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

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

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

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

Interviewer Example

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

Least Squares Estimation (I) – Fixed Blocks

Least Squares Estimation (II)

Least Squares Estimation (III)

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

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

Interviewer Example

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

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:

Car Pricing Example

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

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

Inter-block Estimate

Combined Estimate

Interviewer Example