# Example: Multivariate Analysis of Variance

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 purpose of univariate analyses of variance is to determine the effects of one or more independent variables upon ONE dependent variable. However, there arise situations in which measurements are available on MORE THAN ONE dependent variable. For example, say you are interested in designing an experiment to test the effects of success of street kids exiting street life. Success can be measured in more than one way. It can be measured by both the number of days a kid sleeps at home or the number of days a kid returns to school. In such a case, there are TWO dependent variables involved in the analysis. In analyzing such data, one obvious (and popular) approach is to perform an ANOVA on each dependent variable separately. In our example above, two ANOVA s could be carried out (i.e., one ANOVA with home scores

2 2 of 36 and one ANOVA with school scores). However, it is quite apparent that both the home and school variables are highly correlated. Thus, the two analyses are not independent. One s return to school is most likely positively correlated with one s return home. The independent ANOVA ignores the interrelation between variables. Consequently, substantial information may be lost (SPSS, Release 9.0), and the resultant p values for tests of hypothesis for the two independent ANOVA s are incorrect. The extension of univariate analysis of variance to the case of multiple dependent variables is known as Multivariate Analysis of Variance (MANOVA). MANOVA allows for a direct test of the null hypothesis with respect to ALL the dependent variables in an experiment. In MANOVA, a linear function (y) of the dependent variables in the analysis is constructed, so that inter-group differences on y are maximized. The composite variable y is then treated in a manner somewhat similar to the dependent variable in a univariate ANOVA, with the null hypothesis being accepted or rejected accordingly.

3 3 of 36 Essentially, the multivariate technique known as Discriminant analysis is quite similar to MANOVA. In discriminant analysis we consider the problem of finding the best linear combination of variables for distinguishing several groups. The coefficients (Betas) are chosen in order that the ratio of the between-groups sums of squares to the total sums of squares is as large as possible. MANOVA can be viewed as a problem of first finding linear combinations of the dependent variables that best separate the groups and then testing whether these new variables are significantly different for the groups. In the MANOVA situation, we already know which categories the cases belong to, and thus are not interested in classification BUT in identifying a composite variable (y) which illuminates the differences among the groups. In both procedures the composite variable (y) is known as a discriminant function. In a MANOVA, discriminant function is a weighted sum of the dependent variables, with the weightings chosen such that the distributions of y for the various groups are separated to the greatest possible extent.

4 4 of 36 In a DISCRIMINANT analysis, the discriminant function is used to predict category membership to the greatest extent possible. Thus, both techniques have very much in common and this will be illustrated in the last section of this module. Remember that a MANOVA is used when the dependent variables are CORRELATED. Example Let s see multivariate analysis of variance in action. We borrowed data from Moore and McCabe (1998:Data Appendix 4) in order to illustrate this procedure. The study is described by the researchers as follows: Jim Baumann and Leah Jones of the Purdue University School of Education conducted a study to compare three methods of teaching reading comprehension. The 66 students who participated in the study were randomly assigned to the methods (22 to each). The standard practice of comparing new methods with a traditional one was used in this study. The traditional method is called Basal and the two innovative methods are called DRTA and Strat. (Moore and McCabe, 1998, Data Appendix 4)

5 5 of 36 The 66 subjects in this study are divided into three groups (B, D, S) which we have coded as B=1, D=2, S=3. There are five independent variables which represent either pretest or posttest scores (i.e., pre1, pre2, post1, post2, and post3). Let s imagine that we are interested in discovering whether these variables are significantly different for the three groups. Remember that we are using MANOVA since we suspect that the dependent variables are correlated. To perform a multivariate analysis of variance in SPSS for this reading study, we need to complete the following steps: 1. Pull up the Reading data set. 2. Click Statistics on the main menu bar. 3. Click General Linear Model on the Statistics menu. 4. Select GLM-Multivariate in the General Linear Model submenu. This will open a GLM-Multivariate box similar to that shown on the next page.

