Correlation does not imply causation. It turns out, however, that with some simple ingenious tricks one can, potentially, unveil causal relationships within standard observational data, without having to resort to expensive randomised control trials.<\/p>\n
This post is targeted towards anyone making data driven decisions. The main takeaway message is that causality may be possible by understanding that the story behind the data<\/strong> is as important as the data itself.<\/p>\n By introducing Simpson\u2019s<\/em> and Berkson\u2019s<\/em> Paradoxes<\/em>, situations where the outcome of a population is in conflict with that of its cohorts, I shine a light on the importance of using causal reasoning to identify these paradoxes in data and avoid misinterpretation. Specifically I introduce causal graphs <\/em>as a method to visualise the story behind the data point out that by adding this to your arsenal you are likely to conduct better analyses and experiments.<\/p>\n My ultimate objective is to whet your appetite to explore more on causality, as I believe that by asking data \u201cWhy?\u201d<\/em> you will be able to go beyond correlation calculations and extract more insights, as well as avoid common misjudgement pitfalls.<\/p>\n Note that throughout this gentle intro I do not use equations but demonstrate using accessible intuitive visuals. That said I provide resources for you to take your next step in adding Causal Inference<\/a> to your statistical toolbox so that you may get more value from your data.<\/p>\n In [Deity] We Trust, All Others Bring Data! \u2014 William E. Deming<\/em><\/p>\n<\/blockquote>\n In this digital age it is common to put a lot of faith in data. But this raises an overlooked question: Should we trust data on its own?<\/p>\n Judea Pearl, who is considered the godfather of Causality<\/a>, articulated best:<\/p>\n \u201cThe collection of information is as important as the information itself \u201c \u2014 Judea Pearl<\/em><\/p>\n<\/blockquote>\n In other words the story behind <\/strong>the data is as important as the data itself.<\/p>\n This manifests in a growing awareness of the importance of identifying bias in datasets. By the end of this post I hope that you will appreciate that causality pertains the fundamental tools to best express, quantify and attempt to correct for these biases.<\/p>\n In causality introductions it is customary to demonstrate why \u201ccorrelation does not imply causation\u201d by highlighting limitations of association analysis due to spurious correlations (e.g, shark attacks To understand the importance of the story behind the data we will examine two counter-intuitive (but nonetheless true) paradoxes which are classical situations of data misinterpretation.<\/p>\n In the first we imagine a clinical trial in which patients are given a treatment and that results in a health score. Our objective is to assess the average impact of increased treatment to the health outcome. For pedagogical purposes in these examples we assume that samples are representative (i.e, the sample size is not an issue) and that variances in measurements are minimal.<\/p>\n In the figure above we learn that on average increasing the treatment appears to be beneficial since it results in a better outcome.<\/p>\n Now we\u2019ll color code by age and gender groupings and examine how the treatment increases impacts each cohort.<\/p>\n Track any cohort (e.g, \u201cGirls\u201d representing young females) and you immediately realise that increase in treatment appears adverse.<\/p>\n What is the conclusion of the study? On the one hand increasing the treatment appears to be better for the population at large, but when examining gender-age cohorts it seems disadvantageous. This is Simpson\u2019s Paradox which may be stated:<\/p>\n \u201cTrends can exist in subgroups but reverse for the whole\u201d<\/em><\/p>\n<\/blockquote>\n Below we will resolve this paradox using causality tools, but beforehand let\u2019s explore another interesting one, which also examines made up data.<\/p>\n Imagine that we quantify for the general population their attractiveness and how talented they are as in this figure:<\/p>\n We find no apparent correlation.<\/p>\n Now we\u2019ll focus on an unusual subset \u2014 famous people:<\/p>\n Here we clearly see an anti-correlation that doesn\u2019t exist in the general population.<\/p>\n Should we conclude that Talent<\/em> and Attractiveness<\/em> are independent variables as per the first plot of the general population or that they are correlated as per that of celebrities?<\/p>\n This is Berkson\u2019s Paradox where one population has a trait trend that another lacks.<\/p>\n Whereas an algorithm would identify these correlations, resolving these paradoxes requires a full understanding of the context which normally is not fed to a computer. In other words without knowing the story behind the data results may be misinterpreted<\/strong> and wrong conclusions<\/strong> may be inferred.<\/p>\n Mastering identification and resolution these paradoxes is an important first step<\/strong> to elevating one\u2019s analyses from correlations to causal inference<\/strong>.<\/p>\n Whereas these simple examples may be explained away logically, for the purposes of learning causal tools in the next section I\u2019ll introduce Causal Graphs<\/em>.