Balanced Incomplete Block Design Ford Falcon Prices Quoted

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, " JRSSC (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 § H 0: 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 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