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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 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 more than one independent grouping variable, or factor of interest. Factorial ANOVA is used when we want to consider the effect of more than one factor on differences in the dependent variable. A factorial design is an experimental design in which each level of each factor is paired up or crossed with each level of every other factor. In other words each combination of the levels of the factors is included in the design. This type of design is often depicted in a table. We typically refer to ANOVA designs by the number of factors and/or by the number of levels within a factor. A one-way ANOVA refers to a design with one factor, two-way ANOVA has two factors, three-way ANOVA has three factors, etc. A two-by- three ANOVA is a two-way ANOVA with two levels of the first factor and three levels of the second factor. A three-by-four-by-two ANOVA is a three-way ANOVA with three levels of the first factor, four of the second, and two of the third. Factorial designs allow us to determine if there are interactions between the independent variables or factors considered. An interaction implies that differences in one of the factors depend on differences in another factor. Example Consider a researcher who is interested in determining whether a new mathematics curriculum is better at helping students develop spatial visualization skills. Furthermore, he wonders whether there is a difference between boys and girls, because it is known that males tend to be better at spatial visualization than females. The researcher has the following two-way (two-by-two) factorial design: Females (A 1 ) Factor B: Curriculum New Math Control Curriculum (B (B 1 ) ) Overall Mean (marginal) X X 1 1. Factor A: Gender Males (A ) X 1 Overall Mean (marginal). 1 X X X. X X. X.. Suppose the new curriculum was found to improve spatial visualization scores equally as well for both males and females. Then there would be main effect differences only. Main effect differences reflect differences in the means of one of the factors, ignoring other factors. However, if, for example, the new curriculum worked better for females then there would be an interaction effect. Typically we graph each of the cell means to depict differences obtained in factorial ANOVA. The assumptions underlying the statistical tests associated with factorial ANOVA are the same as those associated with a simple one-way ANOVA. Specifically, it is assumed the dependent variable is normally distributed within each cell, that the population variances are

2 identical within each cell, and that the observations and groups are independent of each other. Conceptually, the way we calculate the statistics associated with factorial ANOVA designs is comparable to what we did for simple one-way ANOVA designs. Basically, we determine the variability associated with different means; there are just more means to deal with now. The SS total in a factorial design is exactly the same as it was in simple ANOVA. It represents the total variability among all observations around the grand mean or ( X X ) In a simple one-way ANOVA the SS within = SS error represented the variability of observations within a particular group. However, now we are partitioning the groups even further so each group is represented by a cell in our table. In other words, the SS error represents the variability of observations within a particular cell of the table. It is the variability that is expected among individuals and can be thought of as an estimate of variability that is common to all cells. In a factorial ANOVA the SS between still represents the variability of the group means from the overall mean. However, now we have to determine which of the variability is due to main effects and which is due to interaction effects. For a two-way ANOVA design, as depicted in the example above, SS between is partitioned into SS A, SS B, and SS AB. SS A represents the variability in the marginal means associated with the different levels of factor A, when compared to the overall mean. In our example, it would represent the variability in the means obtained for boys and girls, ignoring curriculum. It is computed by using the row marginal means and the grand mean. SS B represents the variability in the marginal means associated with the different levels of factor B, when compared to the overall mean. In our example, it would represent the variability in the different curriculum programs, ignoring gender. It is computed by using the column marginal means and the grand mean. SS AB represents the variability in the cell means, after controlling for main effect differences, when compared to the overall mean. It is computed by using the cell means and the overall means, as well as SS A and SS B. Basically, we compute the variability in the cell means and then subtract the variability due to the main effects. Example: Suppose we obtained the following data for the ANOVA design explained previously: Females - New Females - Control Males - New Males - Control

