| Title: | Partially Balanced Incomplete Block Designs |
|---|---|
| Description: | The PBIB designs are important type of incomplete block designs having wide area of their applications for example in agricultural experiments, in plant breeding, in sample surveys etc. This package constructs various series of PBIB designs and assists in checking all the necessary conditions of PBIB designs and the association scheme on which these designs are based on. It also assists in calculating the efficiencies of PBIB designs with any number of associate classes. The package also constructs Youden-m square designs which are Row-Column designs for the two-way elimination of heterogeneity. The incomplete columns of these Youden-m square designs constitute PBIB designs. With the present functionality, the package will be of immense importance for the researchers as it will help them to construct PBIB designs, to check if their PBIB designs and association scheme satisfy various necessary conditions for the existence, to calculate the efficiencies of PBIB designs based on any association scheme and to construct Youden-m square designs for the two-way elimination of heterogeneity. R. C. Bose and K. R. Nair (1939) <http://www.jstor.org/stable/40383923>. |
| Authors: | Parneet Kaur <[email protected]>, Kush Sharma <[email protected]>, Davinder Kumar Garg <[email protected]> |
| Maintainer: | Kush Sharma <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.3 |
| Built: | 2024-11-15 03:50:30 UTC |
| Source: | https://github.com/cran/PBIBD |
The PBIB designs are important type of incomplete block designs having wide area of their applications for example in agricultural experiments, in plant breeding, in sample surveys etc. This package constructs various series of PBIB designs and assists in checking all the necessary conditions of PBIB designs and the association scheme on which these designs are based on. It also assists in calculating the efficiencies of PBIB designs with any number of associate classes. The package also constructs Youden-m square designs which are Row-Column designs for the two-way elimination of heterogeneity. The incomplete columns of these Youden-m square designs constitute PBIB designs. With the present functionality, the package will be of immense importance for the researchers as it will help them to construct PBIB designs, to check if their PBIB designs and association scheme satisfy various necessary conditions for the existence, to calculate the efficiencies of PBIB designs based on any association scheme and to construct Youden-m square designs for the two-way elimination of heterogeneity.
The DESCRIPTION file:
| Package: | PBIBD |
| Type: | Package |
| Title: | Partially Balanced Incomplete Block Designs |
| Version: | 1.3 |
| Date: | 2017-12-20 |
| Author: | Parneet Kaur <[email protected]>, Kush Sharma <[email protected]>, Davinder Kumar Garg <[email protected]> |
| Maintainer: | Kush Sharma <[email protected]> |
| Description: | The PBIB designs are important type of incomplete block designs having wide area of their applications for example in agricultural experiments, in plant breeding, in sample surveys etc. This package constructs various series of PBIB designs and assists in checking all the necessary conditions of PBIB designs and the association scheme on which these designs are based on. It also assists in calculating the efficiencies of PBIB designs with any number of associate classes. The package also constructs Youden-m square designs which are Row-Column designs for the two-way elimination of heterogeneity. The incomplete columns of these Youden-m square designs constitute PBIB designs. With the present functionality, the package will be of immense importance for the researchers as it will help them to construct PBIB designs, to check if their PBIB designs and association scheme satisfy various necessary conditions for the existence, to calculate the efficiencies of PBIB designs based on any association scheme and to construct Youden-m square designs for the two-way elimination of heterogeneity. R. C. Bose and K. R. Nair (1939) <http://www.jstor.org/stable/40383923>. |
| License: | GPL (>= 2) |
| NeedsCompilation: | no |
| Packaged: | 2017-12-21 14:04:21 UTC; ABC |
| Date/Publication: | 2017-12-21 14:23:29 UTC |
| Repository: | https://sharmakush.r-universe.dev |
| RemoteUrl: | https://github.com/cran/PBIBD |
| RemoteRef: | HEAD |
| RemoteSha: | 4358637800ab96f362a4d2d8bc97d3db6434f0d5 |
Index of help topics:
PBIBD-package Partially Balanced Incomplete Block Designs
apbibd Calculates the efficiencies of PBIB designs
with any number of associate classes.
circulant This function generates circulant matrix of
order n.
