Lecture 12 Factorial Design PDF

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Instructor: Karen Lawson...

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Factorial Designs ● Do two or more variables interact to produce an effect bigger than either one could have alone? ● Always include two or more IV (factors) to examine if and how they interact ● Essentially two or more single-factor studies mashed into one ● By testing more than one IV at a time, we can look at the interactive effects of IVs (how they are working together) ● Most IVs in psychology interact w/ other IVs, rarely just a one to one relationship Main Effects and Interactions ● Main Effect = the unique effect of each of the IVs alone on the DV ● Interaction Effect = the combined effect of two or more IV on the DV ● You know you have an interaction when: ○ !!! The influence of one IV depends on the level of of the other IV !!! ○ AKA interactions qualify the main effects ○ => The heart of interaction Example: The Influence of icy roads (IV1) and speeding (IV2) on car accidents (DV) ● Accidents are 3 times more likely to occur when roads are icy ● Accidents are 2 times more likely to occur when a diver is speeding ● If no interaction (IV1 will not depend on IV2) ○ then accidents will be 6 times more likely to occur when a driver is speeding on icy roads ○ Consistent effect in each level of IV (2x or 3x) Factor B / Factor A B1: Not speeding B2: Speeding ●

A1: Normal road condition

A2: Icy Road Condition

5

5 x 3 = 15

5 x 2 = 10

15 x 2 = 30 10 x 3 = 30

If there is an interaction: ○ Accidents are 15 times more likely to occur when a driver is speeding on icy roads ○ The impact of road condition depends on whether a driver is speeding or not ■ If not speeding, 3x ■ If speeding, 7.5x Factor B / Factor A B1: Not speeding B2: Speeding

A1: Normal road condition

A2: Icy road condition

5

5 x 3 = 15

5 x 2 = 10

10 x 7.5 = 75

Example: Children’s fear of the dark ● Research Question ○ Could it be that children’s fear are triggered by a combination of being in a dark

● ●

●

●

● ●

environment and having seen scary images? (Maybe dark is a necessary but not sufficient factor for fear to occur?) ○ Does these two variables (darkness & images) interact to influence children’s fears? Hypothesis ○ Dark condition interacts with images to impact children’s fears Matrix of Cells Factor B / Factor A

A1: Light Condition

A2: Dark Condition

B1: Fearful Images

A1B1

A2B1

B2: Neutral Images

A1B2

A2B2

Method ○ Randomly assign children to each treatment condition (each cell) ○ Between Subjects design - each child can only be in one treatment condition ○ Factorial Notation: this is a 2 x 2 factorial design, w/ 2 IV each has 2 levels ○ To operationally define “fear”, measure heart rate after being exposed to stimuli Main Effects of Factor A / Illumination ○ Compare column mean of A1 & A2 (between light & dark) ○ Pretend the image type wasn’t there at all ○ Like the exact result you would get if you just had one IV: illumination Main Effects of Factor B / Images ○ Compare row mean of B1 & B2 (between fearful & neutral images) Interaction Effect / The 2 x 2 Matrix ○ Analyze not only column mean & row mean, but also each cell means in the matrix ○ Statiscically 3 null hypothesis we have to test: ■ Main effect of factor A: Does illumination impact fear ■ Main effect of factor B: Does image type impact fear ■ Interactive effect: Does impact of illumination depend on impact of image type

Interpreting Factorial Designs (the hardest part) (definitely will be on exams) 1) Map the cells of the design and the total N A1: Light Condition

A2: Dark Condition

B1: Fearful Images

N = 10

N = 10

B2: Neutral Images

N = 10

N = 10

a) 40 in total 2) Check the ANOVA output table for significance Source

df

SS

MS

F

p

Factor A (Illumination)

1

756.62

765.62

7.88

.008

Factor B (Image Type)

1

525.62

525.62

5.41

.026

AB Interaction

1

497.02

497.02

5.12

.030

Error

36

3497.50

97.15

39

5285.78

135.53

Total

a) Source column: source of variation due to Factor A, B, Interaction, and Error b) df column: i) df for A, B interaction= # of levels - 1 = 2 - 1 = 1 (1) Tells us how many levels there are ii) df for Error = # of participants - # of treatment conditions = 40 - 4 = 36 (1) Tells us how many treatment conditions (cells) there are iii) df for total = # of participants - 1 = 40 - 1 = 39 c) Mean Square = SS / df d) F = between / within e) P value: associated w/ each F value, compared w/ the alpha level to make a decision i) p < .05 significant, reject null ii) p > .05 insignificant, fail to reject null f) Interpretation: i) Significant main effect of factor A ii) Significant main effect of factor B iii) Significant interaction effect of A & B iv) !!! If you have a significant interaction effect, you cannot interpret main effect before interpreting the interaction effect first !!! v) Main effects are qualified in interactions 3) Map cell, row, and column means A1: Light Condition

