Crash Course in Causality (Take 1)

Written on January 25, 2019
[ causal-inference  statistical-modeling  easi  ]

Wright [1921] and Neyman [1923] were early pioneers of causal modeling, though the field did not fully mature until the 1970s with Rubin’s causal model (Rubin [1974]). In the 1980s, Greenland and Robins [1986] introduced causal diagrams, Rosenbaum and Rubin [1983] introduced propensity scores, and Robins [1986] introduced time-dependent confounding (followed up in Robins [1997]). Then there the later work on causal diagrams by Pearl [2000] (side note: Pearl is from NJIT!). Post-2000, there are the methods of optimal dynamic treatment strategies (Murphy [2003]; Robins [2004]) and a machine learning approach called targeted learning (van der Laan [2009]).

These methods are primarily for observational studies and natural experiments – things as they are in the world. The ideal is to use randomized control… But you just cannot do that for many things (think weather, astronomy, disease, etc).

This post is based on watching through the first week’s lectures from Coursera’s Crash Course in Causality. Stay tuned for more…

The goals of causal modeling:

  • formal definitions of causal effects
  • assumptions necessary to identify causal effects from data
  • rules about what (confounding) variables need to be controlled for
  • sensitivity analyses to determine the impact of violations of assumptions on conclusions

Potential Outcomes and Counterfactuals

https://www.coursera.org/lecture/crash-course-in-causality/potential-outcomes-and-counterfactuals-0XWFc

Suppose we are interested in the causal effect of some treatment A (e.g., A = {1:vaccine 0:placebo}) on some outcome Y
(e.g., Y = {1:contractsDiseaseAnyTimeAfterTreatment 0:doesNotContractDiseaseAfterTreatment}).
  • Observed outcome: what actually happens
  • Potential outcome: an outcome that we would see given treatment A, e.g., given a binary treatment, each person has two potential outcomes, Y[0] and Y[1] (each of which is a random variable).
    • Notation: Y[a] is the outcome that would be observed if treatment was set to A=a
    • Potential outcomes span the possibilities of treatments

Example 1

Suppose the treatment is the flu shot and the outcome is the time until the individual gets the flu. Then the potential outcomes are:

  • Y[1]: time until the individual would get the flu if they received the flu vaccine
  • Y[0]: time until the individual would get the flu if they did not receive the flu vaccine

Example 2

Suppose the treatment is regional (A=1) versus general (A=0) anesthesia for surgery, and that the outcome (Y) is major pulmonary complications. Then the potential outcomes are:

  • Y[1]: equal to 1 if major pulmonary complications (0 otherwise), if given regional anesthesia
  • Y[0]: equal to 1 if major pulmonary complications (0 otherwise), if not given general anesthesia

Counterfactual outcomes are ones that would have been observed, had the treatment been different

  • Potential outcomes represent what could be, while counterfactual outcomes represent what could have been
    • e.g., if my treatment was A=1, then
      • before treatment, my potential outcomes were Y[1] and Y[0]
      • after treatment, my outcome is Y[1] and counterfactual outcome is Y[0]

Before a treatment decision is made, each patient’s outcome spans the range of potential outcomes. Once a treatment has been given, each patient has an actual/observed outcome and a set of counterfactual outcomes. The idea, ultimately, is that we need a sizable population so that we can estimate all random variables of interest – the range of potential outcomes – despite having “missing data” on any one specific subject.

Hypothetical Interventions

https://www.coursera.org/lecture/crash-course-in-causality/hypothetical-interventions-Lgb6O

We are interested in intervening, or taking action, on a variable in order to reveal causal effects (actually, we are interested in hypothetically intervening, or taking action, on a variable). An “intervention variable” (or just intervention) is just that: a variable that we can “intervene on,” or take an action on (at least hypothetically - e.g., we gave treatment A=1, but we could have given treatment A=0). Intervening on a variable is often referred to as manipulating the variable, and so this type of variable is also known as a manipulable variable.

A quote from Holland [1986] states: “no causation without manipulation.” The point is: if a variable cannot be manipulated, then it is difficult to consider causal effects of that variable

The treatment variables (or treatments) we consider in this course will be interventions: a treatment is a something we can specify/manipulate at random in a hypothetical trial; it is manipulable; it is an intervention:

  • The amount of drug X can be randomly assigned
  • What gender or race a specific subject is cannot be randomly assigned
  • Whether or not a placebo is prescribed is a intervention/treatment
  • Prescribing someone to be aged 45yo at random is not possible, so not an intervention/treatment

Not all variables are manipulable.

