same result as a 2-level 1-way ANOVA) e.g. Comparing the effects of 3 different teaching methods (A, B & C 3 levels of the IV teaching method ) on exam results Independent Variable Teaching Method Method A Method B Method C 2. Factorial score= grand mean + main effects + interactions + noise A factor is an independent variable 2 or more factors each with 2 or more levels each All combinations of each IV occur o Between-group / Within-subject (or repeated measures)

9 o Or both Mixed designs Random versus fixed factors e.g. Comparing the effects of different teaching methods (between group variable) and gender (between-group variable) on exam results Gender Male Female Teaching Method Method A Method B Method C Male Method A Male Method B Male Method C Female Method A Female Method B Female Method C Main effects (Green dashed line in table below): o The effect of a factor averaged across all others Effect of teaching method, averaged over gender Effect of gender, averaged over teaching method o Note that it may be misleading to discuss main effects if there is a significant interaction. Interactions / Simple Main Effects (Blue solid line in table below): o Effect of a particular combination of factors effect of gender at different levels of teaching method e.g. is there a difference between male and female under method A? effect of teaching method at different levels of gender e.g. is there a difference between teaching method A, teaching method B and teaching level C within the males only? Gender Male Female Teaching Method Method A Method B Method C Male Method A Male Method B Male Method C Female Method A Female Method B Female Method C 3. Repeated Measures Each individual is exposed to multiple treatments / multiple levels of an IV

10 Longitudinal studies usually have repeated measures design testing the same person over time. e.g. Each student is tested before, during and after going through each teaching method to see if there s a difference in exam score. 4. Mixed Design A combination of within / between design e.g. Each student is assigned a teaching method (A, B, C), Gender is recorded (Male, Female) - both between-groups variables and test score before and after administration of teaching method (Before, After) within-participants variable. Relationships between factors can be: CROSSED: Each level of each factor is encountered at each level of each other factor (see example above) NESTED: 2 factors are nested if each level of one occurs only with one level of the other: e.g. Include year of study Gender Male Female Teaching Method Method A Method B Method C Male Method A Male Method B Male Method C Year1 Year2 Year 3 Year 4 Year 5 Year 6 Female Method A Female Method B Female Method C Year1 Year2 Year 3 Year 4 Year 5 Year 6 Steps in an ANOVA 1. Check assumptions 2. Visualise data - graphs 3. Find out if there is any difference between any of the groups - ANOVA Omnibus Test; H 0 : µ 1 = µ 2 = µ 3 ; H 1 : there is a difference between at least 2 of the groups not all means are equal. 4. Find out between which groups these differences lie: Planned comparisons or Post Hoc tests. o e.g. H 0: µ 2 = µ3; H 1: µ 2 µ3, or directional: H 0: µ 2 = µ3; H 1: µ 2 > µ3 o use multiple t-tests with Bonferonni comparison adjust for experimentwise error rate o or Fisher s Least Significant Difference (LSD)

11 o Dunnett 2-tailed o Tukey-Kramer (HSD) - preferred o Scheffe s Interpreting and Reporting an ANOVA output In this example : Main effect of diet: F(2,36) = 83.52; p <.01 Main effect of supplement: F(3,36) = 17.82; p <.01 No interaction between diet and supplement; F(6,36) = 0.33; p =.917 NB report the exact p-value for a non-significant outcome, and p <.05 or p <.01 for a significant one. Planned Comparisons and Post Hoc Tests Having found a difference, we need to find where the difference lies

### 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

### Introduction to Analysis of Variance (ANOVA) Limitations of the t-test

Introduction to Analysis of Variance (ANOVA) The Structural Model, The Summary Table, and the One- Way ANOVA Limitations of the t-test Although the t-test is commonly used, it has limitations Can only

### Analysis of numerical data S4

Basic medical statistics for clinical and experimental research Analysis of numerical data S4 Katarzyna Jóźwiak k.jozwiak@nki.nl 3rd November 2015 1/42 Hypothesis tests: numerical and ordinal data 1 group:

