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# What is a Factorial ANOVA?

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What is a Factorial ANOVA?

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### What is a Factorial ANOVA?

1. 1. Factorial Analysis of Variance
3. 3. Having made the jump to sums of squares logic, … (here’s an example of sums of squares calculation:)
4. 4. Having made the jump to sums of squares logic, … (here’s an example of sums of squares calculation:) Scenario 1: Deviatio Person Scores Mean n Squared Bob 1 – 4 = - 3 2 = 9 Sally 4 – 4 = 0 2 = 0 Val 7 – 4 = + 4 2 = 16 Average 4 sum of squares 25
5. 5. Having made the jump to sums of squares logic, … (here’s an example of sums of squares calculation:) Scenario 1: Person Scores Mean Scenario 2: Deviatio n Squared Bob 1 – 4 = - 3 2 = 9 Sally 4 – 4 = 0 2 = 0 Val 7 – 4 = + 4 2 = 16 Average 4 sum of squares 25 Person Scores Mean Deviatio n Squared Bob 3 – 4 = - 1 2 = 1 Sally 4 – 4 = 0 2 = 0 Val 5 – 4 = + 1 2 = 1 Average 4 sum of squares 2
6. 6. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components …
7. 7. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … For example:
8. 8. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … For example: • Explained Sums of Squares component (variation explained by differences between groups) = 30
9. 9. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … For example: • Explained Sums of Squares component (variation explained by differences between groups) = 30 • Unexplained Sums of Squares component (variation explained by differences within groups) = 6
10. 10. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … and having discovered that the ratio of explained to unexplained variance can render a coefficient that can be evaluated for its rarity …
11. 11. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … and having discovered that the ratio of explained to unexplained variance can render a coefficient that can be evaluated for its rarity … Explained Variance (30) Unexplained Variance (6) = 5.0
12. 12. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … and having discovered that the ratio of explained to unexplained variance can render a coefficient that can be evaluated for its rarity … Explained Variance (30) Unexplained Variance (6) = 5.0 Wow, for this data set an F ratio of 5.0 is pretty rare!
13. 13. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … and having discovered that the ratio of explained to unexplained variance can render a coefficient that can be evaluated for its rarity … Explained Variance (30) Unexplained Variance (6) = 5.0 – OR –
14. 14. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … and having discovered that the ratio of explained to unexplained variance can render a coefficient that can be evaluated for its rarity … Explained Variance (30) Unexplained Variance (6) = 5.0 Explained Variance (2) Unexplained Variance (2) = 1.0 – OR –
15. 15. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … and having discovered that the ratio of explained to unexplained variance can render a coefficient that can be evaluated for its rarity … Explained Variance (30) Unexplained Variance (6) = 5.0 Explained Variance (2) Unexplained Variance (2) = 1.0 – OR – Wow, for this data set an F ratio of 1.0 is not rare at all but pretty common!
16. 16. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … and having discovered that the ratio of explained to unexplained variance can render a coefficient that can be evaluated for its rarity … to make decisions about the probability of Type I error when rejecting a null hypothesis, …
17. 17. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … and having discovered that the ratio of explained to unexplained variance can render a coefficient that can be evaluated for its rarity … to make decisions about the probability of Type I error when rejecting a null hypothesis, … Hmm, an F ratio of 5.0 for this data set is so rare that there is a .02 chance that I’m wrong to reject the null hypothesis (this would be a Type I error). I can live with those odds. So I’ll reject the Null hypothesis!
18. 18. Having made the jump to sums of squares logic, … and having observed that the total sums of squares can be partitioned into “explained” and “unexplained” components … and having discovered that the ratio of explained to unexplained variance can render a coefficient that can be evaluated for its rarity … to make decisions about the probability of Type I error when rejecting a null hypothesis, … Hmm, an F ratio of 5.0 for this data set is so rare that there is a .02 chance that I’m wrong to reject the null hypothesis (this would be a Type I error). I can live with those odds. So I’ll reject the Null hypothesis!
19. 19. We can then extend those principles to a wide range of applications.
20. 20. We can then extend those principles to a wide range of applications. sums of squares between groups
21. 21. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups
22. 22. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups degrees of freedom
23. 23. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups degrees of freedom means square
24. 24. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups degrees of freedom means square F ratio & F critical
25. 25. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups degrees of freedom means square F ratio & F critical hypothesis testing
26. 26. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups degrees of freedom means square F ratio & F critical hypothesis testing one-way ANOVA
27. 27. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups degrees of freedom means square F ratio & F critical hypothesis testing factorial ANOVA one-way ANOVA
28. 28. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups degrees of freedom means square F ratio & F critical hypothesis testing factorial ANOVA split plot ANOVA one-way ANOVA
29. 29. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups degrees of freedom means square F ratio & F critical hypothesis testing factorial ANOVA split plot ANOVA repeated measures ANOVA one-way ANOVA
30. 30. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups degrees of freedom means square F ratio & F critical hypothesis testing factorial ANOVA split plot ANOVA repeated measures ANOVA ANCOVA one-way ANOVA
31. 31. We can then extend those principles to a wide range of applications. sums of squares between groups sums of squares within groups degrees of freedom means square F ratio & F critical hypothesis testing one-way ANOVA factorial ANOVA split plot ANOVA repeated measures ANOVA ANCOVA MANOVA
32. 32. Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels
33. 33. Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels One dependent variable
34. 34. Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels One dependent variable Dependent Variable: Amount of pizza eaten
35. 35. Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels One dependent variable Dependent Variable: Amount of pizza eaten One independent variable
36. 36. Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels One dependent variable Dependent Variable: Amount of pizza eaten One independent variable Independent Variable: Athletes
37. 37. Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels One dependent variable Dependent Variable: Amount of pizza eaten One independent variable Independent Variable: Athletes Categorized into several levels
38. 38. Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels One dependent variable Dependent Variable: Amount of pizza eaten One independent variable Independent Variable: Athletes Categorized into several levels Level 1: Football Player
39. 39. Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels One dependent variable Dependent Variable: Amount of pizza eaten One independent variable Independent Variable: Athletes Categorized into several levels Level 1: Football Player Level 2: Basketball Player
40. 40. Thus far we have only considered one dependent variable and one independent variable that was categorized into several levels One dependent variable Dependent Variable: Amount of pizza eaten One independent variable Independent Variable: Athletes Categorized into several levels Level 1: Football Player Level 2: Basketball Player Level 3: Soccer Player
41. 41. We can consider the effect of multiple independent variables on a single dependent variable.
