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Chapter 4 Supplemental Text Material
S4-1. Relative Efficiency of the RCBD
In Example 4-1 we illustrated the noise-reducing property of the randomized complete block design (RCBD). If we look at the portion of the total sum of squares not accounted for by treatments (302.14; see Table 4-4), about 63 percent (192.25) is the result of differences between blocks. Thus, if we had run a completely randomized design, the mean square for error MSE would have been much larger, and the resulting design would not have been as sensitive as the randomized block design.
It is often helpful to estimate the relative efficiency of the RCBD compared to a completely randomized design (CRD). One way to define this relative efficiency is
R
df df df df
b r b r
r b
2 2
where are the experimental error variances of the completely randomized and
randomized block designs, respectively, and are the corresponding error
degrees of freedom. This statistic may be viewed as the increase in replications that is required if a CRD is used as compared to a RCBD if the two designs are to have the same sensitivity. The ratio of degrees of freedom in R is an adjustment to reflect the different number of error degrees of freedom in the two designs.
σ r^2 and σ b^2
b
σ b^2
df (^) r and df
To compute the relative efficiency, we must have estimates of. We can use
the mean square for error MS
σ r^2 and
E from the RCBD to estimate^ , and it may be shown [see Cochran and Cox (1957), pp. 112-114] that
σ b^2
σ^ ^ r (^ b^ )^ MSBlocks^ b a (^^ ) MSE
ab
is an unbiased estimator of the error variance of a the CRD. To illustrate the procedure, consider the data in Example 4-1. Since MSE = 7.33, we have
ˆ 2 7. σ (^) b =
and
ˆ 2 (^ 1)^ (^ 1) 1 (5)38.45 6(3)7. 4(6) 1
Blocks E r
b MS b a MS ab
Therefore our estimate of the relative efficiency of the RCBD in this example is
2 2
b r b r
df df R df df
r b
This implies that we would have to use approximately twice times as many replicates with a completely randomized design to obtain the same sensitivity as is obtained by blocking on the metal coupons.
Clearly, blocking has paid off handsomely in this experiment. However, suppose that blocking was not really necessary. In such cases, if experimenters choose to block, what do they stand to lose? In general, the randomized complete block design has ( a – 1)( b - 1) error degrees of freedom. If blocking was unnecessary and the experiment was run as a completely randomized design with b replicates we would have had a ( b - 1) degrees of freedom for error. Thus, incorrectly blocking has cost a ( b - 1) – ( a - 1 )(b - 1) = b - 1 degrees of freedom for error, and the test on treatment means has been made less sensitive needlessly. However, if block effects really are large, then the experimental error may be so inflated that significant differences in treatment means could possibly remain undetected. (Remember the incorrect analysis of Example 4-1.) As a general rule, when the importance of block effects is in doubt, the experimenter should block and gamble that the block means are different. If the experimenter is wrong, the slight loss in error degrees of freedom will have little effect on the outcome as long as a moderate number of degrees of freedom for error are available.
S4-2. Partially Balanced Incomplete Block Designs
Although we have concentrated on the balanced case, there are several other types of incomplete block designs that occasionally prove useful. BIBDs do not exist for all combinations of parameters that we might wish to employ because the constraint that λ be an integer can force the number of blocks or the block size to be excessively large. For example, if there are eight treatments and each block can accommodate three treatments, then for λ to be an integer the smallest number of replications is r = 21. This leads to a design of 56 blocks, which is clearly too large for most practical problems. To reduce the number of blocks required in cases such as this, the experimenter can employ partially balanced incomplete block designs, or PBIDs, in which some pairs of treatments appear together λ 1 times, some pairs appear together λ 2 times,.. ., and the remaining pairs appear together λ m times. Pairs of treatments that appear together λ i times are called i th associates. The design is then said to have m associate classes.
An example of a PBID is shown in Table 1. Some treatments appear together λ 1 = 2 times (such as treatments 1 and 2), whereas others appear together only λ 2 = 1 times (such as treatments 4 and 5). Thus, the design has two associate classes. We now describe the intrablock analysis for these designs.
A partially balanced incomplete block design with two associate classes is described by the following parameters:
The linear statistical model for the partially balanced incomplete block design with two associate classes is
yij = μ + τ i + β j + ε ij
where μ is the overall mean, τ i is the i th treatment effect, β j is the jth block effect, and ε ij is the NID(0, σ^2 ) random error component. We compute a total sum of squares, a block sum of squares (unadjusted), and a treatment sum of squares (adjusted). As before, we call
=
b
j
i i nijyj k
Q y 1
..
the adjusted total for the i th treatment. We also define
S Qi Q s and i are first associates s
1 (^ )^ =^ ∑ s
∆ = k -2{( rk - r + λ 1 )( rk - r +λ 2 ) + (λ 1 + λ 2 )
c 1 = ( k ∆) - 1[λ 1 ( rk - r + λ 2 ) + (λ 1 - λ 2 )( λ 2 p^112 - λ 1 p^212 )]
c 2 = ( k ∆) - 1[λ 2 ( rk - r + λ 1 ) + (λ 1 - λ 2 )( λ 2 p^112 - λ 1 p^212 )]
The estimate of the i th treatment effect is
( )
τ i [( ) i ( ) ( )]
r k
= k c Q c c S Q −
1 2 1 2 1 i
and the adjusted treatment sum of squares is
SS (^) Treatments adjusted i Qi i
a ( ) =^ =
1
The analysis of variance is summarized in Table 3. To test H 0 : τi = 0, we use F 0 = MSTreatments(adjusted) / MSE.