### Testing Hypotheses using SPSS

Is the mean hourly rate of male workers \$2.00? T-Test One-Sample Statistics Std. Error N Mean Std. Deviation Mean 2997 2.0522 6.6282.2 One-Sample Test Test Value = 2 95% Confidence Interval Mean of the

### ANOVA ANOVA. Two-Way ANOVA. One-Way ANOVA. When to use ANOVA ANOVA. Analysis of Variance. Chapter 16. A procedure for comparing more than two groups

ANOVA ANOVA Analysis of Variance Chapter 6 A procedure for comparing more than two groups independent variable: smoking status non-smoking one pack a day > two packs a day dependent variable: number of

### Week 7 Lecture: Two-way Analysis of Variance (Chapter 12) Two-way ANOVA with Equal Replication (see Zar s section 12.1)

Week 7 Lecture: Two-way Analysis of Variance (Chapter ) We can extend the idea of a one-way ANOVA, which tests the effects of one factor on a response variable, to a two-way ANOVA which tests the effects

### Study Guide for the Final Exam

Study Guide for the Final Exam When studying, remember that the computational portion of the exam will only involve new material (covered after the second midterm), that material from Exam 1 will make

### 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

### UNDERSTANDING THE ONE-WAY ANOVA

UNDERSTANDING The One-way Analysis of Variance (ANOVA) is a procedure for testing the hypothesis that K population means are equal, where K >. The One-way ANOVA compares the means of the samples or groups

### ANOVA MULTIPLE CHOICE QUESTIONS. In the following multiple-choice questions, select the best answer.

ANOVA MULTIPLE CHOICE QUESTIONS In the following multiple-choice questions, select the best answer. 1. Analysis of variance is a statistical method of comparing the of several populations. a. standard

### Analysis of Data. Organizing Data Files in SPSS. Descriptive Statistics

Analysis of Data Claudia J. Stanny PSY 67 Research Design Organizing Data Files in SPSS All data for one subject entered on the same line Identification data Between-subjects manipulations: variable to

### Reporting Statistics in Psychology

This document contains general guidelines for the reporting of statistics in psychology research. The details of statistical reporting vary slightly among different areas of science and also among different

### ANSWERS TO EXERCISES AND REVIEW QUESTIONS

ANSWERS TO EXERCISES AND REVIEW QUESTIONS PART FIVE: STATISTICAL TECHNIQUES TO COMPARE GROUPS Before attempting these questions read through the introduction to Part Five and Chapters 16-21 of the SPSS

### Statistiek II. John Nerbonne. October 1, 2010. Dept of Information Science j.nerbonne@rug.nl

Dept of Information Science j.nerbonne@rug.nl October 1, 2010 Course outline 1 One-way ANOVA. 2 Factorial ANOVA. 3 Repeated measures ANOVA. 4 Correlation and regression. 5 Multiple regression. 6 Logistic

### One-Way ANOVA using SPSS 11.0. SPSS ANOVA procedures found in the Compare Means analyses. Specifically, we demonstrate

1 One-Way ANOVA using SPSS 11.0 This section covers steps for testing the difference between three or more group means using the SPSS ANOVA procedures found in the Compare Means analyses. Specifically,

### SCHOOL OF HEALTH AND HUMAN SCIENCES DON T FORGET TO RECODE YOUR MISSING VALUES

SCHOOL OF HEALTH AND HUMAN SCIENCES Using SPSS Topics addressed today: 1. Differences between groups 2. Graphing Use the s4data.sav file for the first part of this session. DON T FORGET TO RECODE YOUR

### c. The factor is the type of TV program that was watched. The treatment is the embedded commercials in the TV programs.

STAT E-150 - Statistical Methods Assignment 9 Solutions Exercises 12.8, 12.13, 12.75 For each test: Include appropriate graphs to see that the conditions are met. Use Tukey's Honestly Significant Difference

### Inferential Statistics

Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

### INTERPRETING THE REPEATED-MEASURES ANOVA

INTERPRETING THE REPEATED-MEASURES ANOVA USING THE SPSS GENERAL LINEAR MODEL PROGRAM RM ANOVA In this scenario (based on a RM ANOVA example from Leech, Barrett, and Morgan, 2005) each of 12 participants

