3e:null 3f:["$","title",null,{"children":"Bayes factors for comparing variances | Lincoln Colling, PhD | Observable"}] 45:T671,texmd` Again, note that because the likelihood is in terms of variance rather than precision, $\mathcal{H_-}$ above is the same as the $\mathcal{H_+}$ reported by Dablander et al. If we look at the results produced by ~Bayesplay~, we can see that the undirected Bayes factor is the sample as that produced by ~bfvartest~ and the value reported in the paper: $\mathrm{BF}_{10}=${example52_results .get("BF10") .toFixed(7)}$. However, if we look at the directed Bayes factor, the result produced by ~Bayesplay~ is different. For the directed Bayes factor, ~Bayesplay~ gives the result: $\mathrm{BF}_{-0}=${example52_results .get("BF-0") .toFixed(7)}$. Which result is correct? First, we can look at the two directed hypotheses and whether they're consistent with the undirected hypothesis. We should expect that $\mathrm{BF_{10}}=\frac{BF_{+0}+BF_{-0}}{2}$. Checking the numbers produced by ~bfvartest~ we get: $$ \frac{${bfvartest_results.u05_h1plus.toFixed( 3 )}+${bfvartest_results.u05_h1minus.toFixed(3)}}{2}=\frac{${( bfvartest_results.u05_h1plus + bfvartest_results.u05_h1minus ).toFixed(3)}}{2}=${( (bfvartest_results.u05_h1plus + bfvartest_results.u05_h1minus) / 2 ).toFixed(3)} \ne ${bfvartest_results.u05_h10.toFixed(3)} $$ We can also check the numbers produced by ~Bayesplay~: $$ \frac{${example52_results.get("BF-0").toFixed(2)}+${example52_results .get("BF+0") .toFixed(3)}}{2}=\frac{${( example52_results.get("BF-0") + example52_results.get("BF+0") ).toFixed(3)}}{2}=${( (example52_results.get("BF-0") + example52_results.get("BF+0")) / 2 ).toFixed(3)} = ${bfvartest_results.u05_h10.toFixed(3)} $$ `46:T1104,function texmd(strings, ...values) { const id = crypto.randomUUID(); const displayMaths = []; const inlineMaths = []; const codeBlocks = []; const inlineCode = []; const valueMarker = (i) => `\uE000VALUE${i}\uE001`; const valueMarkerRegex = /\uE000VALUE(\d+)\uE001/g; const codeBlockMarker = (i) => `\uE000CODEBLOCK${i}\uE001`; const inlineCodeMarker = (i) => `\uE000INLINECODE${i}\uE001`; const paragraphToSpan = (paragraph) => { const span = document.createElement("span"); while (paragraph.firstChild) { span.appendChild(paragraph.firstChild); } return span; }; const normalizeValue = (value) => { if (!value?.__texmd) { return value; } if (value.nodeType === Node.ELEMENT_NODE && value.tagName === "P") { return paragraphToSpan(value); } if ( value.nodeType === Node.ELEMENT_NODE || value.nodeType === Node.DOCUMENT_FRAGMENT_NODE ) { const children = Array.from(value.childNodes).filter((node) => { return !( node.nodeType === Node.TEXT_NODE && node.textContent.trim() === "" ); }); if ( children.length === 1 && children[0].nodeType === Node.ELEMENT_NODE && children[0].tagName === "P" ) { return paragraphToSpan(children[0]); } } return value; }; let source = strings.raw[0]; for (let i = 0; i < values.length; i++) { source += valueMarker(i) + strings.raw[i + 1]; } const resolveValues = (text) => text.replace(valueMarkerRegex, (_, i) => String(values[Number(i)])); source = source.replace( /~~~([^\n]*)\r?\n([\s\S]*?)~~~/g, (_, language, code) => { code = code.replace(/\r?\n$/, ""); const i = codeBlocks.push({ language: language.trim(), code: resolveValues(code) }) - 1; return codeBlockMarker(i); } ); source = source.replace(/~([^~\n]+)~/g, (_, code) => { const i = inlineCode.push(resolveValues(code)) - 1; return inlineCodeMarker(i); }); source = source.replace(/\$\$([\s\S]*?)\$\$/g, (_, maths) => { const i = displayMaths.push(resolveValues(maths)) - 1; return `\n\n