6 6 of 36 The next step is to specify the dependent variables and the fixed factors. In this example, the five reading scores (i.e. pre1, pre2, post1, post2, post3 ) are the Dependent variables. Group is the Fixed Factor in this MANOVA. To specify the variable as a dependent or fixed factor, simply select the variable in the box on the left hand side, and click the arrow button to the left of the Dependent Variables or Fixed Factor(s) respectively. The GLM-Multivariate dialogue box should now appear as seen on the next page:

7 7 of 36 The commands for the basic MANOVA have been defined through the aforementioned four steps. We now want to include specific criteria for our MANOVA. In order to do this we must complete the following steps: 1. Click the Model pushbutton at the top-right corner of the GLM-Multivariate dialogue box. This will open up the submenu and allow us to further specify our

8 8 of 36 analysis as follows: Click the Custom button. Select group(f) in the Factors & Covariates box. Then move group over to the Model box by clicking the right arrow under build term. Select Main effects in the drop-down window below the Build Term(s) arrow. Click Continue to return to the GLM-Multivariate box.

9 9 of Click the Contrasts pushbutton. This will allow us to further specify our analysis as follows: Select Simple in the Contrast drop-down window. Click the First pushbutton for the Reference Category. Click the Change pushbutton. The Factor submenu should now read Group[Simple[first]] as seen below. Click Continue to return to the GLM-Multivariate box.

10 10 of Click the Plots pushbutton. This will open up this submenu and allow us to make the following selections: Select group in the Factors box and click the right arrow to the Horizontal Axis:. This will move group into the Horizontal Axis:. The Add button on the Plots box is now an available option. Click this Add button. This will now add group to this box, and submenu should appear as follows: Click Continue to return to the GLM-Multivariate box.

11 11 of Click the Post Hoc pushbutton. This will allow us to further specify our analysis as follows: Select group in the Factor(s): box and click the right arrow to the Post Hoc Tests for: box. This will move group into the Post Hoc Tests for: box. Click the Bonferroni box in the Equal Variances Assumed section, and the submenu will appear as follows: Click Continue to return to the GLM-Multivariate box.

12 12 of Click the Save pushbutton. This will allow us to further specify our analysis as follows: In the Predicted Values section, select both the Unstandardized and Standard error boxes. In the Diagnostics section, select both the Cook s distance and Leverage values boxes. In the Residuals section, select both the Unstandardized and Standardized boxes. Click Continue to return to the GLM-Multivariate box.

13 13 of Click the Options pushbutton. This will allow us to further specify our analysis as follows: In the Estimated Marginal Means section, select group. Then click the right arrow button to move group into the Display Means for: box. In the Display section select the following boxes: Descriptive statistics ; Estimates of effect size ; Observed power ; Residual SSCP matrix ; Homogeneity tests ; and Residual plots. Your submenu should now appear as below:

14 14 of 36 Click Continue to return to the GLM-Multivariate box. Click OK to run this SPSS analysis. Congratulations! You have just completed the SPSS MANOVA for this module. Now let s see how to interpret our SPSS Data Output.

15 15 of 36 Multivariate Analysis of Variance SPSS Output Explanation Descriptives SPSS provides us with a general idea of the distribution of scores in each group in the Descriptive Statistics output. Descriptive Statistics Mean Std. Deviation N PRE1 GROUP Basal DRTA Strategies Total PRE2 GROUP Basal DRTA Strategies Total POST1 GROUP Basal DRTA Strategies Total POST2 GROUP Basal DRTA Strategies Total POST3 GROUP Basal DRTA Strategies Total

16 16 of 36 A boxplot of the multiple dependent variables of our MANOVA can be easily created utilizing SPSS. The output is as follows: PRE PRE2 POST POST2-10 POST3 N = Basal DRTA Strategies GROUP

17 17 of 36 It is obvious that there is an even spread within groups for each dependent variable. We can also perform a test of Homogeneity in order to know whether the variances for each of the groups are the same: Univariate Test Results oucontdepenpre1 Variab PRE2 POST1 POST2 POST3 ErrorDepenPRE1 Variab PRE2 POST1 POST2 POST3 Sum of Mean Eta oncen bserve 1 quares df Square F Sig. quared arametpower omputed using alpha =.05

