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)
n is the order of the circulant matrix which is also the number of treatments
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
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