31:T73b,plot1_2 = { let xmin = -5.2; let xmax = 5.2; let w = 800; let alpha; if (pars1_2.side == 'both'){ alpha = pars1_2.sig / 2; }else{ alpha = pars1_2.sig; } let t1 = jstat.normal.inv(alpha, pars1_2.mean, pars1_2.sd ); let t2 = jstat.normal.inv(1-alpha, pars1_2.mean, pars1_2.sd ); let p = plotter({ xDomain: [xmin, xmax], yDomain: [0.5, -0.05], grid: false, width: w, height: w / 2, xLabel: '', yLabel: '', yTickCount: 5, xTickCount: 5, }); let dx = 0.05; let left = [{ x: xmin, y: 0 }] .concat( d3.range(xmin, t1, dx).map(x => ({ x: x, y: jstat.normal.pdf(x, pars1_2.mean, pars1_2.sd ) })) ) .concat([{ x: t1, y: jstat.normal.pdf(t1, pars1_2.mean, pars1_2.sd) },{x: t1, y: 0}]) let right = [{x: t2, y: 0}] .concat( d3.range(t2, xmax, dx).map(x => ({x: x, y:jstat.normal.pdf(x, pars1_2.mean, pars1_2.sd ) })) ) if (pars1_2.side == 'both'){ p.polygon(left, { fill: 'lightblue', width: 0.2 }); p.polygon(right, { fill: 'lightblue', width: 0.2 }); p.text((alpha*100).toFixed(1)+'%', t1*1.5, 0.05, { stroke: 'purple', weight: 600 }); p.text((alpha*100).toFixed(1)+'%', t2*1.5, 0.05, { stroke: 'purple', weight: 600 }); }else if (pars1_2.side == 'left') { p.polygon(left, { fill: 'lightblue', width: 0.2 }); p.text((alpha*100).toFixed(1)+'%', t1*1.5, 0.05, { stroke: 'purple', weight: 600 }); }else if (pars1_2.side == 'right') { p.polygon(right, { fill: 'lightblue', width: 0.2 }); p.text((alpha*100).toFixed(1)+'%', t2*1.5, 0.05, { stroke: 'purple', weight: 600 }); } p.func(x => jstat.normal.pdf(x, pars1_2.mean, pars1_2.sd ), { stroke: 'black', width: 2 }); p.text(((1-pars1_2.sig)*100).toFixed(0)+'%', pars1_2.mean, 0.42, { stroke: 'purple', weight: 600 }); return p.node; }32:T606,

## Hypothesis Test >

${tex` H_0 : \mu = \mu_0`}

${tex` H_A : \mu \neq \mu_0 \Leftarrow H_A`} is what we want to claim

Null and alternative hypotheses should be pre-specified with both verbal statement and mathematical expression
Only parameters and constants can be shown in the hypotheses.

The test statistic

${tex`\large z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}`}

The test statistic compares the data with what is expected under the null hypothesis.
Why pre-specify the test statistic?
Test statistics based on different statistical models could make different statistical inferences.
Significance level ${tex`\alpha`} is the tolerable risk level that we are willing to take in rejecting a true null hypothesis (i.e., Type 1 error rate).
Why pre-specify α?
Because ${tex`\alpha`} establishes a criterion, or "cut-off", for making a decision about the null hypothesis.