42. 42. We can consider the effect of multiple independent variables on a single dependent variable. For example:
43. 43. We can consider the effect of multiple independent variables on a single dependent variable. For example: First Independent Variable: Athletes Level 1: Football Player Level 2: Basketball Player Level 3: Soccer Player
44. 44. We can consider the effect of multiple independent variables on a single dependent variable. For example: First Independent Variable: Athletes Level 1: Football Player Level 2: Basketball Player Level 3: Soccer Player Second Independent Variable: Age
45. 45. We can consider the effect of multiple independent variables on a single dependent variable. For example: First Independent Variable: Athletes Level 1: Football Player Level 2: Basketball Player Level 3: Soccer Player Second Independent Variable: Age Level 1: Adults Level 2: Teenagers
46. 46. We can consider the effect of multiple independent variables on a single dependent variable. For example: the differences in number of slices of pizza consumed (this is the single independent variable) among 3 different athlete groups (Football, Basketball, & Soccer) at two different age levels (Adults & Teenagers).
47. 47. We can consider the effect of multiple independent variables on a single dependent variable. For example: the differences in number of slices of pizza consumed (this is the single independent variable) among 3 different athlete groups (Football, Basketball, & Soccer) at two different age levels (Adults & Teenagers). Now, rather than comparing only 3 groups, we will be comparing 6 groups (3 levels of athlete x 2 levels of age groups).
48. 48. We can consider the effect of multiple independent variables on a single dependent variable. For example: the differences in number of slices of pizza consumed (this is the single independent variable) among 3 different athlete groups (Football, Basketball, & Soccer) at two different age levels (Adults & Teenagers). Now, rather than comparing only 3 groups, we will be comparing 6 groups (3 levels of athlete x 2 levels of age groups). Let’s see what this data set might look like.
49. 49. First we list our three levels of athletes
50. 50. First we list our three levels of athletes Athletes Football Player 1 Football Player 2 Football Player 3 Football Player 4 Football Player 5 Football Player 6 Basketball Player 1 Basketball Player 2 Basketball Player 3 Basketball Player 4 Basketball Player 5 Basketball Player 6 Soccer Player 1 Soccer Player 2 Soccer Player 3 Soccer Player 4 Soccer Player 5 Soccer Player 6
51. 51. Then our two age groups Athletes Football Player 1 Football Player 2 Football Player 3 Football Player 4 Football Player 5 Football Player 6 Basketball Player 1 Basketball Player 2 Basketball Player 3 Basketball Player 4 Basketball Player 5 Basketball Player 6 Soccer Player 1 Soccer Player 2 Soccer Player 3 Soccer Player 4 Soccer Player 5 Soccer Player 6
52. 52. Then our two age groups Athletes Adults Teenagers Football Player 1 Football Player 2 Football Player 3 Football Player 4 Football Player 5 Football Player 6 Basketball Player 1 Basketball Player 2 Basketball Player 3 Basketball Player 4 Basketball Player 5 Basketball Player 6 Soccer Player 1 Soccer Player 2 Soccer Player 3 Soccer Player 4 Soccer Player 5 Soccer Player 6
53. 53. Now we add our dependent variable - pizza consumed Athletes Adults Teenagers Football Player 1 Football Player 2 Football Player 3 Football Player 4 Football Player 5 Football Player 6 Basketball Player 1 Basketball Player 2 Basketball Player 3 Basketball Player 4 Basketball Player 5 Basketball Player 6 Soccer Player 1 Soccer Player 2 Soccer Player 3 Soccer Player 4 Soccer Player 5 Soccer Player 6
54. 54. Now we add our dependent variable - pizza consumed Athletes Adults Teenagers Football Player 1 9 Football Player 2 10 Football Player 3 12 Football Player 4 12 Football Player 5 15 Football Player 6 17 Basketball Player 1 1 Basketball Player 2 5 Basketball Player 3 9 Basketball Player 4 3 Basketball Player 5 6 Basketball Player 6 8 Soccer Player 1 1 Soccer Player 2 2 Soccer Player 3 3 Soccer Player 4 2 Soccer Player 5 3 Soccer Player 6 5
55. 55. The procedure by which we analyze the sums of squares among the 6 groups based on 2 independent variables (Age Group and Athlete Category) is called Factorial ANOVA.
56. 56. The procedure by which we analyze the sums of squares among the 6 groups based on 2 independent variables (Age Group and Athlete Category) is called Factorial ANOVA. sums of squares between groups sums of squares within groups degrees of freedom means square F ratio & F critical hypothesis testing one-way ANOVA factorial ANOVA
57. 57. Factorial ANOVA partitions the total sums of squares into the unexplained variance and the variance explained by the main effects of each of the independent variables and the interaction of the independent variables.
58. 58. Factorial ANOVA partitions the total sums of squares into the unexplained variance and the variance explained by the main effects of each of the independent variables and the interaction of the independent variables. Main Effect Interaction Effect Error Explained Variance Type of Athlete Age group Type of Athlete by Age Group Unexplained Variance Within Groups
59. 59. Continuing our example:
60. 60. Continuing our example: • The type of athlete may have an effect on the number of slices of pizza eaten.
61. 61. Continuing our example: • The type of athlete may have an effect on the number of slices of pizza eaten. • But also the age group might as well have an effect on the number of slices eaten.
62. 62. Continuing our example: • The type of athlete may have an effect on the number of slices of pizza eaten. • But also the age group might as well have an effect on the number of slices eaten. • And the interaction of type of athlete and age group may have an effect on slices eaten as well
63. 63. Continuing our example: • The type of athlete may have an effect on the number of slices of pizza eaten. • But also the age group might as well have an effect on the number of slices eaten. • And the interaction of type of athlete and age group may have an effect on slices eaten as well In other words, some age groups within different athlete categories may consume different amounts of pizza. For example, maybe football and basketball adults eat much more than football and basketball teenagers, while adult soccer players eat much less than teenage soccer players.
