ANOVA table

The results from this experiment are presented in the analysis of variance table.

Source df SS MS F P(F)

𝒜 5 146.85 29.37 10.32^∗∗ .000,005

𝒮 7 52.48 7.50

𝒜𝒮 35 99.65 2.85

Total 47 21,806.50

12 Two factors repeated measures, ^{𝒮 × 𝒜 × ℬ}

12.1 Plungin’

What follows is a replication of Godden and Baddeley’s (1975) experi-ment to show the effects of context on memory. Godden and Baddeley’s hypothesis was that memory should be better when the conditions at test are more similar to the conditions experienced during learning. To oper-ationalize this idea, Godden and Baddeley decided to use a very particu-lar population: deep-sea divers. The divers were asked to learn a list of 40 unrelated words either on the beach or under about 10 feet of water, The divers were then tested either on the beach or undersea. The divers needed to be tested in both environments in order to make sure that any effect observed could not be attributed to a global effect of one of the en-vironments. The rationale behind using divers was twofold. The first rea-son was practical: is it worth designing training programs on dry land for divers if they are not able to recall undersea what they have learned? There is strong evidence, incidently, that the problem is real. The second rea-son was more akin to good principles of experimental design, The differ-ence between contexts undersea and on the beach seems quite important, hence a context effect should be easier to demonstrate in this experiment.

Because it is not very easy to find deep-sea divers (willing in addition to participate in a memory experiment) it was decided to use the small number of divers in all possible conditions of the design. The list of words were randomly created and assigned to each subject. The order of testing was randomized in order to eliminate any possible carry-over effects by confounding them with the experimental error.

The first independent variable is the place of learning. It has 2 levels (on the beach and undersea), and it is denoted 𝒜. The second independent variable is the place of testing. It has 2 levels (on the beach and undersea, like 𝒜), and it is denoted ℬ. Crossing these 2 independent variables gives 4 experimental conditions:

∙ 1. Learning on the beach and recalling on the beach.

∙ 2. Learning on the beach and recalling undersea.

158 12.1 Plungin’

∙ 3. Learning undersea and recalling on the beach.

∙ 4. Learning undersea and recalling undersea.

Because each subject in this experiment participates in all four experi-mental conditions, the factor 𝒮 is crossed with the 2 experimental factors.

Hence, the design can be symbolized as a 𝒮 × 𝒜 × ℬ design. For this (fic-titious) replication of Godden and Baddeley’s (1975) experiment we have been able to convince 𝑆 = 5 (fictitious) subjects to take part in this experi-ment (the original experiexperi-ment had 16 subjects).

The subjects to learn lists made of 40 short words each. Each list has been made by drawing randomly words from a dictionary. Each list is used just once (hence, because we have 𝑆 = 5 subjects and 𝐴 × 𝐵 = 2 × 2 = 4 experimental conditions, we have 5× 4 = 20 lists). The dependent variable is the number of words recalled 10 minutes after learning (in order to have enough time to plunge or to come back to the beach).

The results of the (fictitious) replication are given in Table 12.1. Please take a careful look at it and make sure you understand the way the results are laid out.

Recall that the prediction of the authors was that memory should be better when the context of encoding and testing is the same than when the context of encoding and testing are different. This means that they have a very specific shape of effects (a so-called 𝑋-shaped interaction) in

𝒜 Learning Place

𝑎₁On Land 𝑎₂Underwater ∑

Means

𝑌.1𝑠 𝑀.1𝑠

𝑠1 34 14 48 24

𝑏₁ 𝑠₂ 37 21 58 29

Testing place 𝑠₃ 27 31 58 29

On Land 𝑠4 43 27 70 35

𝑠₅ 44 32 76 38

𝑌₁₁.= 185 𝑌₂₁.= 125 𝑌.1.= 310 𝑀11.= 37 𝑀21.= 25 𝑀.1.= 31

𝑠₁ 18 22 40 20

𝑏2 𝑠2 21 25 46 23

Testing place 𝑠₃ 25 33 58 29

Underwater 𝑠₄ 37 33 70 35

𝑠5 34 42 76 38

𝑌₁₂.= 135 𝑌₂₂.= 155 𝑌.1.= 290 𝑀₁₂.= 27 𝑀₂₂.= 31 𝑀.1.= 29

TABLE 12.1Result of a (fictitious) replication of Godden and Baddeley’s (1975) experiment with deep sea divers (see text for explanation).

