2. What if. . .
You were asked to determine if psychology and sociology majors have significantly different class attendance (i.e., the number of days a person misses class)
You would simply do a two-sample t-test
two-tailed
Easy!

3. But, what if. . .
You were asked to determine if psychology, sociology, and biology majors have significantly different class attendance
You can’t do a two-sample t-test
You have three samples
No such thing as a three sample t-test!

4. One-Way ANOVA ANOVA = Analysis of Variance
This is a technique used to analyze the results of an experiment when you have more than two groups

5. Example
You measure the number of days 7 psychology majors, 7 sociology majors, and 7 biology majors are absent from class
You wonder if the average number of days each of these three groups was absent is significantly different from one another

6. Hypothesis Alternative hypothesis (H1)
H1: The three population means are not all equal

7. Hypothesis
Alternative hypothesis (H1)
socio = bio

8. Hypothesis
Alternative hypothesis (H1)
socio = psych

9. Hypothesis
Alternative hypothesis (H1)
psych = bio

10. Hypothesis
Alternative hypothesis (H1)
psych = bio =  soc

11. Hypothesis Alternative hypothesis (H1)
Notice: It does not say where this difference is at!!

12. Hypothesis Null hypothesis (H0) psych = socio = bio
In other words, all three means are equal to one another (i.e., no difference between the means)

13. Results
X = 3.00
X = 2.00
X = 1.00

14. Logic Is the same as t-tests
1) calculate a variance ratio (called an F; like t-observed)
2) Find a critical value
3) See if the the F value falls in the critical area

15. Between and Within Group Variability
Two types of variability
Between / Treatment
the differences between the mean scores of the three groups
The more different these means are, the more variability!

16. Between Variability
Compute S2 on the means
X = 3.00
X = 2.00
X = 1.00

17. Between Variability
S2 = 1
X = 3.00
X = 2.00
X = 1.00

18. Between Variability
+ 5
X = 3.00
X = 2.00
X = 1.00

19. Between Variability
X = 8.00
X = 2.00
X = 1.00

20. Between Variability
Compute S2 on the means
X = 8.00
X = 2.00
X = 1.00

21. Between Variability
S2 = 14.33
X = 8.00
X = 2.00
X = 1.00

22. Between Group Variability
What causes this variability to increase?
1) Effect of the variable (college major)
2) Sampling error

23. Between and Within Group Variability
Two types of variability
Within / Error
the variability of the scores within each group

24. Within Variability Compute S2 within each group X = 3.00 X = 2.00

25. Within Variability
S2 =.67
S2 =1.67
S2 =.67
X = 3.00
X = 2.00
X = 1.00

26. Within Group Variability
What causes this variability to increase?
1) Sampling error

27. Between and Within Group Variability
Between-group variability
Within-group variability

28. Between and Within Group Variability
sampling error + effect of variable
sampling error

29. Between and Within Group Variability
sampling error + effect of variable
sampling error
Thus, if null hypothesis was true this would result in a value of 1.00

30. Between and Within Group Variability
sampling error + effect of variable
sampling error
Thus, if null hypothesis was not true this value would be greater than 1.00

31. Calculating this Variance Ratio

32. Calculating this Variance Ratio

33. Calculating this Variance Ratio

34. Degrees of Freedom dfbetween dfwithin dftotal
dftotal = dfbetween + dfwithin

35. Degrees of Freedom dfbetween = k - 1 (k = number of groups)
dfwithin = N - k (N = total number of observations)
dftotal = N - 1
dftotal = dfbetween + dfwithin

36. Degrees of Freedom dfbetween = k - 1 3 - 1 = 2
dfwithin = N - k = 18
dftotal = N = 20
20 =

37. Sum of Squares SSBetween SSWithin SStotal
SStotal = SSBetween + SSWithin

38. Sum of Squares
SStotal

39. Sum of Squares
SSWithin

40. Sum of Squares
SSBetween

42. Sum of Squares
Ingredients: X
X2
Tj2 N n

43. To Calculate the SS

44. X
Xs = 21
Xp = 14
XB = 7

45. X
X = 42
Xs = 21
Xp = 14
XB = 7

46. X2
X = 42
Xs = 21
Xp = 14
XB = 7
X2s = 67
X2P = 38
X2B = 11

47. X2 X = 42 X2 = 116 Xs = 21 Xp = 14 XB = 7 X2s = 67 X2P = 38

48. T2 = (X)2 for each group X = 42 X2 = 116 Xs = 21 Xp = 14 XB = 7
T2P = 196
T2B = 49
T2s = 441

49. Tj2 X = 42 X2 = 116 Tj2 = 686 Xs = 21 Xp = 14 XB = 7 X2s = 67
T2P = 196
T2B = 49
T2s = 441

50. N X = 42 X2 = 116 Tj2 = 686 N = 21 Xs = 21 Xp = 14 XB = 7
T2P = 196
T2B = 49
T2s = 441

51. n X = 42 X2 = 116 Tj2 = 686 N = 21 n = 7 Xs = 21 Xp = 14 XB = 7
T2P = 196
T2B = 49
T2s = 441

