30:T561,Now the regression line is pretty flat! There is a much weaker association between speed and height in the larger population. So why is there a stronger negative association when we look at just the professional NBA players? The idea is that to get into the NBA, you need to be a good player. You might be a good enough player if you're very tall or you're very fast. But if you're not very tall you probably need to make up for that by being faster, and if you aren't very fast you probably need to be taller. This selection process means that _among professional players_, you can find a negative association between speed and height, even if in the larger population they are not associated with one another. This phenomenon is referred to as [Berkson's Paradox](https://en.wikipedia.org/wiki/Berkson%27s_paradox): conditioning on a variable can induce a statistical association that isn't present otherwise. (Again, it's good to remember that I've simulated the data specifically to make this phenomenon appear! It's possible that if we got real data on this larger population of players, we'd see a different pattern.) So far, I've only been talking in terms of statistical associations ("height and speed are negatively associated"). Next, we'll take a causal inference perspective and describe how this phenomenon might be an example of causal **collider bias**.31:Td13,If there is a backdoor path between two variables, it mean's we may see a statistical association between them, even though they have no causal relationship. This phenomenon is called **collider bias**. If you condition on a collider, then it is possible to see an association between the variables that point to the collider, even if those variables have no causal relationship to one another. In our simulated data, this is exactly what we see: _Height_ and _Speed_ were simulated such that there is no causal relationship between them (neither causes the other), but _NBA Player_ depends entirely on _Height_ and _Speed_. --- Now let's return to the original tweet that motivated this discussion. ${tweet("1461090535985324039")} Anton actually went and downloaded real data on NBA players in 2020, and found that indeed there is actually a negative association between speed rating and player height. Cool! Next, if we could somehow get our hands on data from the larger population from which NBA players are drawn (and we'd have to think carefully about how to define this population), we could again plot speed rating vs. height in this larger dataset. If there is no association between the two in the larger population, then indeed we have an example of Berkman's Paradox: conditioning on a variable induces a statistical association between two other variables that otherwise are not associated with one another. Even if we did that, though, we couldn't say for sure that this is an example of causal collider bias. Rather, from a causal perspective we have demonstrated that if we assume the causal graph we gave above is true, then it's possible for the observed association between speed rating and height among NBA players to arise _even when they have no causal relationship to one another._ This illustrates an important point about causal inference: we _start_ by positing a particular set of causal relationships, and _then_ we use those assumptions to inform how we analyze and interpret the observed data. Without making strong assumptions, it's impossible to go the other way: to take a dataset and infer what the causal relationships between variables are. You might respond to this by saying "look, if we do somehow get data on this larger population, and there is no association between speed rating and height, then surely they can't be causally related to one another! That would validate our causal graph, which means that the negative association among NBA players is in fact causal collider bias!" The issue with this line of thinking is that two variables can be causally related to one another, and still have no association! Rachel Meager has a nice example of when this might be the case: ${tweet("1395791835063373824")} I'd have to think about a possible scenario in which speed rating and height are causally related to one another, but don't have any association. Again, it's possible that in reality they are causally related! The important point, though, is that the data can't help us distinguish between these different causal scenarios. Overall, what I take away from collider bias from a causal perspective is that if you draw a causal graph that includes collider variables, you shouldn't condition on them because doing so can induce spurious associations, which we call "collider bias". 