18 18 of 36 Univariate Homogeneity of Variance Tests Variable.. POST1 Cochrans C(21,3) =.50564, P =.072 (approx.) Bartlett-Box F(2,8930) = , P =.155 Variable.. POST3 Cochrans C(21,3) =.45641, P =.227 (approx.) Bartlett-Box F(2,8930) =.97142, P =.379 Variable.. POST2 Cochrans C(21,3) =.49688, P =.090 (approx.) Bartlett-Box F(2,8930) = , P =.185 Variable.. PRE2 Cochrans C(21,3) =.50566, P =.072 (approx.) Bartlett-Box F(2,8930) = , P =.148 Variable.. PRE1 Cochrans C(21,3) =.40980, P =.538 (approx.) Bartlett-Box F(2,8930) =.48111, P =.618

19 19 of 36 As seen through the Cochran s C and the Bartlett Box F tests, the significance levels indicate that there is no reason to reject the hypotheses that the variances in the three groups are equal (all values are greater than.05). These tests are UNIVARIATE and are a convenient starting point for examining homogeneity (covariance); however, we also need to simultaneously consider both the variances and the covariances. The Box s M test is such a test. Box s M is based on the determinants of the variancecovariance matrices in each cell, as well as the pooled variance-covariance matrix. Thus Box s M provides us with a multivariate test for homogeneity (SPSS 9.0). Multivariate test for Homogeneity of Dispersion matrices ox's Test of Equality of Covariance Matrice 1 Box's M F df1 df2 Sig Tests the null hypothesis that the observed covariance matrices of the dependent variables are equal across groups. 1. Design: Intercept+GROUP

20 20 of 36 Chi-Square with 30 DF = , P =.488 (Approx.) Given the results of Box s M test (either.493 or.488 depending on the F or Chi-square), we have no reason to suspect that homogeneity has been violated (values greater than.05). Bartlett's Test of Sphericity 1 Likelihood Ratio.000 Approx. Chi-Squa df Sig Tests the null hypothesis that the residual covariance proportional to an identity matrix. 1. Design: Intercept+GROUP Multivariate analysis of variance is a procedure used when the dependent variables are correlated. The Bartlett s Test of Sphericity is an option that is used to test the hypothesis that the population correlation matrix is an identity matrix (all diagonal terms are 1 and all off-diagonal terms are 0). The test is based

21 21 of 36 on the determinant of the error correlation matrix: a determinant that is close to 0 indicates that one or more variables are correlated. Thus, if the determinant is small, independence of the variables are rejected. As seen above, our determinant value is small (-1.17) with a significance value of.000. Therefore, we can reject the hypothesis of independence and conclude that out variables are correlated, hence the need for MANOVA. Statistics for WITHIN+RESIDUAL correlations Log(Determinant) = Bartlett test of sphericity = with 10 D. F. Significance =.000 Another way to visualize variable independence is through the Half-normal plot of the correlation coefficients (see below). This plot is very similar to the normal plot except for the fact that positive and negative values are treated identically. This plot is used to test graphically the hypothesis that the correlation matrix is an identity matrix. The Fischer s Z-transform is the way in which the observed

22 22 of 36 correlation coefficients are transformed. If the correlation matrix is an identity matrix, then the plot should be linear and the line should pass through the origin. Our plot shows deviation from linearity, and thus suggests that the variables are not independent (as indicated by the Bartlett s test).