33:T924,plot1_3 = { let xmin0 = pars1_3.mean0 - 5.2; let xmax0 = pars1_3.mean0 + 5.2; let xmin1 = pars1_3.mean1 - 5.2; let xmax1 = pars1_3.mean1 + 5.2; let xmin = d3.min([xmin0, xmin1]); let xmax = d3.max([xmax0, xmax1]); let w = 800; let labels = [tex`H_0`, tex`H_1`, 'H0'] let alpha = pars1_3.sig; let t0 = jstat.normal.inv(1-alpha, pars1_3.mean0, pars1_3.sd ); let t1 = jstat.normal.inv(1-alpha, pars1_3.mean1, pars1_3.sd ); let t2 = jstat.normal.inv(alpha, pars1_3.mean1, pars1_3.sd ); let p = plotter({ xDomain: [xmin, xmax], yDomain: [0.5, -0.05], grid: false, width: w, height: w / 2, xLabel: '', yLabel: '', yTickCount: 5, xTickCount: 5, y: { axis: null } }); let dx = 0.05; let right0 = [{x: t0, y: 0}] .concat( d3.range(t0, xmax0, dx).map(x => ({x: x, y:jstat.normal.pdf(x, pars1_3.mean0, pars1_3.sd ) })) ); let right1 = [{x: t0, y: 0}] .concat( d3.range(t0, xmax1, dx).map(x => ({x: x, y:jstat.normal.pdf(x, pars1_3.mean1, pars1_3.sd ) })) ); let left1 = [{ x: xmin1, y: 0 }] .concat( d3.range(xmin1, t0, dx).map(x => ({ x: x, y: jstat.normal.pdf(x, pars1_3.mean1, pars1_3.sd ) })) ) .concat([{ x: t0, y: jstat.normal.pdf(t0, pars1_3.mean1, pars1_3.sd) },{x: t0, y: 0}]) p.func(x => jstat.normal.pdf(x, pars1_3.mean0, pars1_3.sd ), { stroke: 'grey', width: 2, opacity: 0.7, dash: '2,2' }); p.func(x => jstat.normal.pdf(x, pars1_3.mean1, pars1_3.sd ), { stroke: 'grey', width: 2, opacity: 0.7, }) p.polygon(right0, { fill: 'red', width: 0.2, opacity: 0.3 }); p.text('α', t0+0.2, 0.03, { stroke: 'red', weight: 600 }); p.polygon(right1, { fill: 'lightblue', width: 0.2, opacity: 0.5 }); p.text('1-β', t1, 0.03, { stroke: 'blue', weight: 600 }); p.polygon(left1, { fill: 'orange', width: 0.2, opacity: 0.5 }); p.text('β', t0-0.2, 0.03, { stroke: 'orange', weight: 600 }); p.text('H0', pars1_3.mean0, 0.4, { stroke: 'grey', weight: 600 }); p.text('H1', pars1_3.mean1, 0.4, { stroke: 'grey', weight: 600 }); p.line_segment(t0, -0.05, t0, 0.4, { stroke: 'grey', width: 1}); p.arrow(t0, -0.05, t0+1, -0.05, { stroke: 'grey'}); p.text(`Reject Null Hypothesis`, t0+2.5, -0.06, { stroke: 'grey' }); return p.node; }34:T513,plot6_2 = Plot.plot({ marginLeft: 30, marginRight: 50, height:400, width:700, style: { background: '#fafafa', }, y: { label: 'Estimated Event-free Survival', // tickSize: 0, margin: 10, domain: [0,1.05], }, x:{ label: 'Month →', grid: true, domain: [0,24], ticks: 8, // axis: 'top', }, marks: [ // Plot.ruleY(data2.plot1, {x: "time", y: "id", fill: "black"}), Plot.dot(data2.plot2.filter(d=>(d.change & d.time <= month)), {x: "time", y: "s", fill: d=>d.pos==1 ? 'white': 'black', stroke:'black', r: d=>d.time == 0 ? 0 : 4} ), Plot.text(data2.plot2.filter(d=>d.change == false & d.time <= month), {x: "time", y: "s", text: d=>'|'}), Plot.ruleY([0]), Plot.ruleX([0]), Plot.text(data2.plot2.filter(d=>d.time == month), { x: d=>d.time == 24 ? d.time -5 : d.time, y: d=>d.pos == 1 ? d.s-0.02 : d.s+0.03, text: d=>d.note, fontSize: 12, textAnchor: 'start', // dy: d=>d.pos == 1 ? -10 : 10, dx: 5 } ), Plot.line(data2.plot2.filter(d=>d.time <= month), {x:'time', y:'s'}) ] })35:T919,text_8_1 = { const interim1 = htl.html`

1st interim look – OBF boundary

Trial begins . 6 ½ months in, event #22 occurs
- Data locked, interim analysis run, DMC meeting held
  - 6 SCDs in ICD arm; 16 SCDs in placebo arm
  - Test-statistic calculated: Z = -2.02; p = 0.0434

` const interim2 = htl.html`

2nd interim look – OBF boundary

Trial continues . 12 months in, event #44 occurs
- Data locked, interim analysis run, DMC meeting held
  - 14 SCDs in ICD arm; 30 SCDs in placebo arm
  - Test-statistic calculated: Z = -2.38; p = 0.0174

` const interim3 = htl.html`

3rd interim look – OBF boundary

Trial continues. 17 months in, event #66 occurs
- Data locked, interim analysis run, DMC meeting held
  - 21 SCDs in ICD arm; 45 SCDs in placebo arm
  - Test-statistic calculated: Z = -2.92; p = 0.0036