### Analysis of variance (ANOVA) is a

Steven F. Sawyer, PT, PhD Analysis of variance (ANOVA) is a statistical tool used to detect differences between experimental group means. ANOVA is warranted in experimental designs with one dependent variable

### An analysis method for a quantitative outcome and two categorical explanatory variables.

Chapter 11 Two-Way ANOVA An analysis method for a quantitative outcome and two categorical explanatory variables. If an experiment has a quantitative outcome and two categorical explanatory variables that

### Inferential Statistics. Probability. From Samples to Populations. Katie Rommel-Esham Education 504

Inferential Statistics Katie Rommel-Esham Education 504 Probability Probability is the scientific way of stating the degree of confidence we have in predicting something Tossing coins and rolling dice

About Single Factor ANOVAs TABLE OF CONTENTS About Single Factor ANOVAs... 1 What is a SINGLE FACTOR ANOVA... 1 Single Factor ANOVA... 1 Calculating Single Factor ANOVAs... 2 STEP 1: State the hypotheses...

### UNDERSTANDING THE TWO-WAY ANOVA

UNDERSTANDING THE e have seen how the one-way ANOVA can be used to compare two or more sample means in studies involving a single independent variable. This can be extended to two independent variables

### How to choose a statistical test. Francisco J. Candido dos Reis DGO-FMRP University of São Paulo

How to choose a statistical test Francisco J. Candido dos Reis DGO-FMRP University of São Paulo Choosing the right test One of the most common queries in stats support is Which analysis should I use There

### One-Way Analysis of Variance (ANOVA) Example Problem

One-Way Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means

### Chapter 12 Statistical Foundations: Analysis of Variance 377. Chapter 12 Statistical Foundations: Analysis of Variance

Chapter 1 Statistical Foundations: Analysis of Variance 377 Chapter 1 Statistical Foundations: Analysis of Variance There are many instances when a researcher is faced with the task of examining three

### COMP6053 lecture: Analysis of variance.

COMP6053 lecture: Analysis of variance jn2@ecs.soton.ac.uk Variances are additive Variance is such a useful measure because the variance of a combined distribution is equal to the sum of the variances

### An example ANOVA situation. 1-Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters)

An example ANOVA situation Example (Treating Blisters) 1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College Subjects: 25 patients with blisters Treatments: Treatment A, Treatment

### Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### 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

### 13: Additional ANOVA Topics. Post hoc Comparisons

13: Additional ANOVA Topics Post hoc Comparisons ANOVA Assumptions Assessing Group Variances When Distributional Assumptions are Severely Violated Kruskal-Wallis Test Post hoc Comparisons In the prior

### SIMULTANEOUS COMPARISONS AND THE CONTROL OF TYPE I ERRORS CHAPTER 6

SIMULTANEOUS COMPARISONS AND THE CONTROL OF TYPE I ERRORS CHAPTER 6 ERSH 8310 Lecture 8 September 18, 2007 Today s Class Discussion of the new course schedule. Take-home midterm (one instead of two) and

### Examining Differences (Comparing Groups) using SPSS Inferential statistics (Part I) Dwayne Devonish

Examining Differences (Comparing Groups) using SPSS Inferential statistics (Part I) Dwayne Devonish Statistics Statistics are quantitative methods of describing, analysing, and drawing inferences (conclusions)

### One-Way Analysis of Variance

Spring, 000 - - Administrative Items One-Way Analysis of Variance Midterm Grades. Make-up exams, in general. Getting help See me today -:0 or Wednesday from -:0. Send an e-mail to stine@wharton. Visit

### The Dummy s Guide to Data Analysis Using SPSS

The Dummy s Guide to Data Analysis Using SPSS Mathematics 57 Scripps College Amy Gamble April, 2001 Amy Gamble 4/30/01 All Rights Rerserved TABLE OF CONTENTS PAGE Helpful Hints for All Tests...1 Tests

### Module 9: Nonparametric Tests. The Applied Research Center

Module 9: Nonparametric Tests The Applied Research Center Module 9 Overview } Nonparametric Tests } Parametric vs. Nonparametric Tests } Restrictions of Nonparametric Tests } One-Sample Chi-Square Test

