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By Hui Bian Office for Faculty Excellence 1

# SPSS Series 1: ANOVA and Factorial ANOVA

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By Hui BianOffice for Faculty Excellence

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K-group between-subjects MANOVA with SPSS

Factorial between-subjects MANOVA with SPSS

How to interpret SPSS outputsHow to report results

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We use 2009 Youth Risk Behavior Surveillance System (YRBSS, CDC) as an example.YRBSS monitors priority health-risk

behaviors and the prevalence of obesity and asthma among youth and young adults.

The target population is high school students

Multiple health behaviors include drinking, smoking, exercise, eating habits, etc.

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MANOVAWe focus on K-group between subjects

design.Assess the effects of one independent

variable (K-group) on two or more dependent variables simultaneously.

Dependent variables are correlated and share a common conceptual meaning.

MANOVA uses Pillai’s trace, Wilks’lambda, Hotelling’s trace, and Roy’s largest root criterion

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Why use MANOVASingle dependent measure seldom captures

completely a phenomenon being studied. MANOVA provides some control over the overall

alpha level or type I error. Multiple univariate t tests or ANOVA can inflate the operational alpha level.

MANOVA considers dependent variable intercorrelations.

MANOVA helps indentify dependent variables that produce the most group separation or distinction.

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When NOT use MANOVAIf the dependent variables are not correlated.If the dependent variables are highly

correlated. It will produce the risk of a multicollinearity condition.Use subscales together with the total scores of

the scale as dependent variablesThe dependent variable is computed from one or

more of the others.Using baseline and posttest scores would create

linear dependence.

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Assumptions Independence: the participants that compose the

levels of an independent variable must be independent of each other.

Homogeneity of covariance matricesBox’s M test from SPSS is used to assess equivalence of

covariance matrices. Homogeneity of variance

When the sample size is fairly equal across the group, violation of homogeneity produces minor consequences.

The group sizes are approximately equal (largest/smallest 1.5).

Multivariate normalityCheck univariate normality for each dependent variable.

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Example: Research design: four-group between-subjects

designResearch question: whether grade levels affect

high school students’ sedentary behaviors.One independent variable: Grade with 4 levels: 9th,

10th, 11th, and 12th grade (Q3r).Two dependent variables: sedentary behaviors: Q80

(physical activity) and Q81: (How many hours watch TV).

Higher score of Q80 = More days of physically active.Higher score of Q81 = More hours on watching TV.

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Initial data screeningStem-and-Leaf Plots: use the original data

values to display the distribution's shape. Normal Q-Q Plots: the straight line in the

plot represents expected values when the data are normally distributed.

Box Plots: is used to identify outliers.

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Select Analyze Descriptive Statistics Explore

Move Q80 and Q81Move Q3rClick Plots

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Stem-and-Leaf Plots (Q80 for 9th grade)

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Stem

Leaves

Stem-and-Leaf Plots (Q81 for 9th grade)

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Normal Q-Q Plots: the straight line in the plot represents expected values when the data are normally distributed.

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Box Plots

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Median

Minimum value

25th percenti

le

75th percenti

le

Kurtosis

Normality of our dependent variablesThe plots obtained from SPSS look reasonably

normal.We judge these variables ready for multivariate

analysis.MANOVA using SPSS

Select Analyze General Linear Model Multivariate

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Options and Post-hoc

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Post hoc tests: A follow-up analysisFollowing a significant multivariate effect.The purpose of post hoc tests is to discover

which specific dependent variables are affected.

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SPPS OutputsDescriptive statistics

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SPSS Outputs

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The non-significant Box’s M indicates homogeneity of

covariance matrices

Significant result indicates sufficient correlation between the dependent variables.

SPSS Outputs

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SPSS Outputs: univariate test results

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SPSS Outputs: estimated marginal means

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SPSS Outputs

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SPSS Outputs

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P values

Plots

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ResultsThe mutivariate analysis of variance (MANOVA) was

conducted to assess grade differences on two sedentary behaviors: physical activity and hours of watching TV and. A non-significant Box’s M test (p = .12) indicates homogeneity of covariance matrices of the dependent variables across the levels of grade.