` return interim == 3 ? interim3 : interim == 2 ? interim2 : interim1 }36:T93d,plot8_1 = { const color = 'orange'; return Plot.plot({ marginLeft: 60, marginRight: 60, height:350, width:600, style: { background: '#fafafa', }, y: { label: '↑ Z', // tickSize: 0, grid: true, margin: 10, domain: [-5,5], }, x:{ label: 'Event →', // grid: true, // type: 'point', domain: [20,90], ticks: [22,44,66,88], }, marks: [ Plot.ruleY([-5]), Plot.ruleX([20]), Plot.dot(data3, { x: "event", y: "b", fill: '#fafafa', stroke: color, fill: color, r: 4} ), Plot.line(data3, { x: "event", y: "b", // fill: '#fafafa', stroke: color, } ), Plot.text([[55,-4], [55,0], [55,4]], { // x: 55, y: -4, text: [['Reject H0'], ['Continue'], ['Reject H0'],], fontSize: '16px', textAnchor: 'start', } ), Plot.text(data3.filter(d=>d.i == interim), { x: 'event' , y: d=> interim === 2 ? d.z+1 : d.z, text: d=> 'Z = ' + d.z + ' p = ' + d.p, fontSize: '12px', textAnchor: 'start', dy: 15, } ), Plot.text(data3.filter(d=>d.i == interim), { x: 'event' , y: d=> interim === 3 ? d.b + 0.5 : d.b - 0.5 , text: d=> 'Z = ' + d.b, fontSize: '12px', textAnchor: 'start', fill: color, } ), Plot.line(data3.filter(d=>d.i < interim+0.1), { x: 'event' , y: 'z', stroke: 'steelblue', } ), Plot.text(data3.filter(d=>d.i < interim+0.1), { x: 'event' , y: 'z', text: [['🧑'],['🧑'],['🧑'],], fontSize: '14px', textAnchor: 'middle', } ), ] }) }37:T5d6,text_12_1 = { const scenario1 = htl.html`

Se: 25%
Sp: 95%
Prevalence: 0.0005

What this means:

Out of 10,000 women, 5 are expected to have breast cancer and about 1 of them will be test positive and 4 will be missed.
Out of the remaining 9,995 women who are expected not to have breast cancer, about 500 of them will be test positive.

` const scenario2 = htl.html`

Se: 80%
Sp: 95%
Prevalence: 0.0005

What this means:

Out of 10,000 women, 5 are expected to have breast cancer and about 4 of them will be test positive and 1 will be missed.
Out of the remaining 9,995 women who are expected not to have breast cancer, about 500 of them will be test positive.

` const scenario3 = htl.html`

Se: 80%
Sp: 95%
Prevalence: 0.5

What this means:

Out of 10,000 women, 5,000 are expected to have Osteopenia and about 4,000 of them will be test positive and 1000 will be missed.
Out of the remaining 5,000 women who are expected not to have Osteopenia, about 250 of them will be test positive.

` return scenario == 3 ? scenario3 : scenario == 2 ? scenario2 : scenario1 }38:T4c5,data2 = ({ plot1: [ {id:'Subject1', time:24, event:'Event free', censor:true}, {id:'Subject2', time:8, event:'Hospitalization', censor:false}, {id:'Subject3', time:3, event:'Death', censor:false}, {id:'Subject4', time:14, event:'Withdrawal', censor:true}, {id:'Subject5', time:16, event:'Hospitalization', censor:false} ], plot2: [ {time:0, s:1, pos:1, note:'beginning of follow-up', change:true}, {time:3, s:1, pos:1, note:'Subject3: died at Month 3', change:true}, {time:3, s:0.8, pos:2, note:'S(3) = 1 x (5-1)/5 = 0.8', change:true}, {time:8, s:0.8, pos:1, note:'Subject3: Hospitalization at Month 8', change:true}, {time:8, s:0.6, pos:2, note:'S(8) = 0.8 x (4-1)/4 = 0.6', change:true}, {time:14, s:0.6, pos:1, note:'Subject4: Withdrawal at Month 14', change:false}, {time:14, s:0.6, pos:2, note:'S(t) does not change', change:false}, {time:16, s:0.6, pos:1, note:'Subject5: Hospitalization at Month 16', change:true}, {time:16, s:0.3, pos:2, note:'S(16) = 0.6 x (2-1)/2 = 0.3', change:true}, {time:24, s:0.3, pos:1, note:'Subject1: censored at Month 24', change:false}, {time:24, s:0.3, pos:2, note:'S(t) does not change', change:false}, ] })39:[] 2f:{"title":"Statistics: Basic Concepts","cells":[{"id":0,"value":"# Statistics: Basic Concepts\n

