Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 207 / 2

Table of contents
- 1 2
  - Solution 1
- 2 2
  - Solution 2
  - 3 2
    - Solution 3
  - 4 2
    - Solution 4
  - 5 2
    - Solution 5
  - 6 2
    - Solution 6
  - 7 2
    - a 7
      - Solution a
    - b 7
      - Solution b

1

Solution

Ignoring factors that do not depend on the parameters, the product of the two binomial likelihoods is

L (ϕ_{0}, ϕ_{1}) = ϕ_{0}^{120} (1 - ϕ_{0})^{80} ϕ_{1}^{110} (1 - ϕ_{1})^{110} .

Including the sampling constants multiplies this by

(120 200) (110 220)

.

2

Solution

The maximum-likelihood estimators are the success proportions

ϕ_{0} = \frac{120}{200} = \frac{3}{5}, ϕ_{1} = \frac{110}{220} = \frac{1}{2} .

The invariance property of maximum likelihood estimation therefore gives

α = ϕ_{1} - ϕ_{0} = \frac{1}{2} - \frac{3}{5} = - \frac{1}{10} .

3

Solution

A complete-case analysis estimates each success probability among clinic attenders. If attendance depends on the unobserved outcome even after conditioning on treatment, the observed success proportions differ systematically from those in the randomized groups. This outcome-dependent dropout causes selection bias and can bias their difference.

4

Solution

For a patient whose

Y

is missing, summing the Bernoulli likelihood contribution over its two possible values gives

ϕ_{z} + (1 - ϕ_{z}) = 1.

Consequently the observed-data likelihood for all 490 patients is

L_{obs} (ϕ_{0}, ϕ_{1}) = ϕ_{0}^{120} (1 - ϕ_{0})^{80} ϕ_{1}^{110} (1 - ϕ_{1})^{110},

up to constants. The 70 missing outcomes contribute no information about

ϕ_{0}

or

ϕ_{1}

in this marginal model, so the estimators and

α = - 1/10

are unchanged. This likelihood calculation alone does not make an outcome-dependent missingness mechanism ignorable.

5

Solution

The data have a monotone missing-data pattern:

Z

is always observed; a missing

Y

always entails a missing later

W

; and

W

may be missing after an observed

Y

.

6

Solution

Under missing at random, dropout before the one-month visit may depend on observed treatment

Z

but, conditional on

Z

, not on the unseen

Y

or

W

. Dropout between the one- and six-month visits may depend on the observed history

(Z, Y)

but, conditional on that history, not on the unseen

W

.

7

a

Solution

The saturated first imputation model reproduces the observed conditional proportions. Thus

ϕ_{0} ⟶ \frac{120}{200} = \frac{3}{5}, α ⟶ \frac{1}{2} - \frac{3}{5} = - \frac{1}{10} .

b

Solution

For

Z = 0

, the fitted conditional success probabilities for

W

are

P (W = 1 ∣ Z = 0, Y = 0) = \frac{20}{40} = \frac{1}{2}, P (W = 1 ∣ Z = 0, Y = 1) = \frac{75}{90} = \frac{5}{6} .

Among all 250 subjects with

Z = 0

, multiple imputation asymptotically assigns

100

to

Y = 0

and

150

to

Y = 1

. Hence

ψ_{0} ⟶ \frac{100 ( 1/2 ) + 150 ( 5/6 )}{250} = \frac{7}{10} .

Using the supplied limit for

ψ_{1}

gives

β ⟶ \frac{25}{48} - \frac{7}{10} = - \frac{43}{240} .

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