anova between and within subjects suppose you are a cell phone designer, and you would like to test...

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.

_ _ (Xj-Xg)2

MSB = ------------- For all X K – 1

Σ

_ (Xij-Xj)2

MSW = ------------- For all X Ng - K

Σ

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

Between Groups Within Groups

Between Subject ANOVA

SSTot=SSB+SSW

_SSTot = (Xij-Xg)2Σ

Sum of Squares partitions

Time phone is used (minutes) Person Flat Flip Fold Telesc.

Jo 70 95 90 101

Bob 81 96 108 117

Sam 89 93 99 85

Jill 107 109 124 125

Mary 79 81 94 90

ANOVA Summary Table as in Chapter 14 Source SS df MS F

Between 1115.75 3 371.92 1.93

Within 3080.80 16 192.55

Total 4196.55 19

Critical value of F (3 and 16 df) = 3.24

ANOVA Between 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.


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.


You enlist volunteers and give everybody a phone with the following schedule

Week 1 – flatWeek 2 – flipWeek 3 – foldedWeek 4 – telescoping

For each week, you record the number of minutes that the phones are used.

You collect the following data:

_ _ (Xj-Xg)2

MSBO = ------------- K – 1

Σ

SSTot-SSBO-SSBS

MSR = ------------- (n-1)(k-1)

SSBO

MSBO = ------------- K – 1

SSBO

MSBO = ------------- dfB

SSR

MSR = ------------- (n-1)(k-1)

SSR

MSR = ------------- dfR

Between Occasions Residual

Within Subject ANOVA

_ _ (Xi-Xg)2

MSBS = ------------- n – 1

Σ

SSBS

MSBS = ------------- n – 1

SSBS

MSBS = ------------- dfBS

Between Subjects

SSTot=SSBO+SSBS+SSR

_SSTot = (Xij-Xg)2Σ


14526.55

10702.95 2125.3 ?


14526.55

10702.95 2125.3 14526.55-10702.95-2125.3=

1698.3


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?


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.

You’ve introduced a CONFOUNDING VARIABLE because the condition of FOLDED is CONFOUNDED with the vacation week.How do we solve this problem?


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..

Week1 2 3 4

Group 1 flat flip folded tel.Group 2 flip folded tel. flatGroup 3 folded tel. flat flipGroup 4 tel. flat flip folded

Notice how each phone design appears in all 4 weeks for differentSubject groups


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?

Let’s look at that design again.


Week1 2 3 4


Anything a little weird here?


Week1 2 3 4


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, there’s way too many combinations to work out. So what do we do?


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 4Group I A B C D Group 2 B D A CGroup 3 C A D BGroup 4 D C B A

Notice how each letter follows and precedes every other letter once.


With our conditions, this looks like…

Week 1 2 3 4Group I flat flip folded tel Group 2 flip tel flat foldedGroup 3 folded flat tel flipGroup 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.

SSResidual = SSTotal - SSSubjects - SSBetween Occasions

=4196.55 - 2389.8 - 1115.75 = 691


Source SS df MS F

Between Occasions 1115.75 3 371.92 6.46

Between Subjects 2389.80 4 597.45

Residual 691.00 12 57.58

Total 4196.55 19

Source SS df MS F

Between 1115.75 3 371.92 1.93

Within 3080.80 16 192.55

Total 4196.55 19

