Basic Concepts of Oneway Analysis of Variance ANOVA

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Basic Concepts of One-wayAnalysis of Variance
(ANOVA)

• Bernardo Aguilar-Gonzalez

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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 v(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 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

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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 SMALL?do 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 gt 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 gt 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 (contd)
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

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Example 2 ? SSw

• Within SS 790

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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 on df 4,?
• Since Fgtvalue 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, lets go to our SPSS manual Ch. 10

• 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.

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Sources

• http//undergrad.biol.yorku.ca/biol2050/laborator
  ies/anovaPresentation_files/anovaPresentation.ppt
  
• http//www.psychology.nottingham.ac.uk/staff/cr1/a
  nova2b.pdf
• http//www.zoology.unimelb.edu.au/Stats/handouts/a
  nova.pdf
• http//www.cc.umb.edu/grc/HOWTO/handouts/spss3mar0
  3-3.pdf
• http//pages.infinit.net/rlevesqu/spss.htm