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Review Compare one sample to another value One sample t-test

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Presentation on theme: "Review Compare one sample to another value One sample t-test"— Presentation transcript:

1

2 Review Compare one sample to another value One sample t-test
Compare two independent samples to each other Two sample independent t-test (equal or unequal n) Compare two dependent samples to each other Two sample dependent t-test Compare two or more independent samples to each other One-way ANOVA

3 Next. . . Notice all of these tests examine one VARIABLE at a time
What if you have two or more VARIABLES?

4 Study You are interested in if people like Pepsi and Coke differently. To examine this you give: 20 people regular Pepsi 20 people regular Coke You then ask them to rate how much they liked the soda (1 = do not like at all, 5 = like a lot). 3 designs we started with are bad

5 What kind of statistic could you use?
Two-sample t-test An ANOVA

6 But what if In addition to brand type you were also interested in examining diet vs. regular soda. To examine this you give: 20 people regular Pepsi 20 people regular Coke 20 people diet Pepsi 20 people diet Coke You then ask them to rate how much they liked the soda (1 = do not like at all, 5 = like a lot).

7 Factorial Design Research design that involves 2 or more Independent Variables Involves all combinations of at least 2 values of 2 or more IVs

8 Factorial Design Pepsi Coke Diet Regular 2 X 2 Factorial Design
Diet Pepsi Diet Diet Coke Regular Pepsi Regular Coke Regular 2 X 2 Factorial Design

9 Factorial Design: Influences on Ratings of Attractiveness
Does individuals’ gender or age influence their ratings of a woman’s attractiveness?

10 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Male Gender Female

11 Factorial Design: Influences on Ratings of Attractiveness
Age 2 X 2 Factorial Design Adult Adolescent Male Gender Female

12 Factorial Design: Influences on Ratings of Attractiveness
Relationship Status Single Dating Married Male Gender Female

13 Factorial Design: Influences on Ratings of Attractiveness
Relationship Status 2 X 3 Factorial Design Single Dating Married Male Gender Female

14 Factorial Design: Influences on Ratings of Attractiveness
Relationship Status Single Dating Married Age Adults Adolescents Male Gender Female

15 Factorial Design: Influences on Ratings of Attractiveness
Relationship Status Single Dating Married Age Adults Adolescents Male Gender Female 2 X 2 X 3 Factorial Design

16 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Male Gender Female 2 X 2 Factorial Design

17 Factorial Design: Influences on Ratings of Attractiveness
Rate the attractiveness of the woman in this picture on a scale from 1-10 (10 is most attractive)

18 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 Male Gender Average score of 9 Average score of 4 Female 2 X 2 Factorial Design

19 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 Male 9 Gender Average score of 9 Average score of 4 Female 6.5

20 Factorial Design: Main Effects
Main effects are the effects of one independent variable in an experiment (averaged over all levels of another independent variable)

21 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 Male 9 Gender Average score of 9 Average score of 4 Female 6.5

22 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 Male 9 Gender Average score of 9 Average score of 4 Female 6.5

23 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 Male 9 Gender Average score of 9 Average score of 4 Female 6.5

24 Factorial Design: Interactions
When the effect of one independent variable depends on the level of another independent variable

25 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 Male 9 Gender Average score of 9 Average score of 4 Female 6.5

26 Factorial Design: Influences on Ratings of Attractiveness
Males Females Adolescents Adults Male Female Age

27 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 Male Gender Average score of 10 Average score of 8 Female 2 X 2 Factorial Design

28 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 9 Male Gender Average score of 10 Average score of 8 Female 9

29 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 9 Male Gender Average score of 10 Average score of 8 Female 9

30 Factorial Design: Influences on Ratings of Attractiveness
Males Females Adolescents Adults Male Female Age

31 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 Male Gender Average score of 6 Average score of 8 Female 2 X 2 Factorial Design

32 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 Male 9 Gender Average score of 6 Average score of 8 Female 7

33 Factorial Design: Influences on Ratings of Attractiveness
Age Adult Adolescent Average score of 8 Average score of 10 Male 9 Gender Average score of 6 Average score of 8 Female 7

34 Factorial Design: Influences on Ratings of Attractiveness NO Interaction
Males Females Adolescents Adults Male Female Age

35 Factorial Design: Another Example
A researcher is interested in studying the effects of relationship status (single, cohabitating, married) and age (30s or 40s) on individuals’ ratings of satisfaction with life

36 Factorial Design: Another Example
A researcher is interested in studying the effects of relationship status (single, cohabitating, married) and age (30s or 40s) on individuals’ ratings of satisfaction with life What is the Dependent Variable? What are the Independent Variables? What kind of a design is this?