Table 2. Relationship of Treatments to 1 and 2
Treatment 1
Treatment 2 1 st^ Associate 2 nd^ Associate 1st associate 2nd associate
Table 3. Analysis of Variance for the Partially Balanced Incomplete Block Design with Two Associate Classes Source of Variation Sum of Squares Degrees of Freedom Treatments (adjusted)
τ i i
i
a Q =
1 a- Blocks 1 2 2 k (^) 1
y
y bk
j j
b .
=
∑ b-
Error Subtraction bk-b-a+ Total y
y bk
ij i j
2
2
∑ ∑ −^
bk-
We may show that the variance of any contrast of the form τ^ u − τ v is
V
k c u v r k ( ) (^ i ) ( )
2
where treatments u and v are i th associates ( i = 1, 2). This indicates that comparisons between treatments are not all estimated with the same precision. This is a consequence of the partial balance of the design.
We have given only the intrablock analysis. For details of the interblock analysis, refer to Bose and Shimamoto (1952) or John (1971). The second reference contains a good discussion of the general theory of incomplete block designs. An extensive table of partially balanced incomplete block designs with two associate classes has been given by Bose, Clatworthy, and Shrikhande (1954).
S4-3. Youden Squares
Youden squares are "incomplete" Latin square designs in which the number of columns does not equal the number of rows and treatments. For example, consider the design shown in Table 4. Notice that if we append the column (E, A, B, C, D ) to this design, the result is a 5 × 5 Latin square. Most of these designs were developed by W. J. Youden, hence their name.
Although a Youden square is always a Latin square from which at least one column (or row or diagonal) is missing, it is not necessarily true that every Latin square with more than one column (or row or diagonal) missing is a Youden square. The arbitrary removal of more than one column, say, for a Latin square may destroy its balance. In general, a
Table 5. The Youden Square Design used in the Example
Day Work Station (Block) (^1 2 3 4) yi..
Treatment totals 1 A=3 B=1 C=-2 D=0 2 y.1.=12 (A) 2 B=0 C=0 D=-1 E=7 6 y.2.=2 (B) 3 C=-1 D=0 E=5 A=3 7 y.3.=-4 (C) 4 D=-1 E=6 A=4 B=0 9 y.4.=-2 (D) 5 E=5 A=2 B=1 C=-1 7 y.5.=23 (E) y..h 6 9 7 9 y…=
Considering this design as a balanced incomplete block, we find a = b = 5, r= k = 4, and k = 3. Also,
SS y
y T (^) N ijh i j h
= ∑ ∑∑^2 − = − =
2 2 183 00
Q 1 = 12 - 1
4
Q 2 = 2 - 1
4
Q 3 = - 4 - 1
4
Q 4 = -2 - 1
4
Q 5 =23 - 1
4
SS
k Q
a T rea tm ents adjusted
i i
a
( ) =^
=
2 1
[( / ) 2 ( / ) 2 ( / ) 2 ( / ) 2 ( / )^2
Also,
SS
y k
y Days N
i i
b = − =
=
2
1
.. ... ( )^ ( )^ ( )^ ( )^ ( )^ (^ )^
SS
y b
y Stations N
h h
k = − =
=
2 2 2 2 2 2
1
and
SSE = SST - SSTreatments (adjusted) - SSDays - SSStations = 134.95 - 120.37 - 6.70 - 1.35 = 6.
Block or day effects may be assessed by computing the adjusted sum of squares for blocks. This yields
Q 1 ' = 2 - 1 4
Q 2 ' = 6 - 1
4
Q 3 ' = 7 - 1
4
Q 4 ' = 9 - 1
4
Q 5 ' = 7 - 1
4
SS
r Q
Days adjusted b
j j
b
( )
=
2 1
[( / ) 2 (5 / ) 2 ( / ) 2 ( / ) 2 ( / ) ]^2
The complete analysis of variance is shown in Table 6. Illumination levels are significantly different at 1 percent.
There are other types of lattice designs that occasionally prove useful. For example, the cubic lattice design can be used for k^3 treatments in k^2 blocks of k runs. A lattice design for k ( k + 1) treatments in k + 1 blocks of size k is called a rectangular lattice. Details of the analysis of lattice designs and tables of plans are given in Cochran and Cox (1957).
Supplemental References
Bose, R. C. and T. Shimamoto (1952). “Classification and Analysis of Partially Balanced Incomplete Block Designs with Two Associate Classes”. Journal of the American Statistical Association , Vol. 47, pp. 151-184.
Bose, R. C. W. H. Clatworthy, and S. S. Shrikhande (1954). Tables of Partially Balanced Designs with Two Associate Classes. Technical Bulletin No. 107, North Carolina Agricultural Experiment Station.
Smith, C. A. B. and H. O. Hartley (1948). “Construction of Youden Squares”. Journal of the Royal Statistical Society Series B , Vol. 10, pp. 262-264.
