ANOVA Between-Subject Design: A conceptual approach

1
ANOVA Between-Subject Design A conceptual
approach

• Chong Ho (Alex) Yu

2
Objective

• Illustrate the purpose, the concept, and the
  application of ANOVA between-subject design
• will NOT walk through the procedure of
  hand-calculation you will use a statistical
  software package to do your exercises.
• By the end of the lesson you will understand the
  meaning of the following concepts
• One-way ANOVA vs. Two-way ANOVA
• Grouping factor and level
• Between-subject and within-subject
• Parametric assumptions
• Variance and F-ratio
• Confidence intervals and diamond plots

3
What is ANOVA?

• Analysis of variance a statistical procedure to
  compare the mean difference
• Null hypothesis all means are not significantly
  different from each other
• Alternate Some means are not equal

4
One-way ANOVA

• There must be three or more groups. If there are
  two groups only, you can use a 2-independent-sampl
  e t-test.
• The independent variable is called the grouping
  factor. The group is called the level. In this
  example, there is one factor and three levels
  (Group 1-3).
• .

5
Two-way ANOVA

• There are two grouping factors.
• Unlike one-way ANOVA, in this design it is
  allowed to have fewer than three levels (groups)
  in each factor.
• In this example, there are two factors A and B.
  In each factor, there are two levels 1 and 2.
  Thus, it is called a 2X2 ANOVA between-subject
  design.
• In this lesson we focus on one-way ANOVA only,
  but you need to know why on some occasions there
  are only two groups in ANOVA.

6
What is between-subject?

• Between-subject The subjects in each level
  (group) are not the same people (independent).

7
What is within-subject?

• Within-subject The subjects in each level are
  the same people (correlated). They are measured
  at different points of time.
• In this lesson we will focus on the
  between-subject ANOVA only

8
Why isnt it called Analysis of means?

• If we want to compare the means, why is it called
  Analysis of Variance, not Analysis of Mean?

9
Why isnt it called Analysis of means?

• In the unreal world, the people in the same group
  have the same response to the treatment
• All people in Group 1 got 10.
• All people in Group 2 got 11.
• All people in Group 3 got 12.
• But in the real world, usually there is
  variability in each group (dispersion). We must
  take the variance into account while comparing
  the means.

10
Parametric assumptions

• Independence The responses to the treatment by
  the subjects in different groups are independent
  from each other.
• Normality The sample data have a normal
  distribution.
• the variances of data in different groups are not
  significantly different from each other.

11
A hypothetical example

• Three different teaching formats (levels) are
  used in three different classes

12
When we look at the means alone
13
F ratio

• F signal /noise(error)
• Between-group variance is the signal we want to
  see whether there is a significant difference
  (variability) between the groups.
• Within-group variance is the noise or the error
  it hinders us from seeing the between-group
  difference when the within-group variances
  overlap.
• F mean square between / mean square within
• MSB Sum of square between / DF between
• MSW Sum of square within/ DF within
• Effect size eta square SS effect (between) /
  SS total

14
ANOVA results

• Mean square between and mean square error ? F
  ratio ? Probability (p value)
• The p value is smaller than .05 and therefore we
  reject the null hypothesis.
• Somewhere there is a difference.
• But, where is the difference? Which group can
  significantly outperform which?
• Many textbooks go into multiple comparison
  procedures or post hoc contrast at this point,
  but lets try something else.

15
Diamond plots

•  
• Grand sample mean represented by a horizontal
  dot line 
• Group means the horizontal line inside each
  diamond is the group means 
• Confidence intervals The diamond is the CI for
  each group

16
Assignment 15

• Download the dataset one_way.jmp from the Ch15
  folder.
• Run a one-way ANOVA with this hypothesis There
  is no significant difference between difference
  academic levels in test performance.
• Use level as the IV and score as the DV
• Use Test of unequal variances to check whether
  the group variances are equal.
• If OK, create a diamond plot.
• Is there any performance gap?