23 23 of 36 Multivariate Tests 4 ypothes Eta Noncent Observed 1 Value F df Error df Sig. SquaredarametePower EffeIntercPillai's Tra Wilks' Lam Hotelling's Roy's Larg GRO Pillai's Tra Wilks' Lam Hotelling's Roy's Larg Computed using alpha = Exact statistic 3. The statistic is an upper bound on F that yields a lower bound on the sig 4. Design: Intercept+GROUP All four tests explore whether the means for each of the groups are the same. The first line contains the values of the parameters (S, M, N) used to discover significant levels in tables of the

24 24 of 36 exact distributions of the statistics. For the first three tests, the value of the test statistic is given, followed by its transformation to a statistic that has approximately an F distribution. The next two columns contain the numerator (hypothesis) and denominator (Error) degrees of freedom for the F statistic. The next column gives us the observed significance levels which are translated as the probability of observing a difference at least as large as the one found in the sample when there is no difference in the populations (SPSS 9.0). Note that the Roy s test value cannot be transformed into a statistic and thus only the value is given In our case, due to the significance values of.000 (sig column) we can conclude that the null hypothesis - that there is no difference is rejected. Therefore, we know that there are significant differences between the three groups on the means of the five variables.

25 Levene's Test of Equality of Error Variances 1 PRE1 F df1 df2 Sig of 36 PRE2 POST1 POST2 POST Tests the null hypothesis that the error variance of the dependent variable is equal across groups. 1. Design: Intercept+GROUP

26 SourceCorrected MoDepende PRE1 Variable PRE2 Intercept GROUP Error Total Depende Variable Depende Variable Depende Variable Depende Variable POST1 POST2 POST3 PRE1 PRE2 POST1 POST2 POST3 PRE1 PRE2 POST1 POST2 POST3 PRE1 PRE2 POST1 POST2 POST3 PRE1 PRE2 POST1 POST2 POST3 Corrected To Depende PRE1 Variable PRE2 POST1 POST2 POST3 1. Computed using alpha = R Squared =.035 (Adjusted R Squared =.004) 3. R Squared =.004 (Adjusted R Squared = -.028) 4. R Squared =.144 (Adjusted R Squared =.117) 5. R Squared =.191 (Adjusted R Squared =.166) 6. R Squared =.125 (Adjusted R Squared =.097) Tests of Between-Subjects Effects Type III Sum of Squares df 26 of Mean Square F Sig. Eta Noncent. Observed Squared Parameter Power 1

27 27 of 36 The table below contains several sets of statistics for the discriminant functions. The eigenvalue represents the ratio of the Between-Groups sum of squares to the Within-Groups sum of squares. We see that our examples consists of two functions that contribute to group differences. The first function has an eigenvalue of.630 and explains 73% of the differences in the three groups (column: cumulative percentage). The final column gives the Canonical Correlation, and the square of this value is the ratio of the Between-groups sum of squares to the Total sum of squares. Therefore about 38.6% ( ) of the variability in the first discriminant score is attributable to between-group differences. Eigenvalues and Canonical Correlations Root No. Eigenvalue Pct. Cum. Pct. Canon Cor Wilks Lambda can be seen as a measure of the proportion

28 28 of 36 of total variability not explained by group differences. In our case (see below) both functions are significant (last columnsignificance values less than.05). The second and third columns represent the numerator and denominator degrees of freedom for the F statistic. In sum, we can conclude that both functions are distinct and significant in discriminating amongst the groups. Dimension Reduction Analysis Roots Wilks L. F Hypoth. DF Error DF Sig. of F 1 TO TO

29 29 of 36 DATA OUTPUT Custom Hypothesis Tests Contrast Results (K Matrix) Dependent Variable PRE1 PRE2 POST1 POST2 POST3 GROUP Simple Contrast 1 Level 2 vs. Level 1 Contrast Estimate Hypothesized Value Difference (Estimate - Hypothesized) Std. Error Sig % Confidence Interval for Difference Lower Bound Upper Bound Level 3 vs. Level 1 Contrast Estimate Hypothesized Value Difference (Estimate - Hypothesized) Std. Error Sig % Confidence Interval for Difference Lower Bound Upper Bound Reference category = 1 Pillai's trac Wilks' lam Hotelling's Value F Multivariate Test Results ypothesi df Error df Sig. Eta Noncent Observed 1 SquaredarametePower Roy's large Computed using alpha = Exact statistic 3. The statistic is an upper bound on F that yields a lower bound on th

