Rules for Sources of Variance, Degrees of Freedom, and F...

Replaces 445-446

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Rules for Sources of Variance, Degrees of Freedom, and F ratios

1. If necessary, identify the sub-sections of the table. In the source column list each factor, including subjects (S)

(a) If there are only between-subjects factors or only within-subject factors, there

are no subsections. (b) If there are both between-subjects and within-subjects factors, the summary

ANOVA table has two separate sections: one for between-subjects effects and one for within-subjects effects.

Subjects goes in the between-subjects section. A between-subjects effect is an effect that involves only between-subjects factors.

A within-subjects effect is an effect that involves only within-subjects factors or both between-subjects and within-subjects factors. 2. If necessary, indicate nesting by using S/ notation. (a) If the design has one or more between-subjects factor, then subjects are nested

in combinations of the between-subjects factors. Indicate this nesting by using S/ notation.

(b) If there are only within-subject factors subjects are not nested 3. List all possible interactions. (a) Only crossed factors can interact. (b) Subjects interacts with any factor with which it is crossed (c) For designs with both within-subjects and between-subjects factors,

interactions involving only between-subjects factors go in the Between part. Interactions involving one or more within-subject factors go in the Within part, even if the interaction also involves a between-subjects factor.

Replaces 445-446

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4. List the degrees of freedom. (a) In designs with between-subjects factors, the degrees of freedom for subjects

nested in cells is the total number of subjects (N) minus number of cells in which they are nested. The latter is the product of the number of levels of all between subjects-factors.

(b) In designs without between-subjects factors the degrees of freedom for

subjects is n 1. (c) The main effect degrees of freedom for all factors other than subjects is one

less than the number of levels of the factor. (d) The degrees of freedom for an interaction is the product of the degrees of

freedom for the factors in the interaction.

5. Construct the F ratios. (a) The error term for an effect involving only between subjects factors is the S/

mean square. (b) The error term for an effect involving a within-subjects factor is the

interaction of subjects with that effect.

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A sample of 113 fifth grade teachers graded a single essay supposedly written by a fifth grade student. The teachers were provided, via a cover letter, with fictitious information about the student's sex, race (black or white), and cognitive ability (high or low). Teachers were randomly assigned to sex-race-ability combinations.

1. What are the factors in the design? The factors are (a) ethnic background: Black and white, (b) gender: male and

female, and (c) IQ: low and high

Note that the three factors are crossed and so each pair of factors enters has a

potential interaction and there is a potential three-way interaction 2. Are these factors within-subjects or between-subjects factors? The factors are between-subjects because there are no repeated measures and

there is no blocking. Because all factors are between subjects, teachers are nested in combinations of ethnic background, gender, and IQ.

3. What are the sources of variance in the ANOVA?

4. What are the degrees of freedom for these sources of variance?

5. How are the F-ratios formed to test the main and interaction effects?

See following pages. With the exception of the table displaying the mean square ratios, in each table new entries are in blue.

High Low Male Female Male Female

Black White

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are no subsections.

Three Between

Source df F

Race (R)

Gender (G)

Ability (A)

S

5

2. If necessary, indicate nesting by using S/ notation. (a) If the has one or more between-subjects factor, then subjects are nested in

combinations of the between-subjects factors. Indicate this nesting by using S/ notation.

Three Between

Source df F

Race (R)

Gender (G)

Ability (A)

S/RGA

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3. List all possible interactions. (a) Only crossed factors can interact.

Three Between

Source df F

Race (R)

Gender (G)

Ability (A)

RA

RG

GA

RGA

S/RGA

7

4.. List the degrees of freedom. (a) In designs with between-subjects factors, the degrees of freedom for subjects


Three Between

Source df F

Race (R)

Gender (G)

Ability (A)

RG

RA

GA

RGA

S/RGA 113 2 2 2 105N JKL

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4.. List the degrees of freedom. (c) The main effect degrees of freedom for all factors other than subjects is one

less than the number of levels of the factor.

Three Between

Source df F

Race (R) 1 2 1 1J

Gender (G) 1 2 1 1K

Ability (A) 1 2 1 1L

RG

RA

GA

RGA

S/RGA 113 2 2 2 105N JKL

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4.. List the degrees of freedom. (d) The degrees of freedom for an interaction is the product of the degrees of


Three Between

Source df F

Race (R) 1 2 1 1J

Gender (G) 1 2 1 1K

Ability (A) 1 2 1 1L

RG 1 1 1J K

RA 1 1 1J L

GA 1 1 1K L

RGA 1 1 1 1J K L

S/RGA 113 2 2 2 105N JKL

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mean square.

