I need to define a design matrix $X$ so that it fits this scenario: For $1\leq i< j \leq 5$, where $i$ are competitors and $j$ are different competitors. Their score is $y_{ij} =$ score of player $i$ minus score of player $j$.
$$Y_{ij} =\beta_i - \beta_j+\varepsilon_{ij}.$$
$\beta_i,\ldots,\beta_5$ are unknown parameters. $\varepsilon \sim N(0,\sigma^2I)$.
The $X$ matrix has to represent only 4 matches: $y_{12}$, $y_{34}$, $y_{25}$, $y_{15}$. and then be usable in general linear form $y=X\beta+\varepsilon$.
A couple challenges:
(a) I've never done this before. I am thinking I should use zeros, ones and negative ones. Am I right in putting ones and negative ones in the two parameter columns for a match, and zeros in the parameter columns that aren't playing in the match?
(b) How do I decide if a couple comparisons can be made, $\beta_1 - \beta_2$ and $\beta_1 - \beta_3$?
$X =$ in the row order $y_{12}$, $y_{34}$, $y_{25}$, $y_{15}$ $$ \begin{bmatrix} 1 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 \\ 0 & 1 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 & -1 \\ \end{bmatrix} $$
I further have $E(y)$ = $$ \begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_3 \\ \mu_4 \\ \end{bmatrix} $$
I know of 2 ways to see if it's estimable. The first is by seeing if a matrix $A$ exists such that $Ae(Y) = AX\beta.$ The other way is the way I need to practice, so where I look to you: By seeing if there is a given linear combination of the elements of beta that can be written as a linear combination of the elements of E(y).