A2: Dark Condition

Row Mean

B1: Fearful Images

98.3

114.1

106.2

B2: Neutral Images

98.1

99.9

99.0

Column Mean

98.2

107.0

a) 2 x 2 design is like 2 studies combined together b) By collapsing across each factor, we can look at main effects c) You can’t jump in and interpret the main effects first without taking into account of the interaction effect d) You can’t compare column mean and just say “Children in the dark have higher heart rate” because that main effect does not hold at each level of the IV image type. That statement is not true under neutral images condition e) Similarly, you can’t say “Children who had seen fearful images have higher heart rate” because it does not hold in the light condition

f) We have to interpret the interaction effect first from examining the cell means 4) Graph the results a) The graph should be a bar graph b/c light and dark is categorical data, instead we use a line across data points for pragmatic ease b) Y axis: DV c) X axis: IV1, colour of line: IV2 d) Variable w/ higher # of levels goes on X axis (just to minimize # of lines in the graph) e) You can see there’s an interaction b/c the lines are not parallel (they don’t have to necessarily cross to have an interaction) f) Visualize the difference 5) Conduct post-hoc tests to interpret interaction a) Compare the effect of lighting across both image conditions: i) Is there a difference between the lighting conditions when a child sees scary images? (1) YES – those in dark room have more fear (98.3 vs 114.1) ii) Is there a difference between the lighting conditions when a child sees neutral images? (1) NO – they have the same amount of fear (98.1 vs 99.9) b) Compare the effect of images across both lighting conditions: i) Is there a difference between the images conditions when a child is in a lighted room? (1) NO – they have the same amount of fear (98.3 vs 98.1) ii) Is there a difference between the images conditions when a child is in a dark room? (1) YES – those who see scary images have more fear (114.1 vs 99.9) c) The impact of lighting conditions depends on the impact of images 6) Determine if interaction qualifies main effects a) The ANOVA table indicated i) Main effect for illumination (1) More fear in dark condition (2) But only when they see scary images ii) Main effect for image (1) More fear in scary image condition (2) But only when they are in the dark 7) Describe the results a) Results suggest that children have more fear when they are in the dark but only when they are also confronted w/ scary images b) Children have more fear when they are confronted w/ scary images but only when they are in the dark ●

Factorial ANOVA ○ ANOVA can analyze an factorial design ○ # of effects will depend on the # of IVs ■ 2 IVs: main effect A, main effect B, interaction AXB

■ ■ ■ ■

Stops here for this class 3 IVs: main effect A, B, C, interactions AB, AC, BC (two-way interactions), ABC (three-way interactions) Limit to interpretation cognitive capacity stops here 4 IVs: 4 main effects, 6 two-way interactions, 3 three-way interactions, 1 four-way interactions

Practice Problems 1. Indicate how many IV and how many levels a. 2 x 2: 2 IV, 2 level each b. 2 x 3: 2 IV, 2 level in 1, 3 level in 2 c. 2 x 3 x 2: 2 IV, 2 level, 3 level, 2 level d. 4 x 3 x 2 x 3: 4 IVs, 4 level, 3 level, 2 level, 2 level 2. Cognitive distraction and the ability to withstand pain for men and women a. IV: sex (2 level), distraction level (4 level), DV: time in cold water, 2 x 4 between subjects design b. 80 participants c. ANOVA Table: Alpha = .05 unless stated otherwise. Significant Main effect for sex, significant interaction effect, no significant main effect for distraction d. The impact of distraction on pain tolerance depends on the impact of sex i. (try to fill in the blank based on the research hypothesis given) ii. Huge implication on how we visualize the data iii. Research question: does the impact of distraction on pain tolerance depends on sex? e. Main effect of A is qualified at each level of the other IV (B), impact of A depends on different levels of B f. Interpretation: i. The impact of distraction on pain tolerance does depend on sex ii. What is that impact? 1. Women become more tolerant to pain w/ increased distraction 2. Men become less tolerant to pain w/ increased distraction 3. Impact of therapy and prozac on emotional well-being a. IV: Therapy (2 level), Prozac (2 level). DV: emotional well-being Placebo

Prozac

Column mean

No Therapy

14

22

18

Therapy

26

18

22

Row mean

20

20

b. Significant main effect of therapy, no main effect of prozac, significant interaction effect c. Graph 2 x 2 design, doesn’t really matter which IV is on the x-axis, usually put the main IV in the x-axis, and the moderator variable (2nd) in coloured lines; in this case, therapy on x-axis, prozac on the lines; graph is really just a

interpretation tool for humans d. The impact of therapy on well-being depends on whether or not a client is taking prozac i. Only When ppl who are taking placebo, therapy seem to be decreasing well being ii. Therapy enhances well being only when ppl are taking a prozac...