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Basic Concepts of One-wayAnalysis of Variance

(ANOVA)Sporiš Goran, PhD.http://kif.hr/predmet/mkihttp://www.science4performance.com/

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Overview

• What is ANOVA?• When is it useful?• How does it work?• Some Examples• Limitations• Conclusions

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Definitions

• ANOVA: analysis of variation in an experimental outcome and especially of a statistical variance in order to determine the contributions of given factors or variables to the variance.

• Remember: Variance: the square of the standard deviation

Remember: RA Fischer, 1919-Evolutionary Biology

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Introduction• Any data set has variability

• Variability exists within groups…

• and between groups

• Question that ANOVA allows us to answer : Is this variability significant, or merely by chance?

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• The difference between variation within a group and variation between groups may help us determine this. If both are equal it is likely that it is due to chance and not significant.

• H0: Variability w/i groups = variability b/t groups, this means that 1 = n

• Ha: Variability w/i groups does not = variability b/t groups, or, 1 n

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Assumptions

• Normal distribution

• Variances of dependent variable are equal in all populations

• Random samples; independent scores

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One-Way ANOVA

• One factor (manipulated variable)

• One response variable

• Two or more groups to compare

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Usefulness

• Similar to t-test

• More versatile than t-test

• Compare one parameter (response variable) between two or more groups

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For instance, ANOVA Could be Used to:

• Compare heights of plants with and without galls

• Compare birth weights of deer in different geographical regions

• Compare responses of patients to real medication vs. placebo

• Compare attention spans of undergraduate students in different programs at PC.

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Why Not Just Use t-tests?• Tedious when many groups are

present

• Using all data increases stability

• Large number of comparisons some may appear significant by chance

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Remember that…

• Standard deviation (s) n

s = √[(Σ (xi – X)2)/(n-1)] i = 1

• In this case: Degrees of freedom (df)

df = Number of observations or groups - 1

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Notation• k = # of groups• n = # observations in each group• xij = one observation in group i

• Y = mean over all groups

• Yi = mean for group i• SS = Sum of Squares• MS = Mean of Squares• λ = Between MS/Within MS

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FYI this is how SS Values are calculated

k ni

• Total SS = Σ Σ (xij – )2 = SStot i=1 j=1

k ni

• Within SS = Σ Σ (xij – i)2 = SSw

i=1 j=1 k ni

• Between SS = Σ Σ ( i – )2 = SSbet i=1 j=1

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and

• SStot = SSw + SSbet

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Calculating MS Values

• MS = SS/df

• For between groups, df = k-1

• For within groups, df= n-k

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Hypothesis Testing & Significance Levels

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F-Ratio = MSBet/MSw

• If:– The ratio of Between-Groups MS:

Within-Groups MS is LARGE reject H0 there is a difference between groups

– The ratio of Between-Groups MS: Within-Groups MS is SMALLdo not reject H0 there is no difference between groups

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p-values

• Use table in stats book to determine p

• Use df for numerator and denominator

• Choose level of significance

• If F > critical value, reject the null hypothesis (for one-tail test)

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Example 1, pp. 400 of your handout• Three groups:

– Middle class sample– Persons on welfare– Lower-middle class sample

• Question: Are attitudes toward welfare payments the same?

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So,

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and

From the table with = 0.05 and df = 2 and 24, we see that if F > 3.40 we can reject Ho. This is what we would conclude in this case.

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Example 2

• Bat cave gates:– Group 1 = No gate (NG)– Group 2 = Straight entrance gate (SE)– Group 3 = Angled entrance gate (AE)– Group 4 = Straight dark zone gate (SD)– Group 5 = Angled dark zone gate (AD)

• Question: Is variation in bat flight speed greater within or between groups? Or Ho = no difference significant difference in means.

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Just leave me alone Max! Go back to

your hockey!

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Example 2 (cont’d)

Group #, i

Gate Type

Mean FS (m/s) sd FS (m/s) ni

1 NG 5.6 0.93 150

2 SE 3.8 1.05 150

3 AE 4.7 0.97 150

4 SD 4.2 1.23 137

5 AD 5.1 1.03 143

Hypothetical data for bat flight speed with various gate arrangements.

FS= Flight speed; sd = standard deviation

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Example 2 SSbet

Between SS = 300

Group #, i

Gate Type

Mean FS (m/s)

sd FS (m/s) ni

1 NG 5.6 0.93 150

2 SE 3.8 1.05 150

3 AE 4.7 0.97 150

4 SD 4.2 1.23 137

5 AD 5.1 1.03 143

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Example 2 SSw

Within SS = 790

Group #, i

Gate Type

Mean FS (m/s)

sd FS (m/s) ni

1 NG 5.6 0.93 150

2 SE 3.8 1.05 150

3 AE 4.7 0.97 150

4 SD 4.2 1.23 137

5 AD 5.1 1.03 143

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Example 2 (cont’d)

• Between MS = 300/4 = 75

• Within MS = 790/(730-5) = 1.09

• F Ratio = 75/1.09 = 68.8

• See Table find p-value based on df= 4,• Since F>value found on the table we

reject Ho.

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What ANOVA Cannot Do• Tell which groups are different

– Post-hoc test of mean differences required

• Compare multiple parameters for multiple groups (so it cannot be used for multiple response variables)

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Some Variations

• Two-Way, Three-Way, etc. ANOVA (will talk about this next class)– 2+ factors

• MANOVA (Multiple analysis of variance)– multiple response variables

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Summary

• ANOVA:– allows us to know if variability in a data

set is between groups or merely within groups

– is more versatile than t-test– can compare multiple groups at once– cannot process multiple response

variables– does not indicate which groups are

different

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Now, let’s go to our SPSS manual

• Perform the sample problem on the effects of attachment styles on the psychology of sleep with the data set from the NAAGE site called Delta Sleep.

• Pay attention to the procedure and the post-hoc tests to determine which groups are significantly different. Perform the Tukey Test at a 5% significance level.

• Look at your output and interpret your results.

• Tell me when you are done.

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So, you had

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Then, following the steps

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You got

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and

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What do all these mean?

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When you are done with this,

• Do practice exercises 1, 4, 6, 7 and 12 from the handout in SPSS.– Create the data sets.– Run the one-way ANOVAS and

interpret your results.