Statistics 400 - Lecture 18

1. Statistics 400 - Lecture 18

2. Last Day: ANOVA Example, Paired Comparisons • Today: Re-visit boy’s shoes...Randomized Block Design

3. Example (Boys Shoes) • Company ran an experiment to determine if a new synthetic material is better than the existing one used for making the soles of boys' shoes • Experiment was run to see if the new, cheaper sole wears at the same rate at which the soles wear out

4. Example (Boys Shoes) • 10 boys were selected at random • Each boy was given a pair of shoes • Each pair had 1 shoe with the old sole (Sole A) and 1 shoe with the new sole (sole B) • For each pair of shoes, the sole type was randomly assigned to the right or left foot

5. Data

7. Can we use a 2-sample t-test or ANOVA here? • Would the 2-sample t-test or ANOVA detect a significant difference?

8. Paired or Matched Pairs T-test • Situation: • Two measurements made on same experimental unit • Compute difference (say B-A) in observation on the same experimental unit • Analyze differences using a 1-sample t-test • Because we analyze the differences using a 1-sample t-test, what must we assume about the difference?

9. Data

10. Analyzing the Data

11. Just looked at comparing means for two treatments applied to the same experimental unit (see boys’ shoes example) • Used a matched pairs T-test and analyzed the differences to see if there was a significant difference in the treatment means • When more than 2 treatments are applied to the same experimental unit, the experiment is called a randomized block experiment

12. Example • An experiment was performed to investigate the impact of soil salinity on the growth of salt marsh plants (C. Schwarz, 2001) • Plots of land at 4 agriculture field stations were used to grow plants in this environment • Six different amounts of salt (in ppm) are to be investigated • The plots of land were divided into 6 smaller plots • Each of the 6 smaller plots were treated with a different amount of salt and the bio-mass at the end of several months recorded

13. Data

14. The application of the 6 treatments to the smaller plots are done randomly • Like the Boys’ Shoes Example, each experimental unit has received more than 1 treatment • Here each unit receives 6 treatments

15. Plot of Bio-mass for Each Treatment

16. Plot of Bio-mass for each plot

17. Observations • Notice that the 4th plot gives smaller results than the other plots • Due to a block (plot effect) • Similar to the way a boy wears his shoes • Are the observations independent?

18. Randomized Block Design • Situation: • Have k treatments • Have b blocks • Each of the k treatments appears in each of the b blocks • The treatments within block are assigned to the within block units in random order

19. Structure of Data • Have k treatments in b blocks • Denote ith treatment from the jth block as yij

20. Model: • Model for comparing k treatments from a randomized block design: • for i =1, 2, …, k and j =1, 2, …, b • where is the overall mean, and • is the i th treatment effect • is the jth block effect • eij has a distribution • Want to test:

22. ANOVA Table for the Bio-Mass Example

23. What are the F-Tests?

24. Why do this?

25. General Rule: Block what you can, randomize what you cannot