spelling and grammar edits for paper

paper/paper.md

@@ -27,11 +27,11 @@ toc-title: Table of contents
 
 Estimating the strength of causal relationships between variables is an
 important problem across many scientific disciplines. A variety of
-statistical methods have been developed to estimate and obtain inference
-about causal quantities, yet few tools readily support the comparison of
-candidate approaches. `CausalTables.jl` provides tools to evaluate and
-compare statistical causal inference methods in Julia. The package
-provides two main functionalities. Firstly, it implements a
+statistical methods have been developed to estimate and obtain
+inferences about causal quantities, yet few tools readily support the
+comparison of candidate approaches. `CausalTables.jl` offers tools to
+evaluate and compare statistical causal inference methods in Julia. The
+package provides two main functionalities. Firstly, it implements a
 `CausalTable` interface for storing data with partially-labeled causal
 structure in a `Tables.jl`-compatible format. Secondly, it introduces a
 `StructuralCausalModel` for randomly generating data with a
@@ -145,8 +145,8 @@ of how `CausalTables.jl` can be used as a benchmarking tool.
 
 The prototypical causal inference problem involves estimating the
 average treatment effect (ATE) of a binary treatment $A$. The ATE
-describes the difference in the counterfactual mean of $Y$ had everyone
-been treated versus no one treated. An example SCM describing this
+describes the difference in the counterfactual mean of $Y$ had every
+unit been treated versus no unit treated. An example SCM describing this
 scenario might be the following: `\begin{align*}
 W &\sim Beta(2, 4) \\
 A &\sim Bernoulli(W) \\
@@ -195,13 +195,13 @@ ate(scm) # average treatment effect
 ```
 
 ::: {.cell-output .cell-output-display execution_count="1"}
-    (μ = 1.001, eff_bound = 1.999)
+    (μ = 1.000, eff_bound = 1.996)
 :::
 ::::
 
 In addition, `CausalTables.jl` provides a low-level interface allowing
 users to (1) apply common interventions to the treatment variable in a
-`CausalTable`, and (2) compute ground-truth conditional densities and
+`CausalTable`, and (2) compute ground truth conditional densities and
 functions of these (e.g., mean, variance), which typically arise as
 nuisance parameters in the construction of estimators in causal
 inference. For example, below, we compute the difference in the
@@ -220,11 +220,11 @@ mean(conmean(scm, treated, :Y) .- conmean(scm, untreated, :Y))
 :::
 ::::
 
-The above recovers an estimate of the ground-truth via plug-in estimates
+The above recovers an estimate of the ground truth via plug-in estimates
 based on the outcome regression (i.e., the conditional expectation of
 the outcome, given treatment and covariates). Alternatively, one can
-also compute an inverse probability weighted (IPW) estimate with
-ground-truth weights using the `propensity` function:
+also compute an inverse probability weighted (IPW) estimate with ground
+truth weights using the `propensity` function:
 
 :::: {.cell execution_count="1"}
 ``` {.julia .cell-code}
@@ -234,7 +234,7 @@ mean(y .* (2 * a .- 1) ./ propensity(scm, ct, :A))
 ```
 
 ::: {.cell-output .cell-output-display execution_count="1"}
-    0.918
+    1.171
 :::
 ::::
 
@@ -251,7 +251,7 @@ mean(y_treated .- y_untreated)
 ```
 
 ::: {.cell-output .cell-output-display execution_count="1"}
-    1.024
+    0.950
 :::
 ::::
 
