Provides computational implementations of models of causal judgment, including the Counterfactual Effect Size (CES) model of Quillien and Lucas (2023)
The causaljudgment package computes the predictions of computational models of causal judgment. Currently two models are implemented: the Counterfactual Effect Size (CES) model (Quillien & Lucas, 2023) and the Necessity-Sufficiency (NS) model (Icard, Kominsky & Knobe, 2017). You can use the package in R (see below) or with the interactive app.
In this file, we give examples of how to compute causal judgments, describe current limitations, and then (for interested readers) give a high-level explanation of how these scripts work. More background about when and why to use this package is given here. For the original scientific papers see here (CES) and here (NS).
First, install and load the package:
install.packages("remotes")
remotes::install_github("tadegquillien/causaljudgment")
library(causaljudgment)
Computing a causal judgment requires three steps:
-Specify a causal model,
-Specify what happened in the actual world,
-Request a causal judgment.
For example, suppose that Alice will graduate if she passes both her History and her Math exam. The Math exam is very difficult and the History exam is very easy.
We specify this information as a causal model:
p_math <- .1 # probability of passing math exam
p_history <- .9 # probability of passing history
causal_rule <- 'history & math' # Alice passes if she passes history and math
# collect the above information in a causal model
causalmodel <- list(graduate=causal_rule, math=p_math, history=p_history)
The string 'history & math' specifies the structural equation for the
graduate variable. The variables history and math each have an associated
prior probability.
Next specify what happens in the actual world; Alice passed both History and Math, and she graduates:
actual_world <- list(graduate=1, math=1, history=1)
Finally we request a causal judgment:
# to what extent did passing the math exam cause Alice to graduate?
compute_judgment('math', 'graduate', causalmodel, actual_world, 'ces', .7)
The first two arguments say that we want to see to what extent math
caused graduate.
The next two arguments specify the causal model and the actual-world value of
the variables (defined above). The fifth argument specifies the computational
model we want to use (here, CES). The last argument specifies the value of the
stability parameter $s$ (how much counterfactual simulation is 'anchored' to
the actual world). This argument is optional, by default $s=0$.
Running this command will return a 'causal score' from -1 to 1 (for CES) or from 0 to 1 (for NS). Higher values indicate higher actual causal strength. For the CES model, a negative value like -.8 indicates a very weak causal score, not something like 'negative' causation. And -.8 is weaker than for example -.4.
More examples are provided in this vignette.
References:
Icard, Kominsky & Knobe (2017). Normality and actual causal strength. Cognition.
Quillien & Lucas (2023). Counterfactuals and the logic of causal selection. Psychological Review.
We currently only support causal models that have binary variables. The value of these variables must be input as 0 or 1.
Structural equations must be specified as strings that use symbols that will be recognized by R as logical symbols.
For example, 'a or not b' will not work, but 'a | !b' will. In principle it should be possible to express any Boolean function this way, although for some of them it might be cumbersome.
Causal models must be expressed as Structural Causal Models (SCMs). Non-deterministic models like Causal Bayes Nets are not supported, but one can always formulate an SCM that emulates a Causal Bayes Net by introducing 'noise' variables. See the 'Noisy-Disjunctive structure' in the vignette for an example.
On the positive side, any causal structure that can be expressed by an SCM with binary variables should be supported. This includes 'deep' chain structures like A $\rightarrow$ B $\rightarrow$ C $\rightarrow$ D $\rightarrow$ E, and confounded models where the effect of the cause C on the effect E is confounded by other variables.
We use an analytical approach to compute causal judgments for the CES and NS models. For background on how these models work see the references above.
Our goal is to compute a causal judgment for the extent to which $C=c$ caused $E=e$. The models work by representing the causal structure as a Structural Causal Model (SCM), and simulating counterfactual worlds from this SCM. Here, instead of explicitly simulating counterfactual worlds by sampling from the SCM, we analytically compute the probability distribution over counterfactual worlds that would follow from this sampling process. This makes computation much faster. First it is useful to describe what this sampling process would look like (this will help understand what the analytical computation is trying to formalize).
For the CES model, we sample each counterfactual world by doing the following:
Sample each exogenous variable according to the Lucas-Kemp process (see Lucas & Kemp, 2015; Quillien & Lucas, 2023). That is, each exogenous variable $X$ is sampled with probability $s\mathbb{I}(X=x)+(1-s)P(X=x)$, where $P(X=x)$ is the prior probability of $X=x$, $\mathbb{I}(X=x)$ is 1 if $X=x$ in the actual world and 0 otherwise, and $s$ is a free 'stability' parameter controlling to what extent the simulation process is biased to actual-world values of the variables. Values around $s=.7$ usually give a good fit to empirical data.
If $C$ is endogenous, sample the value of $C$ by making a random intervention on $C$, where $C$ is sampled from $p(C)$. $p(C)$ is the marginal probability of $C$: it is the probability we would get if we simply generated counterfactual worlds without implementing step 2.
Determine the value of all other variables by applying the relevant structural equations.
The distribution over counterfactual worlds that we wish to compute is the relative frequency of worlds simulated this way, as the number of samples goes to infinity. For the NS model we do the same thing, except that we don't perform step 2, and we typically set $s=0$ in step 1.
To compute this probability distribution analytically, we do the following. We list each possible world (i.e. combination of variable values) as a row in a table. We then assign a probability to each world:
We compute the probability of each exogenous variable (defined by the Lucas-Kemp process).
For each endogenous variable $X$, we compute $p(X=x|\text{pa}(X))$ as 0 or 1, depending on whether $X=x$ is consistent with the value of the variable's parents (i.e. $\text{pa}(X)$ ) in the current world. (For example if we have $A := B$, and in the current world $A=1$ but $B=0$, then $p(B=0|\text{pa}(B))$ is 0). Note that since we are using SCMs with fully observed variables the conditional probabilities here are deterministic.
We compute a probability $p(w)$ for the whole world by using the factorization defined by the network structure, i.e. $p(w)=\prod_{V} p(V|\text{pa}(V))$. Note that if $V$ is exogenous then $p(V|\text{pa}(V)) = p(V|\emptyset) = p(V)$.
If $C$ is an endogenous variable, we must perform an additional step to ensure that the distribution reflects the fact that $C$'s value is set by interventions. We first compute the marginal probability $p(C=c)$ in the distribution we just computed. Then we replace $p(C=c|\text{pa}(C))$ with $p(C=c)$ in every world. Doing this ensures that the probability of $C=c$ is now independent from the value of $C$'s parents, as required by the fact that $C$ is set by interventions. After doing this, we now re-compute the probability of each world.
For the NS model we do the same thing but without step 4.
After we have obtained the probability distribution over counterfactual worlds, it is easy to analytically compute the correlation between $C=c$ and $E=e$ in this distribution (for the CES model). We can also analytically compute Necessity and Sufficiency (for the NS model). See here for more details on these computations.
References:
Lucas & Kemp (2015). An improved probabilistic account of counterfactual reasoning. Psychological Review.
This package would not exist without enthusiastic encouragement and advice from Joshua Knobe. I also thank Shubhamkar Ayare for raising an interesting technical point about CES implementation (see here).
To cite the package in publications please use:
Quillien T (2026). causaljudgment: computational models of causal judgment. R package.