32:T8e1,plot = (data, options) => { options = Object.assign({ height: width < 600 ? width : width / 2, width: width < 600 ? width : width / 2, margin: { top: 50, right: 90, bottom: 50, left: 50 }, xlab: "", ylab: "", xlim: d3.extent(data.map(d => d.X)), ylim: d3.extent(data.map(d => d.Y)) }, options); const svg = d3.select(DOM.svg(options.width, options.height)); const x = d3.scaleLinear().domain(options.xlim).range([options.margin.left, options.width - options.margin.right]); const y = d3.scaleLinear().domain(options.ylim).range([options.height - options.margin.bottom, options.margin.top]); const xAxis = d3.axisBottom(x); const yAxis = d3.axisLeft(y); if(options.legend == true) { svg.append("g") .attr("transform", `translate(${options.width - options.margin.left - 20}, ${options.margin.top})`) .html(Legend(colorScale, { title: "NBA Player", width: 75 }).outerHTML); } svg.append("text") .attr("x", options.width - options.margin.right + 10) .attr("y", options.height - options.margin.bottom + 15) .attr("alignment-baseline", "middle") .attr("font-style", "italic") .text(options.xlab); svg.append("g") .attr("transform", `translate(0, ${options.height - options.margin.bottom + 15})`) .call(xAxis); svg.append("text") .attr("x", options.margin.left - 15) .attr("y", options.margin.top - 10) .attr("text-anchor", "left") .attr("font-style", "italic") .text(options.ylab); svg.append("g") .attr("transform", `translate(${options.margin.left - 15}, 0)`) .call(yAxis); svg.selectAll("circle") .data(data) .enter() .append("circle") .attr("r", 3) .attr("cx", (d) => x(d.X)) .attr("cy", (d) => y(d.Y)) .attr("fill", (d) => colorScale(d.Z)); if(options.regression == true) { const fit = linearRegression(data.map(d => d.X), data.map(d => d.Y)); console.log(fit); svg.append("line") .attr("x1", x(options.xlim[0])) .attr("y1", y(fit.intercept + options.xlim[0] * fit.slope)) .attr("x2", x(options.xlim[1])) .attr("y2", y(fit.intercept + options.xlim[1] * fit.slope)) .attr("stroke", "black") .attr("stroke-dasharray", "4 4"); } return svg.node(); }33:Tc6b,collider_dag = (x, y, z, options) => { options = Object.assign({ height: 400, width: width, fontSize: "1.25em", condition: false, backdoorPath: false }, options); const svg = d3.select(DOM.svg(options.width, options.height)); svg.append("defs") .html(` `); const coordinates = [ { x: options.width / 5, y: options.height / 4 }, // Upper left { x: 4 * options.width / 5, y: options.height / 4 }, // Upper right { x: options.width / 2, y: 3 * options.height / 4 } // Bottom center ]; const labels = [x, y, z]; const lines = [ { x1: coordinates[0].x, y1: coordinates[0].y, x2: coordinates[2].x, y2: coordinates[2].y, shortenStart: 30, shortenEnd: 60, arrow: true, dashed: false }, { x1: coordinates[1].x, y1: coordinates[1].y, x2: coordinates[2].x, y2: coordinates[2].y, shortenStart: 30, shortenEnd: 60, arrow: true, dashed: false }, ]; if(options.backdoorPath == true) { lines.push({ x1: coordinates[0].x, y1: coordinates[0].y, x2: coordinates[1].x, y2: coordinates[1].y, arrow: false, dashed: true , shortenStart: 40, shortenEnd: 40, }); } const shorten = (line, start, end) => { const w = line.x2 - line.x1; const h = line.y2 - line.y1; const a = Math.atan(h / w); return { x1: line.x1 + (a >= 0 ? 1 : -1) * start * Math.cos(a), y1: line.y1 + (a >= 0 ? 