Table 1 Different scenarios for simulations, comparing approaches to combining multiple biomarkers to construct personalised treatment recommendations

1.

Simple linear

Y = 3X₁ + 2Z₁ − 0.5Z_{3. lv2} − 1.5Z_{3. lv3}

+A(2 + 2Z₁ − 1.5Z₂ − 3Z_{3. lv3}) + e

All variables

2.

Linear with weak moderators

Y = 3X₁ + 2Z₁ − 0.5Z_{3. lv2} − 1.5Z_{3. lv3}

+A(2 + 0.5Z₁ − 0.5Z₂ − 0.25Z_{3. lv3}) + e

All variables

3.

Strong unobserved prognostic marker

Y = 12U₁ + 3X₁ + 2Z₁ − 0.5Z_{3. lv2} − 1.5Z_{3. lv3}

+A(2 + 2Z₁ − 1.5Z₂ − 3Z_{3. lv3}) + e

Excluding U₁

4.

Strong unobserved moderator variables

Y = 3X₁ + 2Z₁ − 0.5Z_{3. lv2} − 1.5Z_{3. lv3} + A(2 + 10U₂ + 2Z₁ − 1.5Z₂ − 3Z_{3. lv3}) + e

Excluding U₂

5.

Misspecified prognostic part

Y = 3X₁ + 2Z₁ − 0.5Z_{3. lv2} − 1.5Z_{3. lv3} + 2X₁² + 1.5U₁X₁ − 1.5X₁M₁ + 0.5X₁Z₁Z_{3. lv3}

+A(2 + 2Z₁ − 1.5Z₂ − 3Z_{3. lv3}) + e

All linear terms, excluding U₁

6.

Misspecified moderator part

Y = 3X₁ + 2Z₁ − 0.5Z_{3. lv2} − 1.5Z_{3. lv3}

+A(2 + 2Z₁ − 1.5Z₂ − 3Z_{3. lv3} + 2Z₁Z₂ − 1.5Z₁Z₂U₂ + 0.5Z₁Z₂Z_{3. lv2}) + e

All linear terms, excluding U₂

7.

Non-linear model

Y = (3X₁ + 2Z₁ − 0.5Z_{3. lv2} − 1.5Z_{3. lv3})

+A(2 + 2Z₁ − 1.5Z₂ − 3Z_{3. lv3})/(−2Z₁ + 1.5Z₂ − 1.5Z_{3. lv3}) + e

All as linear terms