@@ -297,7 +297,7 @@ ape(scm, additive_mtp(1)) # average policy effect
 ```
 
 ::: {.cell-output .cell-output-display execution_count="1"}
-    (μ = 2.502, eff_bound = 5.264)
+    (μ = 2.501, eff_bound = 5.270)
 :::
 ::::
 
@@ -306,7 +306,7 @@ regression model and use it to predict the outcome $Y$ under the
 modified treatment policy $d(A, W;\delta): A \to A + \delta$; the
 average difference of these predictions on the observed data yields the
 APE. We can use `CausalTables.jl` to simulate the output of such a
-procedure under the true value of the outcome regression:
+procedure had we known the true value of the outcome regression:
 
 :::: {.cell execution_count="1"}
 ``` {.julia .cell-code}
@@ -316,7 +316,7 @@ mean(conmean(scm, ct_intervened, :Y) .- responsematrix(ct))
 ```
 
 ::: {.cell-output .cell-output-display execution_count="1"}
-    2.558
+    2.551
 :::
 ::::
 

paper/paper.qmd

@@ -27,9 +27,9 @@ engine: julia
 
 Estimating the strength of causal relationships between variables is an
 important problem across many scientific disciplines. A variety of statistical methods have
-been developed to estimate and obtain inference about causal quantities, yet
+been developed to estimate and obtain inferences about causal quantities, yet
 few tools readily support the comparison of candidate approaches.
-`CausalTables.jl` provides tools to evaluate and compare statistical causal
+`CausalTables.jl` offers tools to evaluate and compare statistical causal
 inference methods in Julia. The package provides two main functionalities.
 Firstly, it implements a `CausalTable` interface for storing data with
 partially-labeled causal structure in a `Tables.jl`-compatible format.
@@ -133,7 +133,7 @@ remainder of this section, we will present two example use cases of how
 
 The prototypical causal inference problem involves estimating the average
 treatment effect (ATE) of a binary treatment $A$. The ATE describes the difference in the
-counterfactual mean of $Y$ had everyone been treated versus no one treated. An
+counterfactual mean of $Y$ had every unit been treated versus no unit treated. An
 example SCM describing this scenario might be the following:
 \begin{align*}
 W &\sim Beta(2, 4) \\
@@ -171,7 +171,7 @@ values of common causal estimands such as the ATE (using `ate`). These functions
 also simultaneously estimate the corresponding efficiency bound---the asymptotic 
 lower bound on the variance---for a class of estimators (those that are regular 
 and asymptotically linear) commonly used in causal inference. This facilitates 
-the comparison of candidate estimators not only in  terms of their bias (average 
+the comparison of candidate estimators not only in terms of their bias (average 
 distance from the ground truth) but also their efficiency. Below, we demonstrate 
 these for the example SCM given above.
 
@@ -181,7 +181,7 @@ ate(scm) # average treatment effect
 
 In addition, `CausalTables.jl` provides a low-level interface allowing users to (1)
 apply common interventions to the treatment variable in a `CausalTable`, and
-(2) compute ground-truth conditional densities and functions of these (e.g.,
+(2) compute ground truth conditional densities and functions of these (e.g.,
 mean, variance), which typically arise as nuisance parameters in the
 construction of estimators in causal inference. For example, below, we compute
 the difference in the conditional mean of $Y$ under treatment versus no
@@ -193,10 +193,10 @@ untreated = intervene(ct, treat_none) # CausalTable with no one treated
 mean(conmean(scm, treated, :Y) .- conmean(scm, untreated, :Y))
 ```
 
-The above recovers an estimate of the ground-truth via plug-in estimates based
+The above recovers an estimate of the ground truth via plug-in estimates based
 on the outcome regression (i.e., the conditional expectation of the outcome,
 given treatment and covariates). Alternatively, one can also compute an inverse
-probability weighted (IPW) estimate with ground-truth weights using the
+probability weighted (IPW) estimate with ground truth weights using the
 `propensity` function:
 
 ```{julia}
@@ -256,12 +256,12 @@ One strategy for estimating an APE is to fit a parametric outcome regression
 model and use it to predict the outcome $Y$ under the modified treatment policy 
 $d(A, W;\delta): A \to A + \delta$; the average difference of these predictions 
 on the observed data yields the APE. We can use `CausalTables.jl` to simulate 
-the output of such a procedure under the true value of the outcome regression:
+the output of such a procedure had we known the true value of the outcome regression:
 
 ```{julia}
 ct = rand(scm, 500)  # Randomly draw data
 ct_intervened = intervene(ct, additive_mtp(1))  # apply MTP
-mean(conmean(scm, ct_intervened, :Y) .- responsematrix(ct)) 
+mean(conmean(scm, ct_intervened, :Y) .- responsematrix(ct))
 ```
 
 # Closing remarks