First There Was the t-Test

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# First There Was the t-Test - PowerPoint PPT Presentation

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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Presentation Transcript

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)
• 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
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)
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
Tukey’s HSD Test

Unequal N

Equal N

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?