Download On the efficiency of supplemented designs related to the crop

On the efficiency of supplemented designs related to the crop
rotation experiments.
Tadeusz Przybysz
Department of Applied Mathematics, Agricultural University of Lublin
Akademicka 13, P.O. Box 158
20-934 Lublin, Poland
[email protected]
Urszula Bronowicka-Mielniczuk
Department of Applied Mathematics, Agricultural University of Lublin
Akademicka 13, P.O. Box 158
20-934 Lublin, Poland
[email protected]
Jacek Mielniczuk
Department of Applied Mathematics, Agricultural University of Lublin
Akademicka 13, P.O. Box 158
20-934 Lublin, Poland
[email protected]
1. Problem
The subject of our investigation is a measure of the efficiency of cropping system design
versus orthogonal design as the inverse ratio of the average variances with which the elementary
contrasts are estimated in two designs. Since the study concerns the unequally replicated set-up, we
assume the vector of replications to be the same in both designs. Due to the comparative character
of crop rotation designs, the formula for efficiency factor has been restricted to the elementary
contrasts comprising treatments from distinct systems (Przybysz et al., 1999).
Our main objective is to re-examine the efficiency concepts concerning the block designs with
the incidence matrix of the following form
(1)
 1 ’v 2 ⊗ N r1 
,

N= ’
1v ⊗ N r 
2 
 1
where the binary matrices N r1 , N r2 of dimensions v1 × v1 , v 2 × v 2 respectively, have properties of
being circulant. The subscript denotes the number of ”ones” in the initial row of such the matrix.
The other rows are obtained from the first one by a cyclical rotation. The detailed study concerns the
case of equal rotation lengths ( v1 = v 2 ). The discussion has been concluded with the following
bound
(2)
*
var − var ≥
1 r1 + r2 v − r1 − r2
≥0,
v 2 r1 r2 r1 + r2 − 1
where star superscript stands for the restricted average variance. The counter relations are
exemplified.
2. Application
We give the short description of the practical context within the theoretical considerations are
embedded. Crop rotation (cropping system) is the growing two or more crops sequentially on the
same area in the successive years. We start an experiment with all plants included into the sequence.
An experiment is aimed at comparing the effects of the competing rotations. The scheme of the
experiment is arranged in the following table
7DEOH7KHDUUDQJHPHQWRIH[SHULPHQWZLWKIRXUFRXUVHFURSSLQJV\VWHPV
Year no.
1
wheat
lupine
wheat
potato
1
2
3
4
Plot no. ( system A)
2
3
potato wheat
wheat potato
lupine wheat
wheat lupine
4
lupine
wheat
potato
wheat
Plot no. ( system B)
5
6
7
rape
beans
wheat
wheat wheat rape
beans wheat wheat
rape
beans wheat
8
wheat
beans
rape
wheat
In the approach undertaken, the statistical inference is made on the basis of wheat responses
(test crop). The linear response model specifies the observation as follows
(3)
y N = 1 N + R ’Np +
’
Nv
+ e N + D ’Nb + G ’Nn ϕ +
N
,
where y is the observation vector, is the general mean, , , , ϕ are the vectors of replications,
plots, years and interaction effects, respectively, e, are vectors of random errors, whereas
R, , D, G are the design matrices. The implementation is concerned with the induced model
comprising one replication, i.e., model of incomplete block design. The incidence matrix of such the
design admits representation (1). The incidence submatrices coincide with the patterns of allocation
of plants to the plots. Namely, the rows are related to the plots, the columns correspond to the years,
’’one’’ indicates that the wheat was on the plot, zero refers to the other species.
REFERENCES
Shah, K. R. and Sinha, B. K. (1989). Theory of Optimal Designs. Springer-Verlag.
John, J. A. (1987). Cyclic designs. London, Chapman and Hall.
Przybysz, T., Bronowicka-Mielniczuk, U. and Mielniczuk, J. Efficiency of crop rotation designs
with unequal numbers of replications. To appear in Tatra Mountains Mathematical Publications,
Probastat’98, Mathematical Institute; Slovak Academy of Sciences.
RÉSUMÉ
Nous considerons certains aspects d’effectivitè d’essais rotation culturalèes. Cet article
ètudie les plans avec repetitions inègalèes et suit l’approche presentèe á des articles prècèdentes.
Les formules pour les dèpendances entre les coefficients d’effectivitè sont presentès. L’application
est facile au point de vue de calcul. Les examples de l’application sont demonstrès.