3 3 Calculating the cell and marginal means we obtain the following: Factor B: Curriculum New Math Control Curriculum (B (B 1 ) ) Overall Mean (marginal) Females (A 1 ) X = 6.4 X 1 = 4.0 X 1. = 5. Factor A: Gender Males (A ) X 1 = 8.0 X = 3.4 X. = 5.7 Overall Mean (marginal) X. 1= 7. X. =3.7 X.. = 5.45 The SS error = ( X X j ) = (5 6.4) + (8 6.4) (6 4.0) + (3 4.0) (7 8.0) + (9 8.0) (5 3.4) + (3 3.4) = 7.8 The SS A = n ( X X ) = 0( ) + 0( ) = i. i... = The SS B = n ( X X ) = 0( ) + 0( ) = j. j.. = The SS AB = [ n ij ( X ij X.. ) ] SS A - SS B = [10( ) + 10( ) + 10( ) + 10( ) ] = [ ] 16 = = 1.1 SS total = SS error + SS A + SS B + SS AB = = 09.9 To obtain our F-ratios for each test we need to use the df associated with each main effect and interaction. df A = Number of levels of Factor A 1 = 1 = 1, for our example df B B = Number of levels of Factor B 1 = 1 = 1, for our example df AB = (df A )( df B ) B = 1(1) = 1, for our example df error = N (number of cells) = 40 4 = 36, for our example df total = N 1 (checking this number is a good way to make sure you ve entered your data correctly)

4 4 Using the appropriate df we can obtain the corresponding MS term needed to calculate our F- statistic: MS A = SS A / df A =.5 / 1 =.5, for our example MS B = SS B / df B = 1.5 / 1 = 1.5, for our example MS AB = SS AB / df AB = 1.1 / 1 = 1.1, for our example MS error = SS error / df error = 7.8 / 36 =.0, for our example The null hypothesis for each test is that there is no difference in the means. F A = MS A / MS error =.5 /.0 1.4, (compare to a critical F with 1 and 36 df 4.15) F B = MS B / MS error = 1.5 / , (compare to critical F with 1 and 36 df 4.15) F A = MS AB / MS error = 1.1 / , (compare to critical F with 1 and 36 df 4.15) SPSS Output: Univariate Analysis of Variance - obtained using defaults under "Analyze" and General Linear Model and "Univariate" Between-Subjects Factors sex curriculum 1 1 Value Label female 0 male 0 new program N 0 control 0 Tests of Between-Subjects Effects Dependent Variable: spatial Source Corrected Model Intercept sex curriculum sex * curriculum Error Total Corrected Total Type III Sum of Squares df Mean Square F Sig a a. R Squared =.653 (Adjusted R Squared =.64) Under the model option in SPSS you can choose to use either Type II SS, Type III SS (default) or Type IV SS. It is recommended that you go with the default which adjusts the tests conducted when you have an unequal number of observations in each cell and conducts each test independently of other tests.

5 5 Typically when one finds an interaction they graph it to aid in the interpretation. However, our example wasn t very interesting so let s consider a more interesting example. Suppose a counseling psychologist conducted a study to determine the best type of therapy for various levels of depression and obtained the following data: Tests of Between-Subjects Effects Source Corrected Model Intercept treatment severity treatment * severity Error Total Corrected Total Type III Sum of Squares df Mean Square F Sig a a. R Squared =.634 (Adjusted R Squared =.55) Estimated Marginal Means - obtained under "options" button 3. treatment * severity 95% Confidence Interval treatment hypnosis CBT behavioral severity moderate moderate moderate Mean Std. Error Lower Bound Upper Bound There is a significant interaction in this example and the best way to interpret it is to create separate line graphs for each level of one factor that depicts the cell means for the other factor. This can be done in two different ways as the following demonstrates:

6 Mild Moderate Severe 7 5 Hypnosis CBT Behavioral Hypnosis CBT Behavioral 7 5 Mild Moderate Severe If the interaction was not found to be significant than the lines in the above plots would be parallel. If, for example, we had only compared hypnosis to behavioral therapy then we would not have found a significant interaction. Once we find a significant interaction many methodologists would argue that any significant main effects that are found should not be interpreted. However, this is somewhat dependent on the type of interaction that is obtained. In the example above a disordinal interaction was obtained, because the interaction lines intersect (or move in opposite directions). In this case it is not appropriate to interpret any significant main effects because differences found in different levels of one factor depend on differences in the second factor. However, it is also possible to obtain an ordinal interaction. In this case, the lines would not be parallel, however the lines would not cross or move in different directions. For example,