series1 This function constructs five-associate class
PBIB designs.
series2 This function constructs five-associate class
PBIB designs
series3 This function constructs five-associate class
PBIB designs
series4 This function constructs three-associate class
PBIB designs.
series5 This function constructs three-associate class
PBIB designs.
series6 This function constructs three-associate class
PBIB designs.
verify Verifies all the necessary conditions for the
existence of PBIB designs based on any
association scheme.
ym1 The function constructs Youden-m square
designs. The function provides the parameters
of the PBIB design constituted when the
incomplete columns of this Youden-m square are
taken as blocks.
ym2 The function constructs Youden-m square
designs. The function provides the parameters
of the PBIB design constituted when the
incomplete columns of this Youden-m square are
taken as blocks.
ym3 The function constructs Youden-m square
designs. The function provides the parameters
of the PBIB design constituted when the
incomplete columns of this Youden-m square are
taken as blocks.
This package is currently under intensive development and changes are to be expected in the near future.
Parneet Kaur <[email protected]>
Kush Sharma <[email protected]>
Davinder Kumar Garg <[email protected]>
Maintainer: Kush Sharma <[email protected]>
Dey, A. (1986). Theory of block designs.Wiley Eastern Limited, New Delhi
Garg, D.K., and Singh, G.P. (2015). General solution of normal equations in the intra-block analysis of PBIB designs with any (m>=2) number of associate classes. American Journal of Sustainable Cities and Society, 4(11), 196-202
Kaur, P. and Garg, D. K. (2016). Construction of some higher associate class PBIB designs using symmetrically repeated differences, Arya Bhatta Journal of Mathematics and Informatics, 8(2), 65-78.
Rao, C. R. (1947b). General methods of analysis for incomplete block designs, J. Amer. Statist. Assoc. 42, 541-561
Sharma, K. and Garg, D. K. (2017) Construction of Three associate PBIB designs using some sets of initial blocks. International Journal of Agricultural and Statistical Sciences (IJASS), 13(1): 55-60
Sharma, K. and Garg, D. K. (2017). m-associate PBIB designs using Youden-m Squares. Communications in Statistics-Theory and Methods, In press. DOI: 10.1080/03610926.2017.1324990
This function calculates the different kinds of efficiencies and the overall efficiency factor of Partially Balanced Incomplete Block Designs with any number of associate classes. The total number of treatments i.e. v, replications i.e. r, block size i.e. k, vector l of lambda's (lambda i being the ith element of vector l), vector n of number of associates (n i, i.e. number of ith associates, being the ith element of vector n), a list P of P-matrices of the association scheme of the design (Pi being the ith matrix of the list P) are to be supplied as input to the function.
apbibd(v, r, k, l, n, P)apbibd(v, r, k, l, n, P)
v |
Total number of treatments of the design |
r |
Replication of the treatments in the design |
k |
Block size of the design |
l |
A vector containing lambda 1, lambda 2, lambda 3,..., lambda m as its first, second, third,..., mth elements |
n |
A vector containing n1, n 2, n 3, ..., n m as its first, second, third,..., mth elements |
P |
A list containing P-matrices of the association scheme of the design such that P1 is its first element, P2 is second element, P3 is third element,..., Pm is the mth element |
Returns a list with (m+1) components:
E1 |
Efficiency E1 of the design |
E2 |
Efficiency E2 of the design and so on ... |
Em |
Efficiency Em of the design |
E |
Overall efficiency factor of the design |
Kush Sharma, Davinder Kumar Garg