In the following the numerator means square is in blue and the denominator means square is in red. Note that all F statistics have the same denominator.

Three Between

Source df F

Race (R) 1 2 1 1J /SR RGAMSMS

Gender (G) 1 2 1 1K /SG RGAMSMS

Ability (A) 1 2 1 1L /SA RGAMSMS

RA 1 1 1J K / R ARA S GMSMS

RG 1 1 1J L / R ARG S GMSMS

GA 1 1 1K L / R AGA S GMSMS

RGA 1 1 1 1J K L /RGA S RGAMSMS

S/RGA 113 2 2 2 105N JKL

11

Eighteen children took part in a study: Each child responded to 12 cards on which objects were printed by counting the number of objects. Responses were scored zero-one, with zero indicating an incorrect answer. Six of the cards were shown for five seconds; the remainder were shown for thirty seconds. In each set of six cards, three had homogeneous objects and three had heterogeneous objects printed on them. In each set of three cards, one had three objects, one had five objects, and one had seven objects on it.

1. What are the factors in the design? The factors are (a) array: homogeneous or heterogeneous, (b) number of objects

per card: 3, 5, or 7, and (c) time: 5 or 30.

5 30 3 5 7 3 5 7

Homogeneous Heterogeneous


potential interaction and there is a potential three-way interaction. 2. Are these factors within-subjects or between-subjects factors? The factors are within-subjects because each student counts the objects on each

card. Because all factors are within-subjects, there is no nesting. In addition subjects can be crossed with each of the factors, and therefore subjects can interact with each factor (two-way interactions), with each pair of factors (three-way interactions), and with all three factors (four way interaction). Also subjects are not nested in any factors.





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are no subsections.

Three Within

Source df F

Array (A)

Number (N)

Time (T)

S

13

2. If necessary, indicate nesting by using S/ notation. (b) If there are only within-subject factors subjects are not nested

Three Within

Source df F

Array (A)

Number (N)

Time (T)

S

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3. List all possible interactions. (a) Only crossed factors can interact. (b) Subjects interacts with any factor with which it is crossed.

Three Within

Source df F

Array (A)

SA

Number (N)

SN

Time (T)

ST

AN

SAN

AT

SAT

NT

SNT

ANT

SANT

S

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4.. List the degrees of freedom. (b) In designs without between-subjects factors the degrees of freedom for

subjects is n 1.

Three Within

Source df F

Array (A)

SA

Number (N)

SN

Time (T)

ST

AN

SAN

AT

SAT

NT

SNT

ANT

SANT

S 1n

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Three Within

Source df F

Array (A) 1 2 1 1P

SA

Number (N) 1 3 1 2Q

SN

Time (T) 1 2 1 1R

ST

AN

SAN

AT

SAT

NT

SNT

ANT

SANT

S 1n

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.

Three Within

Source df F

Array (A) 1 2 1 1P

SA 1 1 18 1 2 1 17n P

Number (N) 1 3 1 2Q

SN 1 1 18 1 3 1 34n Q

Time (T) 1 2 1 1R

ST 1 1 18 1 2 1 17n R

AN 1 1 2 1 3 1 2P Q

SAN 1 1 1 18 1 2 1 3 1 34n P Q

AT 1 1 2 1 2 1 1P R

SAT 1 1 1 18 1 2 1 2 1 17n P R

NT 1 1 3 1 2 1 2Q R

SNT 1 1 1 18 1 3 1 2 1 34n Q R

ANT 1 1 1 2 1 3 1 2 1 2P Q R

SANT 1 1 1 1 18 1 2 1 3 1 2 1 34n P Q R

S 1n

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5. Construct the F ratios. (b) The error term for an effect involving a within-subjects factor is the

interaction of subjects with that effect.

In the following subscripts in blue denote within-subjects effect and subscripts in red denote subjects Note that in each F the denominator is the interaction of subject with the within-subjects effect in the numerator.