64. 64. Continuing our example: • The type of athlete may have an effect on the number of slices of pizza eaten. • But also the age group might as well have an effect on the number of slices eaten. • And the interaction of type of athlete and age group may have an effect on slices eaten as well In other words, some age groups within different athlete categories may consume different amounts of pizza. For example, maybe football and basketball adults eat much more than football and basketball teenagers, while adult soccer players eat much less than teenage soccer players.
65. 65. Continuing our example: • The type of athlete may have an effect on the number of slices of pizza eaten. • But also the age group might as well have an effect on the number of slices eaten. • And the interaction of type of athlete and age group may have an effect on slices eaten as well In other words, some age groups within different athlete categories may consume different amounts of pizza. For example, maybe football and basketball adults eat much more than football and basketball teenagers, while adult soccer players eat much less than teenage soccer players.
66. 66. In that case, the soccer players did not follow the trend of the football and basketball players. This would be considered an interaction effect between age group and type of athlete.
67. 67. In that case, the soccer players did not follow the trend of the football and basketball players. This would be considered an interaction effect between age group and type of athlete. Of course, there are 6 (3 x 2) possible combinations of age groups and types of athletes any one of which may not follow the direct main effect trend of age group or type of athlete.
68. 68. In that case, the soccer players did not follow the trend of the football and basketball players. This would be considered an interaction effect between age group and type of athlete. Of course, there are 6 (3 x 2) possible combinations of age groups and types of athletes any one of which may not follow the direct main effect trend of age group or type of athlete. • Adult Football Player • Teenage Football Player • Adult Basketball Player • Teenage Basketball Player • Adult Soccer Player • Teenage Soccer Player
69. 69. You could also order them this way:
70. 70. You could also order them this way: • Adult Football Player • Adult Basketball Player • Adult Soccer Player • Teenage Football Player • Teenage Basketball Player • Teenage Soccer Player
71. 71. You could also order them this way: • Adult Football Player • Adult Basketball Player • Adult Soccer Player • Teenage Football Player • Teenage Basketball Player • Teenage Soccer Player The order doesn’t really matter.
72. 72. When subgroups respond differently under different conditions, we say that an interaction has occurred.
73. 73. When subgroups respond differently under different conditions, we say that an interaction has occurred. Adult Football Players eat 19 slices on average Teenage Football Players eat 12 slices on average
74. 74. When subgroups respond differently under different conditions, we say that an interaction has occurred. Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average
75. 75. When subgroups respond differently under different conditions, we say that an interaction has occurred. Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Do you see the trend here? Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average
76. 76. When subgroups respond differently under different conditions, we say that an interaction has occurred. Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Do you see the trend here? • Football players consume more pizza slices in one sitting than do basketball players Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average
77. 77. When subgroups respond differently under different conditions, we say that an interaction has occurred. Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Do you see the trend here? Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average • Football players consume more pizza slices in one sitting than do basketball players • And adults consume more pizza slices than do teenagers
78. 78. When subgroups respond differently under different conditions, we say that an interaction has occurred. Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Do you see the trend here? • Football players consume more pizza slices in one sitting than do basketball players • And adults consume more pizza slices than do teenagers Now let’s add the soccer players Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average
79. 79. When subgroups respond differently under different conditions, we say that an interaction has occurred. Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Do you see the trend here? • Football players consume more pizza slices in one sitting than do basketball players • And adults consume more pizza slices than do teenagers Now let’s add the soccer players Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 6 slices on average Teenage Soccer Players eat 8 slices on average
80. 80. Because the soccer players do not follow the trend of the other two groups, this is called an interaction effect between type of athlete and age group.
81. 81. So in the case below there would be no interaction effect because all of the trends are the same:
82. 82. So in the case below there would be no interaction effect because all of the trends are the same: Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 8 slices on average Teenage Soccer Players eat 6 slices on average
83. 83. So in the case below there would be no interaction effect because all of the trends are the same: Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average • As you get older you eat more slices of pizza • If you play football you eat more than basketball and soccer players • etc. Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 8 slices on average Teenage Soccer Players eat 6 slices on average
84. 84. But in our first case there is an interaction effect because one of the subgroups is not following the trend:
85. 85. But in our first case there is an interaction effect because one of the subgroups is not following the trend: Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 6 slices on average Teenage Soccer Players eat 8 slices on average
86. 86. But in our first case there is an interaction effect because one of the subgroups is not following the trend: Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average • Soccer players do not follow the trend of the older you are the more pizza you eat. Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 6 slices on average Teenage Soccer Players eat 8 slices on average
87. 87. A factorial ANOVA will have at the very least three null hypotheses. In the simplest case of two independent variables, there will be three.
88. 88. A factorial ANOVA will have at the very least three null hypotheses. In the simplest case of two independent variables, there will be three. Here they are:
89. 89. A factorial ANOVA will have at the very least three null hypotheses. In the simplest case of two independent variables, there will be three. Here they are: • Main Effect for Age Group: There is no significant difference between the amount of pizza slices eaten by adults and teenagers in one sitting.
90. 90. A factorial ANOVA will have at the very least three null hypotheses. In the simplest case of two independent variables, there will be three. Here they are: • Main Effect for Age Group: There is no significant difference between the amount of pizza slices eaten by adults and teenagers in one sitting. • Main Effect for Type of Athlete: There is no significant difference between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting.
91. 91. A factorial ANOVA will have at the very least three null hypotheses. In the simplest case of two independent variables, there will be three. Here they are: • Main Effect for Age Group: There is no significant difference between the amount of pizza slices eaten by adults and teenagers in one sitting. • Main Effect for Type of Athlete: There is no significant difference between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting. • Interaction Effect Between Age Group and Type of Athlete: There is no significant interaction between the amount of pizza eaten by football, basketball and soccer players in one sitting.
92. 92. Let’s begin with the main effect for Age Group
93. 93. Let’s begin with the main effect for Age Group Adults eat 13 slices on average Teenagers eat 11 slices on average
94. 94. Let’s begin with the main effect for Age Group Adults eat 13 slices on average Teenagers eat 11 slices on average So adults eat 2 slices on average more than teenagers. Is this a statistically significant difference? That’s what we will find out using sums of squares logic.
