One-Way ANOVA

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One-Way ANOVA

• Introduction to Analysis of Variance (ANOVA)

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What is ANOVA?

• ANOVA is short for ANalysis Of VAriance
• Used with 3 or more groups to test for MEAN
  DIFFS.
• E.g., caffeine study with 3 groups
• No caffeine
• Mild dose
• Jolt group
• Level is value, kind or amount of IV
• Treatment Group is people who get specific
  treatment or level of IV
• Treatment Effect is size of difference in means

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Rationale for ANOVA (1)

• We have at least 3 means to test, e.g., H0 ??1
  ??2 ??3.
• Could take them 2 at a time, but really want to
  test all 3 (or more) at once.
• Instead of using a mean difference, we can use
  the variance of the group means about the grand
  mean over all groups.
• Logic is just the same as for the t-test.
  Compare the observed variance among means
  (observed difference in means in the t-test) to
  what we would expect to get by chance.

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Rationale for ANOVA (2)
Suppose we drew 3 samples from the same
population. Our results might look like this
Note that the means from the 3 groups are not
exactly the same, but they are close, so the
variance among means will be small.
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Rationale for ANOVA (3)
Suppose we sample people from 3 different
populations. Our results might look like this
Note that the sample means are far away from one
another, so the variance among means will be
large.
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Rationale for ANOVA (4)
Suppose we complete a study and find the
following results (either graph). How would we
know or decide whether there is a real effect or
not?
To decide, we can compare our observed variance
in means to what we would expect to get on the
basis of chance given no true difference in
means.
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Review

• When would we use a t-test versus 1-way ANOVA?
• In ANOVA, what happens to the variance in means
  (between cells) if the treatment effect is large?

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Rationale for ANOVA
We can break the total variance in a study into
meaningful pieces that correspond to treatment
effects and error. Thats why we call this
Analysis of Variance.
Definitions of Terms Used in ANOVA
The Grand Mean, taken over all observations.
The mean of any level of a treatment.
The mean of a specific level (1 in this case) of
a treatment.
The observation or raw data for the ith person.
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The ANOVA Model
A treatment effect is the difference between the
overall, grand mean, and the mean of a cell
(treatment level).
Error is the difference between a score and a
cell (treatment level) mean.
The ANOVA Model
An individuals score
A treatment or IV effect
The grand mean
is

Error

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The ANOVA Model
The grand mean
A treatment or IV effect
Error
The graph shows the terms in the equation. There
are three cells or levels in this study. The IV
effect and error for the highest scoring cell is
shown.
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ANOVA Calculations
Sums of squares (squared deviations from the
mean) tell the story of variance. The simple
ANOVA designs have 3 sums of squares.
The total sum of squares comes from the distance
of all the scores from the grand mean. This is
the total its all you have.
The within-group or within-cell sum of squares
comes from the distance of the observations to
the cell means. This indicates error.
The between-cells or between-groups sum of
squares tells of the distance of the cell means
from the grand mean. This indicates IV effects.
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Computational Example Caffeine on Test Scores
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Total Sum of Squares
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In the total sum of squares, we are finding the
squared distance from the Grand Mean. If we took
the average, we would have a variance.
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Within Sum of Squares
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Within sum of squares refers to the variance
within cells. That is, the difference between
scores and their cell means. SSW estimates error.
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Between Sum of Squares
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The between sum of squares relates the Cell Means
to the Grand Mean. This is related to the
variance of the means.
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ANOVA Source Table (1)
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ANOVA Source Table (2)

• df Degrees of freedom. Divide the sum of
  squares by degrees of freedom to get
• MS, Mean Squares, which are population variance
  estimates.
• F is the ratio of two mean squares. F is another
  distribution like z and t. There are tables of F
  used for significance testing.

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The F Distribution
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F Table Critical Values
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Review

• What are critical values of a statistics (e.g.,
  critical values of F)?
• What are degrees of freedom?
• What are mean squares?
• What does MSW tell us?

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Review 6 Steps

• Set alpha (.05).
• State Null Alternative
• H0
• H1 not all ? are .
• Calculate test statistic F12.5

• Determine critical value F.05(2,12) 3.89
• Decision rule If test statistic gt critical
  value, reject H0.
• Decision Test is significant (12.5gt3.89). Means
  in population are different.

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Post Hoc Tests

• If the t-test is significant, you have a
  difference in population means.
• If the F-test is significant, you have a
  difference in population means. But you dont
  know where.
• With 3 means, could be ABgtC or AgtBgtC or AgtBC.
• We need a test to tell which means are different.
  Lots available, we will use 1.

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Tukey HSD (1)
Use with equal sample size per cell.
HSD means honestly significant difference.
?? is the Type I error rate (.05).
Is a value from a table of the studentized range
statistic based on alpha, dfW (12 in our example)
and k, the number of groups (3 in our example).
Is the mean square within groups (10).
Is the number of people in each group (5).
MSW
Result for our example.
From table
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Tukey HSD (2)
To see which means are significantly different,
we compare the observed differences among our
means to the critical value of the Tukey test.
The differences are 1-2 is 79-84 -5 (say 5 to
be positive). 1-3 is 79-74 5 2-3 is 84-74
10. Because 10 is larger than 5.33, this result
is significant (2 is different than 3). The
other differences are not significant. Review 6
steps.
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Review

• What is a post hoc test? What is its use?
• Describe the HSD test. What does HSD stand for?

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Test

• Another name for mean square is _________.
• standard deviation
• sum of squares
• treatment level
• variance

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Test

• When do we use post hoc tests?
• a. after a significant overall F test
• b. after a nonsignificant overall F test
• c. in place of an overall F test
• d. when we want to determine the impact of
  different factors