2b:null 2c:["$","title",null,{"children":"Samplingfördelningen - linear regression | Mattias Villani | Observable"}] 31:T5a9,### Samplingfördelningen och varför den är viktig. Skattningarna ${tex`b_0`} och ${tex`b_1`} kommer dock **variera från stickprov till stickprov**. Den vanliga situationen är att vi bara har *ett* stickprov som ska analyseras, men det är ändå intressant att veta hur ${tex`b_0`} och ${tex`b_1`} *skulle kunna ha blivit* om vi fått ett annat stickprov. Samplingfördelningen är viktig när vill göra **inferens** för populationsparametrarna ${tex`\beta_0`} och ${tex`\beta_1`}. Antag att vi har ett fått ett datamaterial med en förklarande variabel ${tex`x`} som vi tror kan ha en påverkan på en annan variabel ${tex`y`}. Från datamaterialet skattar vi lutningen till ${tex`b_1=1.238`}. Har variabeln ${tex`x`} en effekt på responsvariabeln ${tex`y`}? Skattningen ${tex`b_1=1.238`} är uppenbarligen inte 0, men det kanske bara är slumpen som spelar oss ett spratt: i populationen kanske ${tex`\beta_1`} faktiskt är noll, men pga osäkerheten i vårt begränsade stickprov får vi en skattning som *verkar* skilja sig från noll. Skattningen ${tex`b_1=1.238`} måste vara **tillräckligt** långt från 0 för att vi ska tro att det verkligen finns en påverkan på ${tex`y`}, dvs att ${tex`\beta\neq 0`}. För att bedöma vad som menas med tillräckligt måste vi veta samplingfördelningens standardavvikelse, det s k **standardfelet**, dvs hur mycket ${tex`b_1`} kan variera från stickprov till stickprov enbart pga slumpen.32:[] 2f:{"id":"4ab3c3f7ade79d7c","slug":"samplingfordelningen-linear-regression","title":"Samplingfördelningen - linear regression","thumbnail":"b6098082b54890a9300cc7d521f8cb5b3ddedd8efa1368f720ae6cef3cc5404c","access_level":"public","create_time":"$D2023-01-01T22:17:35.709Z","update_time":"$D2023-11-04T10:45:17.840Z","view_time":null,"first_public_time":"$D2023-01-01T22:17:35.709Z","fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":"pro_2024"},"version":1659,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null}],"members":[{"type":"individual","id":"56499be8cf43f0ef","login":"mattiasvillani","name":"Mattias Villani","avatar_url":"https://avatars-next.observableusercontent.com/a440aaf39a3abefb86cbcc758fd7a5ab42555ae68d6825b52ec006f385a94d1d","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"subscribed":null,"forks":1,"collection":{"id":"209e13808f44ed11","slug":"sda1","title":"SDA1","owner":{"id":"7592a7ab121f74a0","login":"mattiasvillani","name":"Mattias Villani"}},"type":"notebook","body":{"title":"Samplingfördelningen - linear regression","cells":[{"id":0,"value":"# Samplingfördelningen - linear regression","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":623,"value":"### Modell\n\nDen enkla linära regressionsmodellen är\n\n> ${tex`y = \\beta_0 + \\beta_1 x + \\varepsilon\\,\\,\\,`} där ${tex`\\,\\,\\,\\varepsilon \\sim \\operatorname{N}(0,\\sigma_\\varepsilon^2)`}. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":712,"value":"Modellens parametrar kan skattas med minsta-kvadrat-skattningarna\n\n> ${tex`b_1 = \\hat \\beta_1 = \\frac{s_{xy}}{s_x^2}`} där ${tex`s_x^2 = \\frac{\\sum (x_i-\\bar x)^2}{n-1}`} och ${tex`s_{xy} = \\frac{\\sum (x_i-\\bar x)(y_i-\\bar y)}{n-1}`}\n\noch\n\n> ${tex`b_0 = \\hat \\beta_0 = \\bar y -b_1 \\bar x`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":708,"value":"Variansen för feltermen ${tex`\\sigma^2_\\varepsilon`} kan skattas med **residualvariansen**\n\n> ${tex`s_e^2 = \\sum \\frac{e_i^2}{n-2}`}\ndär ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":724,"value":"> ${tex`e_i = y_i - \\hat y_i`}\n\när **residualen** för den *i*te observationen och \n\n> ${tex`\\hat y_i = b_0 + b_1 x_i`} \n\när **prediktionen** för ${tex`y_i`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":637,"value":"$31","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":661,"value":"### Samplingfördelningen - matematiska formler\n\nI en del enkla modeller, som t ex linjär regression med normalfördelade feltermer ${tex`\\varepsilon`}, kan vi räkna ut matematiskt vilken fördelning ${tex`b_0`} och ${tex`b_1`} har. För lutningen gäller t ex","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":665,"value":">

\nOm ${tex`b_1 \\sim \\operatorname{N}(\\beta_1,s_{b_1}^2)\\,\\,\\,`} där ${tex`\\,\\,\\,s_{b_1}^2=\\frac{s_e^2}{(n-1)s_x^2}`}.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":681,"value":"Variansen ${tex`s_{b_1}^2`} är **samplingvariansen** och ${tex`s_{b_1} = \\sqrt{s_{b_1}^2}`} är standardfelet för ${tex`b_1`}. Formeln för samplingvariansen ${tex`s_{b_1}^2`} visar att: \n- samplingvariansen minskar när residualvariansen ${tex`s^2_{e}`} minskar - ju mindre variation kring regressionslinjen desto lättare 'r det att bestämma ${tex`b_1`}.\n- samplingvariansen minskar när variationen i den förklarande variabeln, ${tex`s^2_{x}`} , ökar - ju mer spridning i *x*-variabeln desto mer \"stabilt\" kan vi bestämma regressionlinjens lutning.\n- sampling variansen minskar när stickprovsstorleken *n* ökar.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":743,"value":"### Samplingfördelning - simulering\n\nEtt sätt att bättre förstå begreppet samplingfördelning är att simulera datamaterial från populationsmodellen. Vi kan då simulera många datamaterial och i varje datamaterial beräkna skattningarna ${tex`b_0`} och ${tex`b_1`}. Vi får då en regressionslinje för varje simulerat datamaterial. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":748,"value":"För att utföra simuleringen måste du först bestämma populationsmodellens parametrar ${tex`\\beta_0`}, ${tex`\\beta_1`} och ${tex`\\sigma_{\\varepsilon}`} och även storleken på varje stickprov från populationen, *n*. Klicka sen på Play-knappen observera hur regressionslinjen i grafen ned förändras för varje nytt stickprov. Du kan dra i slidern brevid Play-knappen om du vill dra ny stickprov i snabbare takt.\n\nI histogrammen nedan kan du se hur samplingfördelningen för ${tex`b_0`} och ${tex`b_1`} byggs upp från de beräknade skattningar från varje stickprov. Grafen visar också den ovan angivna teoretiska samplingfördelningarna som mörkblå täthetsfunktioner. Om du simulerar tillräckligt många stickprov från populationen så kommer histogrammen bli alltmer lika de teoretiska fördelningarna. ","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":33,"value":"#### Populationsmodell som vi simulerar data från","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":67,"value":"viewof n = Inputs.range([20, 1000], {value: 20, step: 1, label: tex`\\text{stickprovsstorlek, }n`});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":27,"value":"viewof beta0 = Inputs.range([-2, 2], {value: -1, step: 0.1, label: tex`\\text{intercept, }\\beta_0`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":29,"value":"viewof beta1 = Inputs.range([-2, 2], {value: 1, step: 0.1, label: tex`\\text{lutning, }\\beta_1`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":70,"value":"viewof sigmaepsilon = Inputs.range([0, 1], {value: 0.5, step: 0.1, label: tex`\\text{sd