Lab Chapter 14: Analysis of Variance

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Lab Topics:

• One-way ANOVA– the F ratio– post hoc multiple comparisons

• Two-way ANOVA– main effects– interaction effects

One-Way ANOVA

• To test hypotheses about the mean on one variable for three or more groups

• Sample hypotheses:– “There are differences in the average income of

sociology, social work, and criminology majors.”– “There are differences in the recidivism rate of

persons convicted of burglary, larceny, forgery, and robbery.”

One-Way ANOVA (cont.)

• The hypotheses:– research hypothesis: at least one group has a

different mean– null hypothesis: all groups have the same mean

• inferential statistic: F ratio– non-directional hypotheses– degrees of freedom:one-way ANOVA: df = (K – 1 and N - K)

One-Way ANOVA (cont.)• Post hoc multiple comparison tests• Tukey, Tukey’s b, and Bonferonni most

common tests– One-way ANOVA can only tell you if F ratio is

significant but not which groups are significantly different form one another

– Post hoc tests can identify pairs of groups that significantly differ

– If F ratio for model not significant, post hoc test not needed

One-Way ANOVA Example• Were there significant differences by region in

the average willingness to allow legal abortion among 1980 GSS young adults?

• DV: Willingness to allow abortion (I-R level)• IV: Region (nominal level – 4 groups)1. State the research and the null hypothesis.• research hypothesis: There were regional differences in

average willingness to allow legal abortion.• null hypothesis: There were no regional differences in

average willingness to allow legal abortion.

One-Way ANOVA Example (cont.)2. Are the sample results consistent with the null

hypothesis or the research hypothesis?Analyze | Compare Means | One-Way ANOVA

(Use Post-Hoc to request Multiple Comparison Test)

Requesting Sample Means and Post Hoc

Multiple Comparisons

Sample Means for Each Region

3. What is the probability of getting the sample results if the null hypothesis is true?

4. Reject or do not reject null hypothesis. p = .000 < α = .05, Reject null hypothesis, there is a significant

difference. Which groups are different?

5. See post-hoc multiple test (next slide)

One-Way ANOVA Example (cont.)

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duplicate

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Two-Way ANOVA

• Tests hypotheses about the mean on one dependent variable for groups created by two or more independent variables (or factors)

• Nominal or ordinal level variables entered as “fixed factors”

• Tests for significant…– interaction effects– main effects

More on Two-way ANOVA• “Fixed Factors” are at the nominal or ordinal

level of measurement and have a limited number of discrete categories

• Note: Two-way ANOVA can also employ IV’s at the I-R level as covariates– In this case the process is known as ANCOVA

(Analysis of Covariance) and the I-R variable is entered into the dialogue box as a covariate.

– However, this is a much more complex analysis and we will be using multiple regression for models that have both nominal/ordinal and I-R level variables

Two-Way ANOVA Example• questions:– Was there a significant interaction effect of marital

status and gender on hours worked among 1980 GSS young adults?

– If the interaction wasn’t significant, did marital status and gender have significant individual net effects?

• Analyze | General Linear Model | Univariate• (Use Plots and Post-Hoc to request Subgroup

Means and a Plot of Subgroup Means)

Producing a Two-Way ANOVA

Requesting Subgroup Means

and a Plot of Subgroup Means

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Significance Tests 1. overall model: significant

2. interaction effect: significant3. main effect: not needed

Answering Questions with Statistics Chapter 14

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Subgroup Means and

Plot

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More on Two-Way ANOVA• If interaction is significant, then interpret it

along with means and plot.– This indicates that the IV’s are not acting

separately from one another in their effect on the DV. Main effect becomes irrelevant.

• If interaction is not significant, interpret main effects. – This indicates that IV effects on DV are

independent of one another and that there is no significant interaction of the two IV’s in the population.

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Example: Effects of Married and Sex on Number of Children (DV)

1. overall model: significant2. interaction effect: not significant3. main effects– MARRIED: significant– SEX: significant

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