2. Six Easy Steps for an ANOVA 1) State the hypothesis 2) Find the F-critical value 3) Calculate the F-value 4) Decision 5) Create the summary table 6) Put answer into words

3. Example Want to examine the effects of feedback on self-esteem. Three different conditions -- each have five subjects 1) Positive feedback 2) Negative feedback 3) Control Afterward all complete a measure of self- esteem that can range from 0 to 10.

4. Example: Question: Is the type of feedback a person receives significantly (.05) related their self-esteem?

5. Results

6. Step 1: State the Hypothesis H 1: The three population means are not all equal H 0 :  pos =  neg =  cont

7. Step 2: Find F-Critical Step 2.1 Need to first find df between and df within Df between = k - 1 (k = number of groups) df within = N - k (N = total number of observations) df total = N - 1 Check yourself df total = Df between + df within

8. Step 2: Find F-Critical Step 2.1 Need to first find df between and df within Df between = 2 (k = number of groups) df within = 12 (N = total number of observations) df total = 14 Check yourself 14 = 2 + 12

9. Step 2: Find F-Critical Step 2.2 Look up F-critical using table F on pages 370 - 373. F (2,12) = 3.88

10. Step 3: Calculate the F-value Has 4 Sub-Steps 3.1) Calculate the needed ingredients 3.2) Calculate the SS 3.3) Calculate the MS 3.4) Calculate the F-value

11. Step 3.1: Ingredients  X  X 2  T j 2 N n

12. Step 3.1: Ingredients

13. XX  X p = 40  X n = 25  X c = 20  X = 85

14. X2X2  X p = 40  X n = 25  X c = 20  X = 85  X 2 = 555  X 2 p = 330  X 2 n = 135  X 2 c = 90

15. T 2 = (  X) 2 for each group  X p = 40  X n = 25  X c = 20  X = 85  X 2 = 555  X 2 p = 330  X 2 n = 135  X 2 c = 90 T 2 p = 1600 T 2 n = 625T 2 c = 400

16. Tj2Tj2  X p = 40  X n = 25  X c = 20  X = 85  X 2 = 555  T j 2 = 2625  X 2 p = 330  X 2 n = 135  X 2 c = 90 T 2 p = 1600 T 2 n = 625T 2 c = 400

17. N  X p = 40  X n = 25  X c = 20  X = 85  X 2 = 555  T j 2 = 2625 N = 15  X 2 p = 330  X 2 n = 135  X 2 c = 90 T 2 p = 1600 T 2 n = 625T 2 c = 400

18. n  X p = 40  X n = 25  X c = 20  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5  X 2 p = 330  X 2 n = 135  X 2 c = 90 T 2 p = 1600 T 2 n = 625T 2 c = 400

19. Step 3.2: Calculate SS  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5 SS total

20. Step 3.2: Calculate SS SS total 555 85 15 73.33  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5

21. Step 3.2: Calculate SS SS Within  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5

22. Step 3.2: Calculate SS SS Within 555 2625 5 30  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5

23. Step 3.2: Calculate SS SS Between  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5

24. Step 3.2: Calculate SS SS Between 2625 5 85 15 43.33  X = 85  X 2 = 555  T j 2 = 2625 N = 15 n = 5

25. Step 3.2: Calculate SS Check! SS total = SS Between + SS Within

26. Step 3.2: Calculate SS Check! 73.33 = 43.33 + 30

27. Step 3.3: Calculate MS

28. 43.33 2 21.67

29. Calculating this Variance Ratio

30. Step 3.3: Calculate MS 30 12 2.5

31. Step 3.4: Calculate the F value

32. 21.67 2.5 8.67 Step 3.4: Calculate the F value

33. Step 4: Decision If F value > than F critical –Reject H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0

34. Step 4: Decision If F value > than F critical –Reject H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0 F value = 8.67 F crit = 3.88

35. Step 5: Create the Summary Table

36. Step 6: Put answer into words Question: Is the type of feedback a person receives significantly (.05) related their self-esteem? H 1: The three population means are not all equal The type of feedback a person receives is related to their self-esteem

37. SPSS

39. Practice You are interested in comparing the performance of three models of cars. Random samples of five owners of each car were used. These owners were asked how many times their car had undergone major repairs in the last 2 years.