95. 95. Now let’s look at main effect for Type of Athlete
96. 96. Now let’s look at main effect for Type of Athlete Football Players eat 15.5 slices on average Basketball Players eat 10 slices on average Soccer Players eat 7slices on average
97. 97. Now let’s look at main effect for Type of Athlete Football Players eat 15.5 slices on average Basketball Players eat 10 slices on average Soccer Players eat 7slices on average So Football Players eat on average 5.5 slices more than Basketball Players; Basketball Players eat 3 more slices on average than Soccer Players; and Football Players eat 8.5 slices on average more than Soccer Players.
98. 98. Now let’s look at main effect for Type of Athlete Football Players eat 15.5 slices on average Basketball Players eat 10 slices on average Soccer Players eat 7slices on average So Football Players eat on average 5.5 slices more than Basketball Players; Basketball Players eat 3 more slices on average than Soccer Players; and Football Players eat 8.5 slices on average more than Soccer Players. Is this a statistically significant difference? That’s what we will find out using sums of squares logic.
99. 99. Finally let’s consider the interaction effect
100. 100. Finally let’s consider the interaction effect Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 6 slices on average Teenage Soccer Players eat 8 slices on average
101. 101. Finally let’s consider the interaction effect Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 6 slices on average Teenage Soccer Players eat 8 slices on average As noted in this example earlier, it appears that there will be an interaction effect between Age Group and Types of Athletes.
102. 102. So how do we test these possibilities statistically?
103. 103. So how do we test these possibilities statistically? Factorial ANOVA will produce an F-ratio for each main effect and for each interaction.
104. 104. So how do we test these possibilities statistically? Factorial ANOVA will produce an F-ratio for each main effect and for each interaction. • Main effect: Age Group
105. 105. So how do we test these possibilities statistically? Factorial ANOVA will produce an F-ratio for each main effect and for each interaction. • Main effect: Age Group – F ratio.
106. 106. So how do we test these possibilities statistically? Factorial ANOVA will produce an F-ratio for each main effect and for each interaction. • Main effect: Age Group – F ratio. • Main effect: Type of Athlete
107. 107. So how do we test these possibilities statistically? Factorial ANOVA will produce an F-ratio for each main effect and for each interaction. • Main effect: Age Group – F ratio. • Main effect: Type of Athlete – F ratio.
108. 108. So how do we test these possibilities statistically? Factorial ANOVA will produce an F-ratio for each main effect and for each interaction. • Main effect: Age Group – F ratio. • Main effect: Type of Athlete – F ratio. • Interaction effect: Age Group by Type of Athlete
109. 109. So how do we test these possibilities statistically? Factorial ANOVA will produce an F-ratio for each main effect and for each interaction. • Main effect: Age Group – F ratio. • Main effect: Type of Athlete – F ratio. • Interaction effect: Age Group by Type of Athlete – F ratio
110. 110. So how do we test these possibilities statistically? Factorial ANOVA will produce an F-ratio for each main effect and for each interaction. • Main effect: Age Group – F ratio. • Main effect: Type of Athlete – F ratio. • Interaction effect: Age Group by Type of Athlete – F ratio Each of these F ratios will be compared with their individual F-critical values on the F distribution table to determine if the null hypothesis will be retained or rejected.
111. 111. Always interpret the F-ratio for the interactions effect first, before considering the F-ratio for the main effects.
112. 112. Always interpret the F-ratio for the interactions effect first, before considering the F-ratio for the main effects. Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 6 slices on average Teenage Soccer Players eat 8 slices on average
113. 113. Always interpret the F-ratio for the interactions effect first, before considering the F-ratio for the main effects. Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 6 slices on average Teenage Soccer Players eat 8 slices on average If the F-ratio for the interaction is significant, the results for the main effects may be moot.
114. 114. If the interaction is significant, it is extremely helpful to plot the interaction to determine where the effect is occurring.
115. 115. If the interaction is significant, it is extremely helpful to plot the interaction to determine where the effect is occurring.
116. 116. If the interaction is significant, it is extremely helpful to plot the interaction to determine where the effect is occurring. Notice how you can tell visually that soccer players are not following the age trend as is the case with football and basketball players.
117. 117. This looks a lot like our earlier image:
118. 118. This looks a lot like our earlier image: Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 6 slices on average Teenage Soccer Players eat 8 slices on average
119. 119. There are many possible combinations of effects that can render a significant F-ratio for the interaction. In our example, one of the 6 groups might respond very differently than the others …
120. 120. There are many possible combinations of effects that can render a significant F-ratio for the interaction. In our example, one of the 6 groups might respond very differently than the others … or 2, or 3, or … it can be very complex.
121. 121. If the interaction is significant, it is the primary focus of interpretation.
122. 122. If the interaction is significant, it is the primary focus of interpretation. However, sometimes the main effects may be significant and meaningful; even the presence of the significant interaction. The plot will help you decide if it is meaningful.
123. 123. If the interaction is significant, it is the primary focus of interpretation. However, sometimes the main effects may be significant and meaningful; even the presence of the significant interaction. The plot will help you decide if it is meaningful. For example, if all players increase in pizza consumption as they age but some increase much faster in than others, both the interaction and the main effect for age may be important.
124. 124. If the interaction is not significant, it can be ignored and the interpretation of the main effects is straightforward,
125. 125. If the interaction is not significant, it can be ignored and the interpretation of the main effects is straightforward, as would be the case in this example:
126. 126. If the interaction is not significant, it can be ignored and the interpretation of the main effects is straightforward, as would be the case in this example: Adult Football Players eat 19 slices on average Adult Basketball Players eat 14 slices on average Teenage Football Players eat 12 slices on average Teenage Basketball Players eat 10 slices on average Adult Soccer Players eat 8 slices on average Teenage Soccer Players eat 6 slices on average
127. 127. You will now see how to calculate a Factorial ANOVA by hand. Normally you will use a statistical software package to do this calculation. That being said, it is important to see what is going on behind the scenes.
128. 128. Here is the data set we will be working with:
136. 136. Then, we’ll round it off with the total sums of squares.
