# Chapter 9 Incomplete Block Designs

## 9.1 Introduction

In this chapter we are going to learn more about block designs. The
block designs we have seen so far were *complete*, meaning that every
block containted *all* treatments. In practice, this is not always
possible. For example, the physical size of a block might be too
small. There are also situations where it is not advisable to have too
many treatments in each block. If we do a food tasting experiment, we
typically want to restrict the number of different “recipes”
(treatments) we want to show to an individual rater. Otherwise, the
response will get unreliable (because the rater might get bored after
some time).

If we cannot “afford” to do a complete design we get a so-called
**incomplete block design (IBD)**. We
have to decide what *subset* of treatments we use on an individual
block. If we don’t make a wise decision, it can happen that the design
is “destroyed” in the sense that certain quantities are not estimable
anymore. This would be a worst case scenario.

Consider the following design with treatments \(A\) to \(F\) and blocks \(1\)
to \(6\) (each *column* corresponds to a block).

1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|

\(A\) | \(A\) | \(B\) | \(D\) | \(D\) | \(E\) |

\(B\) | \(C\) | \(C\) | \(E\) | \(F\) | \(F\) |

E.g., on block 5 we apply the two treatments \(D\) and \(F\). First, we create a fictional data set.

```
set.seed(25) ## for reproducibility
dat <- data.frame(block = factor(rep(1:6, times = 2)),
treat = factor(c("A", "A", "B", "D", "D", "E",
"B", "C", "C", "E", "F", "F")),
y = rnorm(12)) ## no signal, just noise
str(dat)
```

```
## 'data.frame': 12 obs. of 3 variables:
## $ block: Factor w/ 6 levels "1","2","3","4",..: 1 2 3 4 5 6 1 2 3 4 ...
## $ treat: Factor w/ 6 levels "A","B","C","D",..: 1 1 2 4 4 5 2 3 3 5 ...
## $ y : num 0.1217 0.2135 -0.0477 0.5461 -1.5689 ...
```

We can get an overview of the used combinations with `xtabs`

.

```
## block
## treat 1 2 3 4 5 6
## A 1 1 0 0 0 0
## B 1 0 1 0 0 0
## C 0 1 1 0 0 0
## D 0 0 0 1 1 0
## E 0 0 0 1 0 1
## F 0 0 0 0 1 1
```

What happens if we fit our standard main effects model here?

```
options(contrasts = c("contr.treatment", "contr.poly"))
fit <- aov(y ~ block + treat, data = dat)
dummy.coef(fit)
```

```
## Full coefficients are
##
## (Intercept): 0.307025
## block: 1 2 3 4 5
## 0.0000000 -0.2788532 -1.1683252 1.8724763 0.2491680
## treat: A B C D E
## 0.0000000 0.9989451 1.6586102 -1.8792448 -1.8273439
##
## (Intercept):
## block: 6
## 2.2892593
## treat: F
## 0.0000000
```

At first sight, it looks like we were able to fit the specified model.
However, it is strange that *two* of the treatment coefficients are set
to zero and not only *one* as usual when using `contr.treatment`

.

What happens if we want to investige a certain contrast, say treatment \(A\) vs. \(F\)?

`## Error in h(simpleError(msg, call)): error in evaluating the argument 'object' in selecting a method for function 'summary': some linear functions are not estimable`

We get an error message! The reason is that we *cannot* estimate the
difference between treatments \(A\) and \(F\) with this data set (at least
not with a fixed effects model). The reason lies in the “structure” of
the incomplete design. It is an example of a so-called **disconnected
design**. We have the treatment set \(\{A, B, C\}\) in the first three blocks and \(\{D, E, F\}\) in the last three
blocks (check the output from `xtabs`

above). As the two sets are
disjoint, we have no idea about any differences involving treatments
from the two sets because we never see a pair together in the same
block. This is also causing the strange output in `dummy.coef`

above. As
we have that dependency between treatment and block, we cannot
distinguish block and treatment effects anymore (confounding!). Here,
`R`

automatically drops one of the treatment levels out of the model
(which might be dangerous if you are not carefully inspecting the
output).

Hence, performing such a disconnected design was obviously not a good idea from a design of experiment point of view as we are not capable anymore to estimate differences involving treatments from the two disjoint sets. Intuitively, we should have a good “mix” of treatments in every block.

## 9.2 Balanced Incomplete Block Designs

We could simply randomize subsets of treatments to different blocks. This would work well if we have enough blocks. If we only have a small number of blocks, there would be slight chance that we end up with a disconnected design.

