improve wording

Author: drbenvincent

examples/causal_inference/GLM-simpsons-paradox.ipynb

@@ -22,9 +22,9 @@
     "\n",
     "![](https://upload.wikimedia.org/wikipedia/commons/f/fb/Simpsons_paradox_-_animation.gif)\n",
     "\n",
-    "Another way of describing this is that we wish to causally estimate the effect of a predictor variable $x$ on an outcome variable $y$. The seemingly obvious approach of modelling `y ~ 1 + x` will lead us to conclude (in the situation above) that increasing $x$ causes $y$ to decrease (see Model 1 below). However, the relationship between $x$ and $y$ is confounded by a group membership variable $group$. This group membership variable $g$ is not included in the model, and so the relationship between $x$ and $y$ is biased. In some situations (e.g. the image above) this can lead us to completely reverse the sign of our estimate of the causal effect, now estimating that increasing $x$ causes $y$ to _increase_. \n",
+    "Another way of describing this is that we wish to estimate the causal relationship $x \\rightarrow y$. The seemingly obvious approach of modelling `y ~ 1 + x` will lead us to conclude (in the situation above) that increasing $x$ causes $y$ to decrease (see Model 1 below). However, the relationship between $x$ and $y$ is confounded by a group membership variable $group$. This group membership variable is not included in the model, and so the relationship between $x$ and $y$ is biased. If we now factor in the influence of $group$, in some situations (e.g. the image above) this can lead us to completely reverse the sign of our estimate of $x \\rightarrow y$, now estimating that increasing $x$ causes $y$ to _increase_. \n",
     "\n",
-    "In short, this 'paradox' (or simply ommitted variable bias) can be resolved by assuming a causal DAG which includes how the main predictor variable _and_ group membership (the confounding variable) influence the outcome variable. We demonstrate an example where we _don't_ incorporate group membership (so our causal DAG is wrong, or in other words our model is misspecified). We then show 2 ways to resolve this by including group membership as causal influence upon $x$ and $y$. This is shown in an unpooled model (Model 2) and a hierarchical model (Model 3)."
+    "In short, this 'paradox' (or simply ommitted variable bias) can be resolved by assuming a causal DAG which includes how the main predictor variable _and_ group membership (the confounding variable) influence the outcome variable. We demonstrate an example where we _don't_ incorporate group membership (so our causal DAG is wrong, or in other words our model is misspecified; Model 1). We then show 2 ways to resolve this by including group membership as causal influence upon $x$ and $y$. This is shown in an unpooled model (Model 2) and a hierarchical model (Model 3)."
    ]
   },
   {

examples/causal_inference/GLM-simpsons-paradox.myst.md

@@ -25,9 +25,9 @@ kernelspec:
 
 ![](https://upload.wikimedia.org/wikipedia/commons/f/fb/Simpsons_paradox_-_animation.gif)
 
-Another way of describing this is that we wish to causally estimate the effect of a predictor variable $x$ on an outcome variable $y$. The seemingly obvious approach of modelling `y ~ 1 + x` will lead us to conclude (in the situation above) that increasing $x$ causes $y$ to decrease (see Model 1 below). However, the relationship between $x$ and $y$ is confounded by a group membership variable $group$. This group membership variable $g$ is not included in the model, and so the relationship between $x$ and $y$ is biased. In some situations (e.g. the image above) this can lead us to completely reverse the sign of our estimate of the causal effect, now estimating that increasing $x$ causes $y$ to _increase_. 
+Another way of describing this is that we wish to estimate the causal relationship $x \rightarrow y$. The seemingly obvious approach of modelling `y ~ 1 + x` will lead us to conclude (in the situation above) that increasing $x$ causes $y$ to decrease (see Model 1 below). However, the relationship between $x$ and $y$ is confounded by a group membership variable $group$. This group membership variable is not included in the model, and so the relationship between $x$ and $y$ is biased. If we now factor in the influence of $group$, in some situations (e.g. the image above) this can lead us to completely reverse the sign of our estimate of $x \rightarrow y$, now estimating that increasing $x$ causes $y$ to _increase_. 
 
-In short, this 'paradox' (or simply ommitted variable bias) can be resolved by assuming a causal DAG which includes how the main predictor variable _and_ group membership (the confounding variable) influence the outcome variable. We demonstrate an example where we _don't_ incorporate group membership (so our causal DAG is wrong, or in other words our model is misspecified). We then show 2 ways to resolve this by including group membership as causal influence upon $x$ and $y$. This is shown in an unpooled model (Model 2) and a hierarchical model (Model 3).
+In short, this 'paradox' (or simply ommitted variable bias) can be resolved by assuming a causal DAG which includes how the main predictor variable _and_ group membership (the confounding variable) influence the outcome variable. We demonstrate an example where we _don't_ incorporate group membership (so our causal DAG is wrong, or in other words our model is misspecified; Model 1). We then show 2 ways to resolve this by including group membership as causal influence upon $x$ and $y$. This is shown in an unpooled model (Model 2) and a hierarchical model (Model 3).
 
 ```{code-cell} ipython3
 import arviz as az