137. 137. Then, we’ll round it off with the total sums of squares. Once we have all of the sums of squares we can produce an ANOVA table …
138. 138. Then, we’ll round it off with the total sums of squares. Once we have all of the sums of squares we can produce an ANOVA table … Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
139. 139. Then, we’ll round it off with the total sums of squares. Once we have all of the sums of squares we can produce an ANOVA table … Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
140. 140. Then, we’ll round it off with the total sums of squares. Once we have all of the sums of squares we can produce an ANOVA table … Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 … that will make it possible to find the F-ratios we’ll need to determine if we will reject or retain the null hypothesis.
141. 141. Then, we’ll round it off with the total sums of squares. Once we have all of the sums of squares we can produce an ANOVA table … Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 … that will make it possible to find the F-ratios we’ll need to determine if we will reject or retain the null hypothesis.
142. 142. Then, we’ll round it off with the total sums of squares. Once we have all of the sums of squares we can produce an ANOVA table … Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 … that will make it possible to find the F-ratios we’ll need to determine if we will reject or retain the null hypothesis.
143. 143. We begin with calculating Age Group Sums of Squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
144. 144. We begin with calculating Age Group Sums of Squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
145. 145. We begin with calculating Age Group Sums of Squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 Here’s how we do it:
146. 146. We organize the data set with Age Groups in the headers,
147. 147. We organize the data set with Age Groups in the headers, Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9
148. 148. We organize the data set with Age Groups in the headers, then calculate the mean for each age group Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9
149. 149. We organize the data set with Age Groups in the headers, then calculate the mean for each age group Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean
150. 150. We organize the data set with Age Groups in the headers, then calculate the mean for each age group Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78
151. 151. We organize the data set with Age Groups in the headers, then calculate the mean for each age group Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00
152. 152. Then calculate the grand mean (which is the average of all of the data) Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00
153. 153. Then calculate the grand mean (which is the average of all of the data) Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean
154. 154. Then calculate the grand mean (which is the average of all of the data) Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39
155. 155. Then calculate the grand mean (which is the average of all of the data) Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39
156. 156. We subtract the grand mean from each age group mean to get the deviation score Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39
157. 157. We subtract the grand mean from each age group mean to get the deviation score Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score
158. 158. We subtract the grand mean from each age group mean to get the deviation score Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39
159. 159. We subtract the grand mean from each age group mean to get the deviation score Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39
160. 160. Then we square the deviations Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39
161. 161. Then we square the deviations Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39 sq.dev.
162. 162. Then we square the deviations Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39 sq.dev. 1.93
163. 163. Then we square the deviations Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39 sq.dev. 1.93 1.93
164. 164. Then multiply each squared deviation by the number of persons (9). This is called weighting the squared deviations. The more person, the heavier the weighting, or larger the weighted squared deviation values. Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39 sq.dev. 1.93 1.93
165. 165. Then multiply each squared deviation by the number of persons (9). This is called weighting the squared deviations. The more person, the heavier the weighting, or larger the weighted squared deviation values. Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39 sq.dev. 1.93 1.93 wt. sq. dev.
166. 166. Then multiply each squared deviation by the number of persons (9). This is called weighting the squared deviations. The more person, the heavier the weighting, or larger the weighted squared deviation values. Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39 sq.dev. 1.93 1.93 wt. sq. dev. 17.36
167. 167. Then multiply each squared deviation by the number of persons (9). This is called weighting the squared deviations. The more person, the heavier the weighting, or larger the weighted squared deviation values. Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39 sq.dev. 1.93 1.93 wt. sq. dev. 17.36 17.36
168. 168. Finally, sum up the weighted squared deviations to get the sums of squares for age group. Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39 sq.dev. 1.93 1.93 wt. sq. dev. 17.36 17.36
169. 169. Finally, sum up the weighted squared deviations to get the sums of squares for age group. Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39 sq.dev. 1.93 1.93 wt. sq. dev. 17.36 17.36
170. 170. Finally, sum up the weighted squared deviations to get the sums of squares for age group. Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 mean 12.78 10.00 grand mean 11.39 11.39 dev.score 1.39 - 1.39 sq.dev. 1.93 1.93 wt. sq. dev. 17.36 17.36 34.722
171. 171. Note – this is the value from the ANOVA Table shown previously:
172. 172. Note – this is the value from the ANOVA Table shown previously: Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
173. 173. Next we calculate the Type of Player Sums of Squares
174. 174. Next we calculate the Type of Player Sums of Squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
