1.1.1 “Causal inference”

• The term has become synonymous with randomization, or design-based inference

• Massive growth in last 15 years in design-based approaches in political science and economics

  • Randomization by the researcher: standard experiments

  • Randomization by nature: pure natural experiments

  • “As if” randomization, e.g.:

    • Regression-discontinuity designs

1.1.2 The benefit of strong design: assumption-free inference

• Broad, deep skepticism about observational, assumption- (or model-) based inference

  • Gerber, Green, and Kaplan (2014): “…observational findings are accorded zero weight….” if biases unknown

• Randomized assignment allows for “assumption-free” causal inference

  • To get the ATE, just difference the outcomes between treatment groups

  • Liberated from models

1.1.3 But sometimes randomization isn’t enough

• To be clear: we think developments in design-based inference are enormously important

  • If you can randomize to get at your estimand, go for it

• But the “causal inference revolution” has some limits

1.1.4 Limits of randomized designs

• Feasibility: Randomization impossible or unavailable for large set of research and policy questions.
  • Some causal variables that are hard to randomize
    • Institutions
    • Conflict
    • Economic development
    • Party systems

1.1.5 And non-experimental causal inference always requires models

Simple example:

• Assume no randomization

• We want to know if \(X\) causes \(Y\)

• We want to use \(X,Y\) correlations as evidence

  • But this evidence depends on beliefs about how the world works

    • Specifically, we have to believe there is no \(Z\) such that: \(X \leftarrow Z \rightarrow Y\)

• Or maybe we do think there is a confounding \(Z\)

  • So then we control for \(Z\)
  • But this strategy depends on a different model of the world
    • A model in which THIS isn’t happening: \(X \rightarrow Z \rightarrow Y\)

• In sum: beliefs about the world will always play a central role in non-randomized inference

1.1.6 Even with randomization, so many questions we can’t answer without models

Randomization can give us an \(ATE\), but can’t address many other causal estimands of interest

• Distribution of effects
  • Say, we know an aid treatment had an ATE of \(+20\%\) on village income
  • For what proportion of villages did it have a positive effect?
  • What proportion did it hurt?
• How effects occurred (mechanisms)
  • Did aid boost income by enabling capital investments?
  • By boosting schooling?
  • By mitigating conflict?

1.6.1 Models \(\Rightarrow\) more transparent, evaluable qualitative inference

• In qualitative inference too, what counts as evidence for a finding depends on how we think the world works

• By making our beliefs about the world explicit and reasoning systematically from them, we can do much better than if we keep our underlying models implicit

  • Logical consistency
  • Transparency
  • Openness to evaluation

1.7.1 How can we integrate qualitative and quantitative data?

• Data of different forms
  • Quant: a little information on lots of cases; focused on \(X\)’s and \(Y\)
  • Qual: lots of information on one/few cases; focused on process and context
• We want two forms of integration
  • Combination: Arrive at inferences that build on all information
  • Joint gains: Allow one form of evidence to inform inferences from other forms of evidence

1.8.1 Key benefits of doing inference with causal models

Using causal models can allow us to

• Answer a vast range of causal questions, with experimental and observational data

• Draw inferences from evidence in ways logically consistent with our prior beliefs/information

• Show, and allow others to see, precisely how our inferences hinge on our model

• Readily integrate quantitative and qualitative evidence into a single set of findings

• [Though not in this course] Make research design choices in a way systematically informed by what we already know

• \(\Rightarrow\) implications for large-n, small-n, and mixed-method research

A cost: inferences are model-dependent

2.1.3 Potential Outcomes

Idea: A causal claim is (in part) a claim about something that did not happen. This makes it metaphysical.

2.1.4 Potential Outcomes

Now that we have a concept of causal effects available, let’s answer two questions:

• TRANSITIVITY: If for a given unit \(A\) causes \(B\) and \(B\) causes \(C\), does that mean that \(A\) causes \(C\)?

2.1.5 Potential Outcomes

Now that we have a concept of causal effects available, let’s answer two questions:

• TRANSITIVITY: If for a given unit \(A\) causes \(B\) and \(B\) causes \(C\), does that mean that \(A\) causes \(C\)?

• A boulder is flying down a mountain. You duck. This saves your life.

• So the boulder caused the ducking and the ducking caused you to survive.

• So: did the boulder cause you to survive?

2.1.6 Causal claims: Contribution or attribution?

The counterfactual model is about contribution and attribution in a very specific sense.

• Focus is on non-rival contributions
• Focus is on conditional attribution. Not: “what caused \(Y\)?” or “What is the cause of \(Y\)?”, but “did \(X\) cause \(Y\) given all other factors were what they were?”

2.1.7 Causal claims: Contribution or attribution?

Consider an outcome \(Y\) that might depend on two causes \(X_1\) and \(X_2\):

\[Y(0,0) = 0\] \[Y(1,0) = 0\] \[Y(0,1) = 0\] \[Y(1,1) = 1\]

What caused \(Y\)? Which cause was most important?

2.1.8 Causal claims: No causation without manipulation

• Some seemingly causal claims not admissible.
• To get the definition off the ground, manipulation must be imaginable (whether practical or not)
• This renders thinking about effects of race and gender difficult
• What does it mean to say that Aunt Pat voted for Brexit because she is old?

2.1.9 Causal claims: No causation without manipulation

• Some seemingly causal claims not admissible.

