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PSY 307 – Statistics for the Behavioral Sciences

PSY 307 – Statistics for the Behavioral Sciences. Chapter 16 – One-Factor Analysis of Variance (ANOVA). Fisher’s F-Test (ANOVA). Ronald Fisher. Testing Yields in Agriculture. X 1. =. =. X 2. X 1. ANOVA. Analysis of Variance (ANOVA) – a test of more than two population means.

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PSY 307 – Statistics for the Behavioral Sciences

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  1. PSY 307 – Statistics for the Behavioral Sciences Chapter 16 – One-Factor Analysis of Variance (ANOVA)

  2. Fisher’s F-Test (ANOVA) Ronald Fisher

  3. Testing Yields in Agriculture X1 = = X2 X1

  4. ANOVA • Analysis of Variance (ANOVA) – a test of more than two population means. • One-Way ANOVA – only one factor or independent variable is manipulated. • ANOVA compares two sources of variability.

  5. Two Sources of Variability • Treatment effect – the existence of at least one difference between the population means defined by IV. • Between groups variability – variability among subjects receiving different treatments (alternative hypothesis). • Within groups variability – variability among subjects who receive the same treatment (null hypothesis).

  6. F-Test • If the null hypothesis is true, the numerator and denominator of the F-ratio will be the same. • F = random error / random error • If the null hypothesis is false, the numerator will be greater than the denominator and F > 1. • F = random error + treatment effect random error

  7. Difference vs Error • Difference on the top and the error on the bottom: • Difference is the variability between the groups, expressed as the sum of the squares for the groups. • Error is the variability within all of the subjects treated as one large group. • When the difference exceeds the variability, the F-ratio will be large.

  8. F-Ratio • F = MSbetween MSwithin • MS = SS df • SS is the sum of the squared differences from the mean.

  9. F-Ratio • F = MSbetween MSwithin • MSbetween treats the values of the group means as a data set and calculates the sum of squares for it. • MSwithin combines the groups into one large group and calculates the sum of squares for the whole group.

  10. Testing Hypotheses • If there is a true difference between the groups, the numerator will be larger than the denominator. • F will be greater than 1 • Writing hypotheses: H0: m1 = m2 = m3 H1: H0 is false

  11. Formulas for F • Description in words of what is being computed. • Definitional formula – uses the SS, described in the Witte text • Computational formula – used by Aleks and in examples in class.

  12. Formula for SStotal • SStotal is the total Sum of the Squares • It is the sum of the squared deviations of scores around the grand mean. • SStotal = ∑(X – Xgrand)2 • SStotal = ∑(X2 – G2/N) • Where G is the grand total and N is its sample size

  13. Hours of Sleep Deprivation

  14. Formula for SSbetween • SSbetween is the between Sum of the Squares • It is the sum of the squared deviations for group means around the grand mean. • SSbetween = n∑(X – Xgrand)2 • SSbetween = ∑(T2/n – G2/N) • Where T is each group’s total and n is each group’s sample size definition computation

  15. Formula for SSwithin • SSwithin is the within Sum of the Squares • It is the sum of the squared deviations for scores around the group mean. • SSwithin = ∑(X – Xgroup)2 • SSwithin = ∑X2 – ∑T2/n) • Where T is each group’s total and n is each group’s sample size definition computation

  16. Degrees of Freedom • dftotal = N-1 • The number of all scores minus 1 • dfbetween = k-1 • The number of groups (k) minus 1 • dfwithin = N-k • The number of all scores minus the number of groups (k)

  17. Checking Your Work • The SStotal = SSbetween + SSwithin. • The same is true for the degrees of freedom: dftotal = dfbetween + dfwithin

  18. Calculating F (Computational) • SSbetween = ST2 – G2 n N • Where T is the total for each group and G is the grand total • SSwithin = S X2 - ST2 N • SStotal = S X2 – G2/N

  19. F-Distribution Common – retain null Rare – reject null Critical value Look up F critical value in the F table using df for numerator and denominator

  20. ANOVA Assumptions • Assumptions for the F-test are the same as for the t-test • Underlying populations are assumed to be normal with equal variances. • Results are still valid with violations of normality if: • All sample sizes are close to equal • Samples are > 10 per group • Otherwise use a different test

  21. Cautions • The ANOVA presented in the text assumes independent samples. • With matched samples or repeated measures use a different form of ANOVA. • The sample sizes shown in the text are small in order to simplify calculations. • Small samples should not be used.

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