9. basic concepts of one way analysis of variance (anova)

7/30/2019 9. Basic Concepts of One Way Analysis of Variance (ANOVA)

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Basic Concepts of One-wayAnalysis of Variance

(ANOVA)

Spori Goran, PhD.

http://kif.hr/predmet/mki

http://www.science4performance.com/
http://kif.hr/predmet/mkihttp://kif.hr/predmet/mki


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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 inorder to determine the contributionsof given factors or variables to thevariance.

Remember: Variance: the square ofthe 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 toanswer : Is this variability significant, ormerely by chance?


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The difference between variationwithin a group and variation

between groups may help usdetermine this. If both are equal it islikely that it is due to chance andnot significant.

H0: Variability w/i groups =variability b/t groups, this meansthat 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 variableare equal in all populations

Random samples; independentscores


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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) betweentwo or more groups


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For instance, ANOVA

Could be Used to: Compare heights of plants with andwithout galls

Compare birth weights of deer indifferent geographical regions

Compare responses of patients toreal medication vs. placebo

Compare attention spans ofundergraduate students in differentprograms at PC.


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Why Not Just Use t-

tests? Tedious when many groups are

present

Using all data increasesstability

Large number ofcomparisons some may

appear significant by chance


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Remember that Standard deviation (s)

n

s = [( (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 = Between MS/Within MS


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FYI this is how SS

Values are calculatedk ni Total SS = (xij )2 = SStot

i=1 j=1

k ni

Within SS = (xij i)2 = SSwi=1 j=1

k ni

Between SS = ( i )2 = SSbeti=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 rejectH0 there isa difference betweengroups

The ratio of Between-Groups MS:Within-Groups MS is SMALL do notreject H0 there is nodifferencebetween groups


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p-values Use table in stats book to determine

p

Use df for numerator anddenominator

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 towardwelfare 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 thatif 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 flightspeed greater within or betweengroups? Or Ho = no differencesignificant difference in means.


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Max! Go back to

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Example 2 (contd)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 SSbetBetween 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 (contd) 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 ondf= 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 differencesrequired

Compare multiple parametersfor multiple groups (so it cannot

be used for multiple responsevariables)


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Some Variations Two-Way, Three-Way, etc.

ANOVA (will talk about this nextclass)

2+ factors

MANOVA (Multiple analysis ofvariance)

multiple response variables


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Summary ANOVA:

allows us to know if variability in a data

set is between groups or merely withingroups

is more versatile than t-test can compare multiple groups at once

cannot process multiple responsevariables does not indicate which groups are

different


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Now, lets go to our

SPSS manual Perform the sample problem on the effects

of attachment styles on the psychology ofsleep with the data set from the NAAGEsite called Delta Sleep.

Pay attention to the procedure and thepost-hoc tests to determine which groupsare 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 inSPSS.

Create the data sets.

Run the one-way ANOVAS andinterpret your results.

9. basic concepts of one way analysis of variance (anova)

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