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First There Was the t-Test. the Psych Dept was Safe. Just When You Thought. Then Came ANOVA!. 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)

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First There Was the t-Test

the Psych Dept was Safe

Just When You Thought

Then Came ANOVA!


Anova
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
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
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
The Structure of the F-Ratio

Estimation (of SEM)

The Differences (among the sample means) you got

----------------------------------------------------------------

The Differences you could expect to find (If H0 True)

Expectation

F =

Evaluation

(If this doesn’t sound familiar, Bite Me!)


The structure of the f ratio1
The Structure of the F-Ratio

If H0 True:

Average Error of Estimation of Mu by the X-Bars

----------------------------------------------------------------

Variability of Subjects within each Sample

F =

  • 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 ratio2
The Structure of the F-Ratio

Between Group Variance

-------------------------------

Within Group Variance

F =

If H0 True:

Error Variance

------------------

Error Variance

Approximately Equal

With random variation

F =

If a treatment effect (H0 False):

Error plus Treatment Variance

-------------------------------------

Error Variance

Numerator

is

Larger

F =


Probability of f as f exceeds 1
Probability of F  as F Exceeds 1

Between Group Variance

-------------------------------

Within Group Variance

F =

If H0 True:

Error Variance

------------------

Error Variance

Approximately Equal

With random variation

F =

If a treatment effect (H0 False):

Error plus Treatment Variance

-------------------------------------

Error Variance

Numerator

is

Larger

F =


For u visual learners
For U Visual Learners

Sampling

Distributions

H0 True:

H0 False:

Reflects SEM (Error)

Error Plus Treatment



Do these measures depend on what drug you took
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
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
Partitioning the Variance in the Data Set

  • Total Variance (Total Sum of Squared Deviations from Grand Mean)

    • Sum (Xi-Grand Mean)^2

Variance among Samples

Sum (X-Bar – Grand Mean)^2

For all Sample Means

Variance among Subjects

Within each group (sample)

Sum ( Xi – Group mean)^2 for

All subjects in all Groups

SS-Total

SS-Between

SS-Within



Step 2 calculate ss between
Step 2: Calculate SS-Between

  • 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
Step 3: Calculate SS-Within

SS-Total – SSb = SSw

84.91667 – 60.6667 = 24.25

Should Agree with Direct Calculation



Step 4 use ss to compute mean squares f ratio
Step 4: Use SS to ComputeMean Squares & F-ratio

  • 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




What do you do now
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
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 test1
Tukey’s HSD Test

Unequal N

Equal N


Assumptions of anova
Assumptions of ANOVA

  • All Populations Normally distributed

  • Homogeneity of Variance

  • Random Assignment

  • ANOVA is robust to all but gross violations of these theoretical

    • assumptions


Effect size
Effect Size

S = 0.10

M = 0.25

L = 0.40

MStreatment is really MSb

Which is T + E

What’s the Question?


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