37 Factorial Design: Another Example
A researcher is interested in studying the effects of relationship status (single, cohabitating, married) and age (30s or 40s) on individuals’ ratings of satisfaction with life This is the data that is collected (average scores per group with scores ranging from 1 –10, most satisfied): Relationship Status Single Cohab Married 10 8 9 30s Age 9 7 40s 8

38 Factorial Design: Another Example
Age 30s 40s Single Cohab Married Relationship Status 8 9 10 7 9 8 8.5 8.5 8.5

39 Factorial Design: Another Example
Age 30s 40s Single Cohab Married Relationship Status 8 9 10 7 9 8 8.5 8.5 8.5

40 Factorial Design: Another Example
Age 30s 40s Single Cohab Married Relationship Status 8 9 10 7 9 8 8.5 8.5 8.5

41 Factorial Design: Another Example
A researcher is interested in studying the effects of marital status (single, cohabitating, married) and age (30s or 40s) on individuals’ ratings of satisfaction with life This is the data that is collected (average scores per group with scores ranging from 1 –10, most satisfied): 30s 40s single cohab married

42 Practice 2 x 2 Factorial Determine if 1) there is a main effect of A
2) there is a main effect of B 3) if there is an interaction between AB

43 Practice A: NO B: NO AB: NO

44 Practice A: YES B: NO AB: NO

45 Practice A: NO B: YES AB: NO

46 Practice A: YES B: YES AB: NO

47 Practice A: YES B: YES AB: YES

48 Practice A: YES B: NO AB: YES

49 Practice A: NO B: YES AB: YES

50 Practice A: NO B: NO AB: YES

51

52 What if. . . You were asked to determine the effects of both college major (psychology, sociology, and biology) and gender (male and female) on class attendance You now have 2 IVs and 1 DV Can examine Main effect of gender Main effect of college major Interaction between gender and college major

53 Sociology Psychology Biology Female 2.00 1.00 3.00 .00 Males 4.00
n = 3 N = 18

54 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 Main effect of gender

55 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 Main effect of major

56 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 Interaction between gender and major

57 Formulas These formulas are conceptual formulas NOT computational formulas

58 Sum of Squares SS Total The total deviation in the observed scores

59 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SStotal = (2-2.06)2+ (3-2.06) (1-2.06)2 = 30.94 *What makes this value get larger?

60 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SStotal = (2-2.06)2+ (3-2.06) (1-2.06)2 = 30.94 *What makes this value get larger? *The variability of the scores!

61 Sum of Squares SS A Represents the SS deviations of the treatment means around the grand mean

62 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSA = (3*3) (( )2+ ( )2)=1.36 *Note: it is multiplied by nb because that is the number of scores each mean is based on

63 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSA = (3*3) (( )2+ ( )2)=1.36 *What makes these means differ? *Error and the effect of A

64 Sum of Squares SS B Represents the SS deviations of the treatment means around the grand mean

65 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSB = (3*2) (( )2+ ( )2+ ( )2)= 14.16 *Note: it is multiplied by na because that is the number of scores each mean is based on

66 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSB = (3*2) (( )2+ ( )2+ ( )2)= 14.16 *What makes these means differ? *Error and the effect of B

67 Sum of Squares SS Cells Represents the SS deviations of the cell means around the grand mean

68 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSCells = (3) (( )2+ ( ) ( )2)= 24.35

69 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSCells = (3) (( )2+ ( ) ( )2)= 24.35 What makes the cell means differ?

70 Sum of Squares SS Cells What makes the cell means differ? 1) error
2) the effect of A (gender) 3) the effect of B (major) 4) an interaction between A and B

71 Sum of Squares Have a measure of how much cells differ
SScells Have a measure of how much this difference is due to A SSA Have a measure of how much this difference is due to B SSB What is left in SScells must be due to error and the interaction between A and B

72 Sum of Squares SSAB = SScells - SSA – SSB 8.83 =

73 Sum of Squares SSWithin SSWithin = SSTotal – (SSA + SSB + SSAB)
The total deviation in the scores not caused by 1) the main effect of A 2) the main effect of B 3) the interaction of A and B SSWithin = SSTotal – (SSA + SSB + SSAB) 6.59 = – ( )

