Definition of the interaction

• The joint intervention of two or more treatments

• A: heart transplant
• E: vitamins
• 4 treatment combinations: \(Y^{a=0, e=0}, Y^{a=1, e=0}, Y^{a=0, e=1}, Y^{a=1, e=1}\)
• Effect measured on the absolute scale: interaction on the additive scale \(Pr[Y^{a=1, e=1}=1]-Pr[Y^{a=0, e=1}=1] \neq Pr[Y^{a=1, e=0}=1]-Pr[Y^{a=0, e=0}=1]\)
• Example:
  • the causal risk difference for receiving the transplant when everyone gets vitamins \(Pr[Y^{a=1, e=1}=1]-Pr[Y^{a=0, e=1}=1]\)=0.1 and the causal risk difference for receiving the transplant when no one gets vitamins \(Pr[Y^{a=1, e=0}=1]-Pr[Y^{a=0, e=0}=1]\)=0.2
  • 0.1 \(\neq\) 0.2
  • there is an interaction on the additive scale between treatments A and E
  • same implied for the second exposure E: \(Pr[Y^{a=1, e=1}=1]-Pr[Y^{a=1, e=0}=1] \neq Pr[Y^{a=0, e=1}=1]-Pr[Y^{a=0, e=0}=1]\)
  • both exposures have an equal status and require the same assumptions
  • in contrast to the effect measure modification, which deals with the causal effect of only one exposure and the causal contrast is defined in terms of the P.O. \(Y^a\), interaction deals with joint effect of several exposures and the causal contrast defined in terms of the P.O. \(Y^{a, e}\)

Technical point 5.1

• Working with the interaction equation:
  • \(Pr[Y^{a=1, e=1}=1]-Pr[Y^{a=1, e=0}=1] \neq Pr[Y^{a=0, e=1}=1]-Pr[Y^{a=0, e=0}=1]\) -->
  • \(Pr[Y^{a=1, e=1}=1] \neq {Pr[Y^{a=0, e=1}=1]-Pr[Y^{a=0, e=0}=1]}+Pr[Y^{a=1, e=0}=1]\) -->
  • substruct \(Pr[Y^{a=0, e=0}=1]\):
  • \(Pr[Y^{a=1, e=1}=1] - Pr[Y^{a=0, e=0}=1] \neq \\{Pr[Y^{a=0, e=1}=1]-Pr[Y^{a=0, e=0}=1]}+Pr[Y^{a=1, e=0}=1] - Pr[Y^{a=0, e=0}=1]\)
• Multiplicative scale:
  • \(\frac{Pr[Y^{a=1, e=1}=1]}{Pr[Y^{a=0, e=0}=1]} \neq \frac{Pr[Y^{a=1, e=0}=1]}{Pr[Y^{a=0, e=0}=1]} * \frac{Pr[Y^{a=0, e=1}=1]}{Pr[Y^{a=0, e=0}=1]}\)

Identifying interaction

• Exchangeability, positivity, and consistency for both exposures
• Imagine E was assigned marginally (unconditionally) at random \(Pr[Y^{a=1, e=1}=1]\) under exchangeability becomes \(Pr[Y^{a=1}=1|E=1]\)
  • For the full equation: \(Pr[Y^{a=1, e=1}=1]-Pr[Y^{a=0, e=1}=1] \neq Pr[Y^{a=1, e=0}=1]-Pr[Y^{a=0, e=0}=1]\)
  • under exchangeability:
  • \(Pr[Y^{a=1}=1|E=1]-Pr[Y^{a=0}=1|E=1] \neq Pr[Y^{a=1}=1|E=0]-Pr[Y^{a=0}=1|E=0]\)
• When treatment E is randomly assigned, the interaction and effect modification coincide

• A and E are joint inrervention --> combined treatment AE with four possible levels (11, 01, 10, 00)
• Identification of the causal effect of one treatment AE

Counterfactual response types

Counterfactual response types

## # A tibble: 20 x 4
##    greek       Y_a0  Y_a1 type  
##    <chr>      <dbl> <dbl> <chr> 
##  1 Demeter        0     0 immune
##  2 Hades          0     0 immune
##  3 Hestia         0     0 immune
##  4 Hera           0     0 immune
##  5 Rheia          0     1 hurt  
##  6 Zeus           0     1 hurt  
##  7 Leto           0     1 hurt  
##  8 Hephaestus     0     1 hurt  
##  9 Aphrodite      0     1 hurt  
## 10 Cyclope        0     1 hurt  
## 11 Kronos         1     0 helped
## 12 Poseidon       1     0 helped
## 13 Apollo         1     0 helped
## 14 Hermes         1     0 helped
## 15 Hebe           1     0 helped
## 16 Dionysus       1     0 helped
## 17 Artemis        1     1 doomed
## 18 Ares           1     1 doomed
## 19 Athena         1     1 doomed
## 20 Persephone     1     1 doomed

Counterfactual response types

Counterfactual response types

• Type 1 dies regardless of the treatments received
• Type 16 is immune

  type	Y_a=1, e=1	Y_a=0, e=1	Y_a=1, e=0	Y_a=0, e=0
  1	1	1	1	1
  16	0	0	0	0

• Type 4 dies if treated with E regardless of A
• Type 13 dies if NOT treated with E regardless of A
• Type 6 dies if treated with A regardless of E
• Type 11 dies if NOT treated with A regardless of E

Counterfactual response types

• The response types 1, 4, 6, 11, 1, 16: causal effect of A on Y is the same regardless of E and vice versa for the effect of E on Y regardless of A
• If all individuals in the population have these response types, there is no interaction between A and E
  • \(Pr[Y^{a=1, e=1}=1] - Pr[Y^{a=0, e=1}=1] = Pr[Y^{a=1, e=0}=1] - Pr[Y^{a=0, e=0}=1]\)
• Interaction is present when there are individuals of:

• Types 8, 12, 14, 15 (outcome under only one of the four treatment combinations)

• Types 7 and 10

  type	Y_a=1, e=1	Y_a=0, e=1	Y_a=1, e=0	Y_a=0, e=0
  7	1	0	0	1
  10	0	1	1	0

• Types 2, 3, 5, and 9 (outcome under 3 put of 4 combinations)

  type	Y_a=1, e=1	Y_a=0, e=1	Y_a=1, e=0	Y_a=0, e=0
  2	1	1	1	0
  3	1	1	0	1
  5	1	0	1	1
  9	0	1	1	1

Monotonicity

• The causal effects of A and E on Y are monotonic if every individual’s counterfactual outcomes \(Y^{a, e}\) are monotonically increasing in both a and e (technical point 5.2)
  • There are no types \(Y^{a=1, e=1}=0, Y^{a=0, e=1}=1\), etc.

Sufficient causes

• Different responses types indicate that treatment A alone is not enough to always cause the outcome Y
• Minimal sufficient causes are A=1 & U1=1 and A=0 & U2=1 and U0
• Each sufficient cause has components --> sufficient-component causes

Sufficient causes

• 9 possible sufficient causes with treatment components A = 1 only, A = 0 only, E = 1 only, E = 0 only, A = 1 and E = 1, A = 1 and E = 0, A = 0 and E = 1, A = 0 and E = 0, and neither A nor E matter.