ANOVA I (Part 2) Class 14

How Do You Regard Those Who Disclose? EVALUATIVE DIMENSION GoodBad Beautiful;Ugly SweetSour POTENCY DIMENSION StrongWeak LargeSmall HeavyLight ACTIVITY DIMENSION ActivePassive FastSlow HotCold

Birth Order Means

Logic of F Test and Hypothesis Testing Form of F Test: Between Group Differences Within Group Differences Purpose: Test null hypothesis: Between Group = Within Group = Random Error Interpretation: If null hypothesis is not supported ( F > 1) then Between Group diffs are not simply random error, but instead reflect effect of the independent variable. Result: Null hypothesis is rejected, alt. hypothesis is supported (BUT NOT PROVED!)

F Ratio F = Between Group Difference Within Group Differences F = Error + Treatment Effects Error

Birth Order and Ratings of “Activity” Deviation Scores AS Total Between Within (AS – T) = (A – T) +(AS – A) 1.33 (-2.97)= (-1.17) +(-1.80) 2.00(-2.30)=(-1.17) +(-1.13) 3.33(-0.97)=(-1.17) + ( 0.20) 4.33(0.03)=(-1.17) +( 1.20) 4.67(0.37)=(-1.17) + ( 1.54) Level a 1: Oldest Child Level a 2: Youngest Child 4.33 (0.03)= (1.17) +(-1.14) 5.00(0.07)= (1.17) +(-0.47) 5.33(1.03)= (1.17) + (-0.14) 5.67(1.37)= (1.17) +( 0.20) 7.00(2.70)= (1.17) + ( 1.53) Sum: (0) = (0) + (0) Mean scores: Oldest (a 1 ) = 3.13 Youngest ( a 2 ) = 5.47 Total (T) = 4.30

Sum of Squared Deviations Total Sum of Squares = Sum of Squared between-group deviations + Sum of Squared within-group deviations SS Total = SS Between + SS Within

Computing Sums of Squares from Deviation Scores Birth Order and Activity Ratings (continued) SS = Sum of squared diffs, AKA “sum of squares” SS T =Sum of squares., total (all subjects) SS A = Sum of squares, between groups (treatment) SS s/A =Sum of squares, within groups (error) SS T = (2.97) 2 + (2.30) 2 + … + (1.37) 2 + (2.70) 2 = SS A = (-1.17) 2 + (-1.17) 2 + … + (1.17) 2 + (1.17) 2 = SS s/A = (-1.80) 2 + (-1.13) 2 + … + (0.20) 2 + (1.53) 2 = Total (SS A + SS s/A ) = 25.88

1.33(-2.97)=(-1.17)+(-1.80) 2.00(-2.30)=(-1.17)+(-1.13) 3.33(-0.97)=(-1.17)+(0.20) 4.33(0.03)=(-1.17)+(1.20) 4.67(0.37)=(-1.17)+(1.54) 4.33(0.03)=(1.17)+(-1.14) 5.00(0.70)=(1.17)+(-0.47) 5.33(1.03)=(1.17)+(-0.14) 5.67(1.37)=(1.17)+(0.20) 7.00(2.70)=(1.17)+(1.53) ASTotal__ (AS - T) = Between (A - T) + Within (AS - A) Level a1 : Oldest Level a2 : Youngest Birth Order and Activity Ratings: Deviation Scores Sum:(0)= + Mean Scores: Oldest = 3.13 Youngest = 5.47 Total = 4.30 SS T = (2.97) 2 + (2.30) (1.37) 2 + (2.70) 2 = SS A = (-1.17) 2 + (-1.17) (1.17) 2 + (1.17) 2 = SS s/A =(-1.80) 2 + (-1.13) (0.20) 2 + (1.53) 2 = Total= 25.88

df=Number of independent Observations -Number of restraints df=Number of independent Observations -Number of population estimates Degrees of Freedom df = Number of observations free to ??? = 24 Number of observations = n = 5 Number of estimates = 1 (i.e. sum, which = 24) df = n - # estimates = 5 -X = Z = 24

df=Number of independent Observations -Number of restraints df=Number of independent Observations -Number of population estimates Degrees of Freedom df = Number of observations free to vary = 24 Number of observations = n = 5 Number of estimates = 1 (i.e. sum, which = 24) df = n - # estimates = 5 -1 = = X = 24 = 20 + X = 24 = X = 4

Degrees of Freedom for Fun and Fortune Coin flip = __ df? Dice = __ df? Japanese game that rivals cross-word puzzle?

