Causal inference

Learn to say no politely. The primary source of motivation in academics (and in many other places) is guilt. To prevent total insanity you need to:

• Say no to most things and focus
• Do it in a way that doesn't offend people
• Do it in a way that doesn't make you feel guilty

This is an art and requires years to master. Some never do.

Causal inference in statistics, an overview

Some usesful resources

• Cosma Shalizi
  • http://www.stat.cmu.edu/~cshalizi/ADAfaEPoV/ADAfaEPoV.pdf
• Elizabeth Stuart
  • https://github.com/jtleek/jhsph753and4/blob/master/lectures/stuartcausal1.pdf
  • https://github.com/jtleek/jhsph753and4/blob/master/lectures/stuartcausal2.pdf
• Judea Pearl
  • http://ftp.cs.ucla.edu/pub/stat_ser/r350.pdf
• Eric Xing
  • http://www.cs.cmu.edu/~epxing/Class/10708-05/Slides/causality.pdf

In approximate order of difficulty

• Descriptive
• Exploratory
• Inferential
• Predictive
• Causal
• Mechanistic

Goal: To find out what happens to one variable when you make another variable change.

• Usually randomized studies are required to identify causation
• There are approaches to inferring causation in non-randomized studies, but they are complicated and sensitive to assumptions
• Causal relationships are usually identified as average effects, but may not apply to every individual
• Causal models are usually the "gold standard" for data analysis

van Nood et al. (2013) NEJM

Goal: Understand the exact changes in variables that lead to changes in other variables for individual objects.

• Incredibly hard to infer, except in simple situations
• Usually modeled by a deterministic set of equations (physical/engineering science)
• Generally the random component of the data is measurement error
• If the equations are known but the parameters are not, they may be inferred with data analysis

http://www.fhwa.dot.gov/resourcecenter/teams/pavement/pave_3pdg.pdf

1. With randomized trials -> straightforwardish (we'll see why in a minute)
2. With observational data -> much, much harder
   • Main problem is confounding
   • The key idea are the assumptions
3. It is all about missing data
   • Missing confounders
   • Missing counterfactual observations
4. Causal inference is not easily described in standard statistical/probabilistic notation.

• Rubin model:
  • \(Y(0), Y(1)\) or \(Y_x(u)\)
  • Outcome Y, had the individual, u, received treatment 0,1 or X.
• Pearl model:
  • \(Pr(Y = y | do(X=x))\)
  • Distribution when setting X=x
• Graphical models

• Intuitively
  • an arrow \(S \rightarrow L\) means S "influences" L
• Bayes Networks/Graphical models (requires DAG)
  • Conditional probabilities e.g. \(Pr(S,L,A) = Pr(A)Pr(S|A)Pr(L|A)\)
• Causal networks (requires DAG)
  • Variables connected on an unblocked path d-separated
  • Basically as long as two arrows don't run into each other on the path collider)
  • For all distributions represented by the graph if sets of vertices X and Y are d-separated by a set Z in the DAG G, then X and Y are independent conditional on Z in P (this is complicated!)

x = rnorm(100); y = rnorm(100,mean=0.1*x)
plot(x,y,pch=19)

Key concepts:

• Treatments
  • The intervention we could apply or withhold
  • Think heavy drug use versus light drug use
  • Difficult with things you can't assign
• Units
  • Who you apply the treatment to
  • Could be genes, people, schools, etc.
• Potential outcomes
  • What would have happend in both treatment scenarios
  • Typically denoted \(Y(0), Y(1)\)

Slides on this framework via:

https://github.com/jtleek/jhsph753and4/blob/master/lectures/stuartcausal2.pdf

Effect of heavy adolescent drug use (T) on earnings at age 40:

Causal inference as missing data problem:

