Cochran’s Theorem
4.4 Balanced Incomplete Block Designs
4.4.1 Statistical Analysis of the BIBD
The concept of orthogonal pairs of Latin squares forming a Graeco-Latin square can be extended somewhat. A pphypersquareis a design in which three or more orthogonal pp Latin squares are superimposed. In general, up to p1 factors could be studied if a complete set ofp1 orthogonal Latin squares is available. Such a design would utilize all (p1)(p1) p21 degrees of freedom, so an independent estimate of the error variance is necessary. Of course, there must be no interactions between the factors when using hypersquares.
rtimes), and that there are Narbktotal observations. Furthermore, the number of times each pair of treatments appears in the same block is
Ifab, the design is said to be symmetric.
The parameter must be an integer. To derive the relationship for , consider any treat- ment, say treatment 1. Because treatment 1 appears in rblocks and there are k1 other treat- ments in each of those blocks, there are r(k1) observations in a block containing treatment 1.
Theser(k1) observations also have to represent the remaining a1 treatments times.
Therefore,(a1)r(k1).
Thestatistical modelfor the BIBD is
(4.31) whereyijis the ith observation in the jth block,is the overall mean,iis the effect of the ith treatment,jis the effect of the jth block, and ijis the NID (0,2) random error component.
The total variability in the data is expressed by the total corrected sum of squares:
(4.32) Total variability may be partitioned into
where the sum of squares for treatments is adjustedto separate the treatment and the block effects. This adjustment is necessary because each treatment is represented in a different set of r blocks. Thus, differences between unadjusted treatment totals y1.,y2., . . . ,ya. are also affected by differences between blocks.
The block sum of squares is
(4.33) wherey.jis the total in the jth block. SSBlockshasb1 degrees of freedom. The adjusted treat- ment sum of squares is
(4.34) whereQiis the adjusted total for the ith treatment, which is computed as
(4.35) withnij1 if treatment iappears in block j andnij0 otherwise. The adjusted treatment totals will always sum to zero. SSTreatments(adjusted)hasa1 degrees of freedom. The error sum of squares is computed by subtraction as
(4.36) and has Nab1 degrees of freedom.
The appropriate statistic for testing the equality of the treatment effects is
The ANOVA is summarized in Table 4.23.
F0MSTreatments(adjusted)
MSE
SSESSTSSTreatments(adjusted)SSBlocks Qiyi.1
k
j1b nijy.j i1, 2, . . . ,a SSTreatments(adjusted)ki1
a Q2ia SSBlocks1
k
j1b y2.jyN2..SSTSSTreatments(adjusted)SSBlocksSSE SST
ij y2ij yN2..yijijij
r(k1) a1
4.4 Balanced Incomplete Block Designs
169
■ T A B L E 4 . 2 3
Analysis of Variance for the Balanced Incomplete Block Design
Source of Degrees of
Variation Sum of Squares Freedom Mean Square F0
Treatments a1
(adjusted)
Blocks b1
Error SSE(by subtraction) Nab1
Total y2ijy2.. N1
N
SSE Nab1
SSBlocks b1 1
k
y2.jyN2..F0MSTreatments(adjusted)
MSE SSTreatments(adjusted)
a1 k
Q2ia
E X A M P L E 4 . 5
Consider the data in Table 4.22 for the catalyst experiment.
This is a BIBD with a4, b4, k3, r3, 2, and N12. The analysis of this data is as follows. The total sum of squares is
The block sum of squares is found from Equation 4.33 as
To compute the treatment sum of squares adjusted for blocks, we first determine the adjusted treatment totals using Equation 4.35 as
55.00 1
3 [(221)2(207)2(224)2(218)2](870)2 12 SSBlocks1
3j1
4 y2.j12y2..63,156(870)2
12 81.00 SST
i j y2ij12y2..The adjusted sum of squares for treatments is computed from Equation 4.34 as
The error sum of squares is obtained by subtraction as
The analysis of variance is shown in Table 4.24. Because the P-value is small, we conclude that the catalyst employed has a significant effect on the time of reaction.
81.0022.7555.003.25 SSESSTSSTreatments(adjusted)SSBlocks 22.75
3[(9/3)2(7/3)2(4/3)2(20/3)2] (2)(4)
SSTreatments(adjusted) ki1
4 Q2ia
Q4(222)13(221207218)20/3 Q3(216)13(221207224) 4/3 Q2(214)13(207224218) 7/3 Q1(218)13(221224218) 9/3
■ T A B L E 4 . 2 4
Analysis of Variance for Example 4.5
Source of Sum of Degrees of Mean
Variation Squares Freedom Square F0 P-Value
Treatments (adjusted 22.75 3 7.58 11.66 0.0107
for blocks)
Blocks 55.00 3 —
Error 3.25 5 0.65
Total 81.00 11
If the factor under study is fixed, tests on individual treatment means may be of interest. If orthogonal contrasts are employed, the contrasts must be made on the adjusted treatment totals, the {Qi} rather than the {yi.}. The contrast sum of squares is
where {ci} are the contrast coefficients. Other multiple comparison methods may be used to compare all the pairs of adjusted treatment effects, which we will find in Section 4.4.2, are estimated by kQi/(a). The standard error of an adjusted treatment effect is
(4.37) In the analysis that we have described, the total sum of squares has been partitioned into an adjusted sum of squares for treatments, an unadjusted sum of squares for blocks, and an error sum of squares. Sometimes we would like to assess the block effects. To do this, we require an alternate partitioning of SST, that is,
HereSSTreatmentsis unadjusted. If the design is symmetric, that is, if ab, a simple formula may be obtained for SSBlocks(adjusted). The adjusted block totals are
(4.38) and
(4.39) The BIBD in Example 4.5 is symmetric because ab4. Therefore,
and
Also,
A summary of the analysis of variance for the symmetric BIBD is given in Table 4.25.
Notice that the sums of squares associated with the mean squares in Table 4.25 do not add to the total sum of squares, that is,
This is a consequence of the nonorthogonality of treatments and blocks.
SST Z SSTreatments(adjusted)SSBlocks(adjusted)SSE
SSTreatments(218)2(214)2(216)2(222)2
3 (870)2
12 11.67 SSBlocks(adjusted)3[(7/3)2(24/3)2(31/3)2(0)2]
(2)(4) 66.08
Q4(218)13(218214222)0 Q3(207)13(214216222) 31/3 Q2(224)13(218214216)24/3 Q1(221)13(218216222)7/3
SSBlocks(adjusted)
rj1
b (Qj)2b Qjy.j1
4
i1a nijyi. j1, 2, . . . , b SSTSSTreatmentsSSBlocks(adjusted)SSES
kMSaEˆi
SSC
k
i1a ciQi2
ai1
a c2i4.4 Balanced Incomplete Block Designs
171
Computer Output. There are several computer packages that will perform the analy- sis for a balanced incomplete block design. The SAS General Linear Models procedure is one of these and Minitab and JMP are others. The upper portion of Table 4.26 is the Minitab General Linear Model output for Example 4.5. Comparing Tables 4.26 and 4.25, we see that Minitab has computed the adjusted treatment sum of squares and the adjusted block sum of squares (they are called “Adj SS” in the Minitab output).
The lower portion of Table 4.26 is a multiple comparison analysis, using the Tukey method. Confidence intervals on the differences in all pairs of means and the Tukey test are displayed. Notice that the Tukey method would lead us to conclude that catalyst 4 is different from the other three.