1 : -1) * start * Math.sin(a), x2: line.x2 + (a >= 0 ? -1 : 1) * end * Math.cos(a), y2: line.y2 + (a >= 0 ? -1 : 1) * end * Math.sin(a) }; } svg.selectAll("text") .data(coordinates) .enter() .append("text") .attr("text-anchor", "middle") .attr("dominant-baseline", "middle") .attr("font-size", options.fontSize) .attr("font-style", "italic") .attr("x", (d) => d.x) .attr("y", (d) => d.y) .text((d, i) => labels[i]); svg.selectAll("line") .data(lines) .enter() .append("line") .attr("x1", (d) => shorten(d, d.shortenStart, d.shortenEnd).x1) .attr("y1", (d) => shorten(d, d.shortenStart, d.shortenEnd).y1) .attr("x2", (d) => shorten(d, d.shortenStart, d.shortenEnd).x2) .attr("y2", (d) => shorten(d, d.shortenStart, d.shortenEnd).y2) .attr("stroke", "rgba(0, 0, 0, 0.8)") .attr("stroke-width", 2) .attr("stroke-dasharray", d => d.dashed ? "4 4" : null) .attr("marker-end", d => d.arrow ? "url(#arrowhead)" : null); if(options.condition == true) { const conditionWidth = 120; const conditionHeight = 40; svg.append("rect") .attr("x", coordinates[2].x - conditionWidth / 2) .attr("y", coordinates[2].y - conditionHeight / 2) .attr("width", conditionWidth) .attr("height", conditionHeight) .attr("fill", "transparent") .attr("stroke", "black"); } return svg.node(); }34:[] 2e:{"title":"Yes Yes No: Collider Bias","cells":[{"id":0,"value":"# Yes Yes No: Collider Bias\nThe podcast Reply All used to have a segment called \"Yes Yes No\", where they would find a tweet that was completely incomprehensible unless you were well steeped in the minutiae of some specific online subculture, and then spend 20 minutes explaining all the context you needed to understand why the tweet was funny. \n\nI'm going to do something perhaps similar, but for an interesting tweet that I saw this morning that might not be immediately understandable without some background in causal inference:","pinned":false,"hidden":false,"mode":"md"},{"id":369,"value":"tweet(\"1461090535985324039\")","pinned":false,"hidden":false,"mode":"ojs"},{"id":378,"value":"I hadn't heard of this example before, so after some research I decided to put together this notebook to explain the context and why this tweet is interesting.\n\nThe key ideas here (as I see them) are [_Berkman's Paradox_](https://en.wikipedia.org/wiki/Berkson%27s_paradox) and _collider bias_. I'm going to explain what those are by walking through the basketball example, being careful to separate issues of _statistical association_ from _causation_.\n\n---\n\nSay you download the speed ratings and player heights for all the players in the [NBA](https://en.wikipedia.org/wiki/National_Basketball_Association). (The NBA is the professional basketball league in the United States. All you really need to know about the NBA is that you need to be really good at basketball to play in it.) You plot the data, and it looks like this:","pinned":false,"hidden":false,"mode":"md"},{"id":397,"value":"plot(data.filter((d) => d.Z == 1), { xlab: \"Height\", ylab: \"Speed Rating\", legend: false })","pinned":false,"hidden":false,"mode":"ojs"},{"id":642,"value":"viewof N_NBA = Inputs.range([50, 200], {value: 100, step: 25, label: \"Number of NBA Players\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":493,"value":"_(Note that this is simulated data. In fact, you can change the sample size by moving that slider!)_\n\nYou think to yourself, \"wow, it looks like there's a strong association here!\" So you run a linear regression next, and your observation is confirmed:","pinned":false,"hidden":false,"mode":"md"},{"id":499,"value":"plot(data.filter((d) => d.Z == 1), { xlab: \"Height\", ylab: \"Speed\", legend: false, regression: true })","pinned":false,"hidden":false,"mode":"ojs"},{"id":511,"value":"The ${tex`R^2`} of that linear regression is ${tex`${d3.format(\"0.2r\")(fit1.r2)}`}.\n\nAt this point, you could make the following statement, and you would be completely correct: _\"Among NBA players, we observe a negative association between height and speed rating.\"_ This is an accurate descriptive statement about the data you have.\n\nBut where you run into trouble is if you go farther and make a more general statement: _\"in general, taller people are slower.\"_\n\nLet's see why such a statement might be wrong.\n\nSay we could get data on _all basketball players_, not just the professional ones in the NBA. We plot their speed rating versus their height, like before. In the plot below the professional players are colored red:","pinned":false,"hidden":false,"mode":"md"},{"id":548,"value":"plot(data, { xlab: \"Height (m)\", ylab: \"Speed Rating\", xlim: [1.5, 2.2], ylim: [20, 90], legend: true, regression: true })","pinned":false,"hidden":false,"mode":"ojs"},{"id":553,"value":"$30","pinned":false,"hidden":false,"mode":"md"},{"id":627,"value":"\n---\n## Explanation using DAGs\n\nLet's start by positing a [causal graph](https://en.wikipedia.org/wiki/Causal_graph) that encodes the causal relationships between variables (these are