175. 175. We reorder the data so that we can calculate sums of squares for Type of Player
176. 176. We reorder the data so that we can calculate sums of squares for Type of Player Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9
177. 177. Calculate the mean for each Type of Player Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9
178. 178. Calculate the mean for each Type of Player Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67
179. 179. Calculate the grand mean (average of all of the scores) Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67
180. 180. Calculate the grand mean (average of all of the scores) Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67 grand mean 11.4 11.4 11.4
181. 181. Calculate the deviation between each group mean and the grand mean(subtract grand mean from each mean). Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67 grand mean 11.4 11.4 11.4
182. 182. Calculate the deviation between each group mean and the grand mean(subtract grand mean from each mean). Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67 grand mean 11.4 11.4 11.4 dev.score 4.11 0.61 - 4.72
183. 183. Square the deviations Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67 grand mean 11.4 11.4 11.4 dev.score 4.11 0.61 - 4.72
184. 184. Square the deviations Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67 grand mean 11.4 11.4 11.4 dev.score 4.11 0.61 - 4.72 sq.dev. 16.9 0.4 22.3
185. 185. Weight the squared deviations by multiplying the squared deviations by 9 Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67 grand mean 11.4 11.4 11.4 dev.score 4.11 0.61 - 4.72 sq.dev. 16.9 0.4 22.3
186. 186. Weight the squared deviations by multiplying the squared deviations by 9 Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67 grand mean 11.4 11.4 11.4 dev.score 4.11 0.61 - 4.72 sq.dev. 16.9 0.4 22.3 wt. sq. dev. 101.4 2.2 133.8
187. 187. Sum the weighted squared deviations Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67 grand mean 11.4 11.4 11.4 dev.score 4.11 0.61 - 4.72 sq.dev. 16.9 0.4 22.3 wt. sq. dev. 101.4 2.2 133.8
188. 188. Sum the weighted squared deviations Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67 grand mean 11.4 11.4 11.4 dev.score 4.11 0.61 - 4.72 sq.dev. 16.9 0.4 22.3 wt. sq. dev. 101.4 2.2 133.8
189. 189. Sum the weighted squared deviations Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 mean 15.50 12.00 6.67 grand mean 11.4 11.4 11.4 dev.score 4.11 0.61 - 4.72 sq.dev. 16.9 0.4 22.3 wt. sq. dev. 101.4 2.2 133.8 237.444
190. 190. Here is the ANOVA table again:
191. 191. Here is the ANOVA table again: Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
192. 192. Here is how we reorder the data to calculate the within groups sums of squares
210. 210. Sum the squared deviations Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
211. 211. Here is a simple way we go about calculating sums of squares for the interaction between type of athlete and age group
212. 212. Here is a simple way we go about calculating sums of squares for the interaction between type of athlete and age group Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
213. 213. We simply sum up the total sums of squares and then subtract it from the other sums of squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
214. 214. We simply sum up the total sums of squares and then subtract it from the other sums of squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 Total Age Type of Player Error Age * Player – – – =
215. 215. We simply sum up the total sums of squares and then subtract it from the other sums of squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 Total Age Type of Player Error Age * Player 386.278 – – – =
216. 216. We simply sum up the total sums of squares and then subtract it from the other sums of squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 Total Age Type of Player Error Age * Player 386.278 – 34.722 – – =
217. 217. We simply sum up the total sums of squares and then subtract it from the other sums of squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 Total Age Type of Player Error Age * Player 386.278 – 34.722 – 237.444 – =
218. 218. We simply sum up the total sums of squares and then subtract it from the other sums of squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 Total Age Type of Player Error Age * Player 386.278 – 34.722 – 237.444 – 40.667 =
219. 219. We simply sum up the total sums of squares and then subtract it from the other sums of squares Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 Total Age Type of Player Error Age * Player 386.278 – 34.722 – 237.444 – 40.667 = 73.444
220. 220. So here is how we calculate sums of squares:
221. 221. We line up our data in one column: Slices of Pizza Eaten 17 19 21 13 14 15 2 6 8 11 12 13 8 10 12 7 8 9
222. 222. Then we compute the grand mean (which the average of all of the scores) and subtract the grand mean from each of Slices of Pizza Eaten the scores. 17 19 21 13 14 15 2 6 8 11 12 13 8 10 12 7 8 9
223. 223. Then we compute the grand mean (which the average of all of the scores) and subtract the grand mean from each of Slices of Pizza Eaten Grand Mean the scores. 17 – 11.4 19 – 11.4 21 – 11.4 13 – 11.4 14 – 11.4 15 – 11.4 2 – 11.4 6 – 11.4 8 – 11.4 11 – 11.4 12 – 11.4 13 – 11.4 8 – 11.4 10 – 11.4 12 – 11.4 7 – 11.4 8 – 11.4 9 – 11.4
224. 224. This gives us the deviation scores between each score and the grand mean Slices of Pizza Eaten Grand Mean 17 – 11.4 19 – 11.4 21 – 11.4 13 – 11.4 14 – 11.4 15 – 11.4 2 – 11.4 6 – 11.4 8 – 11.4 11 – 11.4 12 – 11.4 13 – 11.4 8 – 11.4 10 – 11.4 12 – 11.4 7 – 11.4 8 – 11.4 9 – 11.4
225. 225. This gives us the deviation scores between each score and the grand mean Slices of Pizza Eaten Grand Mean Deviations 17 – 11.4 = 5.6 19 – 11.4 = 7.6 21 – 11.4 = 9.6 13 – 11.4 = 1.6 14 – 11.4 = 2.6 15 – 11.4 = 3.6 2 – 11.4 = - 9.4 6 – 11.4 = - 5.4 8 – 11.4 = - 3.4 11 – 11.4 = - 0.4 12 – 11.4 = 0.6 13 – 11.4 = 1.6 8 – 11.4 = - 3.4 10 – 11.4 = - 1.4 12 – 11.4 = 0.6 7 – 11.4 = - 4.4 8 – 11.4 = - 3.4 9 – 11.4 = - 2.4
226. 226. Then square the deviations Slices of Pizza Eaten Grand Mean Deviations 17 – 11.4 = 5.6 19 – 11.4 = 7.6 21 – 11.4 = 9.6 13 – 11.4 = 1.6 14 – 11.4 = 2.6 15 – 11.4 = 3.6 2 – 11.4 = - 9.4 6 – 11.4 = - 5.4 8 – 11.4 = - 3.4 11 – 11.4 = - 0.4 12 – 11.4 = 0.6 13 – 11.4 = 1.6 8 – 11.4 = - 3.4 10 – 11.4 = - 1.4 12 – 11.4 = 0.6 7 – 11.4 = - 4.4 8 – 11.4 = - 3.4 9 – 11.4 = - 2.4
