Over a period of 90 days a study was carried out on 1500 women. Its purpose
was to investigate the relation between obstetrical practices and the time
spent in the delivery suite by women giving birth. One thing that greatly
affects this time is whether or not a woman has previously given birth.
Unfortunately this vital information was lost, giving the researchers three
options: (a) abandon the study; (b) go back to the medical records and find
which women had previously given birth (very time-consuming); or (c) for each
day check how many women had previously given birth (relatively quick). The
statistical question arising was whether (c) would recover enough information
about the parameter of interest.
Suppose that a linear model is appropriate for log time in delivery suite, and
that the log time for a first delivery is normally distributed with mean
\(\mu+\alpha\) and variance \(\sigma^{2}\), whereas for subsequent deliveries the
mean time is \(\mu\). Suppose that the times for all the women are independent,
and that for each there is a probability \(\pi\) that the labour is her first,
independent of the others. Further suppose that the women are divided into \(k\)
groups corresponding to days and that each group has size \(m\); the overall
number is \(n=m k\). Under (c), show that the average log time on day \(j,
Z_{j}\), is normally distributed with mean \(\mu+R_{j} \alpha / m\) and variance
\(\sigma^{2} / m\), where \(R_{j}\) is binomial with probability \(\pi\) and
denominator \(m\). Hence show that the overall log likelihood is
$$
\ell(\mu, \alpha)=-\frac{1}{2} k \log \left(2 \pi \sigma^{2} /
m\right)-\frac{m}{2 \sigma^{2}} \sum_{j=1}^{k}\left(z_{j}-\mu-r_{j} \alpha /
m\right)^{2}
$$
where \(z_{j}\) and \(r_{j}\) are the observed values of \(Z_{j}\) and \(R_{j}\) and
we take \(\pi\) and \(\sigma^{2}\) to be known. If \(R_{j}\) has mean \(m \pi\) and
variance \(m \tau^{2}\), show that the inverse expected information matrix is
$$
I(\mu, \alpha)^{-1}=\frac{\sigma^{2}}{n \tau^{2}}\left(\begin{array}{cc}
m \pi^{2}+\tau^{2} & -m \pi \\
-m \pi & m
\end{array}\right)
$$
(i) If \(m=1, \tau^{2}=\pi(1-\pi)\), and \(\pi=n_{1} / n\), where \(n=n_{0}+n_{1}\),
show that \(I(\mu, \alpha)^{-1}\) equals the variance matrix for the two-sample
regression model. Explain why.
(ii) If \(\tau^{2}=0\), show that neither \(\mu\) nor \(\alpha\) is estimable;
explain why.
(iii) If \(\tau^{2}=\pi(1-\pi)\), show that \(\mu\) is not estimable when \(\pi=1\),
and that \(\alpha\) is not estimable when \(\pi=0\) or \(\pi=1\). Explain why the
conditions for these two parameters to be estimable differ in form.
(iv) Show that the effect of grouping, \((m>1)\), is that
\(\operatorname{var}(\widehat{\alpha})\) is increased by a factor \(m\) regardless
of \(\pi\) and \(\sigma^{2}\)
(v) It was known that \(\sigma^{2} \doteq 0.2, m \doteq 1500 / 90, \pi \doteq
0.3\). Calculate the standard error for \(\widehat{\alpha}\). It was known from
other studies that first deliveries are typically 20-25\% longer than
subsequent ones. Show that an effect of size \(\alpha=\log (1.25)\) would be
very likely to be detected based on the grouped data, but that an effect of
size \(\alpha=\log (1.20)\) would be less certain to be detected, and discuss
the implications.
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