• Can’t estimate individual-level causal effects
• ATE: average treatement effect
  • \(ATE = \frac{1}{N} \sum_{i=1}^N Y_i(1) - Y_i(0)\)
  • effect of drug use on everyone, if forced everyone to use drugs
• ATT: average treatement effect among treated
  • \(ATT = \frac{1}{N} \sum_{i=1}^N (Y_i(1) - Y_i(0) | T_i = 1)\)
  • effect of drug use on people who actually use drugs

• In a randomized experiment, units randomly assigned to treatment or control groups
• Mathematically, this means that average of control group outcomes an unbiased estimate of average outcome under control for whole population (and same for the treatment group)
  • \(E(\bar{y}_{T_i = 0}) = \bar{Y(0)}\), \(E(\bar{y}_{T_i = 1}) = \bar{Y(1)}\)
  • So \(E(\bar{y}_{T_i = 0} - \bar{y}_{T_i = 1}) = \bar{Y(0)} - \bar{Y(1)}\)
• Intuition is randomization ensures "balance" of covariates

• People don't do what they're told
  • Noncompliance
• Randomization isn't always feasible
  • Random mating in human populations
• Might have to wait a long time to see outcome
• Randomization may not estimate effects for the group we care about

• Stratification/regression

  • Put people in groups with same covariates
  • But some covariates may be missing
  • Limitations of sample size if many covariates aren't there

• Instrumental variables

  • Find an instrument that affects the treatment of real interest, but does not directly affect the outcomes
  • Vietnam draft lottery as instrument for military service
  • physician prescribing preferences as instrument for taking drug A vs. drug B
  • Need a good instrument
  • Implies a set of other assumptions (monotonicity, exclusion restrictions, etc.)

Propensity scores

• Propensity scores
  • Try to replicate features of randomized experiments (create groups that look only randomly different, don't use the outcome when setting up the design)
  • Basically trying to "balance" the data set

• For each treated find a control with exact same covariates
• Easy with 1 covariate but harder with more
• Software for doing this:
  • http://gking.harvard.edu/cem
  • http://cran.r-project.org/web/packages/MatchIt/index.html
• Want similar covariate distributions in treatment/control after matching
• One diagnostic is "standardization bias"
  • Difference in means, divided by standard deviation
  • Like an effect size for covariates
  • Should hopefully be small

\[e_i = P(T_i = 1 | X_i)\]

• Key features
  • Balancing score - at each value the distribution is the same in the treated/control groups.
    
  • Intuitvely people with similar probabilities should be assigned similar treatments other than randomness.
  • If treatment is independent of potential outcomes given covariates, also independent of ptoential outcomes given propensity score (no unmeasured confounders)
• Can be used to match, instead of on covariates individually
• Central goal is to achieve balance

• Assumes that there are no unobserved differences between the treatment and control groups, given the observed variables
  • Other ways of saying essentially the same thing: No unobservedconfounders, no hidden bias, “ignorable”
  • Could be a problem if, e.g., people start smoking marijuana because they are getting bad grades and we don’t have grades measured
  • Can help make unconfoundedness assumption more realistic if think about it during data collection
  • Can also do sensitivity analyses to assess how sensitive results are toviolation of this assumption

\[P(T | X, Y(0), Y(1)) = P(T | X) \implies P(T |X, Y(0),Y(1)) = P(T|e(X))\]

• This is what allows us to match just on propensity score; don’t need to deal with all the covariates individually

• k-nearest neighbors
  • For each treated unit select k similar controls
• Stratification/binning
  • Group individuals with similar propensity scores
• Weighting adjustments
  • Inverse probability of exposure weights (IPTW)

• You can combine propensity score with regression adjustment
  • regression adjustment in 1:1 matched samples
  • weighted regression adjustment
• The two methods work together and if you get one or the other right you are "ok"
• This is an active area of research

• Sometimes we can't randomize the thing we want
  • Gene expression levels
  • Taking a drug
• But we can randomize something related
  • Genetic background
  • Treatment assignment
• The thing we can randomize is "variable"
• Randomization is the "instrument"
• Good instruments
  • Easy to randomize
  • Closely related to the variable you care about