• To get the definition off the ground, manipulation must be imaginable (whether practical or not)

• This renders thinking about effects of race and gender difficult

• Compare: What does it mean to say that Southern counties voted for Brexit because they have many old people?

• What does it mean to say that Trump won because he is a white man? What exactly do we have to imagine…

5.1.1 Causal statements

The key input to model creation is the causal statement:

• “X -> Y”
• “A -> B -> C <- A <-> C”
• “A -> B; B <- C”

Note:

• Arrows can go both directions
• Double headed arrows means confounding
• Parts of a statement can be separated using ;
• Avoid spaces and special characters in names (., -, _, |, etc.)

5.1.2 Causal statements

CausalQueries will always translate causal statements into a simple collection of connections:

make_model("A -> B -> C <- A <-> C") |>
  inspect("statement")

Causal statement: 
A -> B; A -> C; A <-> C; B -> C

Note:

• Arrows can go both directions
• Double headed arrows means confounding
• Parts of a statement can be separated using ;
• See “cheat sheet” for more guidance

5.1.5 Refine

There are various ways to refine a model:

• Set restrictions: set_restrictions removes particular “nodal types”
• Set parameters: set_parameters provide particular parameter values for a model
• Set priors: set_priors provide priors over model parameters

There are many ways to use these so consult help. e.g. ? set_restrictions

5.1.6 Example 1

Say I have a model of the form A -> B -> C. I want to impose monotonicity (no negative effects), but I also want to assume that the A -> B relation is very strong:

Using natural language statements:

model <- 
  make_model("A -> B-> C") |>
  set_restrictions(statement = decreasing("A", "B")) |>
  set_restrictions(statement = decreasing("B", "C")) |>
  set_parameters(statement = increasing("A", "B"), parameters = .8)

model |>  inspect("parameters_df")

parameters_df
Mapping of model parameters to nodal types: 

  param_names: name of parameter
  node:        name of endogeneous node associated
               with the parameter
  gen:         partial causal ordering of the
               parameter's node
  param_set:   parameter groupings forming a simplex
  given:       if model has confounding gives
               conditioning nodal type
  param_value: parameter values
  priors:      hyperparameters of the prior
               Dirichlet distribution 

   param_names node gen param_set nodal_type given param_value priors
1          A.0    A   1         A          0         0.5000000      1
2          A.1    A   1         A          1         0.5000000      1
3         B.00    B   2         B         00         0.1000000      1
5         B.01    B   2         B         01         0.8000000      1
6         B.11    B   2         B         11         0.1000000      1
7         C.00    C   3         C         00         0.3333333      1
9         C.01    C   3         C         01         0.3333333      1
10        C.11    C   3         C         11         0.3333333      1

5.1.7 Example 1

Say I have a model of the form A -> B -> C. I want to impose monotonicity (no negative effects), but I also want to assume that the A -> B relation is very strong:

Using nodal types:

model <- 
  make_model("A -> B-> C") |>
  set_restrictions(param_names  = 'B.10') |>
  set_restrictions(param_names  = 'C.10') |>
  set_parameters(param_names = 'B.01', parameters = .8)

model |>  inspect("parameters_df")

parameters_df
Mapping of model parameters to nodal types: 

  param_names: name of parameter
  node:        name of endogeneous node associated
               with the parameter
  gen:         partial causal ordering of the
               parameter's node
  param_set:   parameter groupings forming a simplex
  given:       if model has confounding gives
               conditioning nodal type
  param_value: parameter values
  priors:      hyperparameters of the prior
               Dirichlet distribution 

   param_names node gen param_set nodal_type given param_value priors
1          A.0    A   1         A          0         0.5000000      1
2          A.1    A   1         A          1         0.5000000      1
3         B.00    B   2         B         00         0.1000000      1
5         B.01    B   2         B         01         0.8000000      1
6         B.11    B   2         B         11         0.1000000      1
7         C.00    C   3         C         00         0.3333333      1
9         C.01    C   3         C         01         0.3333333      1
10        C.11    C   3         C         11         0.3333333      1

5.2.1 A basic plot:

model <- make_model("X -> Y")

model |> plot_model()

Simple model

6.0.1 ggplot layers

The model that is produced is a ggplot object and additional layers can be added in the usual way.

model |>
  plot_model()  +
  annotate("text", x = c(1, -1) , y = c(1.5, 1.5), label = c("Some text", "Some more text")) +
  coord_flip()

Adding additional ggplot layers

6.0.2 Adding labels

Provide labels in the same order as model nodes.

model <- make_model("A -> B -> C <- A")


# Check node ordering
inspect(model, "nodes")

Nodes: 
A, B, C

# Provide labels
model |>
   plot_model(
     labels = c("This is A", "Here is B", "And C"),
     nodecol = "white", textcol = "black")

Adding labels

6.0.3 Controlling positions

You can manually set positions using the x_coord and y_coord arguments.

You can manually set positions using the x_coord and y_coord arguments.

model |>
  plot(x_coord = 0:2,  y_coord = c(0, 2, 1))

Specifying coordinates

6.0.4 Controlling color

You can manually control node color and text color for all nodes together or separately.

model |>
  plot(x_coord = 0:2,  y_coord = c(0, 2, 1),
       nodecol = c("blue", "orange", "red"),
       textcol = c("white", "red", "blue"))

Controlling colors

6.0.5 Models with unobserved confounding

Unobserved confounding is represented using dashed curves.

make_model('X -> K -> Y <- X; X <-> Y; K <-> Y') |>   plot()

Plot showing confounding