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ANOVA
Analysis of Variance:
•Why do these Sample Means differ as much as they do (Variance)?
•Standard Error of the Mean (“variance” of means) depends upon
Population Variance (/n)
•Why do subjects differ as much as they do from one another?
Many Random causes (“Error Variance”)
or
Many Random causes plus a Specific Cause (“Treatment”)
Making Sample Means More Different than SEM
Why Not the t-Test
If 15 samples are ALL drawn from the Same Populations:
•105 possible comparisons
•Expect 5 Alpha errors (if using p<0.05 criterion)
•If you make your criterion 105 X more conservative
(p<0.0005) you will lose Power
The F-Test
ANOVA tests the Null hypothesis that ALL Samples came from
The Same Population
•Maintains Experiment Wide Alpha at p<0.05
Without losing Power
•A significant F-test indicates that At Least One Sample
Came from a different population
(At least one X-Bar is estimating a Different Mu)
The Structure of the F-Ratio
F =
Estimation (of SEM)
The Differences (among the sample means) you got
---------------------------------------------------------------- Evaluation
The Differences you could expect to find (If H0 True)
Expectation
(If this doesn’t sound familiar, Bite Me!)
The Structure of the F-Ratio
If H0 True:
F =
Average Error of Estimation of Mu by the X-Bars
---------------------------------------------------------------Variability of Subjects within each Sample
Size of Denominator determines size of Numerator
If a treatment effect (H0 False):
Numerator will be larger than predicted by
denominator
The Structure of the F-Ratio
F =
Between Group Variance
------------------------------Within Group Variance
If H0 True:
F =
Error Variance
-----------------Error Variance
Approximately Equal
With random variation
If a treatment effect (H0 False):
Error plus Treatment Variance
F = ------------------------------------Error Variance
Numerator
is
Larger
Probability of F  as F Exceeds 1
F =
Between Group Variance
------------------------------Within Group Variance
If H0 True:
F =
Error Variance
-----------------Error Variance
Approximately Equal
With random variation
If a treatment effect (H0 False):
Error plus Treatment Variance
F = ------------------------------------Error Variance
Numerator
is
Larger
For U Visual Learners
H0 True:
Reflects SEM (Error)
Sampling
Distributions
H0 False:
Error Plus Treatment
Keep the Data, Burn the
Formulas
Do These Measures Depend on
What Drug You Took?
Drug A & B don’t look different, but Drug C looks different
From Drug A & B
Partitioning the Variance
Each Subject’s deviation score can be decomposed into 2 parts:
•How much his Group Mean differs from the Grand Mean
•How he differs from his Group Mean
If Grand Mean = 100:
Score-1 in Group A =117; Group A mean =115
(117 - 100) = (115 - 100) + (117 - 115)
17
=
15
+
2
Score-2 in Group A = 113; Group A mean = 115
(113 – 100) = (115 - 100 + (113 – 115)
13
=
15
2
Partitioning the Variance in the
Data Set
Total Variance (Total Sum of Squared Deviations from Grand Mean)
Sum (Xi-Grand Mean)^2
Variance among Subjects
Within each group (sample)
Sum ( Xi – Group mean)^2 for
All subjects in all Groups
SS-Within
Variance among Samples
Sum (X-Bar – Grand Mean)^2
For all Sample Means
SS-Total
SS-Between
Step 1: Calculate SS-Total
Xi
9
8
7
5
Xi-GM
dev-score
3.583333333
2.583333333
1.583333333
-0.416666667
sq-dev
12.84028
6.673611
2.506944
0.173611
Drug B
9
7
6
5
3.583333333
1.583333333
0.583333333
-0.416666667
12.84028
2.506944
0.340278
0.173611
Drug C
4
3
1
1
-1.416666667
-2.416666667
-4.416666667
-4.416666667
2.006944
5.840278
19.50694
19.50694
Drug A
Grand mean= 5.416667
SStot= 84.91667
Step 2: Calculate SS-Between
XBarA - GM
XBarB - GM
XBarC - GM
7.25
6.75
2.25
-5.416667
-5.416667
-5.416667
dev
sq-dev
1.833333 3.36111
1.333333 1.777777
-3.16667 10.02778
n
4
4
4
SS-Bet=
sq-dev * n
13.44444
7.111108
40.11112
60.66667
Multiply by n (sample size) because:
Each subject’s raw score is composed of:
•A deviation of his sample mean from the grand mean
•(and a deviation of his raw score from his sample mean)
Step 3: Calculate SS-Within
SS-Total – SSb = SSw
84.91667 – 60.6667 = 24.25
Should Agree with Direct Calculation
Direct Calculation of SSw
Xi
Drug A
X-Bar-A=
Drug B
X-Bar-B
Drug C
X-Bar-C
9
8
7
5
7.25
9
7
6
5
6.75
4
3
1
1
2.25
Xi-XBarA
dev-score
sq-dev
1.75
3.0625
0.75
0.5625
-0.25
0.0625
-2.25
5.0625
Xi-XBarB
dev-score
sq-dev
2.25
5.0625
0.25
0.0625
-0.75
0.5625
-1.75
3.0625
xi-XBarC
dev-score
sq-dev
1.75
3.0625
0.75
0.5625
-1.25
1.5625
-1.25
1.5625
SS-Within=
24.25
Step 4: Use SS to Compute
Mean Squares & F-ratio
df-Tot=N-1
11.00
df-B=k-1
2
df-W=dfTot-dfB
9
MSb = SSb/df 60.66667
MSw = SSw/df
24.25
2
9
F=MSb/MSw
30.33334 11.25773
2.694444
The differences among the sample means are over 11 x greater than if:
•All three samples came from the Same population
•None of the drugs had a different effect
Look up the Probability of F with 2 & 9 dfs
•Critical F2,9 for p<0.01 = 8.02
•Reject H0
•Not ALL of the drugs have the same effect
The F-Table
The ANOVA Summary Table
What Do You Do Now?
A Significant F-ratio means at least one Sample came from a
Different Population.
What Samples are different from what other Samples?
Use Tukey’s Honestly Significant Difference (HSD) Test
Tukey’s HSD Test
Can only be used if overall ANOVA is Significant
A “Post Hoc” Test
Used to make “Pair-Wise” comparisons
Structure:
Analogous to t-test
But uses estimated Standard Error of the Mean in the Denominator
Hence a different critical value (HSD) table
Tukey’s HSD Test
Equal N
Unequal N
Assumptions of ANOVA
1. All Populations Normally distributed
2. Homogeneity of Variance
3. Random Assignment
ANOVA is robust to all but gross violations of these theoretical
assumptions
Effect Size
S = 0.10
M = 0.25
L = 0.40
MStreatment is really MSb
Which is T + E
What’s the Question?