Things like age and race are not interventions – they are immutable variables, not manipulable variables. Immutable variables can still have causal effects, but they cannot be dealt with in the same way as manipulable variables – e.g., they do not fit cleanly into the potential outcomes framework. To study immutable variables, one often looks for associated manipulable variables, e.g., one’s name on a resume instead of race or gender (i.e., you can gain insight into the immutable variables of race and gender on hiring practices by using the same resume with different names).

We focus on hypothetical interventions in this course

  • b/c they’re well defined
  • b/c they’re potentially actionable
  • b/c we have a framework to parse out causal effects

So… What is a causal effect?

  • Example: Treatment A has a causal effect on Outcome Y if Y[a=1] differs from Y[a=0]

Example: Ibuprofen

  • A: take Ibuprofen (1) or not (0)
  • Y: headache gone in an hour (1) or not (0)

Often you will hear people improperly reasoning about causality: “I took Ibuprofen an hour ago, and now my headache is gone. Therefore, Ibuprofen works!” This is equivalent to saying “Y[a=1]=1” – it tells us nothing about “Y[a=0]” or even the strength of effect between a=1 and Y=1 (how many times does one take Ibuprofen and still have a headache in an hour?).

FPCI: The Fundamental Problem of Causal Inference

We can only observe one potential outcome for each person. This means we cannot necessarily observe causal effects at the individual level, but it doesn’t mean that we cannot measure causal effects at the population level.

This is important: we are not really going to be looking at things at the individual/unit level.

  • Hopeless: What would have happened if I had not taken Ibuprofen an hour ago?
  • Possible: What would the rate of headache remission be if everyone took Ibuprofen when they had a headache versus if no one did?

SIDE NOTE: an “intervention variable” is not the same thing as an “intervening variable,” which is basically synonymous with a “mediating variable” … An “intervention variable” is a “manipulable variable.”

Causal Effects

https://www.coursera.org/lecture/crash-course-in-causality/causal-effects-Qt0ic

“Correlation does not imply causation, and yet causal conclusions drawn from a carefully designed experiment are often valid.” - Holland [1986]

We want to more formally defined what we mean by “causal effects.”

  • What is an average causal effect?
  • What is the causal effect of a treatment on the treated?

Conditioning vs manipulating

In any experiment or knowledge-seeking trial, we have a population of interest. In an ideal setting (for a binary treatment), we would have two worlds to experiment on, each world having the same people in the population of interest: World 0 where no one is treated (A=0) and World 1 where everyone is treated (A=1). In this scenario, we could compute the mean outcome in W0, mean(Y[A=0]), and the mean outcome in W1, mean(Y[A=1]) – their difference defining the average causal effect.

AvgCausalEff := E{Y[1]-Y[0]} = E{Y[1]} - E{Y[0]} != E{Y|A=1} - E{Y|A=0}

In reality, we cannot compute E{Y[1]-Y[0]} since each individual can only have a specific treatment and outcome. However, we can technically estimate each term in E{Y[1]} - E{Y[0]}, and so estimate AvgCausalEff. The last inequality speaks to a subtlety between setting a variable and conditioning on a variable that is important to understanding causal inference. We talk about this more below.

If Y is binary, then this is a risk difference (the mean of a binary variable is just a probability, or risk, so that the difference makes it a difference in probability). However, for other types of random variables, such as continuous random variables, this does not directly translate into a difference of probabilities (you might still call it a difference in “risk”, but often the word “risk” is synonomous with probability in epidemiological studies).

Example of a Binary Outcome: Anesthesia

Suppose the treatment is regional (A=1) versus general (A=0) anesthesia for surgery, and that the outcome (Y) is major pulmonary complications. Also suppose that we know the causal risk difference: E{Y[1]-Y[0]} = -0.1. This means that taking regional anesthesia is less risky by 10 percentage points compared with general anesthesia. In other words, if both World 0 and World 1 had 1000 people, we would expect 100 fewer people to have pulmonoary complications in World 0 than in World 1. Admittedly, this is relative: we are not really saying whether either of them are safe with this statement, just that regional anesthesia is safer; e.g., if 950 people in World 1 had complications, then 850 people in World 1 did too.

Example of a Continous-Valued Outcome:

Suppose the treatment is binary – patient is given thiazide diuretic (A=1) to treat hypertension, or not (A=0) – and that the outcome Y is systolic blood pressure. Also suppose that we know the causal effect: E{Y[1]-Y[0]} = -20 mm Hg. This says that the patients in World 0 would have lower systolic blood pressure, on average, by 20 mm Hg than those patients in World 1.