7 7 suppose the following results had been obtained from 8 patients at each severity level in each of the 3 therapy groups: Tests of Between-Subjects Effects Source Corrected Model Intercept treatment severity treatment * severity Error Total Corrected Total Type III Sum of Squares df Mean Square F Sig a a. R Squared =.445 (Adjusted R Squared =.379) 3. treatment * severity treatment hypnosis CBT behavioral severity 95% Confidence Interval Mean Std. Error Lower Bound Upper Bound In this case the interaction is significant, but the following interaction graphs would be obtained, which makes it clear that all treatments seemed to work better for ly depressed patients so the researcher may be justified in interpreting the main effect: Hypnosis CBT Behavioral Mild Severe

8 Mild Severe Hypnosis CBT Behavior Whenever an interaction is obtained, one might want to do a test of simple main effects. This test teases apart the interaction. A test of simple main effects is different from simply interpreting the main effects, which ignores different levels of the second factor. Rather a test of simple main effects is a test for main effect differences at each level of the other factor. For example, one might want to test the main effect of treatment within each of the two different levels of depression severity. This is accomplished by obtaining the SS therapy for ly depressed patients and SS therapy for ly depressed patients. We do this using the cell means for the different therapy treatments and the following marginal means for severity of depression.. severity 95% Confidence Interval severity Mean Std. Error Lower Bound Upper Bound SS therapy for depression = n X therapy X = (, ) 8[( ) + ( ) + ( ) ] = 8[ ] = 5.5 SS therapy for depression = n X therapy X = (, ) 8[( ) + ( ) + ( ) ] = Each of these SS have df because they use 3 means in the calculation. The F ratio for testing main effect differences of therapy for patients that are ly depressed is based on MS therapy = 5.5/ =.65 and MS within = so F =.65 / = which needs to be compared to a critical F with and 4 df, which is approximately 3.3. There are obviously no main effect differences for therapy treatments for patients that with

9 9 depression. The F statistic for testing main effect difference of therapy for patients that are ly depressed is (16.33/) / = /5.571 =.338. So there is a difference in depression scores for the different therapy treatments for patients that are ly depressed. One could also test the main effect of depression severity within each of the treatment levels. This is accomplished by obtaining the SS severity within each treatment, using the cell means for the different levels of depression severity and the following marginal means for the different therapy treatments. 1. treatment 95% Confidence Interval treatment hypnosis CBT behavioral Mean Std. Error Lower Bound Upper Bound SS severity for hypnosis = n X severity X = (, hypnosis hypnosis ) 8[( ) + ( ) ] = 8[ ] = SS severity for CBT = n X severity X = (, CBT CBT ) 8[( ) + ( ) ] = SS severity for behavioral therapy = n X severity X = 8[( ) + ( ) ] = 90.5 (, behaviroal behavioral ) Each of these SS have 1 df because they use means in the calculation. The F ratio for testing main effect differences of severity of depression for patients that treated using hypnosis is based on MS = 7.563/1 and MS within = so F = / = which needs to be compared to a critical F with 1 and 4 df, which is approximately There are obviously no main effect differences for severity of depression for patients that are treated with hypnosis therapy. The F statistic for testing main effect difference for severity of depression for patients that are treated with CBT is / = So there are no main effect differences for severity of depression for patients that are treated with CBT. The F statistic for testing main effect differences for severity of depression for patients treated with behavioral therapy is 90.5 / = so there is a main effect difference for severity of depression for patients that are treated with behavioral therapy It should be noted that there is no way to test simple main effects in SPSS without using the syntax window. Rather the syntax window in SPSS must be used. The following SPSS syntax can be used to obtain a test of simple main effects, as well as an ANOVA. The majority of the syntax is what is run when one clicks General Linear Model under the Analyze menu option and then chooses Univariate. The middle lines beginning with /EMMEANS are added when one chooses to get an estimate of the means under the options button. All of this syntax can be obtained by choosing the paste button when

