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 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 = r(k-1)/(g-1) Blocks for some integer

BIBD (II)

• Reasoning for Integer : 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

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: = 7(2-1)/(8-1) = 1

Interviewer ExampleDealer\Interviewer A B C D E F G H Dealer Mean

1 100 125 * * * * * * 112.52 235 * 95 * * * * * 165.03 50 * * 30 * * * * 40.04 133 * * * * 80 * * 106.55 50 * * * 30 * * * 40.06 25 * * * * * 88 * 56.57 140 * * * * * * 150 145.08 * 41 50 * * * * * 45.59 * 180 * 195 * * * * 187.5

10 * 65 * * 75 * * * 70.011 * 50 * * * 100 * * 75.012 * 100 * * * * 96 * 98.013 * 170 * * * * * 150 160.014 * * 75 95 * * * * 85.015 * * 25 * * 55 * * 40.016 * * 132 * 50 * * * 91.017 * * 145 * * * * 96 120.518 * * 100 * * * 152 * 126.019 * * * 99 235 * * * 167.020 * * * 100 * 100 * * 100.021 * * * 50 * * 50 * 50.022 * * * 35 * * * 50 42.523 * * * * 150 163 * * 156.524 * * * * 135 * 150 * 142.525 * * * * 70 * * 138 104.026 * * * * * 50 100 * 75.027 * * * * * 75 * 65 70.028 * * * * * * 100 89 94.5

Interviewer Mean 104.714 104.429 88.857 86.286 106.429 89.000 105.143 105.429 98.786Block Total 1331 1497 1346 1344 1542 1246 1285 1473

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

)1()1(,1

1 1

2^^

1 1

2^^^

0

~

)1()1(

1 :StatisticTest

))1(1(

,...,1,...,1 :Model Reduced

1))1()1(1(

), pairs allNot :(Note ,...,1,...,1 :Model Full

brgg

H

F

FR

F

F

FR

FR

obs

R

g

i

b

jjijR

ijjij

F

g

i

b

jjiijF

ijjiij

F

brgSSEgSSESSE

dfSSEdfdfSSESSE

F

brgbNdfySSE

bjgiy

bgrgbgNdfySSE

jibjgiy

Least Squares Estimation (I) – Fixed Blocks

k

ji

g

iijjijii

i

g

iijjj

ji

g

iijj

g

iijij

a

ijiijij

j

j

b

jijii

b

jijij

b

jjiijij

i

k

jj

g

iiij

g

i

b

jij

g

i

b

jjiijij

g

i

b

jjiijij

g

i

b

jijij

ijijjiijijij

nk

yk

nkkrkrky

nk

yk

bjknkyynynQ

ginrryynynQyNy

krNynynQ

ynnQ

jinbjginyn

1

^

1

^^^

^

1

^^

^^

1

^

1

set

1

^

1

^^

1

set

1

^^

1

^

1

^^

1 1

set

1 1

1 1

2

1 1

2

11

11

,...,102

,...,102

02

otherwise 0Blk in Trt if 1

,...,1;,...,1 :Model

Least Squares Estimation (II)^ ^ ^ ^

''1 ' 1

^ ^

''1 ' 1

1

^ ^

1

1 1

Consider the Last Term:

1) Sum of Block Totals that Trt appears in

2)

gb

i ii ij j i jj i

gb

iij j i jj i

b

ij j ij

b

ijj

ky kr kr k n y nk k

n y k n

n y B i

n k k

^ ^

^ ^ ^2

' '' ' '1 ' 1 1 1 ' 1

'

g g^ ^ ^

'1 ' 1

'

1 if Trts , ' in Blk 3)