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Section 5.2 outlines an example of comparing a single sample variance with a reference variance and computing a Bayes factor for both an undirected and a directed hypothesis.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"texmd`The example is as follows:\n\n> The EPA is testing a new device for measuring PCB concentrations in fish. The measurements of the device are required to have a variance of less than ${\n example52.var0\n} ($\\sigma_0\\leq ${Math.sqrt(example52.var0).toFixed(\n 2\n)}$). Therefore, $\\phi>1$. ${numberToWords(\n example52.n\n)} readings are taken from a sample of fish, and the sample standard deviation is $s=${\n example52.s\n}$. This gives a sample effect size of $\\hat{\\phi}=\\frac{\\sigma_0}{s}=${(\n example52.sigma0 / example52.s\n).toFixed(2)}$.\n\n\n\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":84,"value":"texmd`\nDablander et al compare two hypotheses:\n\n$$\n\\mathcal{H_0}: \\phi = 1 \n$$\n$$\n\\mathcal{H_+}: \\phi > 1\n$$\n\nThe Bayes factor calculated with the accompanying ~R~ package as follows:\n\n~~~r\n> bfvartest::onesd_test(n=7, s=.22, popsd=sqrt(0.1), u = 0.5, logarithm=FALSE, alternative_interval=c(1,Inf))\n[1] 0.509796\n~~~\n\nThis matches the value reported in the paper.\n\nDablander et al also reported the undirected Bayes factor. That is, the Bayes factor from comparing the following hypotheses:\n\n$$\n\\mathcal{H_0}: \\phi = 1 \n$$\n$$\n\\mathcal{H_1}: \\phi \\ne 1\n$$\n\nThis can be calculated with the accompanying ~R~ package as follows:\n\n~~~r\n> bfvartest::onesd_test(n=7, s=.22, popsd=sqrt(0.1), u = 0.5, logarithm=FALSE)\n[1] 0.4104968\n~~~\n\nAgain, this matches the value reported in the paper.\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":99,"value":"texmd`\n## Bayesplay for variance Bayes factors\n\nThe ~Bayesplay~ package also has the functionality to compute Bayes factors for differences between variances. These are parametrised slightly differently in ~Bayesplay~ compared with ~bfvartest~, but they work essentially the same. In ~Bayesplay~, the likelihood is parametrised in terms of ~u~, which is the _variance ratio_ of the sample variance against the reference variance ($u=\\frac{s^2}{\\sigma_0^2}$), and a ~df~ of n-1.\n\nInputting the values from Example 5.2, we get the following:\n\n$$\nu = \\frac{${example52.var.toFixed(3)}}{${example52.var0.toFixed(3)}}=${\n example52.var / example52.var0\n}\n$$\n\nBecause the likelihood is in terms of variance, and not precision, the bounds of the truncated prior are also specified in the opposite direction. \n\nWith these differences in mind, we can specify the models in the ~Bayesplay~ modelling language as follows (both directed hypotheses are included, because the results will become useful below):\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":117,"value":"model = `\n analysis R {\n likelihood variance_ratio(u = 0.484, df = 6)\n\n hypotheses {\n 0: point(1)\n 1: beta_prime(0.5)\n -: beta_prime(0.5) | R < 1\n +: beta_prime(0.5) | R > 1\n }\n }\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":137,"value":"$45","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":288,"value":"texmd`\nWe can get a clue as to what is going wrong with ~bfvartest~ if we change the model we use with ~Bayesplay~ slightly. Instead of defining $\\mathcal{H_{-}}$ as $\\delta < 1$ (equivalent to ~bfvartest~ $\\mathcal{H_+}$ with $\\phi > 1$), we define $\\mathcal{H-}$ as follows:\n\n$$\n\\mathcal{H_-}: \\delta \\le \\frac{1}{\\sigma_0^2}=${1 / example52.var0}\n$$\n\nwe get the following model.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":163,"value":"model2 = `\n analysis R {\n likelihood variance_ratio(u = 0.484, df = 6)\n\n hypotheses {\n 0: point(1)\n 1: beta_prime(0.5)\n -: beta_prime(0.5) | R < 10\n }\n }\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":169,"value":"texmd`With this new model definition, we obtain a $\\mathrm{BF}_{-0}=${example52_results_v2\n .get(\"BF-0\")\n .toFixed(2)}$, which accords with the value reported in the paper.