10 10 running an ANOVA from the point and click menu. The last two lines provide a test of the simple main effects (as well as some extraneous output) and must be typed in by the user. UNIANOVA score BY treatment severity /METHOD = SSTYPE(3) /INTERCEPT = INCLUDE /EMMEANS = TABLES(treatment) /EMMEANS = TABLES(severity) /EMMEANS = TABLES(treatment*severity) /CRITERIA = ALPHA(.05) /DESIGN = treatment severity treatment*severity / EMMEANS = tables (treatment * severity) comp (treatment) / EMMEANS = tables (treatment * severity) comp (severity). All of the multiple comparison procedures discussed in terms of simple one-way ANOVA can be generalized to higher way ANOVA designs and these are easily obtained using the point and click menu options in SPSS. However, it should be noted that these are tests of the main effects that ignore other factors. Therefore, I would not recommend interpreting pair-wise comparisons if a significant interaction is obtained. Power analyses for factorial ANOVA designs can also be conducted, similar to how they were conducted for simple one-way ANOVA designs. For factorial ANOVA designs we simply conduct separate power analyses for each factor individually, ignoring any additional factors that may exist. Once again, statistical significance does not imply differences that are important from a practical perspective. An effect size measure can be estimated by dividing the SS effect by SS total. Although this is conceptually simple, estimates of SS effect and SS total are dependent on knowing how to determine the expected mean squares, which is technically difficult. However, estimates of effect size can be obtained under the options button when running a factorial ANOVA in SPSS. Effect size measures will be printed out in the ANOVA table, next to each of the F-statistics for the main effects and the interaction terms. Having unequal cell sizes in a factorial ANOVA is a complex issue, from a technical perspective, because it results in a dependency among the main effect and interaction estimates of variability. Using the Type III SS, which is the default in SPSS, will provide you with a test of unweighted means, which is usually the appropriate test to conduct with unequal cell sizes. It should be noted that higher-order factorial designs are typical in Social Science research and all of the procedures that relate to a two-way ANOVA can easily be applied to higherorder factorial designs. However, with higher order designs there are more interaction terms to deal with and considering anything above a three-way ANOVA makes interpreting the results extremely difficult. Suppose one had a 3-by-4-by- factorial design. In other words, a three-way factorial design with three levels of factor A, four levels of factor B, and two levels of factor C. The corresponding ANOVA would be a test of the following: (1) Three tests of the Main Effects of Factor A, Factor B, and Factor C; () Three tests of the Two-way Interaction Effects of AB, AC, and BC, and (3) One test of the Three-way Interaction Effect of ABC.

Chapter 5 Analysis of variance SPSS Analysis of variance

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,

More information

Main Effects and Interactions

Main Effects and Interactions Main Effects & Interactions page 1 Main Effects and Interactions So far, we ve talked about studies in which there is just one independent variable, such as violence of television program. You might randomly

More information

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

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

More information

EPS 625 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM

EPS 625 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM EPS 6 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM ANCOVA One Continuous Dependent Variable (DVD Rating) Interest Rating in DVD One Categorical/Discrete Independent Variable

More information

Multivariate analysis of variance

Multivariate analysis of variance 21 Multivariate analysis of variance In previous chapters, we explored the use of analysis of variance to compare groups on a single dependent variable. In many research situations, however, we are interested

More information

Factorial Analysis of Variance

Factorial Analysis of Variance Chapter 560 Factorial Analysis of Variance Introduction A common task in research is to compare the average response across levels of one or more factor variables. Examples of factor variables are income

More information

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

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

More information

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

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

More information

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

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,

More information

Simple Tricks for Using SPSS for Windows

Simple Tricks for Using SPSS for Windows Simple Tricks for Using SPSS for Windows Chapter 14. Follow-up Tests for the Two-Way Factorial ANOVA The Interaction is Not Significant If you have performed a two-way ANOVA using the General Linear Model,

More information

SPSS Tests for Versions 9 to 13

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

More information

UNDERSTANDING THE TWO-WAY ANOVA

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

More information

Mixed 2 x 3 ANOVA. Notes

Mixed 2 x 3 ANOVA. Notes Mixed 2 x 3 ANOVA This section explains how to perform an ANOVA when one of the variables takes the form of repeated measures and the other variable is between-subjects that is, independent groups of participants

More information

Chapter 16 - Analyses of Variance and Covariance as General Linear Models Eye fixations per line of text for poor, average, and good readers:

Chapter 16 - Analyses of Variance and Covariance as General Linear Models Eye fixations per line of text for poor, average, and good readers: Chapter 6 - Analyses of Variance and Covariance as General Linear Models 6. Eye fixations per line of text for poor, average, and good readers: a. Design matrix, using only the first subject in each group:

More information

SPSS Guide: Tests of Differences

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

More information

ANSWERS TO EXERCISES AND REVIEW QUESTIONS

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

More information

Simple Linear Regression One Binary Categorical Independent Variable

Simple Linear Regression One Binary Categorical Independent Variable Simple Linear Regression Does sex influence mean GCSE score? In order to answer the question posed above, we want to run a linear regression of sgcseptsnew against sgender, which is a binary categorical

More information

Contrasts ask specific questions as opposed to the general ANOVA null vs. alternative

Contrasts ask specific questions as opposed to the general ANOVA null vs. alternative Chapter 13 Contrasts and Custom Hypotheses Contrasts ask specific questions as opposed to the general ANOVA null vs. alternative hypotheses. In a one-way ANOVA with a k level factor, the null hypothesis

More information

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

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

More information

Testing Hypotheses using SPSS

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

More information

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 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

More information

Independent t- Test (Comparing Two Means)

Independent t- Test (Comparing Two Means) Independent t- Test (Comparing Two Means) The objectives of this lesson are to learn: the definition/purpose of independent t-test when to use the independent t-test the use of SPSS to complete an independent

More information

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

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,

More information

BST 708 T. Mark Beasley Split-Plot ANOVA handout

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

More information

COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES.

COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. 277 CHAPTER VI COMPARISONS OF CUSTOMER LOYALTY: PUBLIC & PRIVATE INSURANCE COMPANIES. This chapter contains a full discussion of customer loyalty comparisons between private and public insurance companies

More information

Randomized Block Analysis of Variance

Randomized Block Analysis of Variance Chapter 565 Randomized Block Analysis of Variance Introduction This module analyzes a randomized block analysis of variance with up to two treatment factors and their interaction. It provides tables of

More information

ABSORBENCY OF PAPER TOWELS

ABSORBENCY OF PAPER TOWELS ABSORBENCY OF PAPER TOWELS 15. Brief Version of the Case Study 15.1 Problem Formulation 15.2 Selection of Factors 15.3 Obtaining Random Samples of Paper Towels 15.4 How will the Absorbency be measured?

More information

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

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

More information

Regression step-by-step using Microsoft Excel

Regression step-by-step using Microsoft Excel Step 1: Regression step-by-step using Microsoft Excel Notes prepared by Pamela Peterson Drake, James Madison University Type the data into the spreadsheet The example used throughout this How to is a regression

More information

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 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

More information

PSY 216. Assignment 14. a. The mean differences among the levels of one factor are referred to as the main effect of that factor.

PSY 216. Assignment 14. a. The mean differences among the levels of one factor are referred to as the main effect of that factor. Name: PSY 216 Assignment 14 a. The mean differences among the levels of one factor are referred to as the main effect of that factor. b. A(n) interaction between two factors occurs whenever the mean differences

More information

One-Way Analysis of Variance

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

More information

INTERPRETING THE REPEATED-MEASURES ANOVA

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

More information

Chapter 21 Section D

Chapter 21 Section D Chapter 21 Section D Statistical Tests for Ordinal Data The rank-sum test. You can perform the rank-sum test in SPSS by selecting 2 Independent Samples from the Analyze/ Nonparametric Tests menu. The first

More information

IBM SPSS Statistics 23 Part 4: Chi-Square and ANOVA

IBM SPSS Statistics 23 Part 4: Chi-Square and ANOVA IBM SPSS Statistics 23 Part 4: Chi-Square and ANOVA Winter 2016, Version 1 Table of Contents Introduction... 2 Downloading the Data Files... 2 Chi-Square... 2 Chi-Square Test for Goodness-of-Fit... 2 With

More information

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY

CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY CHAPTER 11 CHI-SQUARE: NON-PARAMETRIC COMPARISONS OF FREQUENCY The hypothesis testing statistics detailed thus far in this text have all been designed to allow comparison of the means of two or more samples

More information

Chapter 7. One-way ANOVA

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

More information

SPSS Resources. 1. See website (readings) for SPSS tutorial & Stats handout

SPSS Resources. 1. See website (readings) for SPSS tutorial & Stats handout Analyzing Data SPSS Resources 1. See website (readings) for SPSS tutorial & Stats handout Don t have your own copy of SPSS? 1. Use the libraries to analyze your data 2. Download a trial version of SPSS

More information

PASS Sample Size Software

PASS Sample Size Software Chapter 250 Introduction The Chi-square test is often used to test whether sets of frequencies or proportions follow certain patterns. The two most common instances are tests of goodness of fit using multinomial

More information

SPSS for Exploratory Data Analysis Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav)

SPSS for Exploratory Data Analysis Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav) Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav) Organize and Display One Quantitative Variable (Descriptive Statistics, Boxplot & Histogram) 1. Move the mouse pointer

More information

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. 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

More information

SPSS: Descriptive and Inferential Statistics. For Windows

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...

