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PSYC 2002
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Steven Carroll
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Lecture 12

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Psychology

PSYC 2002

Steven Carroll

Winter

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Lecture 12ANOVAs
What if we want to compare more than two groups?
- Imagine we have 3 groups (placebo, half-dose, full dose)
- If we do t-tests, we have to conduct 3 separate tests:
• Placebo vs. Half dose
• Placebo vs. Full dose
• Half dose vs. Full dose
What if we want to compare more than two groups?
- Imagine we had 4 groups (placebo, ½ , ¾ , full dose)
- If we do t-tests, we have to conduct 6 separate tests:
• Placebo vs. ½
• Placebo vs. ¾
• Placebo vs. full
• ½ vs. ¾
• ½ vs. Full
• ¾ vs. Full
Ok, but so what?
- If we set = .05 for each test, what is:
• P ([no Type I error on Test 1] ᴖ
• [no Type I error on Test 2] ᴖ
• [no Type I error on Test 3] ᴖ
• [no Type I error on Test 4] ᴖ
• [no Type I error on Test 5] ᴖ
• [no Type I error on Test 6])?
There is a better way!
- We call if ANalysis Of Variance: AKA ANOVA
• The nice thing about ANOVA is, provided you use the definitional formulae
I’ve uploaded to WebCT, you can do the whole next section just using the
same formula that I’ve been getting you to use since we started this class!!!! SS
- What is SS?
- The summed, squared deviations of the scores from the mean
- (X- M)²
- SS is a measure of variance:
• A large SS means that the scores are quite different than the mean
• A small SS means the scores are quite similar to the mean
Consider the following scores
Placebo 1/2 Full
3 7 5
1 5 6
2 6 4
- How many days can someone stave off stats anxiety if they take Dr. Steve’s
Magical Stats Medicine?
- How many means can we calculate here?
• 4
• The grand mean, and the means for each of the 3 factor levels
• The grand mean = M.. = X / N = 39 / 9 = 4.33
Now think about this...
- If you were to randomly decide to give someone a half dose of the medicine,
what would be your best guess at how many days they could stave off the stats
anxiety?
• What about a full dose?
• What about a placebo?
Now think about this...
- In other words:
E( X ij) = M i
Consider the following
Placebo 1/2 Full
3 7 5
1 5 6
2 6 4
M1. = 2 M2. = 6 M3. = 5 X
- What is E( 21 )?
- In other words, “what is the expected score for the first person in factor treatment
level 2?”
Consider the following
X1j E(X1j) X2j E(X2j) X3j E(X1j)
3 2 7 6 5 5
1 2 5 6 6 5
2 2 6 6 4 5
M1. = 2 M2. = 6 M3. = 5
- Let’s throw in those expected values
- Everyone has contributed to one of the treatment level means
- These E(X) values are the particular means that each individual has helped to
generate
st
- What do you notice about the expected values for everyone in the 1 level? What
about the 2 level? What about the 3 level?d
The first SS
- We first want to calculate the summed squared difference between everyone’s
expected value and the grand mean
SS =
∑∑ (M −i..)²
i j
- “sum for every group i and every individual j the squared deviation of the
individual’s group mean from the grand mean”
- In other words, for every person in the study square the deviation of the mean of
the group they’re from the grand mean, then add up the results
Shortcut!
- But wait! Since every person in any particular group has the same expected
value (the mean) we can take a SHORTCUT!
∑ ni(M i...)²
i
- Rather than square 9 deviation scores, just do it once per group and multiply by
the number of people in that group - Why? Because all the people in that group share the same mean!
SS between groups
- Since everyone within a group has the same expected value, this SS is called SS
between group
- This is a measure of the variability between the means of the treatment levels
- It is sometimes called “SS treatment” or “SS between treatments”
SS between
Placebo 1/2 Full
3 7 5
1 5 6
2 6 4
M1. = 2 M2. = 6 M3. = 5
- Calculate SS between groups
- SS between
- = 3(2 – 4.33)² + 3(6 – 4.33)² + 3(5 – 4.33)²
- = 3(5.43) + 3(2.79) + 3(.45)
- = 16.47 + 8.37 + 1.35
- = 26.19
Now…
X1j E(X1j) X2j E(X2j) X3j E(X1j)
3 2 7 6 5 5
1 2 5 6 6 5
2 2 6 6 4 5
M1. = 2 M2. = 6 M3. = 5
- Did everyone get the scores that we would have expected them to get if our
treatment had been perfe

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