• Try to infer connections between genes (or genes and phenotype)
• Commonly called "Mendelian randomization"
• Hard to do in humans because instrument (randomization) isn't great: population structure for example

• Study estimating the effect of Vitamin A on child mortality
• Carried out in Indonesia
• Villages randomized to receive vitamin A supplements or not
• Not all children in Vitamin A villages actually got Vitamin A
• No children in control villages got Vitamin A (not available)
• For simplicity, will ignore village clustering and treat as if children randomly assigned (village indicators not available anyway)

• Can think of there being two types of people:
  • Compliers: take the treatment if assigned to, don’t take it if not assigned to
  • Non-compliers: don’t take the treatment either way
• Observe compliance status in treatment group
• Don’t observe compliance status in control group
• We don’t know what the controls would have done if they had been in the treatment group

• Intent to treat (ITT) - ignore compliance
  • Gives unbiased estimates of causal effect of being assigned to group
  • Not what we care about
• As treated - compare people who get treatment who get control
  • Ignores randomization
  • Not a valid estimate of causal effect
• Per protocol analysis
  • Compares people who appeared to comply with their assigned treatment
  • Not a valid estimate of a causal effect

• What we really want: the biologic effect of taking Vitman A on Mortality
• We can use instrumental variables to get at this
• Define the average effect among compliers (CACE) and noncompliers (NACE)
• Then we have that \[ACE = p_c \times CACE + (1-p_c) \times NACE\]
• Here we change what we are interested in to be "complier average causal effect"

• ITT provides unbiased estimate of the total causal effect
• We can estimate the proportions of compliers (missing data problem again!)
  • \(p_c\) is proportion who took treatment (in Zeger/Sommer data 0.8)
• Now we have an estimation problem with two unknowns \[ACE = p_c \times CACE + (1-p_c) \times NACE\] \[-0.0025 = 0.8 \times CACE + 0.2 \times NACE\]
• Assume that being assigned to treatment doesn't effect outcome if you don't take it (NACE =0)
• You can now solve \(CACE = -0.0025/0.8 = -0.0031\)
• This is the IV estimate for CACE

• Suppose that the control group has access to treatment
  • Flu shots - encouragement
  • Diet - information
• Now we have three kinds of "non-compliers"
  • Defiers
  • Always takers
  • Never takers
• Main interest is still the CACE but things get more complicated
• Fundamental idea still the same: \[ ACE = p_c \times CACE + p_d \times DACE + p_{at} \times AACE + p_{nt} \times NACE\]

• Has a causal effect on the treatment of interest (\(p_c > 0\))
  • (there exist some compliers)
• Affects the outcome only through the treatment received
  • (exclusion restrictions)
• Does not share common causes with the outcome
  • instrument (encouragement) assigned randomly

\[ ACE = p_c \times CACE + p_d \times DACE + p_{at} \times AACE + p_{nt} \times NACE\]

• No defiers \(\implies p_d = 0\)
• Exclusion restrictions \(AACE = NACE = 0\)
• People in the treatment goroup must be never takers \(\implies p_{nt} = 1-0.307 = 0.693\)
• People in the control group who get the flu shot must be always takers \(\implies p_{at} = 0.19\)
• The remaining people must be compliers \(p_c = 1 - 0.693 - 0.19 = 0.12\)
• \(ITT\) estimates \(ACE\): \(ITT = -0.014\)
• \(ACE = p_c * CACE \implies CACE = ACE/p_c \implies CACE = -0.014/0.12\)

• Causal mailing list: https://groups.google.com/forum/#!forum/jhu-causal

• 3rd Quarter, course by Elizabeth Stuart, who won Golden Apple award; focus is introducing methods and their application: http://www.jhsph.edu/courses/course/140.664/01/2014/20016/

• 4th Quarter, theoretical foundations to provide background for developing new causal inference methods (taught by Frangakis and Rosenblum): http://www.jhsph.edu/courses/course/140.665/01/2013/18324/