74 Sum of Squares SSWithin

75 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSWithin = ((2-2.67)2+(3-2.67)2+(3-2.67)2) ((1-.33)2 + (0-.33)2 + ( )2 = 6.667

76 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 SSWithin = ((2-2.67)2+(3-2.67)2+(3-2.67)2) ((1-.33)2 + (0-.33)2 + ( )2 = 6.667 *What makes these values differ from the cell means? *Error

77 Compute df Source df SS A 1.36 B 14.16 AB 8.83 Within 6.59 Total 30.94

78 Source df SS A 1.36 B 14.16 AB 8.83 Within 6.59 Total 17 30.94
dftotal = N - 1

79 Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 6.59 Total 17 30.94
dftotal = N – 1 dfA = a – 1 dfB = b - 1

80 Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 6.59 Total 17 30.94
dftotal = N – 1 dfA = a – 1 dfB = b – 1 dfAB = dfa * dfb

81 Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 12 6.59 Total 17 30.94
dftotal = N – 1 dfA = a – 1 dfB = b – 1 dfAB = dfa * dfb dfwithin= ab(n – 1)

82 Compute MS Source df SS A 1 1.36 B 2 14.16 AB 8.83 Within 12 6.59
Total 17 30.94

83 Compute MS Source df SS MS A 1 1.36 B 2 14.16 7.08 AB 8.83 4.42 Within
12 6.59 .55 Total 17 30.94

84 Compute F Source df SS MS A 1 1.36 B 2 14.16 7.08 AB 8.83 4.42 Within
12 6.59 .55 Total 17 30.94

85 Test each F value for significance
Source df SS MS F A 1 1.36 2.47 B 2 14.16 7.08 12.87 AB 8.83 4.42 8.03 Within 12 6.59 .55 Total 17 30.94 F critical values (may be different for each F test) Use df for that factor and the df within.

86 Test each F value for significance
Source df SS MS F A 1 1.36 2.47 B 2 14.16 7.08 12.87 AB 8.83 4.42 8.03 Within 12 6.59 .55 Total 17 30.94 F critical A (1, 12) = 4.75 F critical B (2, 12) = 3.89 F critical AB (2, 12) = 3.89

87 Test each F value for significance
Source df SS MS F A 1 1.36 2.47 B 2 14.16 7.08 12.87* AB 8.83 4.42 8.03* Within 12 6.59 .55 Total 17 30.94 F critical A (1, 12) = 4.75 F critical B (2, 12) = 3.89 F critical AB (2, 12) = 3.89

88 Interpreting the Results
Source df SS MS F A 1 1.36 2.47 B 2 14.16 7.08 12.87* AB 8.83 4.42 8.03* Within 12 6.59 .55 Total 17 30.94 F critical A (1, 12) = 4.75 F critical B (2, 12) = 3.89 F critical AB (2, 12) = 3.89

89 Interpreting the Results
Source df SS MS F A 1 1.36 2.47 B 2 14.16 7.08 12.87* AB 8.83 4.42 8.03* Within 12 6.59 .55 Total 17 30.94 F critical A (1, 12) = 4.75 F critical B (2, 12) = 3.89 F critical AB (2, 12) = 3.89

90 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j 2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06

91 Interpreting the Results
Source df SS MS F A 1 1.36 2.47 B 2 14.16 7.08 12.87* AB 8.83 4.42 8.03* Within 12 6.59 .55 Total 17 30.94 F critical A (1, 12) = 4.75 F critical B (2, 12) = 3.89 F critical AB (2, 12) = 3.89

92 Sociology Psychology Biology Mean Female 2.00 1.00 3.00 .00 Mean1j
2.67 1.67 1.78 Males 4.00 Mean2j Mean.j 3.67 3.17 0.33 2.33 2.06 Want to plot out the cell means

93 Sociology Psychology Biology

94

95 Practice These are sample data from Diener et. al (1999). Participants were asked their marital status and how often they engaged in religious behavior. They also indicated how happy they were on a scale of 1 to 10. Examine the data

96 Frequency of religious behavior
Never Occasionally Often Married 6 3 7 2 8 4 5 9 Unmarried 1

97 Interpreting the Results
Source df SS MS F Married 18.00 Relig 31.00 AB 3.00 Within Total 88.00

98 Interpreting the Results
Source df SS MS F Married 1 18.00 6.00* Relig 2 31.00 15.50 5.17* AB 3.00 1.50 .50 Within 12 36.00 Total 17 88.00 F critical A (1, 12) = 4.75 F critical B (2, 12) = 3.89 F critical AB (2, 12) = 3.89

99

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