What are Two ANCOVA models for Lord's paradox?

In the comment in the blog (Stephen Senn: Rothamsted Statistics meets Lord’s Paradox (Guest Post) | Error Statistics Philosophy), Prof. Senn wrote that we could assume two different ANCOVA models for Lord's paradox.

Although I have not used Genstat® at all or I have not read two Nelder's papers in 1965 yet, I wildly guess that Nelder's model is a hierarchical model while Lord's model is a simple ANOVA model. (Please correct me when I misunderstand...)

I write these model formulas in the below.

The 1st Model: Simple ANCOVA Model

$WF_j$ = the final weight of the j-th student.
$WI_j$ = the initial weight of the j-th student.
$D_j$ = the diet taken by the j-th student. 0 or 1.
$e_j$ = the error for the j-th student.

$WF_j = \beta_0 + \beta_1 WI_j + \beta_D D_j + e_j$

The 2nd Model: Hierarchical ANCOVA Model

$WF_{j(i)}$ = the final weight of the j-th student within the i-th hall.
$WI_{j(i)}$ = the initial weight of the j-th student within the i-th hall.
$D_{i}$ = the diet taken by all students at the i-th hall. 0 or 1.
$e_{j(i)}$ = the within-hall error for the j-th student within the i-th hall.

$WF_{j(i)} = b_{0i} + b_{1i} WI_{j(i)} + \beta_D D_{i} + e_{j(i)}$
where $(b_{0i},b_{1i})$ i.i.d. $\sim$ MVN $({\bf \beta}, {\bf \Sigma})$

${\bf \beta}$ is a $2 \times 1$ vector, and ${\bf \Sigma}$ is a $2 \times 2$ matrix.

Note that $b_{0i}$ and $b_{1i}$ in the 2nd model are random variables.

I assume MVN for $(b_{0i},b_{1i})$ just for simplicity, but the model might be robust for non-normal.

If ${\bf \Sigma}$ in the 2nd ANCOVA model is $[0\ 0, 0\ 0]$ , it becomes the 1st model. We can say that the 2nd ANCOVA model is more general than the 1st one.

We cannot estimate fixed parameters of the 2nd model if there is only one hall for each diet.

Conflicts of Interest

I work for SAS Institute Japan, a commercial statistical software company. The opinions in this article are my personal ones. All responsibilities for this article are attributed only to the author.

-Yusuke Ono (JMP Japan Group, SAS Institute Japan)