Sudoku – The Exciting Degrees of Freedom Game

Degrees of Freedom Formulas for the Single Factor (One Way) ANOVA SourceTypeFormulaMeaning. Groupsdf A a – X df for Tx groups; Between-groups df Scoresdf s/A X (s –1)df for individual scores Within-groups df Totaldf T XY – 1Total df (note: df T = df A + df s/A ) Note: a = # levels in factor A; s = # subjects per condition

Degrees of Freedom Formulas for the Single Factor (One Way) ANOVA SourceTypeFormulaMeaning. Groupsdf A a – 1df for Tx groups; Between-groups df Scoresdf s/A a(s –1)df for individual scores Within-groups df Totaldf T as – 1Total df (note: df T = df A + df s/A ) SourceTypeFormula Semantic Differential Study Groupsdf A a – 1 2 –1 = 1 Scoresdf s/A a(s –1) 2 (5 –1 ) = 8 Totaldf T as – 1 (2 * 5) - 1 = 9 (note: df T = df A + df s/A ) Note: a = # levels in factor A; s = # subjects per condition

Variance CodeCalculationMeaning Mean Square Between Groups MS A SS A df A Between groups variance Mean Square Within Groups MS S/A SS S/A df S/A Within groups variance Variance CodeCalculationDataResult Mean Square Between Groups MS A SS A df A Mean Square Within Groups MS S/A SS S/A df S/A Mean Squares Calculations Note: What happens to MS/W as n increases?

F Ratio Computation F = = 8.78 F = MS A = XXX Variance MS S/A YYYY Variance

F Ratio Computation F = = 8.78 F = MS A = Between Group Variance MS S/A Within Group Variance

ASS A a - 1SS A df A MS A MS S/A S/ASS S/A a (s- 1)SS S/A df S/A TotalSS T as - 1 Source of VariationSum of Squares (SS) dfMean Square (MS) F Ratio Analysis of Variance Summary Table: One Factor (One Way) ANOVA

Between Groups13.61 ? ????.018 Within Groups ???? Total Source of Variation Sum of Squares dfMean Square (MS) FSignificance of F Analysis of Variance Summary Table: One Factor (One Way) ANOVA

Between Groups Within Groups Total Source of Variation Sum of Squares dfMean Square (MS) FSignificance of F Analysis of Variance Summary Table: One Factor (One Way) ANOVA

F Distribution Notation " F (1, 8)" means: The F distribution with: one df in the numerator (1 df associated with treatment groups (= between-group variation)) and 8 degrees of freedom in the denominator (8 df associated with the overall sample (= within-group variation))

F Distribution for (2, 42) df

Criterion F and p Value For F (2, 42) = 3.48

F or F′? If F is correct, then Ho supported: (First born ??? Last born) If F' is correct, then H 1 supported : (First born ??? Last born)

F or F′? If F is correct, then Ho supported: u 1 = u 2 (First born = Last born) If F' is correct, then H 1 supported : u 1  u 2 (First born ≠ Last born)

F’ Distribution

F Distribution Notation " F (1, 8)" means: The F distribution with: ??? df in the numerator (1 df associated with treatment groups/between-group variation) and 8 degrees of freedom in the denominator (8 df associated with the ?????? )

F Distribution Notation " F (1, 8)" means: The F distribution with: one df in the numerator (1 df associated with treatment groups/between-group variation) and 8 degrees of freedom in the denominator (8 df associated with the overall sample/within-group variation)

Decision Rule Regarding F Reject null hypothesis when F observed >  (m,n) Reject null hypothesis when F observed > 5.32 (1, 8). F (1,8) = 8.88 >  = 5.32 Decision: Reject null hypothesis Accept alternative hypothesis Note: We haven't proved alt. hypothesis, only supported it. Format for reporting our result: F (1,8) = 8.88, p <.05 F (1,8) = 8.88, p <.02 also OK, based on our results. Conclusion: First Borns regard help-seekers as less "active" than do Last Borns.

Summary of One Way ANOVA 1. Specify null and alt. hypotheses 2. Conduct experiment 3.Calculate F ratio Between Group Diffs Within Group Diffs 4. Does F support the null hypothesis? i.e., is Observed F > Criterion F, at p <.05? ___ p >.05, accept null hyp. ___ p <.05, accept alt. hyp.