2f:[] 2d:{"title":"Median Splits of Two Predictor Variables: BIAS!","cells":[{"id":0,"value":"# Median Splits of Two Predictor Variables: BIAS!","pinned":false,"hidden":false,"mode":"md"},{"id":512,"value":"This notebooks explores the bias in the estimate of regression coefficients caused by using dummy or contrast codes for median splits. Use the sliders below to change the parameters for generating a random sample of 10,000 observations. You may change rho, the correlation between the two predictors X and Y, and the coefficients for X and Y. Below the sliders are the resulting regression models. And below the numerical representations of the models are bar graphs comparing the the estimated coefficients and their squared multiple correlations. The four models are \"Generating\"--the specified theoretical model, \"Original\"--the multiple regression using the continuous predictor variables, \"Split Correct\"--the multiple regression using the quadrant means as predictors for a median split, and \"Split Dummy\"--the multiple regression using equal predictors (dummy codes or contrast codes) for a median split. ","pinned":false,"hidden":false,"mode":"md"},{"id":298,"value":"viewof rho = Inputs.range([-0.99, 0.99], {value: 0, step: 0.01, label: \"Rho\"})\n","pinned":false,"hidden":false,"mode":"ojs"},{"id":408,"value":"viewof b1 = Inputs.range([-1,1], {value: 0.9, label: \"Coefficient for X\", step: .01})","pinned":false,"hidden":false,"mode":"ojs"},{"id":410,"value":"viewof b2 = Inputs.range([-1,1], {value: 0.1, label: \"Coefficient for Y\", step: .01})","pinned":false,"hidden":false,"mode":"ojs"},{"id":394,"value":"### Model for Generating the Data\n#### Z = 3 + ${b1} X + ${b2} Y + N(0,1) where N(0,1) is random standard normal deviate","pinned":false,"hidden":false,"mode":"md"},{"id":387,"value":"### Regression using original continuous predictors\n#### Z = ${mrCont.coef[0].toFixed(2)} + ${mrCont.coef[1].toFixed(2)} X + ${mrCont.coef[2].toFixed(2)} Y \n#### ${tex`R^2 = `}${mrCont.R2.toFixed(2)}","pinned":false,"hidden":false,"mode":"md"},{"id":369,"value":"### Regression using cell means as codes (\"proper\" median split)\n#### Z = ${mr.coef[0].toFixed(2)} + ${mr.coef[1].toFixed(2)} X + ${mr.coef[2].toFixed(2)} Y \n#### ${tex`R^2 = `}${mr.R2.toFixed(2)}","pinned":false,"hidden":false,"mode":"md"},{"id":372,"value":"### Regression using equal cell codes (traditional median splits)\n#### Z = ${mrDummy.coef[0].toFixed(2)} + ${mrDummy.coef[1].toFixed(2)} X + ${mrDummy.coef[2].toFixed(2)} Y\n#### ${tex`R^2 = `}${mrDummy.R2.toFixed(2)}","pinned":false,"hidden":false,"mode":"md"},{"id":433,"value":"Plot.plot({\n marginLeft: 150,\n marginBottom: 50,\n style: {fontSize: 16, padding: \"5px\"},\n marks: [\n Plot.barX(c1, {x: \"coefficient\", y: \"Model\", fill: \"Model\"}),\n Plot.ruleY([0])\n ],\n caption: \"Estimate of Coefficient for X\",\n color: {type: 'categorical', scheme: 'tableau10'},\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":437,"value":"Plot.plot({\n marginLeft: 150,\n marginBottom: 50,\n style: {fontSize: 16, padding: \"5px\"},\n marks: [\n Plot.barX(c2, {x: 'coefficient', y: 'Model', fill: \"Model\"}),\n Plot.ruleY([0])\n ],\n caption: \"Estimate of Coefficient for Y\",\n color: {type: 'categorical', scheme: 'tableau10'},\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":490,"value":"Plot.plot({\n marginLeft: 150,\n marginBottom: 50,\n style: {fontSize: 16, padding: \"5px\"},\n marks: [\n Plot.barX(rsq, {x: 'rsq', y: 'model', fill: 'model'}),\n Plot.ruleY([0])\n ],\n caption: \"RSq\",\n color: {type: 'categorical' , scheme: 'tableau10'}\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":389,"value":"---\n## Support Functions and Data","pinned":false,"hidden":false,"mode":"md"},{"id":7,"value":"n = 10000","pinned":false,"hidden":false,"mode":"ojs"},{"id":11,"value":"nu = Math.sqrt(1-rho*rho)","pinned":false,"hidden":false,"mode":"ojs"},{"id":5,"value":"temp = {let x=[]; for(let i=0; i<=n-1; i++) x.push(jStat.normal.sample(0,1)); // generate bivariate normal sample of size n\n let d = []; for(let i=0; i<=n-1; i++) d.push({x: x[i], y: rho*x[i] + nu*jStat.normal.sample(0,1)}); return d}","pinned":false,"hidden":false,"mode":"ojs"},{"id":58,"value":"xMed = d3.median(temp, d => d.x)","pinned":false,"hidden":false,"mode":"ojs"},{"id":61,"value":"yMed = d3.median(temp, d=> d.y)","pinned":false,"hidden":false,"mode":"ojs"},{"id":56,"value":"data = {let d = []; \n for(let i=0; i< temp.length; i++) { // use 0 = LL, 1 = LH, 2 = HL, 3 = HH\n let grp = 0; let x = temp[i].x; let y = temp[i].y;\n if(x <= xMed && y > yMed) grp = 1; else if(x > xMed && y <= yMed) grp = 2; else if(x > xMed && y > yMed) grp=3;\n d.push({x: temp[i].x, y: temp[i].y, grp: grp})\n }\n return d;}\n ","pinned":false,"hidden":false,"mode":"ojs"},{"id":104,"value":"wtData = {let d = [];\n for(let i=0; i d.grp == i), d => d.x),\n xLoc: 3*dummyMeans[i].xMean, yLoc: 3*dummyMeans[i].yMean});\n return m\n }","pinned":false,"hidden":false,"mode":"ojs"},{"id":305,"value":"means = {let m = []; //compute the means for the four quadrants defined by median splits\n for(let i=0; i <= 3; i++)\n m.push({xMean: d3.mean(data.filter(d => d.grp == i), d => d.x),\n yMean: d3.mean(data.filter(d => d.grp == i), d=> d.y), grp: i});\n return m\n }","pinned":false,"hidden":false,"mode":"ojs"},{"id":163,"value":"x = {let x = []; // extract x as a vector for use in stat functions. surely there is a better way but this works\n for(let i =0; i < wtData.length; i++) x.push(wtData[i].x); return x;}","pinned":false,"hidden":false,"mode":"ojs"},{"id":165,"value":"y = {let y = []; for(let i=0; // extract y as a vector for use in stat functions. surely there is a better way but this works\n i < wtData.length; i++) y.push(wtData[i].y); return y;}","pinned":false,"hidden":false,"mode":"ojs"},{"id":275,"value":"xd = {let x = []; // extract x as a vector for use in stat functions. surely there is a better way but this works\n for(let i =0; i < dummyData.length; i++) x.push(dummyData[i].x); return x;}","pinned":false,"hidden":false,"mode":"ojs"},{"id":273,"value":"yd = {let y = []; for(let i=0; // extract y as a vector for use in stat functions. surely there is a better way but this works\n i < dummyData.length; i++) y.push(dummyData[i].y); return y;}","pinned":false,"hidden":false,"mode":"ojs"},{"id":17,"value":"jStat = require('jStat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":332,"value":"v = 1 // variance of the model error term","pinned":false,"hidden":false,"mode":"ojs"},{"id":321,"value":"z = {let z=[]; for(let i=0; i< data.length; i++)\n z.push(3 + b1*data[i].x + b2*data[i].y + v*jStat.normal.sample(0,1));\n return z;\n }","pinned":false,"hidden":false,"mode":"ojs"},{"id":319,"value":"mod = {let m = []; for(let i=0; i < wtData.length; i++)\n m.push([1, wtData[i].x, wtData[i].y])\n return m;\n }","pinned":false,"hidden":false,"mode":"ojs"},{"id":336,"value":"modDummy = {let m = []; for(let i=0; i < wtData.length; i++)\n m.push([1, dummyData[i].x, dummyData[i].y])\n return m;\n }","pinned":false,"hidden":false,"mode":"ojs"},{"id":381,"value":"modCont = {let m = []; for(let i=0; i < data.length; i++)\n m.push([1, data[i].x, data[i].y])\n return m;\n }","pinned":false,"hidden":false,"mode":"ojs"},{"id":324,"value":"mr = jStat.models.ols(z,mod)","pinned":false,"hidden":false,"mode":"ojs"},{"id":338,"value":"mrDummy = jStat.models.ols(z,modDummy)","pinned":false,"hidden":false,"mode":"ojs"},{"id":383,"value":"mrCont = jStat.models.ols(z,modCont)","pinned":false,"hidden":false,"mode":"ojs"},{"id":428,"value":"c1 = [{coefficient: b1, Model: \"Generating\"}, {coefficient: mrCont.coef[1], Model: \"Original\"}, {coefficient: mr.coef[1], Model: \"Split Correct\"},{coefficient: mrDummy.coef[1], Model: \"Split Dummy\"}]","pinned":false,"hidden":false,"mode":"ojs"},{"id":431,"value":"c2 = [{coefficient: b2, Model: \"Generating\"}, {coefficient: mrCont.coef[2], Model: \"Original\"}, {coefficient: mr.coef[2], Model: \"Split Correct\"},{coefficient: mrDummy.coef[2], Model: \"Split Dummy\"}]","pinned":false,"hidden":false,"mode":"ojs"},{"id":469,"value":"rsq = [{rsq: mrCont.R2, model: \"Original\"}, {rsq: mr.R2, model: \"Split Correct\"}, {rsq: mrDummy.R2, model: \"Split Dummy\"}]","pinned":false,"hidden":false,"mode":"ojs"},{"id":364,"value":"html``","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W2f","stdlib":"1","titling":"auto"}