<\/p>\n \u201c[From the Simpson\u2019s and Berkson\u2019s Paradoxes we learn that] <\/em>certain decisions<\/em><\/strong> <\/em>cannot be made<\/em><\/strong> based on the basis of<\/em> data alone<\/em><\/strong>, but instead depend on the <\/em>story behind the data<\/em><\/strong>. \u2026 Graph Theory enables these stories to be conveyed\u201d \u2014 Judea Pearl<\/em><\/p>\n<\/blockquote>\n Causal graph models are probabilistic graphical models used to visualise the story behind the data. They are perhaps one of the most powerful tools for analysts that is not taught in most statistics curricula. They are both elegant and highly informative. Hopefully by the end of this post you will appreciate it when Judea Pearl says that this is the missing vocabulary to communicate causality.<\/p>\n To understand causal graph models (or causal graphs for short) we start with the following illustration of an example undirected graph<\/em> with four nodes\/vertices and three edges.<\/p>\n Each node is a variable and the edges communicate \u201cwho is related<\/strong> to whom?\u201d (i.e, correlations, joint probabilities).A directed graph<\/em> is one in which we add arrows as in this figure.<\/p>\n A directed edge communicates \u201cwho listens<\/strong> to whom?\u201d which is the essence of causation.<\/p>\n In this specific example you can notice a cyclical relationship between the C and D nodes.A useful subset of directed graphs are the directed acyclic graphs<\/em> (DAG), which have no cycles as in the next figure.<\/p>\n Here we see that when starting from any node (e.g, A) there isn\u2019t a path that gets back to it.<\/p>\n DAGs are the go-to choice in causality for simplicity as the fact that parameters do not have feedback highly simplifies the flow of information. (For mechanisms that have feedback, e.g temporal systems, one may consider rolling out nodes as a function of time, but that is beyond the scope of this intro.)<\/p>\n Causal graphs are powerful at conveying the cause\/effect relationships between the parameter and hence how data was generated (the story behind the data).<\/p>\n From a practical point of view, graphs enable us to understand which parameters are confounders that need to be controlled for, and, as important, which not to control for, because doing so causes spurious correlations. This will be demonstrated below.<\/p>\n The practice of attempting to build a causal graph enables:<\/p>\n To further motivate the usage of causal graph models we will use them to resolve the Simpson\u2019s and Berkson\u2019s paradoxes introduced above.<\/p>\n For simplicity we\u2019ll examine Simpson\u2019s paradox focusing on two cohorts, male and female adults.<\/p>\n Examining this data we can make three statements about three variables of interest:<\/p>\n According to these we can draw the causal graph as the following:<\/p>\n Notice how each arrow contributes to communicate the statements above. As important, the lack of an arrow pointing into Gender conveys that it is an independent variable.<\/p>\n We also notice that by having arrows pointing from Gender to Treatment and Outcome it is considered a common cause<\/em> between them.<\/p>\n The essence of the Simpson\u2019s paradox is that although the Outcome is effected by changes in Treatment, as expected, there is also a backdoor path <\/em>flow of information via Gender.<\/p>\n As you may have guessed by this stage, the solution to this paradox is that the common cause Gender is a confounding variable that needs to be controlled<\/em>.<\/p>\n Controlling for a variable, in terms of a causal graph, means eliminating the relationship between Gender and Treatment.<\/p>\n This may be done in two manners:<\/p>\n In both pre- and post- data collection the elimination of the Treatment dependency of Gender (i.e, controlling for the Gender) may be done by modifying the graph such that the arrow between them is removed as in the following:<\/p>\n Applying this \u201cgraphical surgery\u201d means that the last two statements need to be modified (for convenience I\u2019ll write all three):<\/p>\n This enables obtaining the causal relationship of interest : we can assess the direct impact of modification Treatment on the Outcome.<\/p>\n The process of controlling for a confounder, i.e manipulation of the data generation process, is formally referred to as applying an intervention<\/em>. That is to say we are no longer passive observers of the data, but we are taking an active role in modification it to assess the causal impact.<\/p>\n How is this manifested in practice?<\/p>\n In the case of RCTs the researcher needs to control for important confounding variables. Here we limit the discussion to Gender (but in real world settings you can imagine other variables such as Age, Social Status and anything else that might be relevant to one\u2019s health).<\/p>\n RCTs are considered the golden standard for causal analysis in many experimental settings thanks to its practice of confounding variables. That said, it has many setbacks:<\/p>\n Observational data on the other hand is much more readily available in the industry and academia and hence much cheaper and could be more representative of actual habits of the individuals. But as illustrated in the Simpson\u2019s diagram it may have confounding variables that need to be controlled.