2f:T424,The number above the distribution is the ratio of the variance between the means of the low and high groups after a median split relative to the original variance for the continuous variable. This is the expected proportion of the squared correlation remaining between a predictor X split at its median and an outcome variable Y for a linear relationship. For a uniform distribution (${tex`\alpha = \beta = 1`}), splitting reduces the variance and squared correlation to 75% of what it would have been using the continuous predictor variable. For beta distributions with the mode not at the extremes (i.e., both ${tex`\alpha > 1`} and ${tex`\beta > 1`}), this reduction is always worse than 70%, even for heavily-skewed distributions. Even for strongly bimodal distributions (e.g., ${tex`\alpha = \beta = 0.5`}, there is still an appreciable reduction to 81% of the continuous value. Except for an extreme experimental design with half the observations at 0 and at 1, using a median split always reduces the expected squared correlation between X and Y. 30:[] 2d:{"title":"Median Splits of Different Distribution Shapes","cells":[{"id":0,"value":"# Median Splits of Different Distribution Shapes\n","pinned":false,"hidden":false,"mode":"md"},{"id":100,"value":"Some researchers claim that median splits might be appropriate when predictor distributions are non-normal, especially when they are skewed. The beta distribution with its two shape parameters provides an appropriate method for investigating this claim. The probability density function for the beta distribution is defined as:\n","pinned":false,"hidden":false,"mode":"md"},{"id":105,"value":"f(X) = {{\\Gamma(\\alpha + \\beta)} \\over {\\Gamma(\\alpha) \\Gamma(\\beta)}} {X^{\\alpha-1} (1-X)^{\\beta-1}}, 0 < X < 1","pinned":false,"hidden":false,"mode":"tex"},{"id":109,"value":"The distribution of ${tex`X`} is symmetric when ${tex`\\alpha = \\beta`} and is skewed otherwise. When ${tex`\\alpha = \\beta = 1`}, the beta distribution is equivalent to the uniform distribution for which every value between 0 and 1 is equally likely. When both ${tex`\\alpha > 1`} and ${tex`\\beta > 1`}, the beta distribution has an inverted U-shape with a single mode. When both ${tex`\\alpha < 1`} and ${tex`\\beta < 1`}, the beta distribution has a U-shape with possibly two modes, one at each extreme. Use the sliders below to change ${tex`\\alpha`} and ${tex`\\beta`} and observe the different distribution shapes. **Note:**Extreme values of alpha and beta sometimes stretch the accuracy limits of the simple integration routines used in this applet.","pinned":false,"hidden":false,"mode":"md"},{"id":9,"value":"viewof alpha = Inputs.range([0.5,7.5],{value: 5, step: 0.1, label: '\t\\u03B1'})","pinned":false,"hidden":false,"mode":"ojs"},{"id":13,"value":"viewof beta = Inputs.range([0.5,7.5],{value: 5, step:0.1, label: \"\t\\u03B2\"})","pinned":false,"hidden":false,"mode":"ojs"},{"id":33,"value":"Plot.plot({\n marks: [\n Plot.areaY(pdf, {x: \"x\", y: \"y\", fill: colors.lightBlue}),\n //Plot.lineY(pdf, {x: \"x\", y: \"y\", stroke: colors.orange, strokeWidth: 4}),\n Plot.text(titleLoc, {\n x: \"x\",\n y: \"y\",\n text: [ratio.toFixed(2)],\n dy: -20,\n fill: colors.red,\n fontSize: 24\n })\n ],\n marginTop: 50\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":213,"value":"$2f","pinned":false,"hidden":false,"mode":"md"},{"id":223,"value":"Next","pinned":false,"hidden":false,"mode":"html"},{"id":218,"value":"## Support Functions and Definitions","pinned":false,"hidden":false,"mode":"md"},{"id":17,"value":"pdf = generatePDF()","pinned":false,"hidden":false,"mode":"ojs"},{"id":19,"value":"function generatePDF() {\n let pdf=[];\n //avoid 0 but begin with an extreme value\n pdf.push({x: 0.005, y: jStat.beta.pdf(0.005, alpha,beta)})\n for(let i=0.01; i < 1; i=i+.01)\n pdf.push({x: i, y: jStat.beta.pdf(i, alpha, beta)})\n // avoid 1 but end with an extreme value\n pdf.push({x: .995, y: jStat.beta.pdf(.995,alpha,beta)})\n return pdf\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":148,"value":"titleLoc = {let tl = []; tl.push({x: 0.5, y: d3.max(pdf, d=> d.y)}); return tl}","pinned":false,"hidden":false,"mode":"ojs"},{"id":75,"value":"step = .0001","pinned":false,"hidden":false,"mode":"ojs"},{"id":45,"value":"lowMean = { // old-fashioned integration with approximting rectangles\n let area = 0;\n for(let x=0.0; x<= median-step; x = x+step*2)\n area = area + (2*step) * jStat.beta.pdf(x+step,alpha, beta) * (x + step)*2\n return area\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":56,"value":"highMean = {\n let area = 0;\n for(let x = median; x < 1; x = x + step*2)\n area = area + (step*2) * jStat.beta.pdf(x+step,alpha, beta) * (x + step)*2\n return area\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":52,"value":"// ratio of variance after split to variance of continuous \n ratio=((Math.pow(lowMean - jStat.beta.mean(alpha,beta),2) + Math.pow(highMean - jStat.beta.mean(alpha,beta),2))/2)/jStat.beta.variance(alpha,beta)","pinned":false,"hidden":false,"mode":"ojs"},{"id":230,"value":"","pinned":false,"hidden":false,"mode":"ojs"},{"id":41,"value":"median = jStat.beta.median(alpha,beta)","pinned":false,"hidden":false,"mode":"ojs"},{"id":82,"value":"colors = ({blue: '#4e79a7', orange: '#f28e2d', red: '#e15759', green: '#59a14f', yellow: '#edc94a', lightBlue: '#6facef'})","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"jStat = require('jStat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":221,"value":"","pinned":false,"hidden":false,"mode":"html"}],"files":"$W30","stdlib":"1","titling":"auto"}