felterm, }\\sigma_\\varepsilon`})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1159,"value":"viewof sampleno = Scrubber(d3.ticks(1, 1000, 1000), {\n autoplay: false,\n loop: false,\n delay: 1000,\n initial: 199,\n format: x => `dataset nummer: ${x.toFixed(0)}`\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":241,"value":"estimates = tex`\\text{Skattat intercept: } b_0 = ${data_estimates[1][sampleno-1].intercept.toPrecision(4)} \\\\ \\text{Skattad lutning: } b_1 =${data_estimates[1][sampleno-1].lutning.toPrecision(4)}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":80,"value":"plt = Plot.plot({\n caption: html`Populationsmodell
Skattad regressionslinje`,\n color: {legend: true},\n x: {domain: [-1, 1]},\n //y: {domain: [-3,3]},\n y: {domain: [beta0 + (-1)*beta1 - 3*sigmaepsilon, beta0 + 1*beta1 + 3*sigmaepsilon]},\n marks: [\n Plot.ruleY([beta0 + (-1)*beta1 - 3*sigmaepsilon]),\n Plot.ruleX([-1]),\n Plot.line(\n [{x: -1, y: beta0 + (-1)*beta1}, {x: 1, y: beta0 + (1)*beta1}], \n {x: \"x\", y: \"y\", stroke: \"orange\", strokeWidth: 2}\n ),\n Plot.dot(data_estimates[0][sampleno-1], {x: \"x\", y: \"y\", strokeOpacity: 0.5, r: 2, fill: colors[0]}),\n Plot.line(\n [{x: -1, y: data_estimates[1][sampleno-1].intercept + (-1)*data_estimates[1][sampleno-1].lutning}, \n {x: 1, y: data_estimates[1][sampleno-1].intercept + (1)*data_estimates[1][sampleno-1].lutning}\n ], \n {x: \"x\", y: \"y\", stroke: \"steelblue\", strokeWidth: 2}\n ),\n ],\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1517,"value":"samplingplots = html`\n \n \n \n

\n ${histintercept}\n

\n ${histlutning}\n

\n \n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":837,"value":"
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n---\n## 🙈 Appendix 🙈\n\nHär följer bara kod och detaljer som du antagligen inte vill läsa.","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":848,"value":"#### Imports","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":185,"value":"import {lm} from \"@chrispahm/linear-models-in-observable-notebooks\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":198,"value":"import {summary} from \"@chrispahm/linear-models-in-observable-notebooks\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":23,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1287,"value":"math = require('mathjs')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1351,"value":"import {Histogram} from \"@d3/histogram\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1169,"value":"import {Scrubber} from \"@mbostock/scrubber\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":854,"value":"#### Functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1084,"value":"function simulate_and_estimate(n, beta0, beta1, sigmaepsilon, nrep){\n const x = d3.range(-1, 1, 2/n);\n const xbar = d3.mean(x)\n const SSx = d3.sum(math.dotPow(math.subtract(x, xbar), 2))\n var data = [];\n var coef = [];\n for (let j = 1; j <= nrep; j++){\n var dataset = x.map(x => ({sample_id: j, x: x, y: beta0 + beta1*x + jstat.normal.sample(0,sigmaepsilon)}))\n data.push(dataset);\n let y = dataset.map(d => d.y)\n let ybar = d3.mean(y)\n let SSxy = d3.sum(math.dotMultiply(math.subtract(x, xbar), math.subtract(y, ybar)) )\n let b1 = SSxy/SSx\n let b0 = ybar - b1*xbar\n //let linear_model = lm('y ~ x', dataset);\n coef.push(\n {\n sample_id: j, \n //intercept: linear_model.coefficients[0],\n //lutning: linear_model.coefficients[1],\n intercept: b0,\n lutning: b1\n }\n )\n }\n return [data, coef]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":850,"value":"#### Data","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1126,"value":"data_estimates = simulate_and_estimate(n, beta0, beta1, sigmaepsilon, 1000)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1472,"value":"b0samp_range = [d3.min(data_estimates[1].map( d => d.intercept)),d3.max(data_estimates[1].map( d => d.intercept))]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1474,"value":"b1samp_range = [d3.min(data_estimates[1].map( d => d.lutning)),d3.max(data_estimates[1].map( d => d.lutning))]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1284,"value":"sampdist_theo = {\n const x = d3.range(-1, 1, 2/n);\n const xbar = d3.mean(x)\n const sumsquares = d3.sum(math.dotPow(math.subtract(x, xbar), 2))\n const stdb0 = sigmaepsilon*math.sqrt((1/n) + ((xbar**2)/sumsquares))\n const stdb1 = sigmaepsilon/math.sqrt(sumsquares)\n const x_b0 = d3.range(beta0 - 4*stdb0, beta0 + 4*stdb0, 1/500);\n const x_b1 = d3.range(beta1 - 4*stdb1, beta1 + 4*stdb1, 1/500);\n var pdfb0 = [];\n var pdfb1 = [];\n for (let j = 1; j <= x_b0.length; j++){\n pdfb0.push({\n x: x_b0[j-1], \n pdf: jstat.normal.pdf(x_b0[j-1], beta0, stdb0)\n })\n }\n for (let j = 1; j <= x_b1.length; j++){\n pdfb1.push({\n x: x_b1[j-1], \n pdf: jstat.normal.pdf(x_b1[j-1], beta1, stdb1),\n })\n }\n return [pdfb0, pdfb1]\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1559,"value":"#### Plots","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1376,"value":"histintercept = Plot.plot({\n caption:html`\n teoretisk samplingfördelning för b₀
\n histogram på skattningar från simulerade dataset
\n🟠 populationens β₀
\n🔵 skattning b₀ för nuvarande datamaterial.\n `,\n y: {axis: false},\n x: {domain: [beta0 -0.5, beta0 + 0.5]},\n marks: [\n Plot.rectY(data_estimates[1].filter((d,i) => i<= (sampleno-1)), Plot.binX({y: d => (d.length/sampleno)/(20/sampleno)}, {x: \"intercept\", thresholds: d3.range(b0samp_range[0], b0samp_range[1], 20/sampleno), fill: \"steelblue\", opacity: 0.5}) ),\n Plot.ruleY([0]),\n Plot.dot([{beta0: beta0}], {x: \"beta0\", y: 0, r: 5, fill: \"orange\"}),\n Plot.dot([data_estimates[1][sampleno-1]], {x: \"intercept\", y: 0, r: 5, fill: \"steelblue\"}),\n Plot.lineY(sampdist_theo[0], {x: \"x\", y: \"pdf\", stroke: \"midnightblue\", strokeWidth: 2.5})\n ],\n}) ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1262,"value":"histlutning = Plot.plot({\n caption:html` \n teoretisk samplingfördelning för b₁
\n histogram på skattningar från simulerade dataset
\n🟠 populationens β₁
\n🔵 skattning b₁ för nuvarande datamaterial.\n `,\n y: {axis: false},\n x: {domain: [beta1 -0.5, beta1 + 0.5]},\n marks: [\n Plot.rectY(data_estimates[1].filter((d,i) => i<= (sampleno-1)), Plot.binX({y: d => (d.length/sampleno)/(20/sampleno)}, {x: \"lutning\", thresholds: d3.range(b1samp_range[0], b1samp_range[1], 20/sampleno), fill: \"steelblue\", opacity: 0.5}) ),\n Plot.ruleY([0]),\n Plot.dot([{beta1: beta1}], {x: \"beta1\", y: 0, r: 5, fill: \"orange\"}),\n Plot.dot([data_estimates[1][sampleno-1]], {x: \"lutning\", y: 0, r: 5, fill: \"steelblue\"}),\n Plot.lineY(sampdist_theo[1], {x: \"x\", y: \"pdf\", stroke: \"midnightblue\", strokeWidth: 2.5})\n ],\n}) ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":843,"value":"#### Style","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":667,"value":"viewof md_colour = Inputs.color({label: \"Highlight color cells\", value: \"#F5F5DC\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":361,"value":"colors = d3.schemePaired","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W32","stdlib":"1","titling":"auto"}} 30:"$2f:body"