30 30 of 36 Univariate Test Results Source Contrast Error Dependent Variable Dependent Variable 1. Computed using alpha =.05 PRE1 PRE2 POST1 POST2 POST3 PRE1 PRE2 POST1 POST2 POST3 Sum of Mean Eta Noncent. Observed Squares df Square F Sig. Squared Parameter Power Since our multivariate results were significant (remember the Pillais, Hotellings, Wilks and Roys tests above), we can now examine our univariate results shown below: EFFECT.. GROUP (Cont.) Univariate F-tests with (2,63) D. F. Variable Hypoth. SS Error SS Hypoth. MS Error MS F Sig. of F POST POST POST PRE PRE This table helps determine which variables contribute to the overall differences. In our case, we can see that the first three variables: Post1, Post 3, and Post2 all yield significant values (p

31 31 of 36 less than.05). It also makes sense that both Pretest 1 and Pretest 2 have no influence upon the three groups, since pretests are in fact supposed to NOT influence respondents. Lastly, the table below contains the standardized discriminant function coefficients for the group term. The magnitude of the coefficients give us some idea of the variables contributing most to group differences. In our case, we can see that Post1 (.963), Post2 (.709), and Pre1 (-.867) have large coefficients and thus could play an important role in group differences within the first function. In the second function, Post1 and Post 2 both yield large coefficients. Standardized discriminant function coefficients Function No. Variable 1 2 POST POST POST PRE PRE

32 32 of 36 In addition, we can get a sense of the spread of group means of each discriminant function (in our case, we are interested in both functions) which was derived from the discriminant analysis procedure: Canonical discriminant functions evaluated at group means (group centroids) Group Func 1 Func *** In summary, let s return to our initial discussion around the similarities between MANOVA and discriminant analysis. As noted earlier, these two procedures are much alike, and here is

33 33 of 36 the test: I ran the same data through a discriminant analysis and look at the results. Do they look familiar at all? Discriminant Results on Same Data: Tests of Equality of Group Means PRE1 PRE2 POST1 POST2 POST3 Wilks' Lambda F df1 df2 Sig Wilks' Lambda (U-statistic) and univariate F-ratio with 2 and 63 degrees of freedom Variable Wilks' Lambda F Significance POST POST POST PRE PRE see page 8 for MANOVA output. Eigenvalues Eigenvalue % of Variance Cumulative % Canonical Correlation Function Wilks' Lambda Wilks' First 2 canonical discriminant functions were used in the analysis. Lambda Chi-square df Sig. Test of Function(s) 1 through

34 34 of 36 Canonical Discriminant Functions Pct of Cum Canonical After Wilks' Fcn Eigenvalue Variance Pct Corr Lambda Chi-square df Sig 1* * * Marks the 2 canonical discriminant functions remaining in the analysis. See page 7 for MANOVA output. tandardized Canonical Discriminant Function Coefficient PRE1 PRE2 POST1 POST2 POST3 Function

35 Standardized canonical discriminant function coefficients Func 1 Func 2 POST POST POST PRE PRE see page 9 for MANOVA output. We see that there are two functions, both are significant in 35 of 36 examining group differences. With our univariate F-tests, we identify the same significant variables. And with our standardized canonical discriminate function coefficients, we see the same variables having strength (or significance) with each particular function. This is the identical test to MANOVA. In conclusion, MANOVA is a procedure utilized when our dependent variables are correlated. Multivariate analysis helps control the error rate, or the chance of getting something significant when in actuality it is not. Very similar in process to discriminant analysis, MANOVA permits a direct test of the

36 36 of 36 null hypothesis with respect to ALL of the analysis dependent variables.

