Lecture 11: Two Way Analysis of Variance

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Tracey Weaver
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1 Lecture 11: Two Way Analysis of Variance Review: Hypothesis Testing o ANOVA/F ratio: comparing variances o F = s variance between treatment effect + chance s variance within sampling error (chance effects) o Remember, null hypotheses assumes there s NO treatment effect o Therefore your equation becomes chance divided by chance What s something divided by itself? 1 Two or More IVs / Factorial Design: more than one manipulation Factor Independent Variable Allows you to consider multiple independent variables that might be interacting with each other and effect your results E.g. Consider, what things could have gotten in the way of you getting to class today? Weather, car troubles, illness, personal problems, apathy, etc There s virtually endless reasons & each one counts as a variable Can handle any number of variables and any number of treatments Gives you a better picture of reality + IVs with + treatments per IV; the most complex design Can handle any number of IVs & any number of treatments for those IVs E.g.: You want to compare different methods of weight loss, diet and exercise. IV: Pritikans DV: weight loss IV: Exercise Aerobics

2 Exercise Exercise Brittany s notes 4/0/17 X Factorial Each represents an IV The number inside each represents the number of treatments for that IV (WT) (AT) AT + WT Pritikans (PR) PR + WT Aerobics (AE) AT + AE PR + AE How do you determine how many groups you need? Multiply _#_ X _#_ diet X exercise: X = 4 groups i.e: AT+WT ; AT+AE ; PR+WT ; PR+AE all possible treatment combinations What if you add another IV (like metabolism)? add another space # X # X # diet X exercise X metabolism What if each has treatments? X X = 8 groups What if diet has 3 treatments, exercise has 3 treatments, and metabolism has treatments? _3_ X _3_ X = 18 groups How do you determining if there is a main effect of the IVs? Compare the averages of the treatments for each IV A yes or no question, asked for each IV. Is there a main effect of diet? Do a column comparison: Add values in columns then divide by the # of treatments you added Pritikans 7 lb loss 10 lb loss Aerobics 15 lb loss 1 lb loss = = 11

3 DV: loss Exercise Brittany s notes 4/0/17 NO, there is not a main effect of diet Becausec the averages are the same If the averages were different, there WOULD be a main effect Is there a main effect of exercise? Do a row comparison Add values in each then divide by the # of treatments you added Pritikins 7 lb loss 10 lb loss Aerobics 15 lb loss 1 lb loss YES, there is a main effect of exercise B/c the averages are different = 8.5 = 13.5 Interaction effect: determining if the IVs are interacting with each other A yes or no question, asked for each possible interaction How do you determine if there s an interaction effect? Graph Choose either IV to label the X axis (it doesn t matter which you choose, the other IV will be plotted as data points) Is there an interaction effect between diet and exercise? Pritikins Pritikins 6 4 Aerobics YES, there is an interaction effect of diet & exercise.

4 When is there they no intersecting/ no interaction effect? When the lines are parallel or coinciding (on top of each other) Hypothesis Testing for ANOVA o Diagram EACH possible treatment effect, including interactions o Hypotheses ALWAYS written as non-directional

5 Class DEMO: Two-way ANOVA (F ratio) Part 1 IV: Pritikans Is there a main effect of diet? yes/no Explanations H1: makes a difference H0: does not make a difference Prob. Calc. outcomes HIGH Probability α =.05 LOW probability decisions Accept H0 Reject H0, accept H1 Is there an interaction effect? yes/no Explanations H1: There is an interaction effect H0: There is not an interaction effect Prob. Calc. outcomes HIGH Probability α =.05 LOW probability decisions Accept H0 Reject H0, accept H1 IV: Exercise Aerobics Is there a main effect of diet? yes/no Explanations H1: Exercise makes a difference H0: Exercise does not make a difference Prob. Calc. outcomes HIGH Probability α =.05 LOW probability decisions Accept H0 Reject H0, accept H1

6 Create Table of Variance Source SS df s or MS F ratio Between Groups 3 A (rows): Exercise 1 B (column): 1 AXB interaction: Exercise X 1 Within Groups 36 Total 39 Determine CRITICAL REGIONS 1. dfbetween = k 1 = 4 1 = 3. dfwithin = (n1 + n + n3 + n4) k = ( ) 4 = 40 4 = 36 denominator 3. dfexercise = (# of levels/treatments of A) 1 = 1 = 1 numerator 4. df = (# of levels/treatments of B) 1 = 1 = 1 numerator 5. dfexercisex consider relationships (refer to formula sheet) a. dfbetween = dfexercise + df + dfexercisex 3 = numerator 6. Total = dfbetween + dfwithin = 39 Now go to statistical table to determine critical F values for each: row, column, & interaction then Graph each

7 Row: F critical (1, 36) = 4.11 Column: F critical (1, 36) = 4.11 Interaction: F critical (1, 36) = 4.11 Continue to Part

Difference in two or more average scores in different groups

ANOVAs Analysis of Variance (ANOVA) Difference in two or more average scores in different groups Each participant tested once Same outcome tested in each group Simplest is one-way ANOVA (one variable as