1. One-Way Analysis of Variance (ANOVA) 2011, 12, 1

2. ANOVA: Compare 3 or More Groups When we wanted to compare differences between two independent groups, we used an independent groups t- test However, if we have 3 or more groups the technique we use is called analysis of variance (ANOVA)  E.g., class standing has 4 levels: Freshman, Sophomore, Junior, Senior Instead of a t-obs, we calculate an F-obs

3. Pain Relief Experiment Aspirin, Tylenol, and a Herb were tested to see how much pain relief each provides. Pain relief was rated on a five-point scale. The differences between our three samples vs. the differences within each single sample. If variance-between > variance within, we know that Aspirin, Tylenol, and Herb have different pain-reducing effects. AspirinTylenolHerb 322 521 343 542 4

4. Step 1: Form Hypotheses H 0 : H 1 :

5. Step 2: Set Decision Criteria  = 0.05 df b = df w = f crit =

7. Step 3: Compute F

8. Aspirin (n 1 = 4)Tylenol (n 2 = 5)Herb (n 3 = 4) Y1Y1 Y12Y12 Y2Y2 Y22Y22 Y3Y3 Y32Y32 392424 5252411 3941639 52541624 4  Y 1 =16  Y 1 2 =68  Y 2 =16  Y 2 2 =56  Y 3 =8  Y 3 2 =18 Y 1 = 4SSW 1 = 4 Y 2 = 3.2SSW 2 = 4.8 Y 3 = 2SSW 3 = 2

9. Compute SSB and MSB (numerator) Y = MSB = Aspirin (n 1 = 4)Tylenol (n 2 = 5)Herb (n 3 = 4) Y1Y1 Y12Y12 Y2Y2 Y22Y22 Y3Y3 Y32Y32 392424 5252411 3941639 52541624 4  Y 1 =16  Y 1 2 =68  Y 2 =16  Y 2 2 =56  Y 3 =8  Y 3 2 =18 Y 1 = 4SSW 1 = 4 Y 2 = 3.2SSW 2 = 4.8 Y 3 = 2SSW 3 = 2

10. Compute SSW and MSw (Denominator) MSW = Aspirin (n 1 = 4)Tylenol (n 2 = 5)Herb (n 3 = 4) Y1Y1 Y12Y12 Y2Y2 Y22Y22 Y3Y3 Y32Y32 392424 5252411 3941639 52541624 4  Y 1 =16  Y 1 2 =68  Y 2 =16  Y 2 2 =56  Y 3 =8  Y 3 2 =18 Y 1 = 4SSW 1 = 4 Y 2 = 3.2SSW 2 = 4.8 Y 3 = 2SSW 3 = 2

11. Step 4: Create ANOVA Source Table SourceSSdfMSF Between8.124.053.75 Within10.08101.08 Total18.912 SST = SSB+SSW = F = MSB/MSW =

12. The distribution of F-ratios with df = 2, 10 4.107.56 1 2 3 4 5 6 7 Step 5: Compare F obs to F crit.