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

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  t
# of Blocks  b
Replicates per Treatment  r < b
Block Size  k < t
Total Number of Units  N = kb = rt
All pairs of Treatments appear together in l = r(k-1)/(t-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 t-1 treatments
Tables of Designs for Various t,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 (t = 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)
Source df SS MS
Blks (Unadj)
b-1
SSB/(b-1)
Trts (Adj)
t-1
SST(Adj)/(t-1)
Error
tr-(b-1)-(t-1)-1
SSE/(t(r-1)-(b-1))
Total
tr-1

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: t = 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 ( = 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