v<-25 r<-9 k<-9 l<-c(5,2,5,2,5) n<-c(2,8,2,8,4) P1<-matrix(c(0,0,1,0,0,0,0,0,4,4,1,0,1,0,0,0,4,0,4,0,0,4,0,0,0),nrow=5,ncol=5) P2<-matrix(c(0,0,0,1,1,0,0,1,3,3,0,1,0,1,0,1,3,1,3,0,1,3,0,0,0),nrow=5,ncol=5) P3<-matrix(c(1,0,1,0,0,0,4,0,4,0,1,0,0,0,0,0,4,0,0,4,0,0,0,4,0),nrow=5,ncol=5) P4<-matrix(c(0,1,0,1,0,1,3,1,3,0,0,1,0,0,1,1,3,0,0,3,0,0,1,3,0),nrow=5,ncol=5) P5<-matrix(c(0,2,0,0,0,2,6,0,0,0,0,0,0,2,0,0,0,2,6,0,0,0,0,0,3),nrow=5,ncol=5) P<-list(P1,P2,P3,P4,P5) apbibd(v,r,k,l,n,P)v<-25 r<-9 k<-9 l<-c(5,2,5,2,5) n<-c(2,8,2,8,4) P1<-matrix(c(0,0,1,0,0,0,0,0,4,4,1,0,1,0,0,0,4,0,4,0,0,4,0,0,0),nrow=5,ncol=5) P2<-matrix(c(0,0,0,1,1,0,0,1,3,3,0,1,0,1,0,1,3,1,3,0,1,3,0,0,0),nrow=5,ncol=5) P3<-matrix(c(1,0,1,0,0,0,4,0,4,0,1,0,0,0,0,0,4,0,0,4,0,0,0,4,0),nrow=5,ncol=5) P4<-matrix(c(0,1,0,1,0,1,3,1,3,0,0,1,0,0,1,1,3,0,0,3,0,0,1,3,0),nrow=5,ncol=5) P5<-matrix(c(0,2,0,0,0,2,6,0,0,0,0,0,0,2,0,0,0,2,6,0,0,0,0,0,3),nrow=5,ncol=5) P<-list(P1,P2,P3,P4,P5) apbibd(v,r,k,l,n,P)
Circulant matrix, which is a special kind of Toeplitz matrix, is a square matrix of order n whose rows are obtained by cyclically rotated versions of a list “l” of length n such that the first row is obtained by cyclically rotating one element toward right the list “l” and each of the other row is the cyclically rotated one element toward the right version of the previous row. This function is used to generate a circulant matrix of order n. The order of the circulant matrix i.e. n is supplied as an argument to the function.
circulant(n)circulant(n)
n |
n is the order of the circulant matrix we want to generate. |
The function returns a circulant matrix c of order n.
Kush Sharma, Davinder Kumar Garg
circulant(7)circulant(7)
Let us consider a module M of residue class mod(5) having elements 0, 1, 2, 3, 4 and all the elements of M are assigned to each of the n >= 2 classes. This function constructs PBIB designs with the following parameters:
v = 5n, b = 5n, r = n+4, k = n+4
lambda 1 = 5, lambda 2 = 2, lambda 3 = 5, lambda 4 = 2, lambda 5 = n
series1(n)series1(n)
n |
n is the number of classes to which the elements of Module M are assigned |
The function returns the required PBIB design with specified parameters
Parneet Kaur, Davinder Kumar Garg
series1(2)series1(2)
Let us consider a module M of residue class mod(5) having elements 0, 1, 2, 3, 4 and all the elements of M are assigned to each of the n >= 2 classes. This function constructs PBIB designs with the following parameters:
v = 5n, b = 5n, r = n+3, k = n+3
lambda 1 = 3, lambda 2 = 1, lambda 3 = 3, lambda 4 = 2, lambda 5 = n
series2(n)series2(n)
n |
n is the number of classes to which the elements of Module M are assigned. |
The function returns the required PBIB design with specified parameters.
Parneet Kaur, Davinder Kumar Garg
series2(4)series2(4)
Let us consider a module M of residue class mod(5) having elements 0, 1, 2, 3, 4 and all the elements of M are assigned to each of the n >= 2 classes. This function constructs PBIB designs with the following parameters:
v = 5n, b = 5n, r = 2(n + 1), k = 2(n + 1)
lambda 1 = n + 2, lambda 2 = n + 2, lambda 3 = 3, lambda 4 = 2, lambda 5 = 2n
series3(n)series3(n)
n |
n is the number of classes to which the elements of Module M are assigned |
The function returns the required PBIB design with specified parameters
Parneet Kaur, Davinder Kumar Garg
series3(5)series3(5)
Let us consider a module M having m elements. To each element of the module, there corresponds n distinct classes, where m >= 5 and n >= 2. With these v = mn treatments following are parameters of the three-associate class PBIB design:
v = mn, b = mn, r = (m+n-1), k = (m+n-1)
lambda 1 = m, lambda 2 = 2, lambda 3 = n
series4(m, n)series4(m, n)
m |
Size of the module M. |
n |
n is the number of classes to which the elements of Module M are assigned. |
This function returns the required three-associate class PBIB design.
