# Balanced Incomplete Block Design

Document Sample

```					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
l = r(k-1)/(g-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 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: l = 7(2-1)/(8-1) = 1
Interviewer Example
Dealer\Interviewer  A         B        C        D         E         F        G         H      Dealer Mean
1         100       125       *         *        *         *        *         *         112.5
2         235        *        95        *        *         *        *         *         165.0
3          50        *        *        30        *         *        *         *          40.0
4         133        *        *         *        *        80        *         *         106.5
5          50        *        *         *        30        *        *         *          40.0
6          25        *        *         *        *         *        88        *          56.5
7         140        *        *         *        *         *        *        150        145.0
8          *         41       50        *        *         *        *         *          45.5
9          *        180       *       195        *         *        *         *         187.5
10          *         65       *         *        75        *        *         *          70.0
11          *         50       *         *        *       100        *         *          75.0
12          *        100       *         *        *         *        96        *          98.0
13          *        170       *         *        *         *        *        150        160.0
14          *         *        75       95        *         *        *         *          85.0
15          *         *        25        *        *        55        *         *          40.0
16          *         *       132        *        50        *        *         *          91.0
17          *         *       145        *        *         *        *         96        120.5
18          *         *       100        *        *         *       152        *         126.0
19          *         *        *        99       235        *        *         *         167.0
20          *         *        *       100        *       100        *         *         100.0
21          *         *        *        50        *         *        50        *          50.0
22          *         *        *        35        *         *        *         50         42.5
23          *         *        *         *       150      163        *         *         156.5
24          *         *        *         *       135        *       150        *         142.5
25          *         *        *         *        70        *        *        138        104.0
26          *         *        *         *        *        50       100        *          75.0
27          *         *        *         *        *        75        *         65         70.0
28          *         *        *         *        *         *       100        89         94.5
Interviewer Mean 104.714   104.429   88.857   86.286   106.429   89.000   105.143   105.429      98.786
Block 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
Full Model : yij     i   j   ij i  1,...,g j  1,...,b (Note : Not all pairs i, j )
2
g   b
         ^   ^     ^

SSEF    yij            i   j               df F  N  (1  ( g  1)  (b  1))  rg  g  b  1
i 1 j 1                      
Reduced Model :         yij     j   ij        i  1,...,g   j  1,...,b
2
g   b
  ^ ^ 
SSER    yij      j                      df R  N  (1  (b  1))  rg  b
i 1 j 1          
 SSER  SSEF     SSER  SSEF 
 df  df                         
                     g 1       
H0
Test Statistic : Fobs          R    F
~    Fg 1, g ( r 1) (b 1)
 SSEF            SSEF          
 df        g (r  1)  (b  1) 
 F                             
Least Squares Estimation (I) – Fixed Blocks
1 if Trt i in Blk j
Model : nij yij  nij    i   j   ij    i  1,...,g ; j  1,...,b nij  
0 otherwise

Q   n    nij  yij     i   j 
g     b             g       b
2                             2
ij ij
i 1 j 1           i 1 j 1

Q
 2 nij  yij     i   j   0 
g   b                       set                    g         b                   ^          g    ^       k       ^

     i 1 j 1
 n
i 1 j 1
ij   yij  N   r   i  k   j
i 1           j 1
^           ^
 y   N     y  
Q
 2 nij  yij     i   j   0 
b                           set             b                               ^        ^              b   ^

 i     j 1
n
j 1
ij   yij  yi  r   r  i   nij  j
j 1
i  1,...,g

Q
 2 nij  yij     i   j   0 
a                          set             g                               ^        g              ^       ^

 j     i 1
n
i 1
ij   yij  y j  k    nij  i  k  j
i 1
j  1,...,b

^  1       ^  1 g    ^
  j  y j     nij  i
k          k i 1
^   ^    1     k ^  1 g    ^ 
 kyi  kr   kr  i  k  nij  y j     nij  i 
j 1  k          k i 1     
Least Squares Estimation (II)
^      1
^      ^  1 g          ^ 
b
kyi  kr   kr  i  k  nij  y j     ni ' j  i ' 
j 1   k         k i '1         
b
        ^        g    ^ 
Consider the Last Term:  nij  y j  k    ni ' j  i ' 
j 1               i '1      
b
1)   n
j 1
ij   y j  Bi                Sum of Block Totals that Trt i appears in

b                   ^           ^                        ^                    ^
2)    n k   k  n
j 1
ij                               i    k  r  kr 