Three Within

Source df F

Array (A) 1 2 1 1P SA AMS MS

SA 1 1 18 1 2 1 17n P Number (N) 1 3 1 2Q

SN NMS MS SN 1 1 18 1 3 1 34n Q Time (T) 1 2 1 1R

ST TMS MS

ST 1 1 18 1 2 1 17n R AN 1 1 2 1 3 1 2P Q

ASAN NMS MS SAN 1 1 1 18 1 2 1 3 1 34n P Q

AT 1 1 2 1 2 1 1P R ASAT TMS MS

SAT 1 1 1 18 1 2 1 2 1 17n P R

NT 1 1 3 1 2 1 2Q R NSNT TMS MS

SNT 1 1 1 18 1 3 1 2 1 34n Q R ANT 1 1 1 2 1 3 1 2 1 2P Q R

ANT ANS TMS MS

SANT

1 1 1 1

18 1 2 1 3 1 2 1 34

n P Q R

S 1n

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Morter (1963) conducted a study on the effects of requiring two responses per card when administering the Holtzman inkblot technique. Standard instructions call for one response to each of 45 cards, but two responses are considered permissible if more productivity is desired. Morter was concerned about the comparability of the two responses and investigated differences between them using fourth and seventh grade students. Students were classified into high and low IQ groups. A total of 30 children took part in the study. Of the ten variables included in Morter's study, only Form Definiteness, corrected for rejections will be used in this example

1. What are the factors in the design? The factors are (a) grade: 4 or 7, (b) IQ: low or high, and (c) response: first or

second

4 7 Low IQ High IQ Low IQ High IQ

First Second


potential interaction and there is a potential three-way interaction. 2. Are these factors within-subjects or between-subjects factors? Grade (4 or 7) and IQ (low or high) are between-subjects. There are different

subjects in each combination of Grade and IQ. Because Grade and IQ are between subjects, children are nested in combinations of Grade and IQ. Response is within-subjects because each child makes two responses. Because subjects is crossed with response, subjects and response can interact.



5. How are the F-ratios formed to test the main and interaction effects? See following pages. With the exception of the tables displaying interactions and

the mean square ratios, in each table new entries are in blue.

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(b) If there are both between-subjects and within-subjects factors, the summary


Two Between, One Within

Source df F

Between

Within

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(b) If there are both between-subjects and within-subjects factors, the summary


Subjects goes in the between-subjects section. A between-subjects effect is an effect that involves only between-subjects factors.

A within-subjects effect is an effect that involves only within-subjects factors or both between-subjects and within-subjects factors.


Source df F

Between

Grade (G)

IQ (I)

S

Within

Response (R)

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Source df F

Between

Grade (G)

IQ (I)

S/GI

Within

Response (R)

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3. List all possible interactions. (a) Only crossed factors can interact. (b) Subjects interacts with any factor with which it is crossed (c) For designs with both within-subjects and between-subjects factors,


In the following interaction in the between section are in red and interactions in

the within section are in blue.


Source df F

Between

Grade (G)

IQ (I)

GI

S/GI

Within

Response (R)

RG

RI

RGI

RS/GI

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Source df F

Between

Grade (G)

IQ (I)

GI

S/GI 30 2 2 26N JK

Within

Response (R)

RG

RI

RGI

RS/GI

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Source df F

Between

Grade (G) 1 2 1 1J

IQ (I) 1 2 1 1K

GI

S/GI 30 2 2 26N JK

Within

Response (R) 1 1P

RG

RI

RGI

RS/GI

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Source df F

Between

Grade (G) 1 2 1 1J

IQ (I) 1 2 1 1K

GI 1 1 1J K

S/GI 30 2 2 26N JK

Within

Response (R) 1 1P

RG 1 1 1P J

RI 1 1 1P K

RGI 1 1 1 1P J K

RS/GI 1 26P N JK

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mean square. In the following the numerator means square is in blue and the denominator means square is in red. Note that all F statistics have the same denominator.


Source df F

Between

Grade (G) 1 2 1 1J / IG S GMSMS

IQ (I) 1 2 1 1K / II S GMSMS

GI 1 1 1J K / IGI S GMSMS

S/GI 30 2 2 26N JK

Within

Response (R) 1 1P

RG 1 1 1P J

RI 1 1 1P K

RGI 1 1 1 1P J K

RS/GI 1 26P N JK

28


interaction of subjects with that effect. In the following subscripts in blue denote within-subjects components of the effect and subscripts in red denote subjects. In each F in the within subjects section of the table, the denominator is the interaction of subjects with the within-subjects effect in the numerator.


Source df F

Between

Grade (G) 1 2 1 1J /G S GIMS MS

IQ (I) 1 2 1 1K /I S GIMS MS

GI 1 1 1J K /GI S GIMS MS

S/GI 30 2 2 26N JK

Within

Response (R) 1 1P /R RS GIMS MS

RG 1 1 1P J /S GRG IRMS MS

RI 1 1 1P K /S GRI IRMS MS

RGI 1 1 1 1P J K /S GIGIR RMS MS

RS/GI 1 26P N JK

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A study is conducted to investigate the effect of outcome of an act and the actor's intent on judgments of the morality of an act. Two groups of children, ages 4-6 and 10-12, are participants in the study. The sample size in each group was five. Each child is exposed to four conditions generated by the four combinations of positive and negative outcome and positive and negative intent. Each child rates the morality of each act on a 0 (low morality) to10 (high morality) scale.