The multivariate effect was significant by grade levels, F(6,31322) = 28.11, p < .01, partial η2 = .01. Univariate tests showed that there were significant differences across the grade levels on physical activity, F(3,15662) = 24.80, p < .01, partial η2 = .01, and hours of watching TV, F(3,15662) = 27.00, p < .01, partial η2 = .01 .

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ResultsTamhane post hoc tests suggested 12th

graders (M = 3.96, SD = 2.53) had less days of physical activity than 9th-11th graders did. However, 9th graders (M = 4.43, SD = 2.61) exercised more than 11th graders (M = 4.24, SD = 2.57).

Tukey HSD tests showed 9th (M = 3.91, SD = 1.76) and 10th (M = 3.83, SD = 1.76) graders spent more hours of watching TV than 11th (M = 3.65, SD = 1.71)and 12th graders (M = 3.61, SD = 1.71)did.

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Two-way MANOVA designThe effects of two independent variables on

several dependent variables are examined simultaneously.

A two-way design enables us to examine the joint effect of independent variables.

Interaction effect means that the effect of one independent variable has on dependent variables is not the same for all levels of the other independent variable.

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Example: Research design: two-way between-subjects

designResearch question: whether grade levels and ever

use cigarettes jointly affect high school students’ sedentary behaviors or whether the grade differences on sedentary behaviors are moderated by ever use. Two independent variable: Grade with 4 levels: 9th,

10th, 11th, and 12th grade (Q3r); ever use cigarettes (Q28) with two levels: female and male.

Two dependent variable: sedentary behaviors: Q80 (physical activity), and Q81 (hours of watching TV).

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Analysis using SPSSSelect Analyze General Linear Model

Multivariate

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Options and Plots

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SPSS Outputs

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SPSS Outputs

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So, we don’t have homogeneity of

variance and covariance matrices

across combination of two independent

variables.

SPSS Outputs: multivariate results

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SPSS Outputs: univariate results

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SPSS Outputs: marginal means

SPSS Outputs: plots

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Post hoc testsIf we use ever use (two levels: Yes and No) as a

moderator, we want to know the relationship patterns of grade and sedentary behaviors from Yes and No groups.

Run one-way MANOVA for Yes group (select cases: Q28 = 1/Yes).

Run one-way MANOVA for No group (select cases: Q28 = 2/No)

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Plots

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Yes No

Other analyses We want to know which combinations of two

independent variables are significantly different from other combinations.

Create a new variable: Grade_Smoke Go to Transform Compute Variable Click If

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Type Q28 = 1 & Q3r = 1 (means Yes/9th grade)

Then click Ok. Now you create a new variable with only one category (Yes to smoking and 9th graders).

Next, you need to continue adding other five categories to the same variable.

Go to Transform Compute Variable

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Use If button to change conditionsType Q28 = 1 & Q3r = 2 (Yes/10th graders)

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After click Continue than OK, you get this small window, click OK.

The same procedure for adding all categories. Type Q28 = 1 & Q3r = 3Type Q28 = 1 & Q3r = 4Type Q28 = 2 & Q3r = 1Type Q28 = 2 & Q3r = 2Type Q28 = 2 & Q3r = 3Type Q28 = 2 & Q3r = 4A new variable with 8 levels.

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Use ANOVA to examine if there is a difference across 8 levels of new variable on Q80.

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Post hoc tests

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P values

ResultsSimilar to the results from one-way MANOVA.But we need to report Pillai’s trace multivariate test result because

we don’t have equal variance and covariance matrices across the groups.

The grade and ever use significantly affected sedentary behaviors.The relationship of grade and sedentary behaviors were moderated

by ever use behavior. 9th and 10th graders who had not ever use cigarettes exercised more

than other students.

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Meyers, L. S., Gamst, G., & Guarino, A. J. (2006). Applied multivariate research: design and interpretation. Thousand Oaks, CA: Sage Publications, Inc.

Stevens, J. P. (2002). Applied multivariate statistics for the social sciences. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

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