However, we can also try to fulfill some optimality criterion. One
candidate would be the criterion that we can estimate all
treatment differences with the **same precision**. For example, this
would mean that all confidence intervals for \(\alpha_i - \alpha_j\) have
the same width (for *any* pair \(i,j\)). The design which fulfills
this criterion is the so-called balanced incomplete block design.

A **balanced incomplete block design (BIBD)**
is an *incomplete* block design where all *pairs* of treatments occur
together in the *same* block equally often (\(= \lambda\)).

We use the following notation:

- \(g\): number of treatments (“number of different chocolate chip cookie brands”)
- \(b\): number of blocks (“number of raters”)
- \(k\): number of units per block (\(k < g\)) (“number of cookie brands each raters gets to see”)
- \(r\): number of replicates per treatment (“how often do we see a certain cookie brand across all raters?”)
- \(N\): total number of units

We have \(N = b \cdot k = g \cdot r\).

For every setting \(k < g\) we can find a BIBD by taking *all* possible
subsets, whereof we have \(g \choose k\) (binomial coefficient,
implemented in the function `choose`

in `R`

). The corresponding design
is called an **unreduced balanced incomplete block design**.
For example, if we have \(g = 6\) treatments and \(k = 3\) units per block we get
\({6 \choose 3} = 20\) blocks. In `R`

we can easily get this with the
function `combn`

.

```
## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13] [,14]
## [1,] 1 1 1 1 1 1 1 1 1 1 2 2 2 2
## [2,] 2 2 2 2 3 3 3 4 4 5 3 3 3 4
## [3,] 3 4 5 6 4 5 6 5 6 6 4 5 6 5
## [,15] [,16] [,17] [,18] [,19] [,20]
## [1,] 2 2 3 3 3 4
## [2,] 4 5 4 4 5 5
## [3,] 6 6 5 6 6 6
```

Every column defines a block in the matrix above. I.e., block 10 would get treatments 1, 5 and 6. Typically, we would randomize the order or the placement of the treatments within a block (as we should always do).

In practice we cannot always afford to do an unreduced BIBD as the required number of blocks might be too large. Whether a BIBD exists for a certain desired setting of number of blocks \(b\), block size \(k\) and number of treatments \(g\), is a combinatorial problem. A necessary (but not sufficient) condition for a BIBD to exist is \[ \frac{r \cdot (k - 1)}{g - 1} = \lambda. \] What is the intuition behind this formula? A treatment appears in a total of \(r\) different blocks. In each of these blocks, there are \(k - 1\) available other units, leading to a total of \(r \cdot (k - 1)\) available “units”. There are \(g - 1\) available (other) treatments that we must divide among them in order to have a balanced design. Hence, it must hold that \(r \cdot (k - 1) = \lambda \cdot (g - 1)\).

In `R`

, package `ibd`

provides some functionality to find (B)IBDs. For
example, function `bibd`

searches for a balanced incomplete block
design.

```
library(ibd)
des.bibd <- bibd(v = 6, b = 10, r = 5, k = 3, lambda = 2)
## arguments of function bibd are:
## - v: number of treatments
## - b: number of blocks
## - r: number of replicates (across all blocks)
## - k: number of units per block
## - lambda: lambda
des.bibd$design ## here, blocks are given by *rows*
```

```
## [,1] [,2] [,3]
## Block-1 1 3 4
## Block-2 1 4 5
## Block-3 1 3 6
## Block-4 4 5 6
## Block-5 1 2 6
## Block-6 3 5 6
## Block-7 2 4 6
## Block-8 2 3 5
## Block-9 1 2 5
## Block-10 2 3 4
```

In `des.bibd$design`

each *row* corresponds to a block.

For a setting where no BIBD is possible, we can try to find a design
that is nearly balanced using the function `ibd`

.

```
des.ibd <- ibd(v = 6, b = 9, k = 3)
## - v: number of treatments
## - b: number of blocks
## - k: number of units per block
des.ibd$design
```

```
## [,1] [,2] [,3]
## Block-1 1 5 6
## Block-2 1 4 5
## Block-3 3 5 6
## Block-4 3 4 6
## Block-5 1 3 4
## Block-6 1 2 6
## Block-7 2 4 6
## Block-8 2 4 5
## Block-9 2 3 5
```

We can find the so-called concurrence matrix of the generated design in
`des.ibd$conc.mat`

. It contains information about how many times any
pair of treatments appears together in the same block.

```
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 4 1 1 2 2 2
## [2,] 1 4 1 2 2 2
## [3,] 1 1 4 2 2 2
## [4,] 2 2 2 5 2 2
## [5,] 2 2 2 2 5 2
## [6,] 2 2 2 2 2 5
```

On the diagonal we see how often every treatment appears across all blocks. In the off-diagonal elements we have the corresponding information about treatment pairs. For example, treatment pair 1 and 4 appear together twice in the same block, while 1 and 3 only once.