12.1 Plungin’ 159

mind. As a consequence, they predict that all of the experimental sums of squares should correspond to the sum of squares of interaction.

12.1.1 [R] code

# ANOVA Two-factor within subjects, SxA

# Plungin’ - Deep Sea Diving Example

# We have 2 Factors, A (Learning), with 2 levels and Factor

# B(Testing) with 2 levels and 5 subjects.

# The 2 levels of Factor A and B are: On Land and Underwater.

# Therefore there are 4 groups with the same 5 observations

#(subjects) per group.

# We collect the data for each subject for all levels of Factor

# A and Factor B for each subject.

b1=c(34,37,27,43,44, 14,21,31,27,32) b2=c(18,21,25,37,34, 22,25,33,33,42)

# We now combine the observations into one long column (score).

score=c(b1,b2)

# We now prepare the labels for the 4x5 scores according to the

# factor levels:

# a1 a2, a1 a2...etc for Factor A

Learning=gl(2,5*1,5*4*1, labels=c("a1","a2"))

# b1 b2, b1 b2... etc for Factor B

Testing=gl(2,2*5*1,5*4*1,labels=c("b1","b2"))

# sub_1 sub_1..., sub_2 sub_2...,sub_3 sub_3 ....,sub_4

# sub_4 ..., sub_5 sub_5...etc for Subjects

Subject=gl(5,1,5*4*1, labels=c("sub_1", "sub_2", "sub_3",

"sub_4", "sub_5"))

# We now form a data frame with the dependent variable and the

# factors.

data = data.frame(score = score, Learning = factor(Learning), Testing = factor(Testing), Subject = factor(Subject))

# We now perform an anova when "Subject" is considered as a random factor.

aov1 = aov(score ˜ (Learning * Testing) + Error(Subject / (Learning * Testing)), data = data)

# We now print the data and all the results summary(aov(score˜Learning*Testing*Subject)) summary(aov1)

print(model.tables(aov(score ˜ Learning * Testing * Subject, data = data), "means"), digits = 3)

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160 12.1 Plungin’

12.1.2 [R] output

> # ANOVA Two-factor within subjects, SxA

> # Plungin’ - Deep Sea Diving Example

> # We have 2 Factors, A (Learning), with 2 levels and Factor

> # B(Testing) with 2 levels and 5 subjects.

> # The 2 levels of Factor A and B are: On Land and Underwater.

> # Therefore there are 4 groups with the same 5 observations

> #(subjects) per group.

> # We collect the data for each subject for all levels of Factor

> # A and Factor B for each subject.

> b1=c(34,37,27,43,44, 14,21,31,27,32)

> b2=c(18,21,25,37,34, 22,25,33,33,42)

> # We now combine the observations into one long column (score).

> score=c(b1,b2)

> # We now prepare the labels for the 4x5 scores according to the

> # factor levels:

> # a1 a2, a1 a2...etc for Factor A

> Learning=gl(2,5*1,5*4*1, labels=c("a1","a2"))

> # b1 b2, b1 b2... etc for Factor B

> Testing=gl(2,2*5*1,5*4*1,labels=c("b1","b2"))

> # sub_1 sub_1..., sub_2 sub_2...,sub_3 sub_3 ....,sub_4

> # sub_4 ..., sub_5 sub_5...etc for Subjects

> Subject=gl(5,1,5*4*1, labels=c("sub_1", "sub_2", "sub_3",

"sub_4", "sub_5"))

> # We now form a data frame with the dependent variable and the

> # factors.

