One-Way ANOVA

Class 16

HANDS ON STATS PRACTICE

SPSS Demo in Computer Lab (Hill Hall Rm. 124)

Tuesday, Nov. 17 5:00 to 7:30Hill Hall, Room 124

Homework:

Extra Credit: 3 Pts full credit, 1 pt partial creditHomework corresponds to Computer Lab

Schedule for Remainder of Semester

1. ANOVA: One way, Two way2. Planned contrasts 3. Correlation and Regression4. Moderated Multiple Regression5. Survey design 6. Non-experimental designs IF TIME PERMITS7. Writing up research

Quiz 2: Nov. 12 -- 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

ANOVAANOVA = 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

Same as

How Do You Regard Those Who Disclose?

EVALUATIVE DIMENSION

Good Bad

Beautiful; Ugly

Sweet Sour

POTENCY DIMENSION

Strong Weak

Large Small

Heavy Light

ACTIVITY DIMENSION

Active Passive

Fast Slow

Hot Cold

Birth Order Means

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

0123456

Youngest Oldest

3.13

3.13

5.47

5.47

4.30

4.30

Do Means Significantly Differ?

Oldest Youngest

Oldest Youngest

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 overallpopulation. 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, 1890-1962

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 (a1) = 3.13 Youngest (a2) = 5.47 Total (T) = 4.30

Why are these "0" sums a problem?

How do we fix this?

Level a1: Oldest Child; A1 = 3.13

Level a2: Youngest Child: A2 = 5.47

  AS1 (AS1 - A) (AS1 -A)2

  1.33 -1.80 3.24  2.00 -1.13 1.28  3.33 0.20 0.04  4.33 1.20 1.44  4.67 1.54 2.37

Average 3.13 = A 0.00 1.67

Average Scores Around the Mean“Oldest Child” Group Only, as Example

AS1 = 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

SSTotal = SSBetween + SSWithin

Computing Sums of Squares from Deviation Scores Birth Order and Activity Ratings (continued)

SS = Sum of squared diffs., AKA “sum of squares”

SST = Sum of squares., total (all subjects)

SSA = Sum of squares, between groups (treatment)

SSs/A = Sum of squares, within groups (error)

SST = (2.97)2 + (2.30)2 + … + (1.37)2 + (2.70)2 = 25.88

SSA = (-1.17)2 + (-1.17)2 + … + (1.17)2 + (1.17)2 = 13.61

SSs/A = (-1.80)2 + (-1.13)2 + … + (0.20)2 + (1.53)2 = 12.27Total (SSA + SSs/A) = 25.88

Hey, Can We Compute F Now? Why the F Not?

F =Estimate Between Group Diffs

Estimate Within Group Diffs

SSA = Total Btwn Diffs = 13.61

SSW = Total Within Diffs = 12.27

F = 13.6112.27 = 1.11 ?Does

NO! Why not?Need AVERAGE estimates of Btwn. Diffs.

variability and Within Diffs. variability.

SSA = Total Btwn Diffs = 13.61

SSW = Total Within Diffs = 12.27

How Do We Obtain AVERAGE Variance Estimates?

Can we get Ave. Between by dividing SSA by number of groups?

Can we get Ave. Within by dividing SSW by number of subjects within each group?

NO

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.

5 + 6 + 4 + 5 + 4 = 24 Number of observations = n = 5Number of estimates = 1 (i.e. sum, which = 24)df = (n - # estimates) = (5 -1) = 4

5 + 6 + 4 + 5 + 4 = 24 5 + 6 + X + 5 + 4 = 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

4 5 2 8

8 5 4 7

1 9

3 4 5 6 8

2 7 9 1 5

3 1

9 6 3 2

7 2 8 6

Sudoku – The Exciting Degrees of Freedom Game

df for just this section?

9 - 4 - 1 = 4

Degrees of Freedom Formulasfor the Single Factor (One Way) ANOVA

Source Type Formula General Meaning .

Groups dfA a – 1 df for Tx groups; Between-groups df

Scores dfs/A a(s –1) df for individual scores Within-groups df

Total dfT as – 1 Total df (note: dfT = dfA + dfs/A)

Source Type Formula “Disclosers” Study

Groups dfA a – 1 2 –1 = 1

Scores dfs/A a(s –1) 2 (5 –1 ) = 8

Total dfT as – 1 (2 * 5) - 1 = 9 (note: dfT = dfA + dfs/A)

Note: a = # levels in factor A; s = # subjects per condition

Variance 

Code Calculation Meaning

Mean Square Between Groups

MSA SSA

dfA

Between groups variance

Mean Square Within Groups

MSS/A SSS/A

dfS/A

Within groups variance

Variance 

Code Calculation Data Result

Mean Square Between Groups

MSASSA

dfA

13.61 1

13.61

Mean Square Within Groups

MSS/ASSS/A

dfS/A

12.278

1.53

Mean Squares (MS) Calculations

Note: What happens to MS/W as n increases?

F Ratio Computation

 

F = 13.611.51

= 8.78

     

F = MSA = Ave. Between Group Variance

MSS/A Ave. Within Group Variance

Thus, between groups difference is 8.78 times greater than random difference.

         

A (Between Groups)

SSA a - 1 SSA

dfA

MSA

MSS/A

         

S/A (Within Groups)

SSS/A a (s- 1) SSS/A

dfS/A

 

         

       

Total SST as - 1    

Source of Variation Sum of Squares

(SS)

df Mean Square (MS)

F Ratio

Analysis of Variance Summary Table:

One Factor (One Way) ANOVA

           

Between Groups 13.61 1 13.61 8.877 .018

           

Within Groups 12.27 8 1.533    

           

Total 25.88 9      

Source of Variation

Sum of Squares

df Mean Square (MS)

F Significance 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: u1 = u2

(First born = Last born)

If F' is correct, then H1 supported: u1 u2

(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. False Alt. Hyp. TrueDecision

Reject Null Incorrect: Correct

Accept Alt. Type I Error

Accept Null Correct Incorrect:

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. communityb. But, OK in some rare applied contexts

2. Increase sample size3. Reduce random error

a. Standardized instructionsb. Train experimentersc. Pilot testing , etc.