## 9.3 Analysis of (B)IBDs

The analysis of an incomplete block design is “as usual”. We use a block factor and a treatment factor leading to \[ Y_{ij} = \mu + \alpha_i + \beta_j + \epsilon_{ij}. \] As we are faced with an unbalanced design we typically use sum of squares for treatment effects that are adjusted for block effects.

We use the data set `taste`

from package `daewr`

as an example. Twelve
different panelists rated four different recipes. We first load the
data and try to get an overview.

```
## 'data.frame': 24 obs. of 3 variables:
## $ panelist: Factor w/ 12 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ recipe : Factor w/ 4 levels "A","B","C","D": 1 1 1 2 2 3 1 1 1 2 ...
## $ score : num 5 7 5 6 6 8 6 5 4 7 ...
```

```
## recipe
## panelist A B C D
## 1 1 1 0 0
## 2 1 0 1 0
## 3 1 0 0 1
## 4 0 1 1 0
## 5 0 1 0 1
## 6 0 0 1 1
## 7 1 1 0 0
## 8 1 0 1 0
## 9 1 0 0 1
## 10 0 1 1 0
## 11 0 1 0 1
## 12 0 0 1 1
```

The output of `xtabs`

tells us that we have an *incomplete* design here.
It is actually a balanced incomplete block design. We can test this
with the function `isGYD`

in package `crossdes`

. For that reason, we
first have to transform the design information to the desired form.

```
##
## [1] The design is a balanced incomplete block design w.r.t. rows.
```

We use the standard model from above. As it is unbalanced data, we use
`drop1`

such that we get the sum of squares of `recipe`

adjusted for
`panelist`

. We could also use `summary`

here because `recipe`

appears
last in the model formula.

```
## Single term deletions
##
## Model:
## score ~ panelist + recipe
## Df Sum of Sq RSS AIC F value Pr(>F)
## <none> 6.875 -0.0039
## panelist 11 7.4583 14.333 -4.3712 0.8876 0.58110
## recipe 3 9.1250 16.000 14.2688 3.9818 0.04649 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

The \(F\)-test of recipe is significant. For illustrational reasons we have a look at all pairwise comparisons without any multiple testing correction.

```
library(multcomp)
contr <- glht(fit, linfct = mcp(recipe = "Tukey"))
summary(contr, test = adjusted("none"))
```

```
##
## Simultaneous Tests for General Linear Hypotheses
##
## Multiple Comparisons of Means: Tukey Contrasts
##
##
## Fit: aov(formula = score ~ panelist + recipe, data = taste)
##
## Linear Hypotheses:
## Estimate Std. Error t value Pr(>|t|)
## B - A == 0 0.750 0.618 1.214 0.2558
## C - A == 0 1.375 0.618 2.225 0.0531 .
## D - A == 0 -0.625 0.618 -1.011 0.3383
## C - B == 0 0.625 0.618 1.011 0.3383
## D - B == 0 -1.375 0.618 -2.225 0.0531 .
## D - C == 0 -2.000 0.618 -3.236 0.0102 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- none method)
```

We are not inspecting the p-values here as it is a multiple testing
problem. However, what we can observe is the fact that all differences
share the *same* standard error (remember the motivation for a BIBD?).

Actually, the largest difference would “survive” a multiple testing correction using Tukey HSD.

```
##
## Simultaneous Tests for General Linear Hypotheses
##
## Multiple Comparisons of Means: Tukey Contrasts
##
##
## Fit: aov(formula = score ~ panelist + recipe, data = taste)
##
## Linear Hypotheses:
## Estimate Std. Error t value Pr(>|t|)
## B - A == 0 0.750 0.618 1.214 0.6341
## C - A == 0 1.375 0.618 2.225 0.1881
## D - A == 0 -0.625 0.618 -1.011 0.7472
## C - B == 0 0.625 0.618 1.011 0.7473
## D - B == 0 -1.375 0.618 -2.225 0.1880
## D - C == 0 -2.000 0.618 -3.236 0.0421 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## (Adjusted p values reported -- single-step method)
```

We see that treatment \(C\) is significantly different from treatment \(D\).