b          g                ^                    b           ^        b           g                 ^                  1 if Trts i, i ' in Blk j
3)     nij ni ' j  i '   n  i   nij ni ' j  i '        2
ij                                                nij ni ' j  
j 1 i '1                                         j 1               j 1 i '1                                         0 otherwise
i ' i
g          ^                                     ^               g     ^             g
Notes: (a)                      
i 1
i    0    i   i '
i '1
(b):  nij ni ' j  l
i '1
i ' i                i ' i
b          g                ^                    b           ^        g       ^           b                  ^             g       ^   ^
  nij ni ' j  i '   nij  i    i '  nij ni ' j  r  i  l   i '   r  l   i
j 1 i '1                                      j 1                 i '1                j 1                            i '1
i ' i                                               i ' i
Least Squares Estimation (III)

kyi  kr   kr  i  Bi  kr   r  l  i
^        ^            ^          ^

 kyi   Bi   i kr  r  l    i r k  1  l  
^                  ^

 i l  g  1  l    i lg
^                        ^

^    kyi   Bi kQi
i               
lg       lg
1
Qi  yi  Bi
k
Analysis of Variance (Fixed or Random Blocks)
2
g
b   ^   ^     ^ Full                                         ^ Full                            1 g    ^
Full Model : SSEF   nij  yij     i   j 
                                                                               y  j  y    nij  i
                                                                                               k i 1
j
i 1 j 1
2
g   b
^  ^ Reduced      ^ Reduced
Reduced Model : SSER   nij  yij     j
                 
    j          y  j  y 
i 1 j 1                  
Difference: SSER  SSEF  SS Trts Adjusted for Blocks  
g
k  Qi2
1                    1
 SST (Adjusted)      i 1
Qi  yi      Bi                   Bi  Sum of Block Means containing Trt i
lg                               k                    k

Source                        df                               SS                                                               MS

2
 ^ Reduced 
b
k  j



j 1            
g

i 1

2
              ^ Full 
Error                         gr-(b-1)-(g-1)-1                                                                                   SSE/(g(r-1)-(b-1))
g       b     ^   ^
 nij  yij     i   j 
                     
i 1 j 1                      

Total                         gr-1                                           n y                                     
g       b
2
ij       ij    y 
i 1    j 1
ANOVA F-Test for Treatment Effects

H 0 : 1  ...   g  0 H A : Not all  i  0
H0
TS : Fobs                      ~     Fg 1, g ( r 1)(b 1)
MSE

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                                                          1
2
13.714
66.214
7.464
64.152
376.163
8768.663
1069.531
6091.320
mu               98.786     3       -58.786      -58.723        6911.520     96.258
4        7.714        -0.723        119.020     652.508
5       -58.786      -63.973        6911.520     5.695
6       -42.286      -62.411        3576.163    1596.125
7        46.214       37.589        4271.520    351.125
A           733          1331      67.5      16.875    1139.063 246.036
B           731          1497      -17.5     -4.375     76.563    222.893      8       -53.286      -44.723        5678.735    150.945
C           622          1346       -51     -12.750     650.250   690.036      9        88.714       99.402       15740.449    381.570
D           604          1344       -68     -17.000    1156.000 1093.750      10       -28.786      -23.348        1657.235     73.508
E           745          1542       -26      -6.500     169.000   408.893     11       -23.786      -21.598        1131.520    1040.820
F           623          1246        0        0.000      0.000    670.321     12        -0.786      -10.286         1.235      504.031
G           736          1285      93.5      23.375    2185.563 282.893       13        61.214       63.214        7494.378    306.281
H           738          1473       1.5       0.375      0.563    308.893     14       -13.786       1.089         380.092     294.031
Sum      5377.000 3923.714
15       -58.786      -52.411        6911.520    148.781
16        -7.786       1.839         121.235     3894.031
17        21.714       27.902        943.020     1929.758
18        27.214       21.902        1481.235    126.008
ANOVA
19        68.214       79.964        9306.378    7875.125
Source                   df             SS          MS          F     P-Value
20        1.214        9.714          2.949      144.500
Trts(Adj)                7            5377.00     768.14     0.5372   0.7967       21       -48.786      -51.973        4760.092    815.070
Error                    21          30030.00     1430.00                          22       -56.286      -47.973        6336.163     2.820
Total                    55          141651.43                                     23        57.714       60.964        6661.878     21.125
24        43.714       35.277        3821.878    110.633
25        5.214        8.277          54.378     1868.133
26       -23.786      -35.473        1131.520    354.445
27       -28.786      -28.973        1657.235     53.820
28        -4.286      -16.161         36.735      72.000