Parneet Kaur, Davinder Kumar Garg
series4(5,2)series4(5,2)
Consider a module M having m elements and there are n classes corresponding to each element of the module. Thus, we have a total of v = mn treatments (m is odd prime). For these v = mn treatments following are the parameters of the three-associate class PBIB design:
v = mn, b = mn, r = (m+n-2), k = (m+n-2)
lambda 1 = m-2, lambda 2 = 2, lambda 3 = n-2
series5(m, n)series5(m, n)
m |
Size of Module M. |
n |
n is the number of classes to which the elements of Module M are assigned. |
The function returns the required three-associate class PBIB design with the parameters specified in the description.
Kush Sharma, Davinder Kumar Garg
series5(5,3)series5(5,3)
Consider a module M having m elements and there are n classes corresponding to each element of the module. Thus, we have a total of v = mn treatments (m is odd prime). For these v = mn treatments following are the parameters of the three-associate class PBIB design:
v = mn, b = m, r = (m-1), k = (m-1)n
lambda 1 = m-2, lambda 2 = m-2, lambda 3 = m-1
series6(m, n)series6(m, n)
m |
Size of Module M. |
n |
n is the number of classes to which the elements of Module M are assigned. |
The function returns the required three-associate class PBIB design with the parameters specified in the description.
Kush Sharma, Davinder Kumar Garg
series6(5,3)series6(5,3)
There exists various necessary conditions for the existence of the PBIB design as well as the association scheme on which the PBIB design is based. This function Verifies all those necessary conditions for the existence of PBIB designs based on any association scheme. The total number of treatments i.e. v, the total number of blocks i.e. b, replications i.e. r, block size i.e. k, vector l of lambda's (lambda i being the ith element of vector l), vector n of number of associates (n i, i.e. number of ith associates, being the ith element of vector n), a list P of P-matrices of the association scheme of the design (Pi being the ith matrix of the list P) are to be supplied as input to the function.
verify(v, b, r, k, l, n, P)verify(v, b, r, k, l, n, P)
v |
Total number of treatments of the design |
b |
Total number of blocks in the design |
r |
Replication of the treatments in the design |
k |
Block size of the design |
l |
A vector containing lambda 1, lambda 2, lambda 3,..., lambda m as its first, second, third,..., mth elements |
n |
A vector containing n 1, n 2, n 3, ..., n m as its first, second, third,..., mth elements |
P |
A list containing P-matrices of the association scheme of the design such that P1 is its first element, P2 is second element, P3 is third element,..., Pm is the mth element |
The function tells if all the necesary conditions for the existence of PBIB design based on some association scheme hold. If not, it highlights all the conditions which do not hold.
Kush Sharma, Davinder Kumar Garg
v<-12 b<-12 r<-5 k<-5 l<-c(1,2,2) n<-c(2,3,6) P1<-matrix(c(1,0,0,0,0,3,0,3,3),nrow=3,ncol=3) P2<-matrix(c(0,0,2,0,2,0,2,0,4),nrow=3,ncol=3) P3<-matrix(c(0,1,1,1,0,2,1,2,2),nrow=3,ncol=3) P<-list(P1,P2,P3) verify(v,b,r,k,l,n,P)v<-12 b<-12 r<-5 k<-5 l<-c(1,2,2) n<-c(2,3,6) P1<-matrix(c(1,0,0,0,0,3,0,3,3),nrow=3,ncol=3) P2<-matrix(c(0,0,2,0,2,0,2,0,4),nrow=3,ncol=3) P3<-matrix(c(0,1,1,1,0,2,1,2,2),nrow=3,ncol=3) P<-list(P1,P2,P3) verify(v,b,r,k,l,n,P)
If we omit the same number of rows, say t rows, from the top and the bottom of the Circulant matrix, such that we are left with atleast two rows, the resulting arrangement of rows is a Youden-m square.
