ANOVA Between and within subjects

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ANOVA Between and within subjects

Suppose you are a cell phone designer, and you
would like to test four phone designs flat,
flip, folded, and telescoping. You're
interested in whether there is a difference
between these phone types in how much they are
actually used. You enlist volunteers and give
each person one phone to use for a week. Each
person only gets one kind of phone. For each
week, you record the number of minutes that the
phones are used.

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Between Subject ANOVA
_ SSTot
(Xij-Xg)2
S
SSTotSSBSSW
Within Groups
Between Groups
_ _ (Xj-Xg)2 MSB
------------- For all X K
1
_ (Xij-Xj)2 MSW
------------- For all X
Ng - K
S
S

SSB MSB ------------- For all X
K 1 SSB MSB
------------- For all X
dfB

SSW MSW ------------- For all X
Ng - K SSW MSW
------------- For all X
dfW
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Sum of Squares partitions
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ANOVA Between subjects

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ANOVA Between and within subjects

After analyzing your data and failing to find a
significant effect. Your boss yells at you and
tells you that you better find a better way of
doing the experiment or you will be fired. After
discussing the situation with your roommate who
aced 100a with that jerk McAuliffe, you realize
that some people are likely to use ANY phone more
than other people because there is a great deal
of VARIABILITY between people. If you could find
a way of getting rid of the variability between
people. Maybe you could find a significant
effect.

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ANOVA Between and within subjects

After 2 Venti cups of coffee, you have a brain
blast what if you could eliminate the subject
variability by having each person take part in
all conditions. Similar to a dependent t-test,
this would reduce variability and might help you
find the effect.

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ANOVA Between and within subjects
You enlist volunteers and give everybody a phone
with the following schedule Week 1 flat Week
2 flip Week 3 folded Week 4
telescoping For each week, you record the number
of minutes that the phones are used. You
collect the following data
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ANOVA Between and within subjects
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Sum of Squares partitions
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Within Subject ANOVA
_ SSTot
(Xij-Xg)2
S
SSTotSSBOSSBSSSR
Residual
Between Occasions
Between Subjects
_ _ (Xj-Xg)2 MSBO
------------- K 1

SSTot-SSBO-SSBS MSR -------------
(n-1)(k-1)
_ _ (Xi-Xg)2 MSBS
------------- n 1

S
S

SSBO MSBO ------------- K
1 SSBO MSBO -------------
dfB


SSR MSR -------------
(n-1)(k-1) SSR MSR
------------- dfR

SSBS MSBS ------------- n
1 SSBS MSBS -------------
dfBS

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Sum of Squares partitions
14526.55
10702.95
2125.3
?
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Sum of Squares partitions
14526.55
10702.95
2125.3
14526.55-10702.95-2125.3 1698.3
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ANOVA Between and within subjects

Your analysis worked. You bring it to your boss
expecting a pat on the back. Unfortunately, he
starts yelling at you again. Why is the boss
yelling at you when your results are
significant?

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ANOVA Between and within subjects

Your boss realizes that you made a big mistake.
Everyone had the folded phone during the 3rd week
of the study, but that was a vacation week when
people had a lot more free time to use the
phone. Youve introduced a CONFOUNDING VARIABLE
because the condition of FOLDED is CONFOUNDED
with the vacation week. How do we solve this
problem?
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ANOVA Between and within subjects
We need to make sure that there is no systematic
effect of when we give people which phone. The
only way to do this is to divide up the subject
into 4 groups and give ¼ of the phones to each
group at each time. For example.. Week 1
2 3 4 Group 1 flat flip folded tel. Group 2
flip folded tel. flat Group 3
folded tel. flat flip Group 4
tel. flat flip folded Notice how each phone
design appears in all 4 weeks for
different Subject groups

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ANOVA Between and within subjects
This is called COUNTERBALANCING and will
eliminate the confound of time (also called an
order effect). Because you are afraid of making
a mistake, you go to your boss before you do the
experiment with your new design. You are sure
this will work perfectly and yet your boss still
starts yelling at you. What could possibly be
wrong with the design this time? Lets look at
that design again.

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ANOVA Between and within subjects

Week 1 2 3 4 Group 1 flat flip folded
tel. Group 2 flip folded tel. flat Group 3
folded tel. flat flip Group 4
tel. flat flip folded Anything a little weird
here?

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ANOVA Between and within subjects
Week 1 2 3 4 Group 1 flat flip folded
tel. Group 2 flip folded tel. flat Group 3
folded tel. flat flip Group 4
tel. flat flip folded Notice that the flip
phone is immediately after the flat phone in 3
out of 4 conditions. If there is some sort of
carry over effect of using a flat phone, that
might affect the usage of the flip phone. So
maybe we could get rid of that carry over, but
how? With 4 phones and 4 weeks, theres way too
many combinations to work out. So what do we do?

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ANOVA Between and within subjects
Time to call in the NERD RESCUE squad. Turns
out, nerds have already solved this problem by
working out a general solution. And it looks like
this Order 1 2 3 4 Group I A B C D
Group 2 B D A C Group 3 C A D B Group 4 D
C B A Notice how each letter follows and
precedes every other letter once.

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ANOVA Between and within subjects

With our conditions, this looks like Week 1
2 3 4 Group I flat flip folded tel Group
2 flip tel flat folded Group 3 folded
flat tel flip Group 4 tel folded flip
flat Notice how each condition follows and
precedes every other condition once. You show it
to your boss and he finally shuts up. You then
collect your data and analyze it.

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SSResidual SSTotal - SSSubjects - SSBetween
Occasions 4196.55 - 2389.8 - 1115.75 691
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ANOVA Between and within subjects

Critical value of F (3 and 12 df) 3.49
Critical value of F (3 and 16 df) 3.24