Analysis of Variance ANOVA

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Analysis of Variance(ANOVA)
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Research Problem

• Does the temperature of this lecture hall affect
  the rate at which Psy 60 students fall asleep
  during class?
• Independent Variable or Factor is Room Temp
• Three Levels 50, 70, 90 degrees
• Dependent Variable is Reaction Time
• How many minutes after the start of lecture does
  the first student fall asleep?

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Raw Data
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ANOVA Terminology

• Single Factor Independent Measures Design
  (a.k.a., One-way Design)
• Factor
• Variable that is independent (manipulated) or
  quasi-independent (non-manipulated, grouping)
• Independent measures
• Designates that it is a between-subjects design
• How many levels does the factor have?
• Levels refers to number of treatments conditions
  (groups)
• k number of levels

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When to Use ANOVA

• You may use ANOVA whenever you have 2 or more
  independent groups
• You must use ANOVA whenever you have 3 or more
  independent groups.
• Why cant we just conduct a series of t-tests
  (one for each pair of sample means)?
• Answer Alpha Inflation

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Testwise and Familywise Error Rates

• Testwise a
• The probability of making a Type I Error on any
  one hypothesis test.
• a is about .05 for each hypothesis test
• Familywise a
• The accumulated probability of making a Type I
  Error when a series of hypothesis tests are
  conducted.
• a is about .15 for 3 t-tests

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The Logic of ANOVA
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The Logic of ANOVA

• With ANOVA, we take the total variance among all
  the scores in our sample (in all conditions), and
  we partition that variance into 2 parts
• 1. Between Groups Variance
• 2. Within Groups Variance

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The Logic of ANOVA

• Between groups variance
• Differences between the group means
• The average amount by which the group means vary
  around the grand mean.
• Within groups variance
• Differences among people within the same group.
• The average amount by which scores within a group
  vary around mean of their group.

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Between Groups Variance

• Recall the means for each of the groups from our
  sleeping in lecture example
• M50 1 M70 4 M90 1
• Two possible explanations for between-groups
  differences
• Treatment Effect The differences are caused
  systematic(non-random) variation due to our
  independent variable manipulation.
• Chance The differences are due to non-systematic
  (random) variation (a) individual differences
• (b) experimental (measurement) error

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Within Groups Variance
Within Groups Variance
Between Groups Variance
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Partitioning Total Variance
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F-Ratio
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F-Ratio
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F-Ratio
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Partitioning Total Variance
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Hypothesis Testing with ANOVA

• Research Question
• Does room temperature affect the rate at which
  students fall asleep in this class?
• State the Statistical Hypothesis
• H0 µ1 µ2 ... µk
• H1 At least two of the population means are
  significantly different from each other

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Hypothesis Testing with ANOVA

• Set Decision Criteria
• To find our critical value we need
• Alpha level (.05)
• Degrees of freedom between groups
• Degrees of freedom within groups
• Decision Rule Reject H0 if observed F exceeds
  critical F

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Hypothesis Testing with ANOVA

• Set Decision Criteria (cont.)
• a .05
• Dfbetween k-1
• k number of groups 3
• For our example
• dfbetween3-12
• Dfwithin N-k
• For our example
• N 15
• dfwithin N-k 15 3 12

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Hypothesis Testing with ANOVA

• Set Decision Criteria (cont.)
• From the F table,
• F(2,15)3.88
• Reject Ho if F obtained 3.88
• Compute F Ratio

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Hypothesis Testing with ANOVA

• Compute the F-Ratio
• First, Compute Summary Statistics Get Organized
  (T, G, SX2, n, M, SS for each group)
• Then
• 1. Compute Sum of Squares SSWithin, SSBetween,
  SSTotal
• 2. Compute degrees of freedom dfWithin,
  dfBetween,dfTotal
• 3. Compute two Mean Squares MSWithin, MSBetween
• 4. Compute the F-Ratio F MSBetween/ MSWithin

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Computing the F-Ratio
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Hypothesis Testing with ANOVA
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Hypothesis Testing with ANOVA Computing SS

• SSwithin
• SSbetween

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Hypothesis Testing with ANOVA Computing SS

• SSTotal

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Hypothesis Testing with ANOVA Compute MS

• 1. Compute Degrees of Freedom
• dfbetween k-1 3-1 2
• dfwithin N-k 15-3 12
• dftotal N-1 15-1 14 or
• dftotal dfbetweendfwithin 2 12 14

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Hypothesis Testing with ANOVA Compute MS

• 2. Compute MS
• MSBetween SSBetween/dfbetween
• MSBetween 30/2 15
• MSWithin SSWithin/dfWithin
• MSWithin 16/12 1.33

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Hypothesis Testing with ANOVA Compute F Ratio

• F Ratio
• F MSBetween / MSWithin
• F 15/1.33 11.28
• Make a Decision
• Reject Ho if Fobtained Fcritical
• Reject Ho 11.283.88
• Room temperature significantly affects the rate
  at which students fall asleep, F(2,12)11.28,
  p

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F Source Table
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Effect Size for ANOVA

• For our data ?2 SSbetween/SStotal
• 30/46 .65