Sum        106244.429   30030.000
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
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:
^     ^   ^     1       k         1
 i     i  y      yi      Bi
rg      lg        lg
^ ^              ^ ^  2k                       ^ ^  2kMSE
2              ^
V   i   j   V  i   j                  V  i   j  
                             lg                             lg
2kMSE                             g
 Bonferroni' s Bij  t / 2C ,                  g (r  1)  (b  1) C   
2
lg                               
^  k a 2
^        g        ^            2
For general Contrast :    wi  i              V       wi
i 1              lg i 1
Car Pricing Example
g  8 r  7 k  2 l  1 MSE  1430
^     ^   ^      1        k         1      1       2     1
i   i        y     yi      Bi     y  yi  Bi
rg       lg        lg      56      8     8
 ^ ^  2k           4 2  2
2
V  i   j                   
             lg       8    2
^
 ^ ^  2kMSE 1430
V  i   j                 715
              lg        2
2kMSE
 Bonferroni' s Bij  t / 2C ,          t.05 / 56, 21 715  3.58(26.7)  95.73
lg
  g (r  1)  (b  1)  8(7  1)  (28  1)  48  27  21
g ( g  1) 8(7)
C                   28
2      2
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
g                      g
          g
                    1 if Trt i occurs in Blk j
y j   nij yij  k    nij i  k  j   nij ij               nij  
i 1              i 1               i 1                          0 otherwise

 j ~ N  0, k 2   k 2 
g
 k    nij i   j                                   2

i 1

This is a multiple regression with g predictors which leads to estimates:
~
~                      ~ B  rk                    g
  y              i  i                  Bi   nij y j
r l                   i 1

~     g                ~
~
    k 2   k 2
2              g
   wi  i               V                        w       2

           r l
i
i 1                                                 i 1

Ng 2
E  MSE    2E  MS  Blks|Trts     2         
 b 1 
^ 2                               b 1 
     MS  Blks|Trts   MSE           
Ng
Combined Estimate
                    
 1 ^         1 ~
 ^              
V         
V                                           ^ ~
~
                                                   V   V   
            
  
                
            
V 

1

 ^   ~
         
 1                                                   V    V  
1                     1         1                 
 ^                              ^             
V    V                    V    V    
~                                 ~
                                    
  
                               
             
where:
^  k a 2
g                  2
^            ^                                     ^        k         1
   wi  i     V       wi                   i        yi      Bi
i 1           lg i 1                             lg        lg
~      g     ~
 
~       k 2   k 2
2           g                   ~      Bi  rk y 
   wi  i     V                      w         2
i 
r l                                        r l
i
i 1                                i 1

^ 2                            b 1                      ^ 2
   MS Blks | Trts  MSE
Ng                         MSE
      
Interviewer Example
Interviewer   alpha-hat   mu+alpha-hat   alpha-tilda   alpha-bar   mu+alpha-bar
ANOVA                                                   A          16.875      115.661         -8.667       11.767      110.552
Source               df         SS         MS           B          -4.375       94.411         19.000       0.300        99.086
Trts(Unadj)          7      3923.714286 560.5306        C         -12.750       86.036         -6.167      -11.433       87.352
D         -17.000       81.786         -6.500      -14.900       83.886
E          -6.500       92.286         26.500       0.100        98.886
Error                21      30030.00     1430          F           0.000       98.786        -22.833       -4.567       94.219
Total                55      141651.43                  G          23.375      122.161        -16.333       15.433      114.219
H           0.375       99.161         15.000       3.300       102.086

^     k                 ^  1430(2)
^     g        ^             2    a          ^                        a                  a
   wi  i       V         wi2        V            wi2  357.5 wi2
i 1                 l g i 1                 1(8) i 1          i 1

 ~  k    k
~     g        ~            2 2        2 g

   wi  i V                        wi2
i 1                  r l        i 1

 28  1 
22  3988.8  1430            2(1430) g
                            56  8 
^  ~                                                              g
V   
                        7 1
 wi  1436.2 wi2
i 1
2

i 1

 1 ^         1 ~                                                     g
2 
g

357.5 wi  1436.2 wi2 
 357.5   1436.2  
                                      

                               286.25 w2
^        ~                                                             g
 
 1         1 
 0.80   0.20             V  

i 1       i 1
 i
2           2
g                g

 357.5  1436.2                                        357.5 wi   1436.2 wi 
i 1

                                                              i 1           i 1  

```
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