40. Results

41. Practice Is there a significant (.05) relationship between the model of car and repair records?

42. Step 1: State the Hypothesis H 1: The three population means are not all equal H 0 :  V =  F =  G

43. Step 2: Find F-Critical Step 2.1 Need to first find df between and df within Df between = 2 (k = number of groups) df within = 12 (N = total number of observations) df total = 14 Check yourself 14 = 2 + 12

44. Step 2: Find F-Critical Step 2.2 Look up F-critical using table F on pages 370 - 373. F (2,12) = 3.88

45. Step 3.1: Ingredients  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

46. Step 3.2: Calculate SS  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5 SS total

47. Step 3.2: Calculate SS SS total 304 60 15 64  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

48. Step 3.2: Calculate SS SS Within  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

49. Step 3.2: Calculate SS SS Within 304 1400 5 24  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

50. Step 3.2: Calculate SS SS Between  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

51. Step 3.2: Calculate SS SS Between 1400 5 60 15 40  X = 60  X 2 = 304  T j 2 = 1400 N = 15 n = 5

52. Step 3.2: Calculate SS Check! SS total = SS Between + SS Within

53. Step 3.2: Calculate SS Check! 64 = 40 + 24

54. Step 3.3: Calculate MS

55. 40 2 20

56. Calculating this Variance Ratio

57. Step 3.3: Calculate MS 24 12 2

58. Step 3.4: Calculate the F value

59. 20 2 10 Step 3.4: Calculate the F value

60. Step 4: Decision If F value > than F critical –Reject H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0

61. Step 4: Decision If F value > than F critical –Reject H 0, and accept H 1 If F value < or = to F critical –Fail to reject H 0 F value = 10 F crit = 3.88

62. Step 5: Create the Summary Table

63. Step 6: Put answer into words Question: Is there a significant (.05) relationship between the model of car and repair records? H 1: The three population means are not all equal There is a significant relationship between the type of car a person drives and how often the car is repaired

65. A way to think about ANOVA Make no assumption about H o –The populations the data may or may not have equal means

66. A way to think about ANOVA 2 4 6

67. A way to think about ANOVA The samples can be used to estimate the variance of the population Assume that the populations the data are from have the same variance It is possible to use the same variances to estimate the variance of the populations

68. S 2 =.50 S 2 = 5.0

69. A way to think about ANOVA

70. Assume about H o is true –The population mean are not different from each other They are three samples from the same population –All have the same variance and the same mean

72. 2,1,2,3,2,5,4,3,4,4,9,6,3,7,5

73. 2 4 6 2,1,2,3,2,5,4,3,4,4,9,6,3,7,5

74. A way to think about ANOVA For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution a N (sample size) get larger. Furthermore, the sampling distribution of the mean will have a mean equal to  and a standard deviation equal to  / N Central Limit Theorem

75. A way to think about ANOVA For any population of scores, regardless of form, the sampling distribution of the mean will approach a normal distribution a N (sample size) get larger. Furthermore, the sampling distribution of the mean will have a mean equal to  and a standard deviation equal to  / N Central Limit Theorem

76. A way to think about ANOVA Central Limit Theorem (remember) The variance of the means drawn from the same population equals the variance of the population divided by the sample size.

77. A way to think about ANOVA Can estimate population variance from the sample means with the formula *This only works if the means are from the same population

78. A way to think about ANOVA 2 4 6 S 2 = 4.00

79. A way to think about ANOVA

80. *Estimates population variance only if the three means are from the same population

81. A way to think about ANOVA *Estimates population variance regardless if the three means are from the same population

82. What do all of these numbers mean?

83. Why do we call it “sum of squares”? SS total SS between SS within Sum of squares is the sum the squared deviations about the mean

84. Why do we use “sum of squares”? SS are additive Variances and MS are only additive if df are the same

85. Another way to think about ANOVA Think in “sums of squares” Represents the SS of all observations, regardless of the treatment.

86. Another way to think about ANOVA Overall Mean= 4

87. Another way to think about ANOVA

88. Think in “sums of squares” Represents the SS deviations of the treatment means around the grand mean Its multiplied by n to give an estimate of the population variance (Central limit theorem)

89. Overall Mean= 4 2 4 6

90. Another way to think about ANOVA

91. Think in “sums of squares” Represents the SS deviations of the observations within each group

92. Overall Mean= 4 2 4 6

93. Another way to think about ANOVA

94. Sum of Squares SS total –The total deviation in the observed scores SS between –The total deviation in the scores caused by the grouping variable and error SS within –The total deviation in the scores not caused by the grouping variable (error)

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

97. 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

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

99. 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

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

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

102. 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?

103. 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

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

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

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

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

111. 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

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

113. 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

114. 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

115. Remember

116. Note

117. Proof CandyGender 5.001.00 4.001.00 7.001.00 6.001.00 4.001.00 5.001.00 1.002.002.00 3.002.00 4.002.00 3.002.002.00

119. t = 2.667 /.641 = 4.16

123. Also, when F has 1 df between

124. 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

125. Note: This formula is for equal n

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

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

130. t crit (20 df) = 2.086

131. 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

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

135. t crit (20 df) = 2.086

136. 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

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

139. Practice To investigate the maternal behavior of lab rats, we move the rat pup a fixed distance from the mother and record the time required for the mother to retrieve the pup. We run the study with 5, 20, and 35 day old pups. Figure out if 5 days is different than 35 days. SPSS Homework (Do the ANOVA analysis in SPSS – use output to answer question above) 5 days151025152018 20 days301520252320 35 days403550434540