52. Ingredients
X = 42
X2 = 116
Tj2 = 686
N = 21
n = 7

53. Calculate SS
X = 42
X2 = 116
Tj2 = 686
N = 21
n = 7
SStotal

54. Calculate SS 42 32 116 21 SStotal X = 42 X2 = 116 Tj2 = 686 N = 21

55. Calculate SS
X = 42
X2 = 116
Tj2 = 686
N = 21
n = 7
SSWithin

56. Calculate SS 686 18 116 7 SSWithin X = 42 X2 = 116 Tj2 = 686 N = 21

57. Calculate SS
X = 42
X2 = 116
Tj2 = 686
N = 21
n = 7
SSBetween

58. Calculate SS 14 686 42 7 21 SSBetween X = 42 X2 = 116 Tj2 = 686

59. Sum of Squares SSBetween SSWithin SStotal
SStotal = SSBetween + SSWithin

60. Sum of Squares
SSBetween = 14
SSWithin = 18
SStotal = 32
32 =

61. Calculating the F value

62. Calculating the F value

63. Calculating the F value14 7 2

64. Calculating the F value7

65. Calculating the F value7 18 1 18

66. Calculating the F value7 7 1

67. How to write it out

68. Significance Is an F value of 7.0 significant at the .05 level?
To find out you need to know both df

69. Degrees of Freedom Dfbetween = k - 1 (k = number of groups)
dfwithin = N - k (N = total number of observations)

70. Degrees of Freedom Dfbetween = k - 1 3 - 1 = 2
dfwithin = N - k = 18
Use F table
Dfbetween are in the numerator
Dfwithin are in the denominator
Write this in the table

71. Critical F Value
F(2,18) = 3.55
The nice thing about the F distribution is that everything is a one-tailed test

72. Decision Thus, if F value > than F critical
Reject H0, and accept H1
If F value < or = to F critical
Fail to reject H0

73. Current Example F value = 7.00 F critical = 3.55
Thus, reject H0, and accept H1

74. Alternative hypothesis (H1)
H1: The three population means are not all equal
In other words, psychology, sociology, and biology majors do not have equal IQs
Notice: It does not say where this difference is at!!

75. How to write it out

76. SPSS

79. Conceptual Understanding
Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at .05.

80. Conceptual Understanding
Fcrit = 3.18
Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at .05.
Fcrit (2, 57) = 3.15

81. Conceptual Understanding
Distinguish between: Between-group variability and within-group variability

82. Conceptual Understanding
Distinguish between: Between-group variability and within-group variability
Between concerns the differences between the mean scores in various groups
Within concerns the variability of scores within each group

83. Between and Within Group Variability
Between-group variability
Within-group variability

84. Between and Within Group Variability
sampling error + effect of variable
sampling error

85. Conceptual Understanding
Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00?

86. Conceptual Understanding
Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00?
F ratio will approach 1.00 when the null hypothesis is true
F ratio will be greater than 1.00 when the null hypothesis is not true

87. Conceptual Understanding
Without computing the SS within, what must its value be? Why?

88. Conceptual Understanding
The SS within is 0. All the scores within a group are the same (i.e., there is NO variability within groups)

91. Example
Freshman, Sophomore, Junior, Senior
Measure Happiness (1-100)

93. ANOVA Traditional F test just tells you not all the means are equal
Does not tell you which means are different from other means

94. Why not Do t-tests for all pairs Fresh vs. Sophomore Fresh vs. Junior
Fresh vs. Senior
Sophomore vs. Junior
Sophomore vs. Senior
Junior vs. Senior

95. Problem What if there were more than four groups?
Probability of a Type 1 error increases.
Maximum value = comparisons (.05)
6 (.05) = .30

96. Chapter 12 A Priori and Post Hoc Comparisons Multiple t-tests
Linear Contrasts
Orthogonal Contrasts
Trend Analysis
Bonferroni t
Fisher Least Significance Difference
Studentized Range Statistic
Dunnett’s Test

97. Multiple t-tests Good if you have just a couple of planned comparisons
Do a normal t-test, but use the other groups to help estimate your error term
Helps increase you df

98. Remember

99. Note

100. Proof
Candy Gender

102. t = / .641 = 4.16

103. t = / .641 = 4.16

104. t = / .641 = 4.16

105. t = / .641 = 4.16

106. Also, when F has 1 df between

107. Within Variability
Within variability of all the groups represents “error”
You can therefore get a better estimate of error by using all of the groups in your ANOVA when computing a t-value

108. Note: This formula is for equal n

109. Hyp 1: Juniors and Seniors will have different levels of happiness
Hyp 2: Seniors and Freshman will have different levels of happiness

110. Hyp 1: Juniors and Seniors will have different levels of happiness

111. Hyp 1: Juniors and Seniors will have different levels of happiness

112. Hyp 1: Juniors and Seniors will have different levels of happiness

113. Hyp 1: Juniors and Seniors will have different levels of happiness
t crit (20 df) = 2.086

114. Hyp 1: Juniors and Seniors will have different levels of happiness
t crit (20 df) = 2.086
Juniors and seniors do have significantly different levels of happiness

115. Hyp 2: Seniors and Freshman will have different levels of happiness

116. Hyp 2: Seniors and Freshman will have different levels of happiness

117. Hyp 2: Seniors and Freshman will have different levels of happiness

118. Hyp 2: Seniors and Freshman will have different levels of happiness
t crit (20 df) = 2.086

119. Hyp 2: Seniors and Freshman will have different levels of happiness
t crit (20 df) = 2.086
Freshman and seniors do not have significantly different levels of happiness

120. Hyp 1: Juniors and Sophomores will have different levels of happiness
Hyp 2: Seniors and Sophomores will have different levels of happiness
PRACTICE!