Analysis of Variance: Part 1

Transcription

1 Analysis of Variance: Part 1 Oneway ANOVA

2 When there are more than two means Each time two means are compared the probability (Type I error) =α.

3 When there are more than two means Each time two means are compared the probability (Type I error) =α. X 1 X 2 P(Type I error)=.05

4 When there are more than two means Each time two means are compared the probability (Type I error) =α. X 1 X 2 X 3 P(Type I error)=

5 When there are more than two means Each time two means are compared the probability (Type I error) =α. X 1 X 2 X 3 P(Type I error)= =.15

6 When there are more than two means Each time two means are compared the probability (Type I error) =α. X 1 X 2 X 3 X 4 P(Type I error)=

7 When there are more than two means Each time two means are compared the probability (Type I error) =α. X 1 X 2 X 3 X 4 P(Type I error)=

8 When there are more than two means Each time two means are compared the probability (Type I error) =α. X 1 X 2 X 3 X 4 P(Type I error)= =.30

9 When there are more than two means Each time two means are compared the probability (Type I error) =α. X 1 X 2 X 3 X 4 X 5 P(Type I error)= =.50

10 Protection level Analysis of Variance protects from inflating Type I errors by making the experiment-wise Probability (Type 1 Error) < α.

11 Partitioning the Variance Total Variance

12 Partitioning the Variance Total Variance Between Groups Within Groups

13 Between Groups Variance The variance in the data that can be attributed to the independent variable. The variance among the means.

14 Within Groups Variance Variance due to all other sources. Subject factors Error variance Residual variance Variances among data and group means in each group.

15 F-Ratio F= Between Groups Variance Within Groups Variance

16 Assumptions Treatments are Independent Dependent Variable is measured on at least an ordinal scale Dependent Variable is normally distributed

17 When to use Between Groups ANOVA Different Subjects are in each treatment. There are 2 means or more to compare. (Can use for 2 groups: t is easier)

18 How to set up the ANOVA Summary Table Source

19 How to set up the ANOVA Summary Table Source Total

20 How to set up the ANOVA Summary Table Source Between Within Total

21 How to set up the ANOVA Summary Table Source Between Sums of Squares SS B Within SS W Total SS T

22 How to set up the ANOVA Summary Table Source Sums of Squares df Between SS B df B Within SS W df W Total SS T df T

23 How to set up the ANOVA Summary Table Source Sums of Squares df Mean Squares Between SS B df B MS B Within SS W df W MS W Total SS T df T

24 How to set up the ANOVA Summary Table Source Sums of Squares df Mean Squares F Between SS B df B MS B F Within SS W df W MS W Total SS T df T

25 Calculating the F Statistic Calculate Sums of Squares Calculate df Calculate Mean Squares Calculate F

26 How to Calculate the Sums of Squares Between Source Sums of Squares Between SS B ( X ) n 2 2 T N Within SS W Total SS T

27 How to Calculate the Sums of Squares Between Source Sums of Squares Sum of scores in each group Between SS B ( X ) n 2 2 T N Within SS W Total SS T

28 How to Calculate the Sums of Squares Between Source Between Within Sums of Squares SS B SS W T=Total= Sum of all scores. ( X ) n 2 2 T N Total SS T

29 How to Calculate the Sums of Squares Between Source Sums of Squares Between SS B ( X ) n 2 2 T N Within SS W n=number of scores/group Total SS T

30 How to Calculate the Sums of Squares Between Source Sums of Squares Between SS B ( X ) n 2 2 T N Within N= Total number of scores SS W Total SS T

31 How to Calculate the Sums of Squares Total Source Sums of Squares Between SS B Within SS W Total SS T X 2 T N 2

32 How to Calculate the Sums of Squares Total Source Between Sums of Squares SS B Within SS W Sum of all scores squared first Total SS T X 2 T N 2

33 How to Calculate the Sums of Squares Within Source Sums of Squares Between SS B ( X ) n 2 2 T N Within SS W SS T -SS B Total SS T X 2 T N 2

34 How to Calculate the Degrees of Freedom Source Sums of Squares df Between SS B df B = k-1 Within SS W df W Total SS T df T

35 How to Calculate the Degrees of Freedom Source Sums of Squares df k is the number of groups Between SS B df B = k-1 Within SS W df W Total SS T df T

36 How to Calculate the Degrees of Freedom Source Sums of Squares df Between SS B df B = k-1 Within SS W df W = k(n-1) Total SS T df T

37 How to Calculate the Degrees of Freedom Source Sums of Squares df Between SS B df B = k-1 Within SS W df W = k(n-1) Total SS T df T = N - 1

38 How to Calculate the Mean Squares Source Sums of Squares df Mean Squares Between SS B df B MS B Within SS W df W MS W Total SS T df T

39 How to set up the ANOVA Summary Table Source Sums of Squares df Mean Squares F Between SS B df B MS B = F MS W Within SS W df W Total SS T df T

40 Determining the Critical F Alpha =.05 Find Column for df Between Find Row for df Within Compare Critical F to Obtained F

41 Statistical Decision Making If Critical F > Obtained F failed to reject null hypothesis If Critical F < Obtained F reject the null hypothesis

42 Interpreting the Results Graph Means Use a multiple Comparison test to determine which means are significantly different

43 Example: The Effects of Mood on Originality Scott Halam s Senior Thesis, 1997 H 1 : Positive mood will facilitate creativity more than negative mood or neutral mood. H 1: X positive > X Neutral =X Negative

44 A Portion 1 Data from Scott s Study Originality Scores Only female participants. 90

45 How to set up the ANOVA Summary Table:Example Source Mood Between Within Sums of Squares SS B SS W Total SS T

46 How to Calculate the Sums of Squares Within: Find the parts Source Sum scores for each Sums of group. Squares Total of all scores. Mood Between Within Total SS B Number in Number SS SS W T -SS each group B All scores squared, SS T ( n X 2 T 2 2 X N 2 ) T N Total then summed.

47 A Portion 1 Data from Scott s Study X n = = = = ( n X ) 2 T N

48 A Portion 1 Data from Scott s Study X n = T = N = = = = T 2 N ( )

49 Finishing SS B Source Sums of Squares Between Mood SS B = = Within SS W Total SS T

50 How to Calculate the Sums of Squares Total: Example Source Sums of Squares Between SS B Within SS W Total SS T X 2 T N 2

51 Finding SS w X T N