<\/p>\n This is where ingenious solutions developed in the causal community in the past few decades are making headway. Detailing them are beyond the scope of this post, but I briefly mention how to learn more at the end.<\/p>\n To resolve for this Simpson\u2019s paradox with the given observational data one<\/p>\n Here we will focus on intuition, but in a future post we will describe the maths behind this solution.<\/p>\n I am sure that many analysts, just like myself, have noticed Simpson\u2019s at some stage in their data and hopefully have corrected for it. Now you know the name of this effect and hopefully start to appreciate how causal tools are useful.<\/p>\n That said \u2026 being confused at this stage is OK <\/strong> I\u2019ll be the first to admit that I struggled to understand this concept and it took me three weekends of deep diving into examples to internalised it. This was the gateway drug to causality for me. Part of my process to understanding statistics is playing with data. For this purpose I created an interactive web application hosted in Streamlit<\/a> which I call Simpson\u2019s Calculator Even if you are confused the main takeaways of Simpson\u2019s paradox is that:<\/p>\n This raises the question \u2014 should we just control for all variables except for the treatment and outcome? Let\u2019s keep this in mind when resolving for the Berkson\u2019s paradox.<\/p>\n As in the previous section we are going to make clear statements about how we believe the data was generated and then draw these in a causal graph.<\/p>\n Let\u2019s examine the case of the general population, for convenience I\u2019m copying the image from above:<\/p>\n Here we understand that:<\/p>\n A causal graph for this is quite simple, two nodes without an edge.<\/p>\n Let\u2019s examine the plot of the celebrity subset.<\/p>\n The cheeky insight from this mock data is that the more likely one is attractive the less they need to be talented to be a celebrity. Hence we can deduce that:<\/p>\n Hence we can draw the causal graph as:<\/p>\n By having arrows pointing into it Celebrity is a collider<\/em> node between Talent and Attractiveness.<\/p>\n Berkson\u2019s paradox is the fact that when controlling for celebrities we see an interesting trend (anti correlation between Attractiveness and Talent) not seen in the general population.<\/p>\n This can be visualised in the causal graph that by confounding for the Celebrity parameter we are creating a spurious correlation between the otherwise independent variables Talent and Attractiveness. We can draw this as the following:<\/p>\n The solution of this Berkson\u2019s paradox should be apparent here: Talent and Attractiveness are independent variables in general, but by controlling for the collider Celebrity node causes a spurious correlation in the data.<\/p>\n Let\u2019s compare the resolution of both paradoxes:<\/p>\n The next figure combines both insights in the form of their causal graphs:<\/p>\n The main takeaway from the resolution of these paradoxes is that controlling for parameters requires a justification. Common causes should be controlled for but colliders should not.<\/p>\n Even though this is common knowledge for those who study causality (e.g, Economics majors), it is unfortunate that most analysts and machine learning practitioners are not aware of this (including myself in 2020 after over 15 years of analysis and predictive modelling experience).<\/p>\n \u201cOddly, statisticians both over- and underrate the importance of confounders<\/em>\u201c \u2014 Judea Pearl<\/em><\/p>\n<\/blockquote>\n The main takeaway from this post is that the story behind the data is as important as the data itself.<\/p>\n Appreciating this will help you avoid result misinterpretation as spurious correlations and, as demonstrated here, in Simpson\u2019s and Berskon\u2019s paradoxes.<\/p>\n Causal Graphs are an essential tool to visualise the story behind the data. By using them to solve for the paradoxes we learnt that controlling for variables requires justification (common causes For those interested in taking the next step in their causal journey I highly suggest mastering Simpson\u2019s paradox. One great way is by playing with data. Feel free to do so with my interactive \u201cSimpson-calculator<\/a>\u201d Loved this post? Unless otherwise noted, all images were created by the author.<\/p>\n Many thanks to Jim Parr<\/a>, Will Reynolds<\/a>, Hedva Kazin<\/a> and Betty Kazin<\/a> for their useful comments.<\/p>\n Wondering what your next step should be in your causal journey? Check out my new article on mastering Simpson\u2019s Paradox<\/a> \u2014 you will never look at data the same way. Here I provide resources that I find useful as well as a shopping list of topics for beginners to learn.<\/p>\n This list is far from comprehensive, but I\u2019m glad to add to it if anyone has suggestions (please mention why the book stands out from the pack).<\/p>\n There are probably a few courses online. I love the One paid course This list is far from comprehensive because the space is rapidly growing:<\/p>\n Causal Wizard app<\/a> also have an article about Causal Diagram tools<\/a>.<\/p>\n Here I highlight a list of topics which I would have found useful when I started my learnings in the field. If I\u2019m missing anything I\u2019d be more than glad to get feedback and adding. I bold face the ones which were briefly discussed here.<\/p>\nThe Era of Data Driven Decision Making<\/strong><\/h2>\n
