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Lecture Overview on ANOVA

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Lecture Overview on ANOVA

- Review
- hypothesis testing; inferential statistics
- z-test, t-test, independent & dependent t-test

- New Stuff
- Power – Ability to reject Ho
- ANOVA
- Analysis of Variance
- Done with 3 or more groups
- Playground Exercise
- Complete SPSS Example

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Power

- Review: Hypothesis Testing Errors
- Wrongly rejecting Ho: Chance of Type I error: α
- Wrongly retaining Ho: Chance of Type II error: β

- Power
- Opposite of β
- Power = 1- β
- Ability to reject Ho (when Ho should be rejected).
- Researchers want Power!
- Want ability to reject Ho; Show you were right to suspect a difference.
- Want to show IV affects your DV.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

α area (where we reject the Ho, and we shouldn’t)

beyond tcritical

under Ho

β area (where we retain the Ho, and we shouldn’t)

inside tcritical

under Ha

α

α

tc

tc

tc

β

Error AreasHo: μ=55

Ha: μ>55

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

#1: Increase Treatment: Increase difference between groups (μ’s)

H0: μ=55

H0: μ=55

Reality: μ=57

Reality: μ=72

β

tc

- #2: Decrease Sampling Error: Decrease differences within groups.

tc

tc

tc

β

β

β

Increasing PowerDr. Sinn, PSYC301, The joy of 1-way ANOVA

(Increase BG differences)

Rat study

0,3,or 6 mg

0,10,or 20 mg

Therapy study

10 therapy sessions

1 therapy session

#2 Decrease Sampling Error

(Decrease WG differences)

Rat study

Different strains of rats

Same strain of rat

Rats allowed to eat freely

Rats all unfed for 24 hours

Therapy study

Diff. types of Therapy

Same type of Therapy

Examples of increasing powerRat Study: IV:Caffeine Level DV:Amt. Food Found

Therapy Study: IV:Therapy (drug, talk, drug+talk, or control) DV: Improvement]

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

1-Way ANOVA

- ANOVA
- Analysis of Variance
- 1-way means 1 Independent Variable (IV)

- Purpose:
- ANOVA allows hypothesis testing with 3+ sample means
- Imagine study on interventions to help frosh make friends
- Three IV levels: Standard courses, interactive courses, clustered courses.

- ANOVA allows hypothesis testing with 3+ sample means

- ANOVA uses F-test
- Strategy: Compare variability within group to variability between groups.
- F is ratio between two values:

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Matching Exercise

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Draw Conclusions from Playground

- What does a large F mean?
- What two things will make F large?

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Partitioning Variance

- Partition
- fancy word for “divide up”
- ANOVA partitions variance (MS means variance)

- Types of variance
- Total variance = MSWG + MSBG
- MSWG= sampling error (background noise)
- MSBG = sampling error + treatment (includes effect of Independent Variable)

- If just error F tends toward 1.0
- If treatment effectF gets larger

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Example of 1-way ANOVA

- Studying effect of caffeine on productivity
- Does caffeine help or hurt?
- IV: Level of Caffeine: 0, 10, 20 mg
- DV: Number of Food Pellets Found

Number of Food Pellets Found

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

SPSS Data Entry

IV

DV

Label levels of IV so output is easier to read.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

SPSS Analysis

- Go to Analyze, Compare Means, & select One-way ANOVA

Put DV here.

Put IV here.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

SPSS Analysis, Part #2

Select this to get descriptive statistics like sample means & standard deviations.

Alpha level still set to .05, just like it was with t-tests.

Gives you a line graph of the sample means

Conducts “after the fact” test to compare all pairs of sample means.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

SPSS Output

Sample means from 3 groups, plus mean amount of food found overall.

Source of Variation Table

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Where does F come from?

- MSWG = SSWG/dfWG = Sum of Squares / degrees of freedom
- MSBG = SSBG/dfBG = Sum of Squares / degrees of freedom
- Degrees of freedom
- dfWG: NT – K (Total # of subjects - # of groups)
- dfBG: K-1(# of groups – 1)
- dfTOTAL: NT – 1 (Total # of subjects – 1)

- Expectations:
- If I give you df and SS, you can calculate F
- You don’t have to get any SS by hand.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

SPSS Output –Post Hoc Test

No Sig. Diff. Between 0 & 10mg

Rats at 20 mg found significantly more food than rats on 0 or 10 mg of caffeine.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

SPSS Output– Practical Significance

- η2 (“eta squared”)
- Effect size statistic – indicates % of variance explained
- Measures impact of IV on DV
- We can explain 68% of the variance in how much food a rat finds if we know the level of caffeine.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Hypothesis Testing Steps