\n\nWe can also update the prior, to use the informed prior reported in the paper. In the ~Bayesplay~ modelling language, this gives the following model:\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":310,"value":"model3 = `\n analysis R {\n likelihood variance_ratio(u = 0.484, df = 6)\n\n hypotheses {\n 0: point(1)\n 1: beta_prime(2.16)\n -: beta_prime(2.16) | R < 10\n }\n }\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":319,"value":"texmd`\nWith this model definition, we get the a result for $\\mathrm{BF}_{-0}=${example52_results_v3\n .get(\"BF-0\")\n .toFixed(2)}$, which again accords with the result in the paper.\n\nThese findings taken together suggests that there is an error with ~bfvartest~ where the directed priors are truncated relative to $\\sigma_0$ rather than being truncated at 1.\n\nTo confirm this, we can modify the model again. We can change our data slightly so that we're measuring our data on a slightly different scale, for example, parts per 100,000 rather than parts per million. For ~Bayesplay~ the model is unchanged, so the result are unchanged.\n\nFor ~bfvartest~ we'd update the function call as follows:\n\n~~~r\n> bfvartest::onesd_test(n=7, s=2.2, popsd=3.2, u = 0.5, logarithm=FALSE, alternative_interval=c(1,Inf))\n[1] 7.897717e-05\n~~~\n\nWe should not expect our Bayes factor to change just because we change our units of measurement. Again, pointing to an error with ~bfvartest~.\n\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":438,"value":"{\n const multiplier = 10;\n const s = 0.22 * multiplier;\n const sigma0 = Math.sqrt(0.1) * multiplier;\n const var0 = sigma0 ** 2;\n const u = s ** 2 / var0;\n const bound = 1 / var0;\n const model = `\n analysis R {\n likelihood variance_ratio(u = ${u}, df = 6)\n\n hypotheses {\n 0: point(1)\n 1: beta_prime(0.5)\n -: beta_prime(0.5) | R < ${1 / var0}\n }\n }\n`;\n const bf = compute(model).get(\"BF-0\").toExponential(1);\n return texmd`\n We can attempt to replicate this result with ~Bayesplay~ by truncating the prior at $\\frac{1}{\\sigma_0^2}=${bound.toFixed(\n 2\n )}$ rather than $1$. Doing this, produces a $\\mathrm{BF_{-0}}$ of ${bf}, which is the same order of magnitude as the result of ~bfvartest~.\n`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":431,"value":"texmd`\n### The issue with ~bfvartest~\n\nIf we have a look at the code for ~bfvartest~, specifically the function ~.compute_logml_restr_k1~ in ~helpers.R~ we can see where the error is. If we correct this function, then ~bfvartest~ produces the correct results.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":356,"value":"texmd`### Try ~Bayesplay~ yourself:\n\nYou can try ~Bayesplay~ yourself by inputting the values below. The default values correspond to the example in Section 5.2 of Dablander et al.\n\nChanging the values with automatically update the **Results** section below.\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":390,"value":"#### Likelihood","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":358,"value":"viewof sample_sd = Inputs.number({\n label: \"Sample Standard Deviation\",\n value: 0.22\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":362,"value":"viewof ref_sd = Inputs.number({\n label: \"Reference Standard Deviation\",\n value: Math.sqrt(0.1)\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":365,"value":"viewof sample_size = Inputs.number({ label: \"Sample size\", value: 7 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":392,"value":"#### Prior","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":394,"value":"viewof