(Tests in Aalen's additive hazards model) Consider \(n\) i.i.d.
$$
\left(N_{i}(t), Y_{i}(t), X_{i}\right)
$$
with \(X_{i}=\left(X_{i 1}, \ldots, X_{i p}\right)\) so that \(N_{i}(t)\) has
intensity
$$
\lambda_{i}(t)=Y_{i}(t)\left(\beta_{0}(t)+\beta_{1}(t) X_{i
1}+\cdots+\beta_{p}(t) X_{i p}\right),
$$
that is, Aalen's additive intensity model is assumed.
(a) Show that the unweighted least squares estimator of \(B(t)\) may be written
as
$$
\hat{B}(t)=\int\left(X_{Y}(t)^{T} G(t) X_{Y}(t)\right)^{-1} X_{Y}(t)^{T} G(t)
d N(t),
$$
where \(X_{Y}(t)\) is the \(n \times p\)-matrix with \(i\) th row
\(Y_{i}(t)\left(X_{i 1}, \ldots, X_{i p}\right)\), and \(G(t)=I-1(t) 1^{-}(t)\)
with \(1(t)=\left(Y_{1}(t), \ldots, Y_{n}(t)\right)^{T}\) assuming in these
expressions that the inverses exist.
Tests of the hypothesis
$$
H_{0}: \beta_{j}(t)=0 \quad \text { for all } t
$$
and for \(j \in J \subseteq\\{1, \ldots, p\\}\) with \(|J|=q(\leq p)\) may be
based on
$$
Z_{j}(t)=\int_{0}^{t} L_{j}(s) d \hat{B}_{j}(s)
$$
where \(L_{j}(t)\) denotes a (predictable) weight process. One choice of weight
function is to take \(L_{j}(t)\) equal to the reciprocal of the \((j+1)\) th
diagonal element of \(\left(X(t)^{T} X(t)\right)^{-1}\), where \(X(t)\) denotes
the usual design matrix of the Aalen model.
(b) Derive, under \(H_{0}\), the asymptotic distribution of
$$
T(t)=Z(t)^{T} \hat{\Sigma}(t)^{-1} Z(t)
$$
where \(Z(t)=\left(Z_{1}, \ldots Z_{p}(t)\right)^{T}\),
$$
\hat{\Sigma}(t)=\int_{0}^{t} \operatorname{diag}(L(s)) d[\hat{B}](s)
\operatorname{diag}(L(s))
$$
with \(\operatorname{diag}(L(t))\) the diagonal matrix with \(j\) th diagonal
element equal to \(L_{j}(t)\).
Consider now the situation where we have three treatments and want to judge
their effects on survival. We then have \(p=2\) and the covariates are indicator
variables of two of the three treatments. A somewhat unpleasant property is
that the value of the above test depends on which group is taken as the
baseline if the applied weight functions vary with groups.