More information

General Guidelines about SPSS. Steps needed to enter the data in the SPSS

General Guidelines about SPSS. Steps needed to enter the data in the SPSS General Guidelines about SPSS The entered data has to be numbers and not letters. For example, in the Gender section, we can not write Male and Female in the answers, however, we must give them a code.

More information

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

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

More information

Hypothesis Testing. Male Female

Hypothesis Testing. Male Female Hypothesis Testing Below is a sample data set that we will be using for today s exercise. It lists the heights for 10 men and 1 women collected at Truman State University. The data will be entered in the

More information

Research Methods & Experimental Design

Research Methods & Experimental Design Research Methods & Experimental Design 16.422 Human Supervisory Control April 2004 Research Methods Qualitative vs. quantitative Understanding the relationship between objectives (research question) and

More information

Two Related Samples t Test

Two Related Samples t Test Two Related Samples t Test In this example 1 students saw five pictures of attractive people and five pictures of unattractive people. For each picture, the students rated the friendliness of the person

More information

1.5 Oneway Analysis of Variance

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

More information

Frequency Tables. Chapter 500. Introduction. Frequency Tables. Types of Categorical Variables. Data Structure. Missing Values

Frequency Tables. Chapter 500. Introduction. Frequency Tables. Types of Categorical Variables. Data Structure. Missing Values Chapter 500 Introduction This procedure produces tables of frequency counts and percentages for categorical and continuous variables. This procedure serves as a summary reporting tool and is often used

More information

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( ) Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

More information

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

KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management KSTAT MINI-MANUAL Decision Sciences 434 Kellogg Graduate School of Management Kstat is a set of macros added to Excel and it will enable you to do the statistics required for this course very easily. To

More information

6 Comparison of differences between 2 groups: Student s T-test, Mann-Whitney U-Test, Paired Samples T-test and Wilcoxon Test

6 Comparison of differences between 2 groups: Student s T-test, Mann-Whitney U-Test, Paired Samples T-test and Wilcoxon Test 6 Comparison of differences between 2 groups: Student s T-test, Mann-Whitney U-Test, Paired Samples T-test and Wilcoxon Test Having finally arrived at the bottom of our decision tree, we are now going

More information

DEPARTMENT OF HEALTH AND HUMAN SCIENCES HS900 RESEARCH METHODS

DEPARTMENT OF HEALTH AND HUMAN SCIENCES HS900 RESEARCH METHODS DEPARTMENT OF HEALTH AND HUMAN SCIENCES HS900 RESEARCH METHODS Using SPSS Session 2 Topics addressed today: 1. Recoding data missing values, collapsing categories 2. Making a simple scale 3. Standardisation

More information

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

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

More information

Module 9: Nonparametric Tests. The Applied Research Center

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

More information

IBM SPSS Statistics 20 Part 4: Chi-Square and ANOVA

IBM SPSS Statistics 20 Part 4: Chi-Square and ANOVA CALIFORNIA STATE UNIVERSITY, LOS ANGELES INFORMATION TECHNOLOGY SERVICES IBM SPSS Statistics 20 Part 4: Chi-Square and ANOVA Summer 2013, Version 2.0 Table of Contents Introduction...2 Downloading the

More information

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

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

More information

Data Analysis in SPSS. February 21, 2004. If you wish to cite the contents of this document, the APA reference for them would be

Data Analysis in SPSS. February 21, 2004. If you wish to cite the contents of this document, the APA reference for them would be Data Analysis in SPSS Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Heather Claypool Department of Psychology Miami University