### Contrasts and Post Hoc Tests for One-Way Independent ANOVA Using SPSS

Contrasts and Post Hoc Tests for One-Way Independent ANOVA Using SPSS Running the Analysis In last week s lecture we came across an example, from Field (2013), about the drug Viagra, which is a sexual

### Quantitative Data Analysis: Choosing a statistical test Prepared by the Office of Planning, Assessment, Research and Quality

Quantitative Data Analysis: Choosing a statistical test Prepared by the Office of Planning, Assessment, Research and Quality 1 To help choose which type of quantitative data analysis to use either before

### Analysis of Variance. MINITAB User s Guide 2 3-1

3 Analysis of Variance Analysis of Variance Overview, 3-2 One-Way Analysis of Variance, 3-5 Two-Way Analysis of Variance, 3-11 Analysis of Means, 3-13 Overview of Balanced ANOVA and GLM, 3-18 Balanced

### For example, enter the following data in three COLUMNS in a new View window.

Statistics with Statview - 18 Paired t-test A paired t-test compares two groups of measurements when the data in the two groups are in some way paired between the groups (e.g., before and after on the

### ANOVA must be modified to take correlated errors into account when multiple measurements are made for each subject.

Chapter 14 Within-Subjects Designs ANOVA must be modified to take correlated errors into account when multiple measurements are made for each subject. 14.1 Overview of within-subjects designs Any categorical

### SPSS Guide: Tests of Differences

SPSS Guide: Tests of Differences 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

### Class 19: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.1)

Spring 204 Class 9: Two Way Tables, Conditional Distributions, Chi-Square (Text: Sections 2.5; 9.) Big Picture: More than Two Samples In Chapter 7: We looked at quantitative variables and compared the

### Statistics for Management II-STAT 362-Final Review

Statistics for Management II-STAT 362-Final Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. The ability of an interval estimate to

### E205 Final: Version B

Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random

### Chapter 7 Section 7.1: Inference for the Mean of a Population

Chapter 7 Section 7.1: Inference for the Mean of a Population Now let s look at a similar situation Take an SRS of size n Normal Population : N(, ). Both and are unknown parameters. Unlike what we used

### Intro to Parametric Statistics,

Descriptive Statistics vs. Inferential Statistics vs. Population Parameters Intro to Parametric Statistics, Assumptions & Degrees of Freedom Some terms we will need Normal Distributions Degrees of freedom

### 10. Analysis of Longitudinal Studies Repeat-measures analysis

Research Methods II 99 10. Analysis of Longitudinal Studies Repeat-measures analysis This chapter builds on the concepts and methods described in Chapters 7 and 8 of Mother and Child Health: Research methods.

### January 26, 2009 The Faculty Center for Teaching and Learning

THE BASICS OF DATA MANAGEMENT AND ANALYSIS A USER GUIDE January 26, 2009 The Faculty Center for Teaching and Learning THE BASICS OF DATA MANAGEMENT AND ANALYSIS Table of Contents Table of Contents... i

### The Statistics Tutor s

statstutor community project encouraging academics to share statistics support resources All stcp resources are released under a Creative Commons licence Stcp-marshallowen-7 The Statistics Tutor s www.statstutor.ac.uk

### Statistics and research

Statistics and research Usaneya Perngparn Chitlada Areesantichai Drug Dependence Research Center (WHOCC for Research and Training in Drug Dependence) College of Public Health Sciences Chulolongkorn University,

### The Statistics Tutor s Quick Guide to

statstutor community project encouraging academics to share statistics support resources All stcp resources are released under a Creative Commons licence The Statistics Tutor s Quick Guide to Stcp-marshallowen-7

### Regression in SPSS. Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology

Regression in SPSS Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology John P. Bentley Department of Pharmacy Administration University of

### UNDERSTANDING ANALYSIS OF COVARIANCE (ANCOVA)

UNDERSTANDING ANALYSIS OF COVARIANCE () In general, research is conducted for the purpose of explaining the effects of the independent variable on the dependent variable, and the purpose of research design

### Profile analysis is the multivariate equivalent of repeated measures or mixed ANOVA. Profile analysis is most commonly used in two cases:

