Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 207 / 1

Table of contents
- a 1
  - i a
    - Solution i
  - ii a
    - Solution ii
  - iii a
    - Solution iii
  - iv a
    - Solution iv
- b 1
  - i b
    - Solution i
  - ii b
    - Solution ii

The permitted transitions in the continuous-time multi-state model are

1 q_{12} 2, 1 q_{13} 3, 1 q_{15} 5, 2 q_{24} 4, 2 q_{25} 5, 3 q_{34} 4, 3 q_{35} 5, 4 q_{45} 5.

(1)

State 5 is absorbing. The absence of arrows from dementia states 2 and 4 back to nondementia states expresses irreversibility; stroke likewise remains in the history after entry to states 3 or 4.

Solution

 0  0

Write

P_{rs} (t; q)

for the transition probabilities generated by the constant intensity matrix. The panel-observed multi-state likelihood contributions are:

For patient 1, observation in state 2 at year 2 followed by a direct death from state 2 one year later contributes
$P_{12} (2; q) P_{22} (1; q) q_{25} .$
(1)

Patient 2 may have entered the stroke state either without or with unobserved dementia. The exactly timed stroke contributes
$P_{11} (0.5; q) q_{13} + P_{12} (0.5; q) q_{24} .$
(2)
With no information after admission, the later contribution is one.

For patient 3, sum over the two possible states immediately after the stroke, propagate to observed state 4 over the following $1.5$ years, and then require death directly from state 4:
${P_{11} (0.5; q) q_{13} P_{34} (1.5; q) + P_{12} (0.5; q) q_{24} P_{44} (1.5; q)} P_{44} (1; q) q_{45} .$
(3)

The homogeneous Markov assumption can fail because mortality and disease incidence depend on age, calendar time, and duration since stroke or dementia; duration dependence would call for a Semi-Markov multi-state model. The common-intensity assumption can fail through patient heterogeneity in sex, genetics, vascular risk, treatment, and frailty. In addition, clinic attendance may be informative: patients with worsening cognition may attend sooner or may be too ill to attend, violating an observation process independent of the latent disease process.

iv

 0  0

Solution

 0  0

After stroke, absence of a dementia effect on the death intensity is the restriction

H_{0} : q_{35} = q_{45} .

(1)

Fit the unrestricted model and the nested model subject to this equality, then use a likelihood-ratio test. Under regularity conditions, twice the log-likelihood difference is asymptotically chi-squared with one degree of freedom.

b

 0  0

i

 0  0

Solution

 0  0

Because the measurement can cross the threshold in both directions while the model declares dementia irreversible, one needs the diagnostic rule's sensitivity

P (X < 10 ∣ true dementia)

(1)

and specificity

P (X \geq 10 ∣ no true dementia) .

(2)

These quantify false negative and false positive state classifications and permit a hidden-state or misclassification model instead of treating every threshold crossing as the true irreversible onset.