1. What are the factors in the design? The factors are (a) Intent: positive and negative, (b) outcome: positive and

negative and (c) age: 4-6 and 10-12. 4-6 10-12 O+ O- O+ O-

I+ I-


potential interaction and there is a potential three-way interaction 2. Are these factors within-subjects or between-subjects factors? Age is between-subjects therefore children are nested in age. Intent and Outcome

are within-subjects. Because subjects is crossed with Intent and Outcome, subjects can interact with each of these factors (two-way interactions) and with the Intent and Outcome pair (three-way interaction).





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1. If necessary, identify the sub-sections of the table. In the source column list each factor, including subjects (S) or blocks (Bl).

(b) If there are both between-subjects and within-subjects (or within-blocks) factors, the summary ANOVA table has two separate sections: one for between-subjects effects and one for within-subjects effects.

One Between, Two Within

Source df F

Between

Within

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1. If necessary, identify the sub-sections of the table. In the source column list each factor, including subjects (S) or blocks (Bl).

(b) If there are both between-subjects and within-subjects (or within-blocks)

factors, the summary ANOVA table has two separate sections: one for between-subjects effects and one for within-subjects effects.

Subjects (or blocks) goes in the between-subjects section. A between-subjects effect is an effect that involves only between-subjects factors.

A within-subjects effect is an effect that involves only within-subjects factors or both between-subjects and within-subjects factors.


Source df F

Between

Age (A)

S

Within

Intent (I)

Outcome (O)

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Source df F

Between

Age (A)

S/A

Within

Intent (I)

Outcome (O)

33

3. List all possible interactions. (a) Only crossed factors can interact. (b) Subjects interacts with any factor with which it is crossed. (c) For designs with both within-subjects and between-subjects factors,



Source df F

Between

Age (A)

S/A

Within

Intent (I)

AI

IS/A

Outcome (O)

AO

OS/A

IO

AIO

IOS/A

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Source df F

Between

Age (A)

S/A 10 2 8N J

Within

Intent (I)

AI

IS/A

Outcome (O)

AO

OS/A

IO

AIO

IOS/A

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Source df F

Between

Age (A) 1 2 1 1J

S/A 10 2 8N J

Within

Intent (I) 1 2 1 1P

AI

IS/A

Outcome (O) 1 2 1 1Q

AO

OS/A

IO

AIO

IOS/A

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Source df F

Between

Age (A) 1 2 1 1J

S/A 10 2 8N J

Within

Intent (I) 1 2 1 1P

AI 1 1 2 1 2 1 1J P

IS/A 1 2 1 10 2 8P N J


AO 1 1 2 1 2 1 1J Q

OS/A 1 2 1 10 2 8Q N J

IO 1 1 2 1 2 1 1P Q

AIO 1 1 1 2 1 2 1 2 1 1J P Q

IOS/A 1 1 2 1 2 1 10 2 8Q P N J

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mean square.

In the following the numerator means square is in blue and the denominator means square is in red.


Source df F

Between

Age (A) 1 2 1 1J /S AAMS MS

S/A 10 2 8N J

Within

Intent (I) 1 2 1 1P

AI 1 1 2 1 2 1 1J P

IS/A 1 2 1 10 2 8P N J


AO 1 1 2 1 2 1 1J Q

OS/A 1 2 1 10 2 8Q N J

IO 1 1 2 1 2 1 1P Q

AIO 1 1 1 2 1 2 1 2 1 1J P Q

IOS/A 1 1 2 1 2 1 10 2 8Q P N J

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interaction of subjects with that effect. In the following subscripts in blue denote within-subjects components of the effect and subscripts in red denote subjects. In each F in the within subjects section of the table, the denominator is the interaction of subjects with the within-subjects effect in the numerator.


Source df F

Between

Age (A) 1 2 1 1J /A S AMS MS

S/A 10 2 8N J

Within

Intent (I) 1 2 1 1P /S AI IMS MS

AI 1 1 2 1 2 1 1J P /SAI AIMS MS

IS/A 1 2 1 10 2 8P N J

Outcome (O) 1 2 1 1Q /S AO OMS MS

AO 1 1 2 1 2 1 1J Q /SAO AOMS MS

OS/A 1 2 1 10 2 8Q N J

IO 1 1 2 1 2 1 1P Q /IO OS AIMS MS

AIO 1 1 1 2 1 2 1 2 1 1J P Q /S AIOA IOMS MS

IOS/A 1 1 2 1 2 1 10 2 8Q P N J

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