Basic Concepts of One-way Analysis of Variance (ANOVA) - PowerPoint PPT Presentation

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Basic Concepts of One-way Analysis of Variance (ANOVA)

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... 7 and 12 from the handout in SPSS. Create the data sets. ... (will talk about this next class) 2+ factors MANOVA (Multiple analysis of variance) ... – PowerPoint PPT presentation

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Title: Basic Concepts of One-way Analysis of Variance (ANOVA)


1
Basic Concepts of One-wayAnalysis of Variance
(ANOVA)
Sporiš Goran, PhD. http//kif.hr/predmet/mki http
//www.science4performance.com/
2
Overview
  • What is ANOVA?
  • When is it useful?
  • How does it work?
  • Some Examples
  • Limitations
  • Conclusions

3
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
4
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?

5
  • 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

6
Assumptions
  • Normal distribution
  • Variances of dependent variable are equal in all
    populations
  • Random samples independent scores

7
One-Way ANOVA
  • One factor (manipulated variable)
  • One response variable
  • Two or more groups to compare

8
Usefulness
  • Similar to t-test
  • More versatile than t-test
  • Compare one parameter (response variable) between
    two or more groups

9
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.

10
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

11
Remember that
  • Standard deviation (s)
  • n
  • s v(S (xi X)2)/(n-1)
  • i 1
  • In this case Degrees of freedom (df)
  • df Number of observations or groups - 1

12
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

13
FYI this is how SS Values are calculated
  • k ni
  • Total SS S S (xij )2 SStot
  • i1 j1
  • k ni
  • Within SS S S (xij i)2 SSw
  • i1 j1
  • k
    ni
  • Between SS S S ( i )2 SSbet
  • i1
    j1

14
and
  • SStot SSw SSbet

15
Calculating MS Values
  • MS SS/df
  • For between groups, df k-1
  • For within groups, df n-k

16
Hypothesis Testing Significance Levels
17
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 SMALL?do not reject H0? there is no difference
    between groups

18
p-values
  • Use table in stats book to determine p
  • Use df for numerator and denominator
  • Choose level of significance
  • If F gt critical value, reject the null hypothesis
    (for one-tail test)

19
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?

20
So,
21
and
From the table with ? 0.05 and df 2 and 24,
we see that if F gt 3.40 we can reject Ho. This is
what we would conclude in this case.
22
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.

23
Just leave me alone Max! Go back to your hockey!
24
Example 2 (contd)
Hypothetical data for bat flight speed with
various gate arrangements.
FS Flight speed sd standard deviation
25
Example 2 ? SSbet
  • Between SS 300

26
Example 2 ? SSw
  • Within SS 790

27
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 on df 4,?
  • Since Fgtvalue found on the table we reject Ho.

28
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)

29
Some Variations
  • Two-Way, Three-Way, etc. ANOVA (will talk about
    this next class)
  • 2 factors
  • MANOVA (Multiple analysis of variance)
  • multiple response variables

30
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

31
Now, lets 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.

32
So, you had
33
Then, following the steps
34
(No Transcript)
35
You got
36
and
37
What do all these mean?
38
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.
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