### 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,

### 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

### 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

### 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,

### 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. 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

### 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

### 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

### DISCRIMINANT FUNCTION ANALYSIS (DA)

DISCRIMINANT FUNCTION ANALYSIS (DA) John Poulsen and Aaron French Key words: assumptions, further reading, computations, standardized coefficents, structure matrix, tests of signficance Introduction Discriminant

### Canonical Correlation

Chapter 400 Introduction Canonical correlation analysis is the study of the linear relations between two sets of variables. It is the multivariate extension of correlation analysis. Although we will present

### 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

### 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

### SPSS Guide: Regression Analysis

SPSS Guide: Regression Analysis I put this together to give you a step-by-step guide for replicating what we did in the computer lab. It should help you run the tests we covered. The best way to get familiar

### 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

### 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

### Simple Linear Regression in SPSS STAT 314

Simple Linear Regression in SPSS STAT 314 1. Ten Corvettes between 1 and 6 years old were randomly selected from last year s sales records in Virginia Beach, Virginia. The following data were obtained,

### Multivariate Analysis of Variance (MANOVA): I. Theory

Gregory Carey, 1998 MANOVA: I - 1 Multivariate Analysis of Variance (MANOVA): I. Theory Introduction The purpose of a t test is to assess the likelihood that the means for two groups are sampled from the

### 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

### 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

Mgt 540 Research Methods Data Analysis 1 Additional sources Compilation of sources: http://lrs.ed.uiuc.edu/tseportal/datacollectionmethodologies/jin-tselink/tselink.htm http://web.utk.edu/~dap/random/order/start.htm

### Chapter 6: t test for dependent samples

Chapter 6: t test for dependent samples ****This chapter corresponds to chapter 11 of your book ( t(ea) for Two (Again) ). What it is: The t test for dependent samples is used to determine whether the

### 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

### Factor Analysis. Principal components factor analysis. Use of extracted factors in multivariate dependency models

Factor Analysis Principal components factor analysis Use of extracted factors in multivariate dependency models 2 KEY CONCEPTS ***** Factor Analysis Interdependency technique Assumptions of factor analysis

### Simple Linear Regression, Scatterplots, and Bivariate Correlation

1 Simple Linear Regression, Scatterplots, and Bivariate Correlation This section covers procedures for testing the association between two continuous variables using the SPSS Regression and Correlate analyses.

### Multivariate analyses

14 Multivariate analyses Learning objectives By the end of this chapter you should be able to: Recognise when it is appropriate to use multivariate analyses (MANOVA) and which test to use (traditional

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### 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

### 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

### T-test & factor analysis

Parametric tests T-test & factor analysis Better than non parametric tests Stringent assumptions More strings attached Assumes population distribution of sample is normal Major problem Alternatives Continue

### This chapter will demonstrate how to perform multiple linear regression with IBM SPSS

CHAPTER 7B Multiple Regression: Statistical Methods Using IBM SPSS This chapter will demonstrate how to perform multiple linear regression with IBM SPSS first using the standard method and then using the

### Simple linear regression

Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

### Chapter 7. Comparing Means in SPSS (t-tests) Compare Means analyses. Specifically, we demonstrate procedures for running Dependent-Sample (or

1 Chapter 7 Comparing Means in SPSS (t-tests) This section covers procedures for testing the differences between two means using the SPSS Compare Means analyses. Specifically, we demonstrate procedures

### 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

### 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,

### 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

### 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

### THE KRUSKAL WALLLIS TEST

THE KRUSKAL WALLLIS TEST TEODORA H. MEHOTCHEVA Wednesday, 23 rd April 08 THE KRUSKAL-WALLIS TEST: The non-parametric alternative to ANOVA: testing for difference between several independent groups 2 NON

### Factor Analysis. Chapter 420. Introduction

Chapter 420 Introduction (FA) is an exploratory technique applied to a set of observed variables that seeks to find underlying factors (subsets of variables) from which the observed variables were generated.

### 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,

### 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?

### SPSS Guide How-to, Tips, Tricks & Statistical Techniques

SPSS Guide How-to, Tips, Tricks & Statistical Techniques Support for the course Research Methodology for IB Also useful for your BSc or MSc thesis March 2014 Dr. Marijke Leliveld Jacob Wiebenga, MSc CONTENT

### Multiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear.

Multiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear. In the main dialog box, input the dependent variable and several predictors.

