I have a matrix $$m = \begin{bmatrix} 0 & 2 & 1 & 4 & 3 \\ 1 & 0 & 3 & 2 & 4 \\ 3 & 1 & 0 & 2 & 4 \\ 4 & 3 & 1 & 0 & 2 \\ 4 & 3 & 1 & 2 & 0 \end{bmatrix}$$
Where the rows represent judges and columns represent persons ranked. There are three transformation rules -
- For $j<i$; place in the $(i,j)$ cell the observed rank of the person expected to be ranked j-th by the i-th judge
- For i = j; $(i,j) = 0$
- For $j>i$; place in the $(i,j)$ cell the observed rank of the person expected to be ranked j - 1.
the transformed matrix is:
$$T = \begin{bmatrix} 0 & 2 & 1 & 4 & 3 \\ 1 & 0 & 3 & 2 & 4 \\ 1 & 2 & 0 & 3 & 4 \\ 1 & 2 & 3 & 0 & 4 \\ 2 & 1 & 3 & 4 & 0 \end{bmatrix}.$$
I think I may be misinterpreting this transformation - for example - in $m$ for $i=5$ and $j=1$ we have $i>j$ so the rule says to place in the $(i,j)$-th cell which would be $(5,1)$, the observed rank of the person expected to be ranked $j-th$ by the $i-th$ judge so this would be entry $m(5,1) = 4$ but $T(5,1) = 2$.
I am simply trying to reproduce this example in this paper - page 6-7 http://www.statpt.com/correlation/lewis_johnson_1971.pdf
It seems that you missed the footnote: “transformed according to Homans’ hypothesis” - for example - in $m$ for $i=5$ and $j=1$ we have $i>j$ so the rule says to place in the $(i,j)$-th cell which would be $(5,1)$, the observed rank of the person expected to be ranked $j$-th by the $i$-th judge, that is of the person expected to be ranked $1$-th by the $5$-th judge, so, according to Homans’ hypothesis, this is the $4$-th person which has the observed rank $2$ by $5$-th judge, so this would be entry $m(5,1)=2=T(5,1)$.