227. 227. Then square the deviations Slices of Pizza Eaten Grand Mean Deviations Squared 17 – 11.4 = 5.6 2 = 31.5 19 – 11.4 = 7.6 2 = 57.9 21 – 11.4 = 9.6 2 = 92.4 13 – 11.4 = 1.6 2 = 2.6 14 – 11.4 = 2.6 2 = 6.8 15 – 11.4 = 3.6 2 = 13.0 2 – 11.4 = - 9.4 2 = 88.2 6 – 11.4 = - 5.4 2 = 29.0 8 – 11.4 = - 3.4 2 = 11.5 11 – 11.4 = - 0.4 2 = 0.2 12 – 11.4 = 0.6 2 = 0.4 13 – 11.4 = 1.6 2 = 2.6 8 – 11.4 = - 3.4 2 = 11.5 10 – 11.4 = - 1.4 2 = 1.9 12 – 11.4 = 0.6 2 = 0.4 7 – 11.4 = - 4.4 2 = 19.3 8 – 11.4 = - 3.4 2 = 11.5 9 – 11.4 = - 2.4 2 = 5.7
228. 228. And sum the deviations Slices of Pizza Eaten Grand Mean Deviations Squared 17 – 11.4 = 5.6 2 = 31.5 19 – 11.4 = 7.6 2 = 57.9 21 – 11.4 = 9.6 2 = 92.4 13 – 11.4 = 1.6 2 = 2.6 14 – 11.4 = 2.6 2 = 6.8 15 – 11.4 = 3.6 2 = 13.0 2 – 11.4 = - 9.4 2 = 88.2 6 – 11.4 = - 5.4 2 = 29.0 8 – 11.4 = - 3.4 2 = 11.5 11 – 11.4 = - 0.4 2 = 0.2 12 – 11.4 = 0.6 2 = 0.4 13 – 11.4 = 1.6 2 = 2.6 8 – 11.4 = - 3.4 2 = 11.5 10 – 11.4 = - 1.4 2 = 1.9 12 – 11.4 = 0.6 2 = 0.4 7 – 11.4 = - 4.4 2 = 19.3 8 – 11.4 = - 3.4 2 = 11.5 9 – 11.4 = - 2.4 2 = 5.7
229. 229. And sum the deviations Slices of Pizza Eaten Grand Mean Deviations Squared 17 – 11.4 = 5.6 2 = 31.5 19 – 11.4 = 7.6 2 = 57.9 21 – 11.4 = 9.6 2 = 92.4 13 – 11.4 = 1.6 2 = 2.6 14 – 11.4 = 2.6 2 = 6.8 15 – 11.4 = 3.6 2 = 13.0 2 – 11.4 = - 9.4 2 = 88.2 6 – 11.4 = - 5.4 2 = 29.0 8 – 11.4 = - 3.4 2 = 11.5 11 – 11.4 = - 0.4 2 = 0.2 12 – 11.4 = 0.6 2 = 0.4 13 – 11.4 = 1.6 2 = 2.6 8 – 11.4 = - 3.4 2 = 11.5 10 – 11.4 = - 1.4 2 = 1.9 12 – 11.4 = 0.6 2 = 0.4 7 – 11.4 = - 4.4 2 = 19.3 8 – 11.4 = - 3.4 2 = 11.5 9 – 11.4 = - 2.4 2 = 5.7 total sums of squares
230. 230. And sum the deviations Slices of Pizza Eaten Grand Mean Deviations Squared 17 – 11.4 = 5.6 2 = 31.5 19 – 11.4 = 7.6 2 = 57.9 21 – 11.4 = 9.6 2 = 92.4 13 – 11.4 = 1.6 2 = 2.6 14 – 11.4 = 2.6 2 = 6.8 15 – 11.4 = 3.6 2 = 13.0 2 – 11.4 = - 9.4 2 = 88.2 6 – 11.4 = - 5.4 2 = 29.0 8 – 11.4 = - 3.4 2 = 11.5 11 – 11.4 = - 0.4 2 = 0.2 12 – 11.4 = 0.6 2 = 0.4 13 – 11.4 = 1.6 2 = 2.6 8 – 11.4 = - 3.4 2 = 11.5 10 – 11.4 = - 1.4 2 = 1.9 12 – 11.4 = 0.6 2 = 0.4 7 – 11.4 = - 4.4 2 = 19.3 8 – 11.4 = - 3.4 2 = 11.5 9 – 11.4 = - 2.4 2 = 5.7 total sums of squares 386.28
231. 231. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
232. 232. And that’s how we calculate the total sums of squares along with the interaction between Age Group and Type of Player.
233. 233. And that’s how we calculate the total sums of squares along with the interaction between Age Group and Type of Player. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 Total Age Type of Player Error Age * Player 386.278 – 34.722 – 237.444 – 40.667 = 73.444
234. 234. And that’s how we calculate the total sums of squares along with the interaction between Age Group and Type of Player. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 Total Age Type of Player Error Age * Player 386.278 – 34.722 – 237.444 – 40.667 = 73.444
235. 235. And that’s how we calculate the total sums of squares along with the interaction between Age Group and Type of Player. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 Total Age Type of Player Error Age * Player 386.278 – 34.722 – 237.444 – 40.667 = 73.444
236. 236. We then determine the degrees of freedom for each source of variance: Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
237. 237. We then determine the degrees of freedom for each source of variance: Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
238. 238. We then determine the degrees of freedom for each source of variance: Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
239. 239. Why do we need to determine the degrees of freedom?
240. 240. Why do we need to determine the degrees of freedom? Because this will make it possible to test our three null hypotheses:
241. 241. Why do we need to determine the degrees of freedom? Because this will make it possible to test our three null hypotheses: • Main effect for Age Group: There is NO significant difference between the amount of pizza slices eaten by adults and teenagers in one sitting.
242. 242. Why do we need to determine the degrees of freedom? Because this will make it possible to test our three null hypotheses: • Main effect for Age Group: There is NO significant difference between the amount of pizza slices eaten by adults and teenagers in one sitting. • Main effect for Type of Player: There is NO significant difference between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting.
243. 243. Why do we need to determine the degrees of freedom? Because this will make it possible to test our three null hypotheses: • Main effect for Age Group: There is NO significant difference between the amount of pizza slices eaten by adults and teenagers in one sitting. • Main effect for Type of Player: There is NO significant difference between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting. • Interaction effect between Age Group and Type of Athlete: There is NO significant interaction between the amount of pizza slices eaten by football, basketball, and soccer players in one sitting.
244. 244. By dividing the sums of squares by the degrees of freedom we can compute a mean square from which we can compute an F ratio which can be compared to the F critical.
245. 245. By dividing the sums of squares by the degrees of freedom we can compute a mean square from which we can compute an F ratio which can be compared to the F critical. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
246. 246. By dividing the sums of squares by the degrees of freedom we can compute a mean square from which we can compute an F ratio which can be compared to the F critical. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
247. 247. By dividing the sums of squares by the degrees of freedom we can compute a mean square from which we can compute an F ratio which can be compared to the F critical. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
248. 248. By dividing the sums of squares by the degrees of freedom we can compute a mean square from which we can compute an F ratio which can be compared to the F critical. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
249. 249. By dividing the sums of squares by the degrees of freedom we can compute a mean square from which we can compute an F ratio which can be compared to the F critical. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 If the F ratio is greater than the F critical, we would reject the null hypothesis and determine that the result is statistically significant.