0 otherwise

Notes: (a) 0 (b):

i

g gb b b

i i iij i j ij ij i j ij i jj i j j i

i i

i i i iji i

i i

n k r kr

i i jn n n n n n n

n

'' 1'

g^ ^ ^ ^ ^ ^

' ' '' '1 ' 1 1 ' 1 1 ' 1

' '

g

i jii i

g gb b b

i i i i i iij i j ij ij i jj i j i j i

i i i i

n

n n n n n r r

Least Squares Estimation (III)

iii

iiii

ii

iiii

iiii

Bk

yQ

gkQ

gBky

gg

krrkrBky

rkrBkrkrky

1

1

1

^

^^

^^

^^^^

Analysis of Variance (Fixed or Random Blocks)

iBk

Bk

yQg

QkSST

SSSSESSE

yyynSSE

nk

yyynSSE

iiii

g

ii

FR

j

g

i

b

jjijijR

i

g

iijj

g

i

b

jjiijijF

j

j

Trt containing MeansBlock of Sum 11Adjusted)(

Blocksfor Adjusted Trts :Difference

:Model Reduced

1 :Model Full

1

2

Reduced^

1 1

2Reduced^^

^

1

Full^

1 1

2Full^^^

Source df SS MS

Blks (Unadj) b-1 SSB/(b-1)

Trts (Adj) g-1 SST(Adj)/(g-1)

Error gr-(b-1)-(g-1)-1 SSE/(g(r-1)-(b-1))

Total gr-1

b

jjk

1

2Reduced^

gQkg

ii

1

2

g

i

b

jjiijij yn

1 1

2Full^^^

g

i

b

jijij yyn

1 1

2

ANOVA F-Test for Treatment Effects

)1()1(,1

10

0

~)Adj(:

0 allNot :0...:

brgg

H

obs

iAg

FMSE

MSTFTS

HH

Note: This test can be obtained directly from the Sequential (Type I) Sum of Squares When Block is entered first, followed by Treatment

Interviewer Examplemu 98.786

Dealer Beta(j)Red Beta(j)Full SSB(Unadj) SSE(Full)1 13.714 7.464 376.163 1069.5312 66.214 64.152 8768.663 6091.3203 -58.786 -58.723 6911.520 96.2584 7.714 -0.723 119.020 652.5085 -58.786 -63.973 6911.520 5.6956 -42.286 -62.411 3576.163 1596.1257 46.214 37.589 4271.520 351.1258 -53.286 -44.723 5678.735 150.9459 88.714 99.402 15740.449 381.570

10 -28.786 -23.348 1657.235 73.50811 -23.786 -21.598 1131.520 1040.82012 -0.786 -10.286 1.235 504.03113 61.214 63.214 7494.378 306.28114 -13.786 1.089 380.092 294.03115 -58.786 -52.411 6911.520 148.78116 -7.786 1.839 121.235 3894.03117 21.714 27.902 943.020 1929.75818 27.214 21.902 1481.235 126.00819 68.214 79.964 9306.378 7875.12520 1.214 9.714 2.949 144.50021 -48.786 -51.973 4760.092 815.07022 -56.286 -47.973 6336.163 2.82023 57.714 60.964 6661.878 21.12524 43.714 35.277 3821.878 110.63325 5.214 8.277 54.378 1868.13326 -23.786 -35.473 1131.520 354.44527 -28.786 -28.973 1657.235 53.82028 -4.286 -16.161 36.735 72.000

Sum 106244.429 30030.000

ANOVASource df SS MS F P-ValueBlocks (Unadj) 27 106244.43 3934.98Trts(Adj) 7 5377.00 768.14 0.5372 0.7967Error 21 30030.00 1430.00Total 55 141651.43

Interviewer y(i*) B(i) Q(i) alpha(i) SST(Adj) SST(Unadj)A 733 1331 67.5 16.875 1139.063 246.036B 731 1497 -17.5 -4.375 76.563 222.893C 622 1346 -51 -12.750 650.250 690.036D 604 1344 -68 -17.000 1156.000 1093.750E 745 1542 -26 -6.500 169.000 408.893F 623 1246 0 0.000 0.000 670.321G 736 1285 93.5 23.375 2185.563 282.893H 738 1473 1.5 0.375 0.563 308.893

