Observational studies as conditionally randomized experiments

• If three assumptions hold
  • Consistency: well-defined intervention (or all versions of the treatment are captured)
  • Exchangeability: conditional probability of receiving each level of the treatment depends only on measured covariate(s), L
  • Positivity: the probability of receiving each level of treatment conditional on L is greater than zero, i.e., positive
  • Non-interference: PO outcomes of one individual is independent of PO of other individuals
• These conditions are identifiability conditions
  • Causal interpretation = data + assumptions
  • Identifiability assumptions can be tracked on a DAG
• In ideal randomized experiments the identifiability conditions hold by design

Instrumental variables

• Demand different assumptions and different set of identifiability criteria

Exchageability

• $Y^a\perp \perp A$
• Had the treated be untreated their risk of PO $Y^a$ would have been the same
• Confounding is a lack of exchangeability
• Confounders are variables, which when adjusted for, restore exchangeability, or remove confounding
• Untestable

# association
greek_gods_condrand %>% 
  group_by(A) %>% 
  count(Y_obs) %>% 
  mutate(
    denominator = sum(n),
    risk = round(n/sum(n), digits = 2)
  ) %>% 
  filter(Y_obs == 1)

## # A tibble: 2 x 5
## # Groups:   A [2]
##       A Y_obs     n denominator  risk
##   <dbl> <dbl> <int>       <int> <dbl>
## 1     0     1     3           7  0.43
## 2     1     1     7          13  0.54

# when controlling confounding by L using stratification
greek_gods_condrand %>% 
  group_by(L, A) %>% 
  count(Y_obs) %>% 
  mutate(
    denominator = sum(n),
    risk = round(n/sum(n), digits = 2)
  ) %>% 
  filter(Y_obs == 1)

## # A tibble: 4 x 6
## # Groups:   L, A [4]
##       L     A Y_obs     n denominator  risk
##   <dbl> <dbl> <dbl> <int>       <int> <dbl>
## 1     0     0     1     1           4  0.25
## 2     0     1     1     1           4  0.25
## 3     1     0     1     2           3  0.67
## 4     1     1     1     6           9  0.67

Conditionally randomized experiment

• If L is the only source of confounding and conditional exchangeability holds $Y^a\perp \perp A|L$
  • this is "an observational study in which the probability of treatment A = 1 is 0.75 among those with L = 1 and 0.50 among those with A = 0"
  • this is "a (non blinded) conditionally randomized experiment in which investigators randomly assigned treatment A = 1 with probability 0.75 to those with L = 1 and 050 to those with L= 0"

greek_gods_condrand %>% 
  group_by(L) %>% 
  count(A) %>% 
  mutate(
    denominator = sum(n),
    pr_A = round(n/sum(n), digits = 2)
  ) %>% 
  filter(A == 1)

## # A tibble: 2 x 5
## # Groups:   L [2]
##       L     A     n denominator  pr_A
##   <dbl> <dbl> <int>       <int> <dbl>
## 1     0     1     4           8  0.5 
## 2     1     1     9          12  0.75

Expert knowledge

• Since exchangeability is untestable, domain knowledge is necessary to guide our inferences on whether or not exchangeability assumption may or may not hold

Positivity

• Positive probability of observing each level of treatment in each strata of L
• $Pr(A=a|L=l>0)$ for all $a$ and $l$
• Only relevant for variables L required for exchangeability
• Can be empirically verified (see chapter 12)

Consistency

• We observe PO - the one under actually received treatment
  • $Pr[Y^{a=1}|A=1]=Pr[Y=1|A=1]$
• Unpacking consistency
  • definition of $Y^{a=1}$ via detailed $a$ (given value of treatment)
  • linking the observed and the counterfactual outcome

Well-defined intervention paradigm

$Y^a$

• Treatment as several versions of the intervention
  • Are all observed and measured?
  • Do all versions of the treatment have the same causal effect?
  • Not well-defined values of $a$ lead to not well-defined PO $Y^a$ under the levels of treatment and the causal contrast $Pr[Y^{a=1}=1]-Pr[Y^{a=0}=1]$ is not well-defined
  • Obesity/weight-loss example : duration, frequency, intensity, and type of the intervention of being "less obese"
  • Challenging causal questions involving biological and social constructs/SES (p. 34)
  • Sufficiently well-defined, meaning in the detail enough for causal inference
  • Domain knowledge
  • Communication of the results

Counterfactuals and observed data

• "Hypothetical intervention" must be linked to actually observed version of treatment, otherwise mathematical notation of consistency $Pr[Y^{a=1}|A=1]=Pr[Y=1|A=1]$ cannot be translated into the "real world" and no causal inference is possible

• Data granularity

• When dealing with treatments with multiple versions --> assuming treatment variation irrelevance

• Transparency

Target trial

• Causal effect

  • a contrast between average counterfactual outcomes under different treatment values

• Imagine a (hypothetical) randomized experiment to quantify it

• Components of the "protocol"

  • Eligibility criteria
  • Interventions (or treatment strategies)
  • Outcome(s)
  • Follow-up
  • Causal contrast
  • Statistical analysis

• Oversimplified analysis example

  • Contrasting the risk of death in obese vs non-obese individuals means emulating a target trial in which obese individuals are instantaneously become to non-obese

Causal effect, Pearl 2018

• Does Obesity Shorten Life? Or is it the Soda? On Non-manipulable Causes
• Factors not fitting into experimentalist concept of causation

44% of U.S. adults could have been obese by 2030, compared to 35.7% in 2012

• $66 billion a year in obesity-related medical costs
• New York City adopted a regulation banning the sale of sugary drinks in containers larger than 16 ounces at restaurants
• Is obesity well-defined and "obesity-related medical costs" exist?

Arguments against obesity as an exposure

• BMI used to measure degree of obesity is a proxy for it

• Obesity is not well-defined "intervention" --> consistency does not hold

• Ill-defined intervention may undermine the exchangeability logic

  • If we cannot define the exposure, we cannot define what may confound its effect on the outcome

• Ill-defined intervention may may threaten positivity

  • Restricting data to confounders, within whose levels the positivity holds may result in a population different from the original one

Arguments for obesity as an exposure (in short)

• Practical interventions may have side effects --> yet, are deemed well-defined

• Root of obesity being ill-defined "intervention"

  • consequences of obesity depend on how we "manipulate" it

• At the same time, the quantity $Pr\left(mortality|do\right(obesity\left)\right)$ (notation of PO using do-operator; means the same as $Pr[mortalit{y}^{obesity=1}|obesity=1]$) describes an intervention set by nature (via complex processes)

• Causal effects of anatomical/physiological conditions may be described in terms of their presence/absence not necessarily via the means they can be manipulated

Take home messages

• Define causal question
• Define the exposure. Does it it have one version or several? Inference still possible?
• Can conditional exchangeability be reached given current domain knowledge?
• Is prediction a better target when exposure cannot be sufficiently well-defined?