Profile Analysis Introduction Profile analysis is the multivariate equivalent of repeated measures or mixed ANOVA. Profile analysis is most commonly used in two cases: ) Comparing the same dependent variables

### UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates

UCLA STAT 13 Statistical Methods - Final Exam Review Solutions Chapter 7 Sampling Distributions of Estimates 1. (a) (i) µ µ (ii) σ σ n is exactly Normally distributed. (c) (i) is approximately Normally

### Background to ANOVA. One-Way Analysis of Variance: ANOVA. The One-Way ANOVA Summary Table. Hypothesis Test in One-Way ANOVA

Background to ANOVA One-Way Analysis of Variance: ANOVA Dr. J. Kyle Roberts Southern Methodist University Simmons School of Education and Human Development Department of Teaching and Learning Recall from

### SPSS Tests for Versions 9 to 13

SPSS Tests for Versions 9 to 13 Chapter 2 Descriptive Statistic (including median) Choose Analyze Descriptive statistics Frequencies... Click on variable(s) then press to move to into Variable(s): list

### Data Analysis. Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) SS Analysis of Experiments - Introduction

Data Analysis Lecture Empirical Model Building and Methods (Empirische Modellbildung und Methoden) Prof. Dr. Dr. h.c. Dieter Rombach Dr. Andreas Jedlitschka SS 2014 Analysis of Experiments - Introduction

### SPSS: Descriptive and Inferential Statistics. For Windows

For Windows August 2012 Table of Contents Section 1: Summarizing Data...3 1.1 Descriptive Statistics...3 Section 2: Inferential Statistics... 10 2.1 Chi-Square Test... 10 2.2 T tests... 11 2.3 Correlation...

### Multivariate Analysis of Variance (MANOVA)

Multivariate Analysis of Variance (MANOVA) Aaron French, Marcelo Macedo, John Poulsen, Tyler Waterson and Angela Yu Keywords: MANCOVA, special cases, assumptions, further reading, computations Introduction

### Chi Square Analysis. When do we use chi square?

Chi Square Analysis When do we use chi square? More often than not in psychological research, we find ourselves collecting scores from participants. These data are usually continuous measures, and might

### Aus: Statnotes: Topics in Multivariate Analysis, by G. David Garson (Zugriff am

Aus: Statnotes: Topics in Multivariate Analysis, by G. David Garson http://faculty.chass.ncsu.edu/garson/pa765/anova.htm (Zugriff am 20.10.2010) Planned multiple comparison t-tests, also just called "multiple

### ACTM State Exam-Statistics

ACTM State Exam-Statistics For the 25 multiple-choice questions, make your answer choice and record it on the answer sheet provided. Once you have completed that section of the test, proceed to the tie-breaker

### Allelopathic Effects on Root and Shoot Growth: One-Way Analysis of Variance (ANOVA) in SPSS. Dan Flynn

Allelopathic Effects on Root and Shoot Growth: One-Way Analysis of Variance (ANOVA) in SPSS Dan Flynn Just as t-tests are useful for asking whether the means of two groups are different, analysis of variance

### Multivariate Analysis of Variance. The general purpose of multivariate analysis of variance (MANOVA) is to determine

2 - Manova 4.3.05 25 Multivariate Analysis of Variance What Multivariate Analysis of Variance is The general purpose of multivariate analysis of variance (MANOVA) is to determine whether multiple levels

### 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.

### 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

### 1. Complete the sentence with the correct word or phrase. 2. Fill in blanks in a source table with the correct formuli for df, MS, and F.

Final Exam 1. Complete the sentence with the correct word or phrase. 2. Fill in blanks in a source table with the correct formuli for df, MS, and F. 3. Identify the graphic form and nature of the source

### Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures

Testing Group Differences using T-tests, ANOVA, and Nonparametric Measures Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone:

### HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

### 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

### Analysing Questionnaires using Minitab (for SPSS queries contact -) Graham.Currell@uwe.ac.uk

Analysing Questionnaires using Minitab (for SPSS queries contact -) Graham.Currell@uwe.ac.uk Structure As a starting point it is useful to consider a basic questionnaire as containing three main sections:

### Experimental Designs (revisited)

Introduction to ANOVA Copyright 2000, 2011, J. Toby Mordkoff Probably, the best way to start thinking about ANOVA is in terms of factors with levels. (I say this because this is how they are described

### SPSS Explore procedure

SPSS Explore procedure One useful function in SPSS is the Explore procedure, which will produce histograms, boxplots, stem-and-leaf plots and extensive descriptive statistics. To run the Explore procedure,

### Post-hoc comparisons & two-way analysis of variance. Two-way ANOVA, II. Post-hoc testing for main effects. Post-hoc testing 9.

