1. Stat 470-3 Today: Will consider the one-way ANOVA model for comparing means of several treatments

2. Example Issue – shelf-life of pre-packaged meat Objective – Compare four different packaging methods. Are there differences? Which packaging is best? : 1.T1. commercial plastic wrap 2.T2. vacuum package 3.T3. 1%CO, 40%O 2, 59%N 4.T4. 100% CO 2 –Factor: Packaging –Experimental units: 12 steaks –Experimental Design: randomly assign 3 steaks to each packaging condition balanced completely randomized design with a = 4, n = 3 –Response: count of bacteria after 9 days at 4 o C (39 o F); y = log(bacteria count/cm 2 )

3. Data Analysis 1: Plot the Data Notation: k = 4 Treatments n i = 3 reps per Treatment N = 12 total observations Eyeball Analysis: Does it look like all of these data could come from the same distribution? Or from four different distributions?

4. Experiments with a Single Factor: Completely Random Design Objective: –Determine if the mean response of a factor is the same at all levels –If there is a difference, which levels differ? Method: –Have a single factors with k levels –N experimental units available for the experiment –N = n 1 + n 2 +…+n k –Randomly assign treatments to different experimental units –Conduct experiment –Results: y ij, i=1,…,k; j=1,…,n i

5. Experiments with a Single Factor: Completely Random Design Model:

6. Sums of Squares

7. Test Statistic

8. ANOVA Table

9. Summary We have found a statistic (F) which: –compares the variance among treatment means to the variance within treatments –has a known distribution when all the treatment means are equal By comparing this F statistic to the F(k-1, N-k) distribution, we evaluate the strength of the evidence against the assumption of equal underlying treatment means

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11. Interpretation:

12. Comment When a = 2 (two treatments), F for testing for no difference among treatments is equal to t 2 in the two-sample (unpaired) t-test –Out-of-Class Exercise. Demonstrate this equality by doing an ANOVA on the data in tomato plant problem. Compare percentiles in F and t tables…what do you observe? –For the mathematically inclined, demonstrate this equality algebraically

13. NOTE: It All Adds Up! It can be shown algebraically that Total SS = Treatment SS + Error SS Also, the degrees of freedom add up: N-1 = (k-1) + (N-k)

14. Exercise: Out-of-Class By using the formulas in the ANOVA table, verify the above ANOVA table for the meat packaging data

15. Estimation of Model Parameters: Constraints The model is over-parameterized Have k types of observation Have (k+1) parameters in the model –k for the treatment effects –1 for the grand mean Need to impose constraints to get solution

16. Constraints Sum to Zero Constraint: Interpretation:

17. Constraints Baseline Constraint: Interpretation:

18. Multiple Comparisons In previous example, we saw that there was a significant treatment effect…so what? If an ANOVA is conducted and the analysis suggests that there is a significant treatment effect, then a reasonable question to ask is

19. Multiple Comparisons Would like to see if there is a difference between treatments i and j Can use two-sample t-test statistic to do this For testing reject if Perform many of these tests

20. Multiple Comparisons Perform many of these tests Error rate must be controlled

21. Tukey Method Tests: Confidence Interval:

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