\n
\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.37.02%25E2%2580%25AFPM.png?resize=828%2C860&ssl=1\")
![\"\ud83e\udd88\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f988.png?ssl=1\")
and ice-cream sales ![\"\ud83c\udf66\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f366.png?ssl=1\")
). In an attempt to reduce the length of this post I defer this aspect to an older one of mine<\/a>. Here I focus on two mind boggling paradoxes ![\"\ud83e\udd2f\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f92f.png?ssl=1\")
and their resolution via causal graphs <\/em>to make a similar point.<\/p>\nParadoxes in Analysis<\/strong><\/h2>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.37.16%25E2%2580%25AFPM-1024x872.png?resize=1024%2C872&ssl=1\")
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.37.30%25E2%2580%25AFPM-1024x871.png?resize=1024%2C871&ssl=1\")
\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.39.45%25E2%2580%25AFPM.png?resize=676%2C600&ssl=1\")
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.37.52%25E2%2580%25AFPM.png?resize=866%2C740&ssl=1\")
Causal Graphs\u2014 Visualising The Story Behind The Data<\/strong><\/h2>\n
\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.38.09%25E2%2580%25AFPM-1024x623.png?resize=1024%2C623&ssl=1\")
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.38.17%25E2%2580%25AFPM-1024x943.png?resize=1024%2C943&ssl=1\")
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.38.25%25E2%2580%25AFPM-1024x949.png?resize=1024%2C949&ssl=1\")
\n
![\"\ud83d\udc8a\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f48a.png?ssl=1\")
Causal Graph Resolution of Simpson\u2019s Paradox<\/strong><\/h2>\n![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.38.37%25E2%2580%25AFPM-1024x864.png?resize=1024%2C864&ssl=1\")
\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.38.59%25E2%2580%25AFPM-1024x876.png?resize=1024%2C876&ssl=1\")
\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.39.21%25E2%2580%25AFPM-1024x908.png?resize=1024%2C908&ssl=1\")
\n
\n
\n
![\"\ud83d\ude15\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f615.png?ssl=1\")
<\/p>\n![\"\ud83e\uddee\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f9ee.png?ssl=1\")
. I\u2019ll write a separate post for this in the future.<\/p>\n\n
![\"\ud83e\udd9a\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f99a.png?ssl=1\")
Causal Graph Resolution of Berkson\u2019s Paradox<\/strong><\/h2>\n\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.39.57%25E2%2580%25AFPM-1024x417.png?resize=1024%2C417&ssl=1\")
\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.40.26%25E2%2580%25AFPM-1024x994.png?resize=1024%2C994&ssl=1\")
\n<\/figcaption><\/figure>\n![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.40.40%25E2%2580%25AFPM-1024x945.png?resize=1024%2C945&ssl=1\")
Paradoxes Summary<\/strong><\/h1>\n
\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.40.50%25E2%2580%25AFPM-1024x483.png?resize=1024%2C483&ssl=1\")
\n
Summary<\/strong><\/h2>\n
![\"\u2705\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/2705.png?ssl=1\")
, colliders ![\"\u26d4\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/26d4.png?ssl=1\")
).<\/p>\n.<\/p>\n![\"\ud83d\udc8c\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f48c.png?ssl=1\")
Join me on LinkedIn<\/a> or ![\"\u2615\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/2615.png?ssl=1\")
Buy me a coffee<\/a>!<\/p>\nCredits<\/strong><\/h2>\n
![\"\ud83d\udd0e\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f50e.png?ssl=1\")
<\/p>\nUseful Resources<\/strong><\/h2>\n
\n
![\"\ud83d\udcda\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f4da.png?ssl=1\")
Books<\/strong>
\n<\/h3>\n![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.41.04%25E2%2580%25AFPM-685x1024.png?resize=685%2C1024&ssl=1\")
\n
![\"\ud83d\udc0d\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f40d.png?ssl=1\")
.<\/li>\n![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.41.18%25E2%2580%25AFPM-1024x317.png?resize=1024%2C317&ssl=1\")
<\/figure>\n\n
![\"\ud83d\udd0f\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f50f.png?ssl=1\")
Courses<\/strong>
\n<\/h3>\n![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.41.30%25E2%2580%25AFPM-1024x741.png?resize=1024%2C741&ssl=1\")
![\"\ud83c\udd93\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f193.png?ssl=1\")
one of Brady Neil bradyneal.com\/causal-inference-course<\/strong><\/a>.<\/strong><\/p>\n\n
![\"\ud83d\udcb0\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f4b0.png?ssl=1\")
that is targeted to practitioners is Altdeep<\/a>.<\/p>\n\n
![\"\ud83d\udcbe\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f4be.png?ssl=1\")
Software<\/strong>
\n<\/h3>\n![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/Screenshot-2025-02-14-at-1.41.40%25E2%2580%25AFPM-1024x769.png?resize=1024%2C769&ssl=1\")
\n
![\"\ud83d\udc3e\"](\"https:\/\/i0.wp.com\/s.w.org\/images\/core\/emoji\/15.0.3\/72x72\/1f43e.png?ssl=1\")
Suggested Next Steps In The Causal Journey<\/strong><\/h3>\n