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

Transcription

1 1 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

2 2 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. Replaces

3 3 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 Black White High Low Male Female Male Female 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.

4 4 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. Three Between Race (R) Gender (G) Ability (A) S

5 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 Race (R) Gender (G) Ability (A) S/RGA

6 6 3. List all possible interactions. (a) Only crossed factors can interact. Three Between Race (R) Gender (G) Ability (A) RA RG GA RGA S/RGA

7 7 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. Three Between Race (R) Gender (G) Ability (A) RG RA GA RGA S/RGA N JKL

8 8 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 Race (R) J Gender (G) K Ability (A) L RG RA GA RGA S/RGA N JKL

9 9 4.. List the degrees of freedom. (d) The degrees of freedom for an interaction is the product of the degrees of freedom for the factors in the interaction. Three Between Race (R) J Gender (G) K Ability (A) L RG J K RA J L GA K L RGA J K L S/RGA N JKL

10 10 5. Construct the F ratios. (a) The error term for an effect involving only between subjects factors is the S/ 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 Race (R) J MS R MSS / Gender (G) Ability (A) RA RG GA RGA RGA K MS G MSS / RGA L MS A MSS / RGA J 1 K 1 1 MS RA MSS / RGA J 1 L 1 1 MS RG MSS / RGA K L MS GA MSS / RGA J 1 K 1 L 1 1 MS RGA MS / S RGA S/RGA N JKL

11 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. Homogeneous Heterogeneous 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 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. 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.

12 12 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. Three Within Array (A) Number (N) Time (T) S

13 13 2. If necessary, indicate nesting by using S/ notation. (b) If there are only within-subject factors subjects are not nested Three Within Array (A) Number (N) Time (T) S

14 14 3. List all possible interactions. (a) Only crossed factors can interact. (b) Subjects interacts with any factor with which it is crossed. Three Within Array (A) SA Number (N) SN Time (T) ST AN SAN AT SAT NT SNT ANT SANT S

15 List the degrees of freedom. (b) In designs without between-subjects factors the degrees of freedom for subjects is n 1. Three Within Array (A) SA Number (N) SN Time (T) ST AN SAN AT SAT NT SNT ANT SANT S n 1

16 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 Within Array (A) P SA Number (N) Q SN Time (T) R ST AN SAN AT SAT NT SNT ANT SANT S n 1

17 List the degrees of freedom. (d) The degrees of freedom for an interaction is the product of the degrees of freedom for the factors in the interaction. Three Within. Array (A) P SA n P Number (N) Q SN n Q Time (T) R ST n R AN P Q SAN n P Q AT P R SAT n P R NT Q R SNT n Q R ANT P Q R SANT n P Q R S n

18 18 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 Array (A) P SA n P MS MS A SA Number (N) Q SN n Q Time (T) R MS MS N MS MS T SN ST ST n R AN P Q SAN n P Q AT P R SAT n P R NT Q R SNT n Q R ANT P Q R SANT n 1 P 1 Q 1 R MS MS AN MS MS AT MS MS NT SAN SAT SNT MSANT MS S AN T S n 1

19 19 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 First Second 4 7 Low IQ High IQ Low IQ High IQ 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? 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 withinsubjects because each child makes two responses. Because subjects is crossed with response, subjects and response can interact. 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 tables displaying interactions and the mean square ratios, in each table new entries are in blue.

20 20 1. If necessary, identify the sub-sections of the table. In the source column list each factor, including subjects (S) (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. Two Between, One Within Between Within

21 21 1. If necessary, identify the sub-sections of the table. In the source column list each factor, including subjects (S) (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. Two Between, One Within Between Grade (G) IQ (I) S Within Response (R)

22 22 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. Two Between, One Within Between Grade (G) IQ (I) S/GI Within Response (R)

23 23 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. In the following interaction in the between section are in red and interactions in the within section are in blue. Two Between, One Within Between Grade (G) IQ (I) GI S/GI Within Response (R) RG RI RGI RS/GI

24 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. Two Between, One Within Between Grade (G) IQ (I) GI S/GI Within Response (R) RG RI RGI RS/GI N JK

25 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. Two Between, One Within Between Grade (G) J IQ (I) K GI S/GI Within Response (R) P 1 1 RG RI RGI RS/GI N JK

26 List the degrees of freedom. (d) The degrees of freedom for an interaction is the product of the degrees of freedom for the factors in the interaction. Two Between, One Within Between Grade (G) J IQ (I) K GI J K S/GI N JK Within Response (R) P 1 1 RG P J RI P K RGI P J K RS/GI P N JK 1 26

27 27 5. Construct the F ratios. (a) The error term for an effect involving only between subjects factors is the S/ 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. Two Between, One Within Between Grade (G) J MS G MSS / G I IQ (I) GI K MS I MSS / GI J 1 K 1 1 MS GI MSS / GI S/GI N JK Within Response (R) P 1 1 RG P J RI P K RGI P J K RS/GI P N JK 1 26

28 28 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 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. Two Between, One Within Between Grade (G) J MSG MS / IQ (I) K MSI MS / GI J K S GI S GI MSGI MS / S GI S/GI N JK Within Response (R) P 1 1 RG P J RI P K RGI P J K MS MS R MS MS RG MS MS RI MS MS RGI RS / GI RS / GI RS / GI RS / GI RS/GI P N JK 1 26

29 29 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 I+ I O+ O- O+ O- 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? 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). 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.

30 30 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 Between Within

31 31 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. One Between, Two Within Between Age (A) S Within Intent (I) Outcome (O)

32 32 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. One Between, Two Within Between Age (A) S/A Within Intent (I) Outcome (O)

33 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, 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. One Between, Two Within Between Age (A) S/A Within Intent (I) AI IS/A Outcome (O) AO OS/A IO AIO IOS/A

34 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. One Between, Two Within Between Age (A) S/A N J Within Intent (I) AI IS/A Outcome (O) AO OS/A IO AIO IOS/A

35 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. One Between, Two Within Between Age (A) J S/A N J Within Intent (I) P AI IS/A Outcome (O) Q AO OS/A IO AIO IOS/A

36 List the degrees of freedom. (d) The degrees of freedom for an interaction is the product of the degrees of freedom for the factors in the interaction. One Between, Two Within Between Age (A) J S/A N J Within Intent (I) P AI J P IS/A P N J Outcome (O) Q AO J Q OS/A Q N J IO P Q AIO J P Q IOS/A Q P N J

37 37 5. Construct the F ratios. (a) The error term for an effect involving only between subjects factors is the S/ mean square. In the following the numerator means square is in blue and the denominator means square is in red. One Between, Two Within Between Age (A) J MSA MS / S/A N J Within Intent (I) P AI J P IS/A P N J Outcome (O) Q AO J Q OS/A Q N J IO P Q AIO J P Q IOS/A Q P N J S A

38 38 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 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. One Between, Two Within Between Age (A) J MSA MS / S/A N J Within Intent (I) P MS MS / AI J P S A I IS A MSAI MS IS / IS/A P N J Outcome (O) Q MS MS / AO J Q A O OS A MSAO MS OS / OS/A Q N J IO P Q MSIO MS I / AIO J P Q A OS A MSAIO MSIO / IOS/A Q P N J S A