### 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

### 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

### 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

### 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

### 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

### 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

### 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,

### Descriptive Statistics

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

### 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:

### 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

### 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

### Data Analysis for Marketing Research - Using SPSS

North South University, School of Business MKT 63 Marketing Research Instructor: Mahmood Hussain, PhD Data Analysis for Marketing Research - Using SPSS Introduction In this part of the class, we will learn

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

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

: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

### 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

### 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

### 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

### 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

### 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

### 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

### Didacticiel - Études de cas

1 Topic Linear Discriminant Analysis Data Mining Tools Comparison (Tanagra, R, SAS and SPSS). Linear discriminant analysis is a popular method in domains of statistics, machine learning and pattern recognition.

### 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

### MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

### 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

### 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

### 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

### 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

### 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

### research/scientific includes the following: statistical hypotheses: you have a null and alternative you accept one and reject the other

1 Hypothesis Testing Richard S. Balkin, Ph.D., LPC-S, NCC 2 Overview When we have questions about the effect of a treatment or intervention or wish to compare groups, we use hypothesis testing Parametric

### SPSS Advanced Statistics 17.0

i SPSS Advanced Statistics 17.0 For more information about SPSS Inc. software products, please visit our Web site at http://www.spss.com or contact SPSS Inc. 233 South Wacker Drive, 11th Floor Chicago,

### 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

### 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

### 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

### Chapter 15. Mixed Models. 15.1 Overview. A flexible approach to correlated data.

Chapter 15 Mixed Models A flexible approach to correlated data. 15.1 Overview Correlated data arise frequently in statistical analyses. This may be due to grouping of subjects, e.g., students within classrooms,

### 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

### 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

### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

### Hatice Camgöz Akdağ. findings of previous research in which two independent firm clusters were

Innovative Culture and Total Quality Management as a Tool for Sustainable Competitiveness: A Case Study of Turkish Fruit and Vegetable Processing Industry SMEs, Sedef Akgüngör Hatice Camgöz Akdağ Aslı

### FACTOR ANALYSIS NASC

FACTOR ANALYSIS NASC Factor Analysis A data reduction technique designed to represent a wide range of attributes on a smaller number of dimensions. Aim is to identify groups of variables which are relatively

### " Y. Notation and Equations for Regression Lecture 11/4. Notation:

Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

### ANOVA - Analysis of Variance

ANOVA - Analysis of Variance ANOVA - Analysis of Variance Extends independent-samples t test Compares the means of groups of independent observations Don t be fooled by the name. ANOVA does not compare

### Doing Quantitative Research 26E02900, 6 ECTS Lecture 2: Measurement Scales. Olli-Pekka Kauppila Rilana Riikkinen

Doing Quantitative Research 26E02900, 6 ECTS Lecture 2: Measurement Scales Olli-Pekka Kauppila Rilana Riikkinen Learning Objectives 1. Develop the ability to assess a quality of measurement instruments

### Unit 29 Chi-Square Goodness-of-Fit Test

Unit 29 Chi-Square Goodness-of-Fit Test Objectives: To perform the chi-square hypothesis test concerning proportions corresponding to more than two categories of a qualitative variable To perform the Bonferroni

### Linear Regression in SPSS

Linear Regression in SPSS Data: mangunkill.sav Goals: Examine relation between number of handguns registered (nhandgun) and number of man killed (mankill) checking Predict number of man killed using number

### 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,

### 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

### 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,

### Linear Models in STATA and ANOVA

Session 4 Linear Models in STATA and ANOVA Page Strengths of Linear Relationships 4-2 A Note on Non-Linear Relationships 4-4 Multiple Linear Regression 4-5 Removal of Variables 4-8 Independent Samples

### 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

### 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

### Notes for STA 437/1005 Methods for Multivariate Data

Notes for STA 437/1005 Methods for Multivariate Data Radford M. Neal, 26 November 2010 Random Vectors Notation: Let X be a random vector with p elements, so that X = [X 1,..., X p ], where denotes transpose.

### 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

### Directions for using SPSS

Directions for using SPSS Table of Contents Connecting and Working with Files 1. Accessing SPSS... 2 2. Transferring Files to N:\drive or your computer... 3 3. Importing Data from Another File Format...

### 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.

### 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:

### 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