Setting, Not Conditioning

Above, we wrote down an equation that might appear confusing at first:

E{Y[1]} - E{Y[0]} != E{Y|A=1} - E{Y|A=0}

In certain settings, the two sides of the equation would actually be equal – but that is the exception, not the rule. Bear with me here… Our equation represented two worlds, W0 and W1, which were parallel universes with the same people in the population of interest. In each world, everyone was assigned the same treatment, and we were able to compute the causal effect by comparing the averages between worlds. The right hand side of the above equation is usually interpreted differently than this: E{Y|A=1} often reads “the mean of Y given A=1”, but it is not asking “what is the mean of Y given A=1 for the entire population?” The term E{Y|A=1} is better read as “the mean of Y when restricting to the subpopulation where A=1.” The keyword is subpopulation: often the subpopulation receiving treatment is much different than the population not receiving treatment.

So, in other words, the term E{Y|A=1} - E{Y|A=0} actually means “the average difference in outcome between subpopulations defined by treatment group.”

Recap 1:

  • E{Y A=1}: the mean outcome among the subpopulation with A=1
    • often read as the mean outcome given A=1
  • E{Y[1]}: the mean outcome if the entire population was given treatment A=1
    • read as “the mean outcome when setting A=1” to differentiate from conditionals

Recap 2:

  • E{Y A=1} - E{Y A=0}: a difference between two subpopulations in the same universe; not necessarily a causal effect
  • E{Y[1]} - E{Y[0]}: a counterfactual difference between two parallel universes, i.e., a causal effect

Other Causal Effects

  • E{Y[1]/Y[0]}
    • for a binary variable, this is the causal relative risk
  • E{Y[1]-Y[0] A=1}
    • this one is confusing at first, but only b/c we defined the worlds homogenously above
    • consider a binary treatment given heterogeneously in W0, then W1 is just defined as the flip of this (those who got treatment in W0 do not get treatment in W1, and vice versa)
    • now it should be more clear what the causal effect above means: “given the subpopulation treated with A=1, what if they were treated with A=0”?
    • point is: there are two worlds we are interested in for a binary treatment: the actual and counterfactual worlds

Causal Assumptions

https://www.coursera.org/lecture/crash-course-in-causality/causal-assumptions-f5LPB

We want to understand what causal assumptions are necessary to link potential outcomes to actual data such that we can identify causal effects. This is known as “identifiability,” which basically says that a parameter of interest is “identifiable” if it can be estimated from data. Identifiability genderally requires some condtions to hold – you might call such conditions “assumptions” or “axioms.” Regardless, the idea is that you can estimate some conceptual parameter in a real data set given certain conditions, or assumptions.

Sometimes in causal inference, these assumptions might be untestable – in which case they are referred to as causal assumptions. Though untestable, such assumptions are often reasonable enough in various circumstances to give confidence to their applicability in those circumstances.

In causal inferences, often there are the following 4 assumptions in place. These assumptions pertain to the outcome Y, the treatment A, and a set of pre-treatment covariates denoted X. The pre-treatment covariates can be anything that might also have correlation with the treatment or outcome, such as age, race, gender, blood pressure, etc.

  • SUTVA
    • Stable Unit Treatment Value Assumption
    • No interfernece:
      • interference is also known as “spillover” or “contagion”
      • Units do not interfere w/ each other, where a unit is, say, a person from your target population; this means that treatment assignment is randomized among units – the treatment assignment of one unit has no impact on the treatment assignment of another unit
      • example: in a behavioral study, there could be interference/spillover if the patients interact with each other – the treatment/intervention that one patient got can “spill over” into their interaction and affect the treatment/intervention that another patient gets
      • more advanced causal methods can hanlde interference, but we will not be covering them in this course
    • One version of treatment
      • important for linking potential outcomes to observed data
      • if there are different versions of treatment, it becomes hard to define what a causal effect means
    • Relevance: SUTVA allows us to write the potential outcomes for the ith person in terms of only the ith person’s treatments
      • this is like when modeling volumes of gas and assuming no interactions between particles
    • In many situations, this is a reasonable assumption
    • Violation examples (source)
      • Job training for too many people may flood the market with qualified job applicants (interference)
      • Some patients get extra-strength aspirin (variation in treatment)
  • Consistency
    • Definition: the potential outcome under treatment A=a, Y[a], is equal to the observed outcome if the actual treatment is A=a
    • Equation: Y = Y[a] if A=a for all a
      • My interpretation (which I’m pretty sure is equivalent): Y A=a = Y[a] for all a (…?)
  • Ignorability
    • aka the assumption of “no unmeasured confounders”
    • Given pre-treatment covariates X, treatment assignment is independent from the potential outcomes
      • Y[A=0], Y[A=1] ∐ A|X
      • where ∐ means “independence”
    • The point here is that treatment A may not be random over entire population of interest, but when accounting for pre-treatment covariates level-by-level it is
      • Example: say X = binnedAge = {old|young}, and that treatment is frequently given to the “old” by nature of the condition being treated for (e.g., Y={1 if hip fracture, else 0}), and possibly that the outcome is more likely for the “old” (i.e., “old” people are more likely to get a hip fracture independent of treatment), while treatment is infrequently given to the young; it would appear that treatment was not independent of potential outcomes over entire population, but not if looking at the two levels separately
    • the idea is if we have accounted for all pre-treatment covariates, then treatment assignment is ignorable (it is (effectively randomized)
    • More on ignorability: http://www.fragilefamilieschallenge.org/causal-inference/
    • Example of violations
      • Omitted Variable Bias
  • Positivity
    • Idea: everybody has nozero probability of receiving any level of treatment
    • Eqn: P(A=a X=x) > 0 ∀ a and x
    • The idea is that we need to learn from data: we need an actual so we can calculate a counterfactual, etc
    • At every level X, there is nonzero probabilty of receiving treat A=a