Two-way ANOVA, II Post-hoc comparisons & two-way analysis of variance 9.7 4/9/4 Post-hoc testing As before, you can perform post-hoc tests whenever there s a significant F But don t bother if it s a main

### 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

### Outline. Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test

The t-test Outline Definitions Descriptive vs. Inferential Statistics The t-test - One-sample t-test - Dependent (related) groups t-test - Independent (unrelated) groups t-test Comparing means Correlation

### Inferences About Differences Between Means Edpsy 580

Inferences About Differences Between Means Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Inferences About Differences Between Means Slide

### II. DISTRIBUTIONS distribution normal distribution. standard scores

Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### T adult = 96 T child = 114.

Homework Solutions Do all tests at the 5% level and quote p-values when possible. When answering each question uses sentences and include the relevant JMP output and plots (do not include the data in your

### Lecture 7: Binomial Test, Chisquare

Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two

### 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

### Unit 31: One-Way ANOVA

Unit 31: One-Way ANOVA Summary of Video A vase filled with coins takes center stage as the video begins. Students will be taking part in an experiment organized by psychology professor John Kelly in which

### QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS

QUANTITATIVE METHODS BIOLOGY FINAL HONOUR SCHOOL NON-PARAMETRIC TESTS This booklet contains lecture notes for the nonparametric work in the QM course. This booklet may be online at http://users.ox.ac.uk/~grafen/qmnotes/index.html.

### Factor B: Curriculum New Math Control Curriculum (B (B 1 ) Overall Mean (marginal) Females (A 1 ) Factor A: Gender Males (A 2) X 21

1 Factorial ANOVA The ANOVA designs we have dealt with up to this point, known as simple ANOVA or oneway ANOVA, had only one independent grouping variable or factor. However, oftentimes a researcher has

### Statistics II Final Exam - January Use the University stationery to give your answers to the following questions.

Statistics II Final Exam - January 2012 Use the University stationery to give your answers to the following questions. Do not forget to write down your name and class group in each page. Indicate clearly

### Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2

Chapter 11: Linear Regression - Inference in Regression Analysis - Part 2 Note: Whether we calculate confidence intervals or perform hypothesis tests we need the distribution of the statistic we will use.

### BST 708 T. Mark Beasley Split-Plot ANOVA handout

BST 708 T. Mark Beasley Split-Plot ANOVA handout In the Split-Plot ANOVA, three factors represent separate sources of variance. Two interactions also present independent sources of variation. Suppose a

### Using SPSS version 14 Joel Elliott, Jennifer Burnaford, Stacey Weiss

Using SPSS version 14 Joel Elliott, Jennifer Burnaford, Stacey Weiss SPSS is a program that is very easy to learn and is also very powerful. This manual is designed to introduce you to the program however,

### CHAPTER 3 COMMONLY USED STATISTICAL TERMS

CHAPTER 3 COMMONLY USED STATISTICAL TERMS There are many statistics used in social science research and evaluation. The two main areas of statistics are descriptive and inferential. The third class of

### Statistics Review PSY379

Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

### Some Critical Information about SOME Statistical Tests and Measures of Correlation/Association

Some Critical Information about SOME Statistical Tests and Measures of Correlation/Association This information is adapted from and draws heavily on: Sheskin, David J. 2000. Handbook of Parametric and

### 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,

### 15. Analysis of Variance

15. Analysis of Variance A. Introduction B. ANOVA Designs C. One-Factor ANOVA (Between-Subjects) D. Multi-Factor ANOVA (Between-Subjects) E. Unequal Sample Sizes F. Tests Supplementing ANOVA G. Within-Subjects