Block system (field test)

A block system is the special form of an agricultural field test to test the performance of different types of seeds , pesticides or fertilizers . With block systems the attempt is made to obtain well-founded statements about the performance characteristics of the test objects by arranging the parcels and evaluating the results using stochastics ( probability theory , statistics ).

history

Test facilities were set up at the beginning of the last century, especially in agricultural field testing in connection with variety tests in the field. One center was Rothamsted Experimental Station near London, where the statistical department was under the direction of Fisher (1926). It was there that one of Fisher's first books on experimental design was written (1935). Since the nature and quality of the soil fluctuated greatly in the test fields, the field was divided into so-called blocks, which were broken down into sections. It was assumed that the soil within the blocks is relatively homogeneous, so that differences in the yields of varieties that were grown on the sections of a block were solely due to the varieties and not the soil differences. To ensure the homogeneity of the soil within the blocks, the blocks could not be too big. On the other hand, the pieces for harvesting (especially with machines) had to be of a certain size. As a result, there were only a limited number of pieces within the block and only a limited number of grades could be tested in a block. If all varieties could be grown in each of the blocks, you had a complete block system. Often, however, the number of types was greater than the number of pieces in the block. This led to the development of incomplete block systems, including above all fully balanced incomplete block systems, which guaranteed that all variety differences can be estimated with the same variance using models of the analysis of variance .

In cases in which disruptive influences in two directions had to be taken into account (e.g. moisture gradient from north to south and changes in soil fertility from west to east), so-called line-column systems were developed, and Latin squares were mainly used .

Balanced incomplete block systems

Definition 1

The assignment of a given number N> 1 of experimental units to the levels of p Prüffaktoren and the step of q confounding factors (block factors) is p- factorial system with q block factors. If p = 1, then the one-factorial test facility is called a simple test facility , if p > 1, one also speaks of a factorial test for short. If q = 0, one speaks of a completely randomized or simple test installation.

A block system is a finite incidence structure , consisting of an incidence matrix, a finite set of v elements, called treatments, and a finite set of b sets, called blocks; they are the levels of the disruptive factor. The levels of the block factor are called blocks.

Definition 2

The elements of the incidence matrix with v rows and b columns indicate how often the i- th treatment representing the i- th row occurs in the j- th block defining the j- th column . If all elements of the incidence matrix n _{ij are} either 0 or 1, then the incidence matrix and the block structure corresponding to it are called binary. The b column sums k _{j of} the incidence matrix are called block sizes. The v row sums r _{i of} the incidence matrix are called repetitions. A block system is called complete if the elements of the incidence matrix are all positive ( n _ij > 1). A block layout is called incomplete if the incidence matrix has at least one zero. Blocks are called incomplete if there is at least one zero in the corresponding column of the incidence matrix.

In block systems, the randomization is to be carried out as follows: The test units in each block are randomly assigned to the treatments that occur in this block. The randomization is applied individually for each block. For complete block systems with v test units per block, each of which is assigned to exactly one of the v treatments, the randomization is thus ended. The situation is different in the case k < v . In the case of incomplete block systems, the abstract blocks, as they arise from the mathematical construction, are randomly assigned to the real blocks.

In the case of incomplete binary block layouts in particular, it makes sense to use a compact notation for characterization instead of the incidence matrix. Each block corresponds to an expression in brackets containing the numbers of the treatments contained in the block.

example

A block system with v = 4 treatments and b = 6 blocks is given by the following compact notation

{(1.3), (2.4), (1.3), (2.4), (2.4), (2.4)}

Are defined. For example, the first bracket represents block 1 in which treatments 1 and 3 occur.

Definition 3

A block layout with a symmetrical incidence matrix is ​​called a symmetrical block layout. If all treatments occur equally often in a block system, i. i.e. , if the number of repetitions r _i = r , then this system is called repetitive. If in a block system the number of test units per block is the same, i. i.e. , if k _j = k, then this system is called the same block.

It is true that both the sum of all repetitions r _i and the sum of all block sizes k _{j must be} equal to the number N of test units in a block system. The following applies to every block system:

${\displaystyle \sum _{i=1}^{v}r_{i}=\sum _{j=1}^{b}k_{j}=N}$

Specifically, it follows for block systems that are repeated and identical in blocks ( r _i = r and k _j = k ):

vr = bk .

In symmetrical block systems, b = v and r _i = k _i ( i = 1, ..., v ).

Definition 12.8

A (fully) balanced incomplete block system ( BUB ) is an incomplete block system that is identical in blocks and repetitive with the additional property that each pair of treatments occurs in an equal number, say in λ , blocks. If a BUB has v treatments with r repetitions in b blocks of size k < v , we call it the B ( v, k , λ ) system. A BUB for a pair ( v, k ) is called elementary if it cannot be broken down into at least two BUB for this pair ( v, k ). A BUB for a pair ( v, k ) is called the smallest BUB for this pair ( v, k ) if r (and thus also b and l ) is minimal.

Only three of the five parameters v, b, k, r , λ of a BUB occur in the symbol B ( v, k , λ ) . This is sufficient because only three of the five parameters can be freely selected, the other two are then automatically set. This happens through the two necessary conditions for a balanced, incomplete block system:

${\displaystyle \sum _{i=1}^{v}r_{i}=\sum _{j=1}^{b}k_{j}=N}$

and λ v -1) = r ( k -1).

The three conditions that are necessary for a BUB to exist are not always sufficient. The values

v = 16, r = 3, b = 8, k = 6, λ = 1

fulfill the necessary conditions because 16 · 3 = 8 · 6 and 1 · 15 = 3 · 5, nevertheless there is no BUB with this parameter combination.

There is another necessary condition, Fisher's inequality, according to which always

b ≥ v must apply.

But even if all three conditions apply, a BUB does not always have to exist, e.g. B. this is for

v = 22, k = 8, b = 33, r = 12, λ = 4

and

v = 34, r = 12, b = 34, k = 12, λ = 4

the case. The smallest BUB for

v = 22, k = 8 and v = 34 and k = 12

exist, have the parameters

v = 22, k = 8, b = 66, r = 24, λ = 8 or v = 34, r = 18, b = 51, k = 12, λ = 6.

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