(A) For even-ordered Circulant matrices with order v >= 4, the columns of the Youden-m squares so obtained constitute the PBIB designs with the following parameters:
v >= 4 and even, b = v, r = k = v-2t
lambda 1 = v - 2(t + 1), lambda m-i = v - 2t - 1 - 2i ; i = 0, 1, ..., t-1
lambda t = lambda t+1 = ... = lambda m-t = v - 4t. If t >=3 then, lambda i = v - 2(t + i); i = 2, 3, ..., t-1
(B)For odd-ordered Circulant matrices with order v >= 5, the columns of the Youden-m squares so obtained constitute the PBIB designs with the following parameters:
v >=5 and odd, b = v, r = k = v-2t
lambda 1 = v - 2t - 1, lambda m-i = v - 2(t + 1) - i; i = 0, 1, ..., t - 1, lambda m-(t-1)-i = lambda m-(t-1) - i ; i = 0, 1, 2, ..., t-1
and lambda 2 = lambda 3= ... = lambda (m-2t+1) = lambda (m-2t+2)
ym1(n, t)ym1(n, t)
n |
n is the order of the circulant matrix which is also the number of treatments |
t |
t is the number of rows you want to omit from both ends of the circulant matrix |
The function returns the required Youden-m square design. It also returns the parameters of the PBIB design constituted by taking the incomplete columns of the Youden-m square as blocks.
Kush Sharma, Davinder Kumar Garg
ym1(6,1)ym1(6,1)
By omitting the middle 2t (t = 1, 2, ...) rows of any even ordered Circulant matrix with order v >= 6 and considering only those rows which lie either above or below the omitted 2t rows, the resulting arrangement of rows gives a new type of Youden-m square. The columns of these Youden-m squares constitute m-associate class PBIB designs, with the following parameters:
v >= 6 and even, b = v, r = k = (v/2)-t, lambda 1 = r-2, lambda m = lambda 1+1
If m is even, then lambda i+1 = lambda i - 2; i = 1, 2, ..., m/2 and lambda i-1 = lambda i - 2; i = m, m-1, ..., (m/2)+1
If m is odd, then lambda i+1 = lambda i - 2; i = 1, 2, ..., (m+1)/2 and lambda i-1 = lambda i - 2; i = m, m-1, ..., ((m+1)/2) + 1
ym2(n, t)ym2(n, t)
n |
n is the order of the circulant matrix which is also the number of treatments |
t |
t is the number of rows you want to omit from both ends of the circulant matrix |
The function returns the required Youden-m square design. It also returns the parameters of the PBIB design constituted by taking the incomplete columns of the Youden-m square as blocks.
Kush Sharma, Davinder Kumar Garg
ym2(8,1)ym2(8,1)
By omitting the middle and equal number of rows, say t rows, from both ends of any odd-ordered Circulant matrix with order v >= 7and considering those rows that lie either between the middle omitted row and omitted rows from the top or between the middle omitted row and the omitted rows from the bottom of the Circulant matrix. Then this arrangement of rows gives us a Youden-m square. These Youden-m square designs are the designs for two-way elimination of heterogenity. The columns of this Youden-m square constitue PBIB design with the following parameters:
v >= 7 and odd, b = v, r = k = ((v+1)/2) -1-t
lambda 1 = v - 6 - (m - 4 + t), lambda i = 0 ; i = 2, 3, ..., t+2
if m > 3 then, lambda j = lambda j-1+1 ; j = t+3, t+4, ...,m
ym3(n, t)ym3(n, t)
n |
n is the order of the circulant matrix which is also the number of treatments |
t |
t is the number of rows you want to omit from both ends of the circulant matrix |
The function returns the required Youden-m square design. It also returns the parameters of the PBIB design constituted by taking the incomplete columns of the Youden-m square as blocks.
Kush Sharma, Davinder Kumar Garg
ym3(7,1)ym3(7,1)