Linking observed data and potential outcomes

Basically, by using the causal assumptions, we can back out potential outcomes from the observed data. That is, we can estimate counterfactuals and causal effects.

  1. We start with data on the outcome Y, treatment A, and set of pre-treatment covariates X.
  2. With this data alone, we can compute the expected value of Y for the subpopulation where treatment A=a and pre-treatment covariates X=x: E{Y|A=a,X=x}.
    • this is all observed data so far
    • we have not yet estimated potential outcomes
  3. By the consistency assumption, we can write: E{Y[a]|A=a,X=x} = E{Y|A=a,X=x}
    • that is, we can write an expression for potential outcomes
    • consistency says that the outcome we observe when A=a is the same as the potential outcome, Y[a]
  4. By the ignorability assumption, we can simplify this expressions: E{Y[a]|X=x} = E{Y[a]|A=a,X=x} = E{Y|A=a,X=x}
    • ignorability allows us to drop the conditioning on treatment
  5. Finally, to estimate a marginal causal effect, we can average of the pre-treatment covariates X

Stratification

https://www.coursera.org/lecture/crash-course-in-causality/stratification-xEcaf

Standardization

  • stratify on important variables, then averaging over the distribution of those variables
  • that is, standardization is a combination of conditioning and marginalizing
  • conditioning means to stratify (remember: whenever we say “given”, we mean “restrict to”)
  • marginalize means to average over

Previously, we found that we can estimate information on potential outcomes by putting a few assumptions into place. Namely, by consistency and ignorability, we can link potential outcomes to actual data: E{Y[a]|X=x} = E{Y|A=a,X=x}.

We needed to condition on X in order to link actual outcomes to potential outcomes. To get the marginal causal effect, we will then need to average over the distribution of X:

  • Let X be a single categorical variable
  • Expected potential outcome for A=a: E{Y[a]} = Σ{X} E{Y|A=a,X=x}P(X=x)
    • this is standardization (condition/stratify, then marginalize)
    • the conditional expectation is the conditioning (or stratification) component
    • the sum is the marginalization

Ultimately the goal is to remove confounding effects – and the conceptually straightforward way to do this is through standardization. That is, to stratify and marginalize. Basically, the stratification step is how to control for confounding variables, and the marginalization step then finds the average effect across strata. In practice, the stratification step can become nigh impossible – there can be many variables to control for, resulting in highly rarefied strata and likely even “empty” strata. There is also the challenge of knowing what to control for. The rest of the course looks at ways of simulating/estimating the stratification step (e.g., propensity score matching, inverse probability of treatment weighting, etc).

Some Further Reading

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Next Up in the Crash Course

  • Module 2: https://www.coursera.org/lecture/crash-course-in-causality/confounding-VT3H4
    • disjunctive cause criterion can also be called “disconnective criterion” or “simply disconnect criterion” since “disjunctive” means “lacking connection” and the criterion basically says “only worry about disconnecting nearest neighbor nodes that flow directly into A or Y” (btw, doesn’t always work, but good rule of thumb)

  • Module 3: https://www.coursera.org/lecture/crash-course-in-causality/observational-studies-V6pDQ

  • Module 4: https://www.coursera.org/lecture/crash-course-in-causality/intuition-for-inverse-probability-of-treatment-weighting-iptw-nrrCT

  • Module 5: https://www.coursera.org/lecture/crash-course-in-causality/introduction-to-instrumental-variables-ueIMD