One application of incomplete block designs is to marketing studies in which v versions of a commodity are to be compared and each shop is to hold k < v versions. The design ensures that each version is seen in comparison with each pair of other possibilities the same number of times. Give such a design for v=6, k=4, and suggest how the responses might be analyzed.
The aim of this study is to design an Incomplete Block Design experiment for a marketing study. There are six versions (v) of a commodity; stores only have 4 versions (k) of the commodity. In this experiment, we are blocking by store (b).
“v” represents the versions of the commodity sold in stores. There are six versions, and they will be denoted as A, B, C, D, E, and F.
v <- c("A", "B", "C", "D", "E", "F")
v## [1] "A" "B" "C" "D" "E" "F"Each store holds four versions of the commodity. To design the incomplete block design, combinations of four varieties of the commodity must be selected to form a block.
incomplete_block_design = combn(v, 4)
incomplete_block_design## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "B" "B" "B"
## [2,] "B" "B" "B" "B" "B" "B" "C" "C" "C" "D" "C" "C" "C"
## [3,] "C" "C" "C" "D" "D" "E" "D" "D" "E" "E" "D" "D" "E"
## [4,] "D" "E" "F" "E" "F" "F" "E" "F" "F" "F" "E" "F" "F"
## [,14] [,15]
## [1,] "B" "C"
## [2,] "D" "D"
## [3,] "E" "E"
## [4,] "F" "F"The blocks are then randomized, first by row, then by column.
randomized_incomplete_block_design1 = incomplete_block_design[sample(nrow(incomplete_block_design)),]
randomized_incomplete_block_design1## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] "B" "B" "B" "B" "B" "B" "C" "C" "C" "D" "C" "C" "C"
## [2,] "A" "A" "A" "A" "A" "A" "A" "A" "A" "A" "B" "B" "B"
## [3,] "C" "C" "C" "D" "D" "E" "D" "D" "E" "E" "D" "D" "E"
## [4,] "D" "E" "F" "E" "F" "F" "E" "F" "F" "F" "E" "F" "F"
## [,14] [,15]
## [1,] "D" "D"
## [2,] "B" "C"
## [3,] "E" "E"
## [4,] "F" "F"randomized_incomplete_block_design2 = incomplete_block_design[,sample(ncol(incomplete_block_design))]
randomized_incomplete_block_design2## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
## [1,] "B" "A" "A" "A" "A" "A" "A" "A" "C" "A" "A" "B" "A"
## [2,] "C" "C" "B" "B" "B" "C" "B" "D" "D" "B" "B" "C" "C"
## [3,] "D" "D" "C" "D" "C" "E" "D" "E" "E" "C" "E" "E" "D"
## [4,] "E" "E" "E" "F" "D" "F" "E" "F" "F" "F" "F" "F" "F"
## [,14] [,15]
## [1,] "B" "B"
## [2,] "D" "C"
## [3,] "E" "D"
## [4,] "F" "F"To confirm that this design works correctly, each version (A-F) must be combined with other pairs of versions the same number of times. Checking the 15 blocks reveals that each combination occurs 3 times; thus, this design is correct. This experiment utilizes a balanced incomplete block design, where any two treatments occur together the same number of times [1].
As a preliminary analysis, main effect and interaction effects may be calculated and boxplots may be made. It is appropriate to statistically analyze this data using an ANOVA. As described by Montgomery, the steps are as follows:
N is the total number of observations. \(N=a*r=b*k\) where r is the number of replicates, a is the number of treatments, b is the number of blocks, and k is the number of treatments included in a block.
For the statistical model, \(y_{ij}\) is ith observation in the jth block. \(y_{ij}=\mu+\tau_{i}+\beta_{j}+\epsilon_{ij}\) where \(\mu\) is the mean, \(\tau_{i}\) is the effect of the ith treatment, \(\beta_{j}\) is the effect of the jth block, \(\epsilon_{ij}\) is the effect of random error.
N and \(y_{ij}\) are used to calculate the corrected sum of squares, \(SS_{T}\).
\(SS_{T}= \sum_{i}\sum_{j} y_{ij}^2-\left(\frac{y^2}{N}\right)\)
\(SS_{Total} = SS_{Treatments, adjusted} + SS_{Blocks} + SS_{Error}\)
\(SS_{Blocks}= \left(\frac{1}{k}\right)\sum_{j=1}^b y_{j}^2-\left(\frac{y^2}{N}\right)\)
\(\lambda\) is the number of times a pair of treatments appears in a given block. \(\lambda = r\left(\frac{k-1}{a-1}\right)\) where r is the number of times each treatment occurs.
\(Q_{i}\) is the adjusted total.\(Q_{i} = y_{i}-\left(\frac{1}{k}\right)\sum_{j=1}^b n_{ij}y_{j}\)
\(SS_{Treatments, adjusted} = k*\left(\frac{\sum_{j=1}^a Q_{i}^2}{\lambda*a}\right)\)
Once \(SS_{Total}\), \(SS_{Treatments, adjusted}\), and \(SS_{Blocks}\) have been calculated, \(SS_{Error}\) can be calculated.
Degrees of freedom can be calculated as follows:
\(DoF_{Treatments}=a-1\)
\(DoF_{Blocks}=b-1\)
\(DoF_{Error}=N-a-b+1\)
\(DoF_{Total}=N-1\)
The Sum of Squares and DoF for Treatments, Blocks, and Error can be used to calculate the Mean Square for each (SS/DoF). Finally, the F statistic, \(F_{0}\), can be calculated. \(F_{0} = \frac{MS_{Treatments, adjustd}}{MS_{Error}}\).
If F>>1, the explained difference is greater than the unexplained difference, and we can assume the variance is due to more than randomization alone.