Sum 5377.000 3923.714

Car Pricing Example The GLM Procedure Dependent Variable: price Sum of Source DF Squares Mean Square F Value Pr > F Model 34 111621.4286 3282.9832 2.30 0.0241 Error 21 30030.0000 1430.0000 Corrected Total 55 141651.4286 Source DF Type I SS Mean Square F Value Pr > F dlr_blk 27 106244.4286 3934.9788 2.75 0.0101 intrvw_trt 7 5377.0000 768.1429 0.54 0.7967 Source DF Type III SS Mean Square F Value Pr > F dlr_blk 27 107697.7143 3988.8042 2.79 0.0093 intrvw_trt 7 5377.0000 768.1429 0.54 0.7967

Recall: Treatments: g = 8 Interviewers, r = 7 dealers/interviewer Blocks: b = 28 Dealers, k = 2 interviewers/dealer = 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:

a

ii

g

iii

Cij

jijiji

iiii

wgkVw

gCbrg

gkMSEtB

gkMSEV

gkVV

Bg

ygky

rg

1

22^

1

^^

,2/

^^^2^^^^

^^^

:Contrast generalFor

2)1()1(2 s'Bonferroni

22

11

Car Pricing Example

282

)7(82

)1(212748)128()17(8)1()1(

73.95)7.26(58.37152 s'Bonferroni

7152

1430228

42

81

82

56111

14301278

21,56/05.,2/

^^^

222^^

^^^

ggC

brg

tg

kMSEtB

gkMSEV

gkV

ByyBg

ygky

rg

MSEkrg

Cij

ji

ji

iiiiii

Car Pricing Example – Adjusted Means The GLM Procedure Least Squares Means intrvw_ trt price LSMEAN 1 115.660714 2 94.410714 3 86.035714 4 81.785714 5 92.285714 6 98.785714 7 122.160714 8 99.160714

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

1 1 1

2 2 2

1

1 if Trt occurs in Blk 0 otherwise

~ 0,

This is a multiple regression with predictors which leads to esti

g g g

j ij ij ij i j ij ij iji i i

g

ij i j ji

i jy n y k n k n n

k n N k k

g

~~ ~

1

2 2 2~ ~ ~2

1 1

2 2 2

2^

mates:

Blks|Trts1

1Blks|Trts

gi

i i ij ji

g g

ii ii i

B rky B n y

rk k

w V wr

N gE MSE E MSb

bMS MSEN g

Combined Estimate

MSEgN

bMSEMS

ryrkBw

rkk

Vw

Bg

ygkw

gkVw

VV

VV

VV

V

VV

VV

ii

g

ii

g

iii

iii

a

ii

g

iii

2^2^

~

1

2222~

1

~~

^

1

22^

1

^^

~^

~^

~^~^

~

~

^

^

1Trts|Blks

1

:where

11

1

11

11

Interviewer ExampleANOVASource df SS MSTrts(Unadj) 7 3923.714286 560.5306Blocks(Adj) 27 107697.71 3988.804Error 21 30030.00 1430Total 55 141651.43

2^ ^ ^ ^ ^2 2 2

1 1 1 1

2 2 2~ ~ ~2

1 1

2^ ~

2 2

1 1

1430(2) 357.51(8)

28 12 3988.8 1430 2(1430)56 8 1436.2

7 1

13

g a a a

ii i i ii i i i

g g

ii ii i

g g

i ii i

kw V w V w wg

k kw V w

r

V w w

^ ~ 2 2

^ ~1 1 2

12 2

1 1

1 357.5 1436.257.5 1436.2 0.80 0.20 286.25

1 1357.5 1436.2

357.5 1436.2

g g

i i gi i

ig gi

i ii i

w wV w

w w

Interviewer alpha-hat mu+alpha-hat alpha-tilda alpha-bar mu+alpha-barA 16.875 115.661 -8.667 11.767 110.552B -4.375 94.411 19.000 0.300 99.086C -12.750 86.036 -6.167 -11.433 87.352D -17.000 81.786 -6.500 -14.900 83.886E -6.500 92.286 26.500 0.100 98.886F 0.000 98.786 -22.833 -4.567 94.219G 23.375 122.161 -16.333 15.433 114.219H 0.375 99.161 15.000 3.300 102.086