often just called \"DAGs\", for \"Directed Acyclic Graphs\", in the causal inference world). \n\nAssume there isn't actually any causal relationship between height and speed, but that both of them cause whether you become an NBA player. \n\nWe can encode this assumption about the causal relationships with the following graph:","pinned":false,"hidden":false,"mode":"md"},{"id":372,"value":"collider_dag(\"Height\", \"Speed\", \"NBA Player\", { width: width < 600 ? width : width / 3, height: 300 })","pinned":false,"hidden":false,"mode":"ojs"},{"id":280,"value":"In this graph, the _NBA Player_ variable is called a **collider** because it has two arrows pointing to it (_Height_ and _Speed_), but those two variables don't have an arrow between them.\n\nNow, what happens if we condition on _NBA Player_? (Subsetting the data to look at only NBA players, like we did above, is one way to condition on _NBA Player_.)\n\nAccording to the causal analysis theory, if we condition on the _NBA Player_ variable then a _backdoor path_ is opened between _Height_ and _Speed_.\n\nIn a DAG, we show that a variable has been conditioned on by drawing a box around it, and we indicate a backdoor path with a dotted line. ","pinned":false,"hidden":false,"mode":"md"},{"id":320,"value":"collider_dag(\"Height\", \"Speed\", \"NBA Player\", { width: width < 600 ? width : width / 3, height: 300, condition: true, backdoorPath: true })","pinned":false,"hidden":false,"mode":"ojs"},{"id":636,"value":"$31","pinned":false,"hidden":false,"mode":"md"},{"id":738,"value":"---\n## More Examples\n\n- I based the simulated data exposition on these two tweets by Darren Dahly, which explain the issue quite succinctly:\n${tweet(\"1117512192801296384\")}\n\n- [Griffith et al. 2020](https://www.nature.com/articles/s41467-020-19478-2) describes in detail various examples of collider bias that can arise in analyzing Covid-19 disease outcomes.\n\n- [Cole et al. 2009](https://academic.oup.com/ije/article/39/2/417/680407) give two more hypothetical examples of collider bias.\n---","pinned":false,"hidden":false,"mode":"md"},{"id":209,"value":"$32","pinned":false,"hidden":false,"mode":"ojs"},{"id":194,"value":"colorScale = d3.scaleOrdinal([1, 0], [\"#d35400\", \"#7f8c8d\"])","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"$33","pinned":false,"hidden":false,"mode":"ojs"},{"id":104,"value":"simulate_data = (N) => {\n const data = [];\n let num_nba = 0;\n while(num_nba < N_NBA) {\n const X = rtruncnorm(1.85, 0.1, 1.5, 2.2);\n const Y = rtruncnorm(55, 10, 20, 90);\n const Z = ((Y >= -75 * X + 210) ? 1 : 0);\n\n num_nba += Z;\n \n data.push({X, Y, Z});\n }\n return data;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":273,"value":"rtruncnorm = (mean, sd, a, b) => {\n if(a > b) return 0;\n let x = 0;\n let iter = 0;\n do {\n x = rnorm(mean, sd);\n iter++;\n } while(iter < 50 && (x < a || x > b));\n return x;\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":109,"value":"data = {\n return simulate_data(N_NBA)\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":515,"value":"fit1 = linearRegression(data.filter(d => d.Z == 1).map(d => d.X), data.filter(d => d.Z == 1).map(d => d.Y))","pinned":false,"hidden":false,"mode":"ojs"},{"id":502,"value":"// Adapted from https://stackoverflow.com/questions/6195335/linear-regression-in-javascript\nlinearRegression = (x,y) => {\n const n = y.length;\n let sum_x = 0;\n let sum_y = 0;\n let sum_xy = 0;\n let sum_xx = 0;\n let sum_yy = 0;\n\n for (var i = 0; i < y.length; i++) {\n sum_x += x[i];\n sum_y += y[i];\n sum_xy += (x[i]*y[i]);\n sum_xx += (x[i]*x[i]);\n sum_yy += (y[i]*y[i]);\n } \n const slope = (n * sum_xy - sum_x * sum_y) / (n*sum_xx - sum_x * sum_x);\n const intercept = (sum_y - slope * sum_x)/n;\n const r2 = Math.pow((n*sum_xy - sum_x*sum_y)/Math.sqrt((n*sum_xx-sum_x*sum_x)*(n*sum_yy-sum_y*sum_y)),2);\n\n return { slope, intercept, r2 };\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":102,"value":"import {rnorm} from \"@herbps10/statistical-functions\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":202,"value":"import {Legend, Swatches} from \"@d3/color-legend\"","pinned":false,"hidden":false,"mode":"ojs"},{"id":366,"value":"import {tweet} from \"@mbostock/tweet\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W34","stdlib":"1","titling":"auto"}