More information

January 26, 2009 The Faculty Center for Teaching and Learning

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

More information

Section 13, Part 1 ANOVA. Analysis Of Variance

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

More information

A Basic Guide to Analyzing Individual Scores Data with SPSS

A Basic Guide to Analyzing Individual Scores Data with SPSS A Basic Guide to Analyzing Individual Scores Data with SPSS Step 1. Clean the data file Open the Excel file with your data. You may get the following message: If you get this message, click yes. Delete

More information

Practice 3 SPSS. Partially based on Notes from the University of Reading:

Practice 3 SPSS. Partially based on Notes from the University of Reading: Practice 3 SPSS Partially based on Notes from the University of Reading: http://www.reading.ac.uk Simple Linear Regression A simple linear regression model is fitted when you want to investigate whether

More information

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 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

More information

10. Comparing Means Using Repeated Measures ANOVA

10. Comparing Means Using Repeated Measures ANOVA 10. Comparing Means Using Repeated Measures ANOVA Objectives Calculate repeated measures ANOVAs Calculate effect size Conduct multiple comparisons Graphically illustrate mean differences Repeated measures

More information

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

Doing Multiple Regression with SPSS. In this case, we are interested in the Analyze options so we choose that menu. If gives us a number of choices: Doing Multiple Regression with SPSS Multiple Regression for Data Already in Data Editor Next we want to specify a multiple regression analysis for these data. The menu bar for SPSS offers several options:

More information

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

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

More information

Example: Multivariate Analysis of Variance

Example: Multivariate Analysis of Variance 1 of 36 Example: Multivariate Analysis of Variance Multivariate analyses of variance (MANOVA) differs from univariate analyses of variance (ANOVA) in the number of dependent variables utilized. The major

More information

Point-Biserial and Biserial Correlations

Point-Biserial and Biserial Correlations Chapter 302 Point-Biserial and Biserial Correlations Introduction This procedure calculates estimates, confidence intervals, and hypothesis tests for both the point-biserial and the biserial correlations.

More information

15. Analysis of Variance

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

More information

Mind on Statistics. Chapter 16 Section Chapter 16

Mind on Statistics. Chapter 16 Section Chapter 16 Mind on Statistics Chapter 16 Section 16.1 1. Which of the following is not one of the assumptions made in the analysis of variance? A. Each sample is an independent random sample. B. The distribution

More information

The general form of the PROC MEANS statement is

The general form of the PROC MEANS statement is Describing Your Data Using PROC MEANS PROC MEANS can be used to compute various univariate descriptive statistics for specified variables including the number of observations, mean, standard deviation,

More information

Multivariate Analysis of Variance (MANOVA)

Multivariate Analysis of Variance (MANOVA) Chapter 415 Multivariate Analysis of Variance (MANOVA) Introduction Multivariate analysis of variance (MANOVA) is an extension of common analysis of variance (ANOVA). In ANOVA, differences among various

More information

One-Way Analysis of Variance (ANOVA) Example Problem

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

More information

UNDERSTANDING ANALYSIS OF COVARIANCE (ANCOVA)

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

More information

The Chi-Square Goodness-of-Fit Test, Equal Proportions

The Chi-Square Goodness-of-Fit Test, Equal Proportions Chapter 11 Chi-Square Tests 1 Chi-Square Tests Chapter 11 The Chi-Square Goodness-of-Fit Test, Equal Proportions A hospital wants to know if the proportion of births are the same for each day of the week.

More information

PSYCHOLOGY 320L Problem Set #3: One-Way ANOVA and Analytical Comparisons

PSYCHOLOGY 320L Problem Set #3: One-Way ANOVA and Analytical Comparisons PSYCHOLOGY 30L Problem Set #3: One-Way ANOVA and Analytical Comparisons Name: Score:. You and Dr. Exercise have decided to conduct a study on exercise and its effects on mood ratings. Many studies (Babyak

More information

Two-Sample T-Test from Means and SD s

Two-Sample T-Test from Means and SD s Chapter 07 Two-Sample T-Test from Means and SD s Introduction This procedure computes the two-sample t-test and several other two-sample tests directly from the mean, standard deviation, and sample size.