- Comparison: cf. three sample means.
- Hypothesis: Ho: μ1= μ2 = μ3 Ha: Not all μ’s equal
- Set-up: α= .05 , dfbg= K-1= 2,dfwg= NT-K = 16-3=13, Fcrit = 3.80
- Fobt = 13.653
- Reject Ho.
- The hypothesis was largely supported. Rats found sig. more food on 20mg of caffeine (M=4.33) than on 0mg (M=2.40) or 10mg (M=1.80), F(2,13) = 13.653, p <=.05. Caffeine has a large effect on food finding behavior, accounting for about 68% of the variance, η2 = .6775.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Lab #8: 1-way ANOVA

- TV Problem: The hypothesis was supported. LightTV users provided more community service (M = 6.13) than did moderateusers (M = 4.00), who provided more than heavyusers (M = 1.75), F(2,21) = 15.963, p ≤ .05. TV accounts for about 60% of the variance in community service, η2 = .6032.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Follow-up Questions

- Q1: Variance within group? MSwg = 2.399
- Q2: Variance between groups? MSbg=38.292
- Q3: Replacing heavy scores with 4,5,4,5,6,5,4,3 would decrease the difference between groups because the heavy users would then difference less from the other groups.
- Q4: Decreasing between group differences (decreasing treatment) would decrease F.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Problem #2: Post Hoc Explanation

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Problem #2: Post Hoc Explanation

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Problem #2:

- The hypothesis was supported. People commuting 0 minutes participated significantly more (M=3.4 hours) than people commuting 45 (M=1.2) or60 minutes (M=1.6), F (3,16) = 7.256, p≤.05. Commuting accounted for a large amount of variance in community involvement, η2 = .5764.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Follow-up Questions

- Q1: Variance within group? MSwg = .650
- Q2: Variance between groups? MSbg=4.717
- Q3: Replacing 30 minute commuting scores with 1,4,1,4,3 would increase the within group variability.
- Q4: Increasing sampling error would decrease F.

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Study: Does alcohol affect reaction time?

Identify the treatment effect in this case.

Explain how sampling error might arise.

μna=?? μ2b=?? μ4b=?? Population Means

Review Partitioning14 23 26 Sample Means

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Study: Does alcohol affect reaction time?

What accounts for variability within groups?

What accounts for variability between groups?

What’s the Formula for F?

Review PartitioningDr. Sinn, PSYC301, The joy of 1-way ANOVA

Study: Does alcohol affect reaction time?

If the alcohol content of the beers is not held constant, what happens to F?

increases

decreases

neither

Review Partitioning- If the alcohol content of the beers is not held constant, what happens?
- error increases
- error decreases
- treatment effect increases
- treatment effect decreases

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Hypothesis Testing Steps

- Comparison: cf. three sample means.
- Hypothesis: Ho: μ1= μ2 = μ3 Ha: Not all μ’s equal
- Set-up: α= .05 , dfbg=K-1=3-1=2,dfwg=NT-K=12-3=9, Fcrit = 4.26
- now do one-way ANOVA on SPSS

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

SPSS Output - Charts

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

SPSS Output - Graphs

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

Hypothesis Testing Steps

- Comparison: cf. three sample means.
- Hypothesis: Ho: μ1= μ2 = μ3 Ha: Not all μ’s equal
- Set-up: α= .05 , dfbg=K-1=3-1=2,dfwg=NT-K=12-3=9, Fcrit = 4.26
- Fobt = 2.633
- Retain Ho.
- The hypothesis was not supported. The reaction times following no alcohol (M=13.75), two beers (M=22.50), and four beers (M=26.25) did not differ significantly, F(2,9) = 2.633, n.s..

Dr. Sinn, PSYC301, The joy of 1-way ANOVA

MSbg: _______

With. Group Varib: L M H

MSwg: _______

Numb. of Words Recalled: Dataset ADr. Sinn, PSYC301, The joy of 1-way ANOVA

MSbg: _______

With. Group Varib: L M H

MSwg: _______

Numb. of Words Recalled: Dataset BDr. Sinn, PSYC301, The joy of 1-way ANOVA

MSbg: _______

With. Group Varib: L M H

MSwg: _______

Numb. of Words Recalled: Dataset CDr. Sinn, PSYC301, The joy of 1-way ANOVA

MSbg: _______

With. Group Varib: L M H

MSwg: _______

Numb. of Words Recalled: Dataset DDr. Sinn, PSYC301, The joy of 1-way ANOVA

MSbg: _______

With. Group Varib: L M H

MSwg: _______

Numb. of Words Recalled: Dataset EDr. Sinn, PSYC301, The joy of 1-way ANOVA

MSbg: _______

With. Group Varib: L M H

MSwg: _______

Numb. of Words Recalled: Dataset FDr. Sinn, PSYC301, The joy of 1-way ANOVA

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