One-Way ANOVA Class 16

HANDS ON STATS PRACTICE SPSS Demo in Computer Lab (Hill Hall Rm. 124) Tuesday, Nov. 17 5:00 to 7:30 Hill Hall, Room 124 Homework : Extra Credit: 3 Pts full credit, 1 pt partial credit Homework corresponds to Computer Lab

Schedule for Remainder of Semester 1. ANOVA: One way, Two way 2. Planned contrasts 3. Correlation and Regression 4. Moderated Multiple Regression 5. Survey design 6. Non-experimental designs IF TIME PERMITS 7. Writing up research Quiz 2: Nov up to and including one-way ANOVA Quiz 3:Dec. 3 – What we’ve covered by Dec. 3 Class Assignment: Assigned Dec. 1, Due Dec. 10

ANOVA ANOVA = Analysis of Variance Next 4-5 classes focus on ANOVA and Planned Contrasts One-Way ANOVA – tests differences between 2 or more independent groups. (t-test only 2 groups) Goals for ANOVA series : 1. What is ANOVA, tasks it can do, how it works. 2. Provide intro to SPSS for Windows ANOVA 3. Objective: you will be able to run ANOVA on SPSS, and be able to interpret results. Notes on Keppel reading: 1. Clearest exposition on ANOVA 2. Assumes no math background, very intuitive 3. Language not gender neutral, more recent eds. are.

Basic Principle of ANOVA Amount Distributions Differ Amount Distributions Overlap Amount Distinct Variance Amount Shared Variance Amount Treatment Groups Differ Amount Treatment Groups the Same Same as

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

Activity Ratings of People Who Disclose Emotions As a Function of Birth Order Activity Rating

Do Means Significantly Differ? OldestYoungest OldestYoungest

Logic of Inferential Statistics: Is the null hypothesis supported? Null Hypothesis Different sub-samples are equivalent representations of same overall population. Differences between sub-samples are random. “First Born and Last Born rate disclosers equally” Alternative Hypothesis Different sub-samples do not represent the same overall population. Instead each represent distinct populations. Differences between them are systematic, not random. “First Born rate disclosers differently than do Last Born ”

Logic of F Test and Hypothesis Testing Form of F Test: Between Group Differences Within Group Differences Meaningful Differences Random Differences Purpose: Test null hypothesis: Between Group = Within Group = Random Error Interpretation: If null hypothesis is not supported 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

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

F Ratio if Null True, VS. if Alt. True Null Hyp true: F = (Treatment Effects = 0) + Error Error Null Hyp true: F = Error = Error Alt. Hyp true: F = (Treatment Effects > 0) + Error Error Alt. Hyp true: F = (Treatment Effects) + Error = Error 1 >1

ANOVA JOB: Estimate Magnitude of Variances NEED TWO MEASURES OF VARIABILTY TO ANSWER THIS QUESTION 1.Treatment effects (Between Group Var.) 2. Random diffs between subjects (Within Group Var.) Thus, ANOVA = Analysis of Variances How much do systematic (meaningful) diffs. between experimental conditions exceed random error?

Key Point : Each score contains both group effect and random error

Rating made by Sub. 1, Oldest Group

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) 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 Why are these "0" sums a problem? How do we fix this? Level a 1: Oldest Child; A 1 = 3.13 Level a 2: Youngest Child: A 2 = 5.47

AS 1 (AS 1 - A)(AS 1 -A) Average 3.13 = A Average Scores Around the Mean “Oldest Child” Group Only, as Example AS 1 = individual scores in condition 1 (Oldest: 1.33, 2.00…) A = Mean of all scores in a condition (e.g., 3.13) (AS - A) 2 = Squared deviation between individual score and condition mean

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

Hey, Can We Compute F Now? Why the F Not? F = Estimate Between Group Diffs Estimate Within Group Diffs SS A = Total Btwn Diffs = SS W = Total Within Diffs = F = = 1.11 ? Does NO!Why not? Need AVERAGE estimates of Btwn. Diffs. variability and Within Diffs. variability.

SS A = Total Btwn Diffs = SS W = Total Within Diffs = How Do We Obtain AVERAGE Variance Estimates? Can we get Ave. Between by dividing SS A by number of groups? Can we get Ave. Within by dividing SS W by number of subjects within each group? NO Why not? Why must life be so hard and complicated? Because we need est. of average of scores that can vary, not average of all scores.

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? 1 5

Sudoku – The Exciting Degrees of Freedom Game df for just this section? = 4

Degrees of Freedom Formulas for the Single Factor (One Way) ANOVA SourceTypeFormula General Meaning. 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 “Disclosers” 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 (MS) Calculations Note: What happens to MS/W as n increases?

F Ratio Computation F = = 8.78 F = MS A = Ave. Between Group Variance MS S/A Ave. Within Group Variance Thus, between groups difference is 8.78 times greater than random difference.

A (Between Groups) SS A a - 1SS A df A MS A MS S/A S/A (Within Groups) SS 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 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 Note: Totals = Between + Within

Analysis of Variance Summary Table: SPSS

F Distribution Notation " F (1, 8)" means: The F distribution with: 1 df in the numerator (1 df associated with treatment groups (= between-group variation)) and 8 df 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 and F' Distributions (from Monte Carlo Experiments)

Which Distribution Do Data Support: 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)

Critical Values for F (1, 8) What must our F be in order to reject null hypothesis? ≥ 5.32

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 or Accept null hypothesis? Reject or Accept alternative hypothesis? Have we proved alt. hypothesis? 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. No, we supported it. There's a chance (p <.05), that we are wrong.

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.

TYPE I AND TYPE II ERRORS

Reality Null Hyp. True Null Hyp. False Alt. Hyp. FalseAlt. Hyp. True Decision Reject Null Incorrect:Correct Accept Alt. Type I Error Accept Null CorrectIncorrect: Reject Null Type II Error Errors in Hypothesis Testing Type I Error Type II Error

Avoiding Type I and Type II Errors Avoiding Type I error: 1. Reduce the size of the Type I rejection region (i.e., go from p <.05 to p <.01 for example). Avoiding Type II error 1. Reduce size of Type II rejection region, BUT a. Not permitted by basic sci. community b. But, OK in some rare applied contexts 2. Increase sample size 3. Reduce random error a. Standardized instructions b. Train experimenters c. Pilot testing, etc.