More information

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

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

More information

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

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

More information

ANOVA Analysis of Variance

ANOVA Analysis of Variance 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,

More information

Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

More information

individualdifferences

individualdifferences 1 Simple ANalysis Of Variance (ANOVA) Oftentimes we have more than two groups that we want to compare. The purpose of ANOVA is to allow us to compare group means from several independent samples. In general,

More information

Chapter 2 Probability Topics SPSS T tests

Chapter 2 Probability Topics SPSS T tests Chapter 2 Probability Topics SPSS T tests Data file used: gss.sav In the lecture about chapter 2, only the One-Sample T test has been explained. In this handout, we also give the SPSS methods to perform

More information

Introduction to Hypothesis Testing. Copyright 2014 Pearson Education, Inc. 9-1

Introduction to Hypothesis Testing. Copyright 2014 Pearson Education, Inc. 9-1 Introduction to Hypothesis Testing 9-1 Learning Outcomes Outcome 1. Formulate null and alternative hypotheses for applications involving a single population mean or proportion. Outcome 2. Know what Type

More information

SPSS BASICS. (Data used in this tutorial: General Social Survey 2000 and 2002) Ex: Mother s Education to eliminate responses 97,98, 99;

SPSS BASICS. (Data used in this tutorial: General Social Survey 2000 and 2002) Ex: Mother s Education to eliminate responses 97,98, 99; SPSS BASICS (Data used in this tutorial: General Social Survey 2000 and 2002) How to do Recoding Eliminating Response Categories Ex: Mother s Education to eliminate responses 97,98, 99; When we run a frequency

More information

DDBA 8438: The t Test for Independent Samples Video Podcast Transcript

DDBA 8438: The t Test for Independent Samples Video Podcast Transcript DDBA 8438: The t Test for Independent Samples Video Podcast Transcript JENNIFER ANN MORROW: Welcome to The t Test for Independent Samples. My name is Dr. Jennifer Ann Morrow. In today's demonstration,

More information

SPSS Explore procedure

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,

More information

Analysis of Covariance

Analysis of Covariance Analysis of Covariance 1. Introduction The Analysis of Covariance (generally known as ANCOVA) is a technique that sits between analysis of variance and regression analysis. It has a number of purposes

More information

ID X Y

ID X Y Dale Berger SPSS Step-by-Step Regression Introduction: MRC01 This step-by-step example shows how to enter data into SPSS and conduct a simple regression analysis to develop an equation to predict from.

More information

Analysis of numerical data S4

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:

More information

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

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)

More information

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

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

More information

HOW TO USE MINITAB: INTRODUCTION AND BASICS. Noelle M. Richard 08/27/14

HOW TO USE MINITAB: INTRODUCTION AND BASICS. Noelle M. Richard 08/27/14 HOW TO USE MINITAB: INTRODUCTION AND BASICS 1 Noelle M. Richard 08/27/14 CONTENTS * Click on the links to jump to that page in the presentation. * 1. Minitab Environment 2. Uploading Data to Minitab/Saving

More information

13: Additional ANOVA Topics. Post hoc Comparisons

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

More information

Statistics and research

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,

More information

Data Analysis Tools. Tools for Summarizing Data

Data Analysis Tools. Tools for Summarizing Data Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool

More information

Latin Square Design Analysis

Latin Square Design Analysis SPSS NOV for Latin Square esign Latin Square esign nalysis Goal: omparing the performance of four different brands of tires (,,, and ). ackground: There are four cars available for this comparative study

More information

The Chi-Square Test. STAT E-50 Introduction to Statistics

The Chi-Square Test. STAT E-50 Introduction to Statistics STAT -50 Introduction to Statistics The Chi-Square Test The Chi-square test is a nonparametric test that is used to compare experimental results with theoretical models. That is, we will be comparing observed

More information

IBM SPSS Statistics for Beginners for Windows

IBM SPSS Statistics for Beginners for Windows ISS, NEWCASTLE UNIVERSITY IBM SPSS Statistics for Beginners for Windows A Training Manual for Beginners Dr. S. T. Kometa A Training Manual for Beginners Contents 1 Aims and Objectives... 3 1.1 Learning

More information

UNDERSTANDING THE INDEPENDENT-SAMPLES t TEST

UNDERSTANDING THE INDEPENDENT-SAMPLES t TEST UNDERSTANDING The independent-samples t test evaluates the difference between the means of two independent or unrelated groups. That is, we evaluate whether the means for two independent groups are significantly

More information