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First There Was the t-TestPowerPoint Presentation

First There Was the t-Test

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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

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-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

- Numerator will be larger than predicted by

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

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

Sampling

Distributions

H0 True:

H0 False:

Reflects SEM (Error)

Error Plus Treatment

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 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

- 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)

- Each subject’s raw score is composed of:

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 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?

- 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

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

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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