prior_alpha = Inputs.number({ label: \"Prior alpha\", value: 0.5 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":405,"value":"viewof hminus_bound = Inputs.number({ label: \"Bound of H-\", value: 1 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":400,"value":"viewof hplus_bound = Inputs.number({ label: \"Bound of H+\", value: 1 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":380,"value":"model_def = {\n const s = sample_sd;\n const sigma0 = ref_sd;\n const n = sample_size;\n const u = s ** 2 / sigma0 ** 2;\n\n return `\n analysis R {\n likelihood variance_ratio(u = ${u}, df = ${n - 1})\n\n hypotheses {\n 0: point(1)\n 1: beta_prime(${prior_alpha})\n -: beta_prime(${prior_alpha}) | R < ${hminus_bound}\n +: beta_prime(${prior_alpha}) | R > ${hplus_bound}\n }\n }\n`;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":419,"value":"texmd`\n\n### Results\n\n$$\n\\mathrm{BF10} = ${custom.get(\"BF10\").toFixed(4)}\n$$\n\n$$\n\\mathrm{BF_{-0} = ${custom.get(\"BF-0\").toFixed(4)}}\n$$\n\n$$\n\\mathrm{BF_{+0} = ${custom.get(\"BF+0\").toFixed(4)}}\n$$\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"### Dependencies","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":239,"value":"bfvartest_results = {\n return {\n u05_h1minus: 0.0005227100611299864,\n u05_h1plus: 0.50979604134540923432,\n u05_h10: 0.41049679276607298206\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"bayesplay = await import(\n \"https://cdn.jsdelivr.net/gh/bayesplay/bayesplay-js@0ffabea3e15245704e8454a1359d084c9a42b3bd/dist/index.js\"\n)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":10,"value":"$46","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":39,"value":"numberToWords = {\n const ones = [\n \"zero\",\n \"one\",\n \"two\",\n \"three\",\n \"four\",\n \"five\",\n \"six\",\n \"seven\",\n \"eight\",\n \"nine\",\n \"ten\",\n \"eleven\",\n \"twelve\",\n \"thirteen\",\n \"fourteen\",\n \"fifteen\",\n \"sixteen\",\n \"seventeen\",\n \"eighteen\",\n \"nineteen\"\n ];\n\n const tens = [\n \"\",\n \"\",\n \"twenty\",\n \"thirty\",\n \"forty\",\n \"fifty\",\n \"sixty\",\n \"seventy\",\n \"eighty\",\n \"ninety\"\n ];\n\n function convert(n) {\n if (n < 20) return ones[n];\n\n if (n < 100) {\n return tens[Math.floor(n / 10)] + (n % 10 ? `-${ones[n % 10]}` : \"\");\n }\n\n if (n < 1000) {\n return (\n `${ones[Math.floor(n / 100)]} hundred` +\n (n % 100 ? ` and ${convert(n % 100)}` : \"\")\n );\n }\n\n if (n < 1_000_000) {\n return (\n `${convert(Math.floor(n / 1000))} thousand` +\n (n % 1000 ? ` ${convert(n % 1000)}` : \"\")\n );\n }\n\n throw new RangeError(\"Number too large\");\n }\n\n return convert;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":142,"value":"compute = (model) => {\n const results = bayesplay.parse_analysis(model);\n const keys = Array.from(results).map((x) => x.comparison);\n const values = Array.from(results).map((x) => x.bf);\n const map = new Map(keys.map((key, i) => [key, values[i]]));\n return map;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":134,"value":"example52_results = compute(model)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"example52 = {\n const data = {\n var0: 0.1,\n n: 7,\n s: 0.22\n };\n\n return {\n ...data,\n sigma0: Math.sqrt(data.var0),\n var: data.s ** 2\n };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":167,"value":"example52_results_v2 = compute(model2)","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":322,"value":"example52_results_v3 = compute(model3)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":414,"value":"custom = compute(model_def)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W47","stdlib":"2"}} 44:"$43:body"