> data = data.frame(score = score, Learning = factor(Learning), Testing = factor(Testing), Subject = factor(Subject))

> # We now perform an anova when "Subject" is considered as a random factor.

> aov1 = aov(score ˜ (Learning * Testing) + Error(Subject / (Learning * Testing)), data = data)

> # We now print the data and all the results

> print(data)

---score Learning Testing Subject

---1 34 a1 b1 sub_1

2 37 a1 b1 sub_2

3 27 a1 b1 sub_3

4 43 a1 b1 sub_4

5 44 a1 b1 sub_5

6 14 a2 b1 sub_1

12.1 Plungin’ 161

7 21 a2 b1 sub_2

8 31 a2 b1 sub_3

9 27 a2 b1 sub_4

10 32 a2 b1 sub_5

11 18 a1 b2 sub_1

12 21 a1 b2 sub_2

13 25 a1 b2 sub_3

14 37 a1 b2 sub_4

15 34 a1 b2 sub_5

16 22 a2 b2 sub_1

17 25 a2 b2 sub_2

18 33 a2 b2 sub_3

19 33 a2 b2 sub_4

20 42 a2 b2 sub_5

---> summary(aov(score˜Learning*Testing*Subject))

---Df Sum Sq Mean Sq

---Learning 1 80 80

Testing 1 20 20

Subject 4 680 170

Learning:Testing 1 320 320

Learning:Subject 4 160 40

Testing:Subject 4 32 8

Learning:Testing:Subject 4 64 16

---> summary(aov1)

Error: Subject

---Df Sum Sq Mean Sq F value Pr(>F) ---Residuals 4 680 170

---Error: Subject:Learning

---Df Sum Sq Mean Sq F value Pr(>F)

---Learning 1 80 80 2 0.2302

Residuals 4 160 40

---Error: Subject:Testing

---Df Sum Sq Mean Sq F value Pr(>F)

---Testing 1 20 20 2.5 0.189

Residuals 4 32 8

---Error: Subject:Learning:Testing

---Df Sum Sq Mean Sq F value Pr(>F)

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162 12.1 Plungin’

---Learning:Testing 1 320 320 20 0.01106 *

Residuals 4 64 16

--

---Signif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’

0.1 ’ ’ 1

> print(model.tables(aov(score ˜ Learning * Testing * Subject, data = data), "means"), digits = 3)

Tables of means Grand mean

30 Learning ---a1 a2 ---32 28 ---Testing ---b1 b2 ---31 29

---Subject

---sub_1 sub_2 sub_3 sub_4 sub_5

---22 26 29 35 38

---Learning:Testing

---Testing ---Learning b1 b2

---a1 37 27 a2 25 31 ---Learning:Subject

---Subject

---Learning sub_1 sub_2 sub_3 sub_4 sub_5

---a1 26 29 26 40 39

a2 18 23 32 30 37

---12.1 Plungin’ 163

Testing:Subject

---Subject

---Testing sub_1 sub_2 sub_3 sub_4 sub_5

---b1 24 29 29 35 38

b2 20 23 29 35 38

---Learning:Testing:Subject

, , Subject = sub_1

---Testing ---Learning b1 b2 ---a1 34 18 a2 14 22

---, ---, Subject = sub_2

---Testing ---Learning b1 b2 ---a1 37 21 a2 21 25

---, ---, Subject = sub_3

---Testing ---Learning b1 b2 ---a1 27 25 a2 31 33

---, ---, Subject = sub_4

---Testing ---Learning b1 b2 ---a1 43 37 a2 27 33

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164 12.1 Plungin’

---, ---, Subject = sub_5

---Testing ---Learning b1 b2 ---a1 44 34 a2 32 42

No documento Lynne J. Williams, (páginas 164-172)