250. 250. By dividing the sums of squares by the degrees of freedom we can compute a mean square from which we can compute an F ratio which can be compared to the F critical. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 If the F ratio is greater than the F critical, we would reject the null hypothesis and determine that the result is statistically significant. If the F ratio is smaller than the F critical then we would fail to reject the null hypothesis.
251. 251. Most statistical packages report statistical significance. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
252. 252. Most statistical packages report statistical significance. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
253. 253. Most statistical packages report statistical significance. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 This means that if we took 1000 samples we would be wrong 1 time. We just don’t know if this is that time.
254. 254. Most statistical packages report statistical significance. But it is important to know where this value came from. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17 This means that if we took 1000 samples we would be wrong 1 time. We just don’t know if this is that time.
255. 255. So let’s calculate the number of degrees of freedom beginning with Age_Group.
256. 256. So let’s calculate the number of degrees of freedom beginning with Age_Group. When determining the degrees of freedom for main effects, we take the number of levels and subtract them by one.
257. 257. So let’s calculate the number of degrees of freedom beginning with Age_Group. When determining the degrees of freedom for main effects, we take the number of levels and subtract them by one. How many levels of age are there?
258. 258. So let’s calculate the number of degrees of freedom beginning with Age_Group. When determining the degrees of freedom for main effects, we take the number of levels and subtract them by one. How many levels of age are there? Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9
259. 259. So let’s calculate the number of degrees of freedom beginning with Age_Group. When determining the degrees of freedom for main effects, we take the number of levels and subtract them by one. How many levels of age are there? Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9
260. 260. So let’s calculate the number of degrees of freedom beginning with Age_Group. When determining the degrees of freedom for main effects, we take the number of levels and subtract them by one. How many levels of age are there? Adults Teens 17 11 19 12 21 13 13 8 14 10 15 12 2 7 6 8 8 9 2 – 1 = 1 degree of freedom for age
261. 261. So let’s calculate the number of degrees of freedom beginning with Age_Group. When determining the degrees of freedom for main effects, we take the number of levels and subtract them by one. How many levels of age are there? Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
262. 262. Now we determine the degrees of freedom for Type of Player.
263. 263. Now we determine the degrees of freedom for Type of Player. How many levels of Type of Player are there?
264. 264. Now we determine the degrees of freedom for Type of Player. How many levels of Type of Player are there? Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9
265. 265. Now we determine the degrees of freedom for Type of Player. How many levels of Type of Player are there? Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9
266. 266. Now we determine the degrees of freedom for Type of Player. How many levels of Type of Player are there? Football Basketball Soccer 17 13 2 19 14 6 21 15 8 11 8 7 12 10 8 13 12 9 3 – 1 = 2 degrees of freedom for type of player
267. 267. Now we determine the degrees of freedom for Type of Player. How many levels of Type of Player are there? Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
268. 268. To determine the degrees of freedom for the interaction effect between age and type of player you multiply the degrees of freedom for age by the degrees of freedom for type of player.
269. 269. To determine the degrees of freedom for the interaction effect between age and type of player you multiply the degrees of freedom for age by the degrees of freedom for type of player. 1 * 2 = 2 degrees of freedom for interaction effect
270. 270. To determine the degrees of freedom for the interaction effect between age and type of player you multiply the degrees of freedom for age by the degrees of freedom for type of player. Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
271. 271. We now determine the degrees of freedom for error.
272. 272. We now determine the degrees of freedom for error. Here we take the number of subjects (18) and subtract that number by the number of subgroups (6):
273. 273. We now determine the degrees of freedom for error. Here we take the number of subjects (18) and subtract that number by the number of subgroups (6): • Adult Football Player • Adult Basketball Player • Adult Soccer Player • Teenage Football Player • Teenage Basketball Player • Teenage Soccer Player
274. 274. We now determine the degrees of freedom for error. Here we take the number of subjects (18) and subtract that number by the number of subgroups (6): • Adult Football Player • Adult Basketball Player • Adult Soccer Player • Teenage Football Player • Teenage Basketball Player • Teenage Soccer Player 18 – 6 = 12 degrees of freedom for error
275. 275. We now determine the degrees of freedom for error. Here we take the number of subjects (18) and subtract that number by the number of subgroups (6): Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
276. 276. To determine the total degrees of freedom we simply add up all of the other degrees of freedom
277. 277. To determine the total degrees of freedom we simply add up all of the other degrees of freedom Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
278. 278. To determine the total degrees of freedom we simply add up all of the other degrees of freedom Tests of Between-Subjects Effects Dependent Variable: Pizza_Slices Source Type III Sum of Squares df Mean Square F Sig. Age_Group 34.722 1 34.722 10.25 0.01 Type of Player 237.444 2 118.722 35.03 0.00 Age_Group * Type of Player 73.444 2 36.722 10.84 0.00 Error 40.667 12 3.389 Total 386.278 17
279. 279. We now calculate the mean square.
280. 280. We now calculate the mean square. The reason this value is called mean square because it represents the average squared deviation of scores from the mean.
281. 281. We now calculate the mean square. The reason this value is called mean square because it represents the average squared deviation of scores from the mean. You will notice that this is actually the definition for variance.
282. 282. So the mean square is a variance.
283. 283. So the mean square is a variance. • The mean square for Age_Group is the variance between the two ages (adult and teenager) and the grand mean. (This is explained variance or variance explained by whether you are an adult or a teenager)
284. 284. So the mean square is a variance. • The mean square for Age_Group is the variance between the two ages (adult and teenager) and the grand mean. (This is explained variance or variance explained by whether you are an adult or a teenager) • The mean square for Type of Player is the variance between the three types of player (football, basketball, and soccer) and the grand mean. (This is explained variance or variance explained by whether you are a football, basketball, or soccer player)
285. 285. So the mean square is a variance. • The mean square for Age_Group is the variance between the two ages (adult and teenager) and the grand mean. (This is explained variance or variance explained by whether you are an adult or a teenager) • The mean square for Type of Player is the variance between the three types of player (football, basketball, and soccer) and the grand mean. (This is explained variance or variance explained by whether you are a football, basketball, or soccer player) • The mean square for the interaction effect represents the variance between each subgroup and the grand mean. (This is explained variance or variance explained by the interaction between Age and Type of Player effects)