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Detailed Guide to ANOVA and Planned Comparisons

Learn about planned comparisons in ANOVA, effect size interpretation, ANOVA in research articles, and conduct ANOVA analysis using SPSS with step-by-step instructions. Understand follow-up analyses including Bonferroni correction and interpreting overall ANOVA test results.

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Detailed Guide to ANOVA and Planned Comparisons

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  1. Chapter 9 Introduction to the Analysis of Variance Part 2: Oct. 24, 2013

  2. Planned Comparisons • When we reject null hypothesis in ANOVA, we conclude group means are not all the same • But exactly which groups differ? Not known from overall F test • Planned comparisons • Allow us to determine which groups differ • Often have a priori (theoretical) reason to predict that certain pairs of groups may differ, can test this

  3. Procedure for planned comparisons: • Estimate within-groups population variance – in same way as for overall F test • Next, estimate between-groups population variance • For overall F test, we used all groups means for this estimate, here… • Use only the two means of interest (see formula next) • Figure F in usual way – same as overall F test

  4. Bonferroni correction • Potential problem when making multiple planned comparisons: • Want overall alpha = .05, but if doing 3 separate comparisons each at alpha = .05, then overall alpha = 3(.05) = .15 (which means a 15% Type 1 error rate) • Not acceptable to increase alpha, need some control • Use more stringent significance level for each separate comparison, not .05 • Divide overall alpha level by # comparisons you plan to make (.05 / 3 = .017) •  use .017 as alpha for each separate comparison and you keep overall alpha at .05

  5. ANOVA Effect size • In ANOVA, total effect size for the mean differences among all the groups is R2 Unlike t-test effect sizes, R2 cannot be negative

  6. Effect size (cont.) • R2also known as η2(eta squared) • Ranges from 0-1 (make sure it’s pos.) • Cohen’s standards for size of effect for R2 or η2 in ANOVA: • small R2 = .01 • medium R2 = .06 • large R2 = or > .14

  7. ANOVA in Research Articles • F(3, 67) = 5.81, p < .01 • F (df between groups, df within groups) = F obtained, significance level (here, signif using alpha of .01) • Means given in a table or in the text • Follow-up analyses • Planned comparisons – report those F tests • See SPSS example for APA format…

  8. SPSS Example for 1-way ANOVA • Harassment data set with school district employees • “School” variable indicates work setting • 1=elementary school • 2=middle school • 3=high school • “Harassment in 1997” indicates har experiences from ’96-’97 • Does the work setting influence harassment experiences?

  9. In SPSS menus: Analyze  Compare Means  One-way ANOVA • Then, “Dependent List” can indicate as many dependent variables as you’d like…here “Harassment in 1997” • In “Factor” indicate the ‘grouping’ variable on which you’ll compare Harassment means… here “School”

  10. Click the “Options” button at bottom, click the box for Descriptives under “Statistics”, hit continue… • Click the “Post Hoc” button at bottom, click the box for “Bonferroni”, hit continue… • (this will give you output for follow-up comparisons in case your overall ANOVA is signif  if it’s not, you’ll ignore these comparisons) • Now hit “OK” to run the analysis

  11. Output • You’ll have 3 sections of output… • The 1st reports the group harassment means for elementary, middle, and high school employees • You’ll need to look back at this to help w/interpretation in case your ANOVA is signif! • 2nd gives the overall ANOVA F test – for the null hypothesis of “no group differences” • Notice the MSBetween and MS Within, then the F statistic is your F obtained value, next to that is the “Sig” value • If “Sig” value is < .05 (or .01 – depends on alpha)  Reject Null and conclude there are significant group differences

  12. Overall ANOVA test results • Is there a significant difference in the group means? • APA-formatted summary: • Bonferroni post-hoc tests?

  13. 3rd section gives follow-up comparisons (Bonferroni – but remember to use .05/3 = .017 as your new comparison alpha level) • Check each row for which pairs are being compared, then its “sig” value • If “sig” < .017 (or whatever your new alpha is)  Reject null of equal group means; conclude those 2 group means differ • Which schools significantly differ in harassment experiences? • APA-formatted summary:

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