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

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

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

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

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).

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

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

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

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

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

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

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

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

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

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

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)

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

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 8.5 7

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

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 8.5 7

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 8.5 7

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 8.5 7

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

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 8.5 7

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

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

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 9 9

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 9 9

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

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

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 7 9

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 7 9

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

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

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?

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

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

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

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

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

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

Practice A: NO B: NO AB: NO

Practice A: YES B: NO AB: NO

Practice A: NO B: YES AB: NO

Practice A: YES B: YES AB: NO

Practice A: YES B: YES AB: YES

Practice A: YES B: NO AB: YES

Practice A: NO B: YES AB: YES

Practice A: NO B: NO AB: YES

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

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

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

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

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

Formulas These formulas are conceptual formulas NOT computational formulas

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

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)2+ . . . . (1-2.06)2 = 30.94 *What makes this value get larger?

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)2+ . . . . (1-2.06)2 = 30.94 *What makes this value get larger? *The variability of the scores!

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

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) ((1.78-2.06)2+ (2.33-2.06)2)=1.36 *Note: it is multiplied by nb because that is the number of scores each mean is based on

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) ((1.78-2.06)2+ (2.33-2.06)2)=1.36 *What makes these means differ? *Error and the effect of A

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

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) ((3.17-2.06)2+ (2.00-2.06)2+ (1.00-2.06)2)= 14.16 *Note: it is multiplied by na because that is the number of scores each mean is based on

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) ((3.17-2.06)2+ (2.00-2.06)2+ (1.00-2.06)2)= 14.16 *What makes these means differ? *Error and the effect of B

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

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) ((2.67-2.06)2+ (1.00-2.06)2+. . . + (0.33-2.06)2)= 24.35

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) ((2.67-2.06)2+ (1.00-2.06)2+. . . + (0.33-2.06)2)= 24.35 What makes the cell means differ?

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

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

Sum of Squares SSAB = SScells - SSA – SSB 8.83 = 24.35 - 14.16 - 1.36

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 = 30.94 – (14.16 +1.36 + 8.83)

Sum of Squares SSWithin

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 + (0-2..33)2 = 6.667

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 + (0-2..33)2 = 6.667 *What makes these values differ from the cell means? *Error

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

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

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

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

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)

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

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

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

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.

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

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

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

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

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

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

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

Sociology Psychology Biology

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

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

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

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