Balanced Incomplete Block Design

1. Balanced Incomplete Block Design Ford Falcon Prices Quoted by 28 Dealers to 8 Interviewers (2 Interviewers/Dealer) Source: A.F. Jung (1961). "Interviewer Differences Among Automile Purchasers," JRSS-C (Applied Statistics), Vol 10, #2, pp. 93-97

2. Balanced Incomplete Block Design (BIBD) • Situation where the number of treatments exceeds number of units per block (or logistics do not allow for assignment of all treatments to all blocks) • # of Treatments  g • # of Blocks  b • Replicates per Treatment  r < b • Block Size  k < g • Total Number of Units  N = kb = rg • All pairs of Treatments appear together in l = r(k-1)/(g-1) Blocks for some integer l

3. BIBD (II) • Reasoning for Integer l: • Each Treatment is assigned to r blocks • Each of those r blocks has k-1 remaining positions • Those r(k-1) positions must be evenly shared among the remaining g-1 treatments • Tables of Designs for Various g,k,b,r in Experimental Design Textbooks (e.g. Cochran and Cox (1957) for a huge selection) • Analyses are based on Intra- and Inter-Block Information

4. Interviewer Example • Comparison of Interviewers soliciting prices from Car Dealerships for Ford Falcons • Response: Y = Price-2000 • Treatments: Interviewers (g = 8) • Blocks: Dealerships (b = 28) • 2 Interviewers per Dealership (k = 2) • 7 Dealers per Interviewer (r = 7) • Total Sample Size N = 2(28) = 7(8) = 56 • Number of Dealerships with same pair of interviewers: l = 7(2-1)/(8-1) = 1

5. Interviewer Example

6. Intra-Block Analysis • Method 1: Comparing Models Based on Residual Sum of Squares (After Fitting Least Squares) • Full Model Contains Treatment and Block Effects • Reduced Model Contains Only Block Effects • H0: No Treatment Effects after Controlling for Block Effects

7. Least Squares Estimation (I) – Fixed Blocks

8. Least Squares Estimation (II)

9. Least Squares Estimation (III)

10. Analysis of Variance (Fixed or Random Blocks)

11. ANOVA F-Test for Treatment Effects Note: This test can be obtained directly from the Sequential (Type I) Sum of Squares When Block is entered first, followed by Treatment

12. Interviewer Example

13. Car Pricing Example Recall: Treatments: g = 8 Interviewers, r = 7 dealers/interviewer Blocks: b = 28 Dealers, k = 2 interviewers/dealer l= 1 common dealer per pair of interviewers

14. Comparing Pairs of Trt Means & Contrasts • Variance of estimated treatment means depends on whether blocks are treated as Fixed or Random • Variance of difference between two means DOES NOT! • Algebra to derive these is tedious, but workable. Results are given here:

15. Car Pricing Example

16. Car Pricing Example – Adjusted Means Note: The largest difference (122.2 - 81.8 = 40.4) is not even close to the Bonferroni Minimum significant Difference = 95.7

17. Recovery of Inter-block Information • Can be useful when Blocks are Random • Not always worth the effort • Step 1: Obtain Estimated Contrast and Variance based on Intra-block analysis • Step 2: Obtain Inter-block estimate of contrast and its variance • Step 3: Combine the intra- and inter-block estimates, with weights inversely proportional to their variances

18. Inter-block Estimate

19. Combined Estimate

20. Interviewer Example