PASW & Hand Calculations for ANOVA
Gravetter & Wallnau Chapter 13, Problem 6 One possible reason that some birds migrate and others don t is intelligence. Birds with small brains relative to their body size might not be smart enough to find food during the winter. Birds with larger brains might be creative enough to find food even when the weather turns harsh. The next slide has the relative brain size for individual birds in each sample. Are their significant difference among the groups? Compute η. If appropriate, calculate Tukey s HSD to determine which groups are sig. different.
Gravetter & Wallnau Chapter 13, Problem 6 Non-Migrating 18 13 19 1 16 1 Short-Distance Migrants 6 11 7 9 8 13 Long-Distance Migrants 4 9 5 6 5 7
Step 1: H 0, H 1 & α H 0 : µ Non-Migrating µ Short-Distance Migrants µ Long-Distance Migrants H 1 : not H 0 α.05
Step : Critical Value Skip use the p value from the ANOVA output F(, 15) 3.68 From table of critical F values df between # levels of factor 1 3 1 dfwithin Σ(# of people in a condition 1) (6 1) + (6 1) + (6 1) 15
Step 3: Calculate F observed Load PASW Create two variables One indicates the level of the factor Name: migrate, Label: Migration Status, Values: 1 Non-Migrating, Short-Distance Migrant, 3 Long-Distance Migrant, Measure: Nominal One is the DV Name: brain, Label: Relative Brain Size, Measure: Scale Type in the data
Step 3: Calculate F observed Click on Analyze General Linear Model Univariate Move the DV (brain size) into the Dependent Variable box Move the factor (migrate) into the Fixed Factor(s) box
Step 3: Calculate F observed Click on Post Hoc Move the factor (migrate) into the Post Hoc Tests for box Select Tukey as the desired test Click Continue Click Options Select Descriptive statistics, Estimates of effect size Click Continue Click OK
Step 3: Calculate F observed F(, 15) 19.86, p.000, MS error 6.533, η.70
Step 3: Calculate F observed Non-Migrating Short-Distance Migrants Long-Distance Migrants 18 13 19 1 16 1 6 11 7 9 8 13 4 9 5 6 5 7 N 18 (total number of scores) G 180 (sum of all scores 90 + 54 + 36) ΣX 150 (square each score and sum the squares) T 90 18+13+19+1+16+1 T 54 6 + 11 + 7 + 9 + 8 + 13 T 36 4 + 9 + 5 + 6 + 5 + 7 M 15 90 / 6 M 9 54 / 6 M 6 36 / 6 SS 48 (18-15) + (13-15) + (19-15) + (1-15) + (16-15) + (1-15) SS 34 (6-9) + (11-9) + (7-9) + (9-9) + (8-9) + (13-9) SS 16 (4-6) + (9-6) + (5-6) + (6-6) + (5-6) + (7-6)
Step 3: Calculate F observed 350 18 180 150 N G X SS total 98 16 34 48 SS SS ment inside each treat within + + 5 1800 16 486 1350 18 180 6 36 6 54 6 90 N G n T SS between + + + + 17 1 18 1 N df total 15 3 18 k N df within 1 3 1 k df between 16 5 df SS MS between between betweeen 6.53 15 98 df SS MS within within within
Step 3: Calculate F observed MS F MS between within 16 6.533 19.86
Step 4: Decide If p α then reject H 0 otherwise fail to reject H 0 p.000, α.05 therefore, reject H 0 It is unlikely that all the means in the population are equal If F observed F critical then reject H 0 otherwise fail to reject H 0 F observed 19.86, F critical 3.68 therefore, reject H 0
η η SS between / SS total 5 / 350 0.70
Tukey HSD H 0 : µ Non-Migrating µ Short-Distance Migrant H 1 : µ Non-Migrating µ Short-Distance Migrant H 0 : µ Non-Migrating µ Long-Distance Migrant H 1 : µ Non-Migrating µ Long-Distance Migrant H 0 : µ Long-Distance Migrant µ Short-Distance Migrant H 1 : µ Long-Distance Migrant µ Short-Distance Migrant α.05
Tukey HSD Pairs that are reliably different (non-migrating vs. short-distance migrant and non-migrating vs. long-distance migrant) p α Pair that is not reliably different (short-distance migrant vs. longdistance migrant) p > α
Tukey HSD From the table of the Studentized Range Statistic with α.05, k 3 (three levels) and df within 15 HSD q 3.675 3.835 6.533 6 MS withintreatment α,k,df withintreatment n Any pair of means that is at least 3.835 apart will be reliably different. Means are likely different Insufficient evidence to state that means are likely different
Writing The means and standard deviations are presented in Table 1. The analysis of variance revealed a significant effect of migration style on relative brain size, F(, 15) 19.86, p.000, MS error 6.533, η.70. Tukey s post hoc tests revealed that nonmigrating birds had a different relative brain size than short-distance migrants (p.003) and longdistance migrants (p.000). There was insufficient evidence to suggest that the long- and short-distance migrants had different brain sizes (p.138).
Writing Table 1 Relative Brain Size of Three Types of Birds Type of Bird M SD Non-migrating 15.00 3.10 Short-distance migrant 9.00.61 Long-distance migrant 6.00 1.79