","pinned":false,"hidden":false,"mode":"md"},{"id":493,"value":"## Bias and Variability\n>

Sample statistic is a point estimate of population parameter

Two measures for a point estimate:

Bias: systematic error from the truth: ${tex`\\bar{x} - \\mu`}.
Variability (${tex`s^2`}): measures the dispersion of the sampling data, i.e., it describes the spread of the sampling distribution\n

Bias is the difference between the sample mean ${tex`\\bar{x}`} and the population mean ${tex`\\mu`}

Variability describes the spread of the sampling distribution\n

","pinned":false,"hidden":false,"mode":"md"},{"id":490,"value":"## Confidence Interval\n>

Standard deviation (SD) ${tex`\\sigma`} measures the amount of variability, or data dispersion, from the individual data values to the population mean ${tex`\\mu`}.

Standard error of mean (SEM) ${tex`\\bar{x}`} , which is ${tex`\\frac{\\sigma}{\\sqrt{n}}`}, measures how far the sample mean ${tex`\\bar{x}`} is likely to be from the population mean ${tex`\\mu`}.

SEM (${tex`\\frac{\\sigma}{\\sqrt{n}}`}) is always smaller than SD (${tex`\\sigma`} )

Sample size increases ${tex`\\Rightarrow`} SEM decreases (i.e., more precision for the mean).

\n\n
","pinned":false,"hidden":false,"mode":"md"},{"id":23,"value":"viewof pars1_1 = Inputs.form({\n n: Inputs.range([50, 200], {step: 5, value: 100, label: tex`\\text{Sample Size: }(n)`}),\n mean: Inputs.range([-2, 2], {step: 0.01, value: 0, label: tex`\\text{Mean: }(\\mu)`}),\n sd: Inputs.range([0.3, 1], {step: 0.1, value: 0.5, label: tex`\\text{SD: }(\\sigma)`})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":184,"value":"legend1_1 = html`

${['Normal Distribution of Sample Means', 'Distribution of Individuals'].map((d,i) => htl.html`

${d.toUpperCase()}

`)}

`","pinned":false,"hidden":false,"mode":"ojs"},{"id":60,"value":"plot1_1 = Plot.plot({\n inset: 10,\n grid: true,\n style: {\n background: '#fafafa',\n },\n x: {\n domain: [-2, 2]\n },\n y: {\n domain: [0, 10]\n },\n color: {\n legend: true\n },\n marks: [\n Plot.line(data1_1, {x: \"x\", y: \"sd\", stroke: 'blue'}),\n Plot.line(data1_1, {x: \"x\", y: \"sem\", stroke: 'green'}),\n // Plot.text(data1_1, {filter: (d,i) => i === Math.round(pars.n * 0.5), x: \"x\", y: \"sem\", text: d => `Normal Distribution \\n of Smaple Means`, dx: 18, dy:-10, fill: 'red'}),\n // Plot.text(data1_1, {filter: (d,i) => i === Math.round(pars.n * 0.5), x: \"x\", y: \"sd\", text: d => `Distribution of \\n Individuals`, dx: 18, dy:-10, fill: 'blue'}),\n ]\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":512,"value":">

A two-sided (1 – ${tex`\\alpha`}) x 100% confidence interval, (lower limit, upper limit), for a population mean ${tex`\\mu`} is

${tex`\\large \\bar{x}\\pm z_{\\alpha/2} \\frac{\\sigma}{\\sqrt{n}} `}

What does (1 – ${tex`\\alpha`}) x 100% C.I. mean?
\n We have (1 – ${tex`\\alpha`}) x 100% confidence that the population mean ${tex`\\mu`} falls in this interval, by assuming no bias introduced in the study.\n

${tex`\\large P\\left(\\bar{x} - z_{\\alpha/2} \\frac{\\sigma}{\\sqrt{n}} < \\mu < \\bar{x} + z_{\\alpha/2} \\frac{\\sigma}{\\sqrt{n}} \\right) = 1 - \\alpha`}

For a fixed confidence level, e.g., 95%,
\n Larger sample size ${tex`\\Rightarrow`} narrower confidence interval ${tex`\\Rightarrow`} more precise information on the value of ${tex`\\mu`}\n

","pinned":false,"hidden":false,"mode":"md"},{"id":268,"value":"viewof pars1_2 = Inputs.form({\n sig: Inputs.range([0.01, 1], {step: 0.01, value: 0.05, label: tex`\\alpha`}),\n mean: Inputs.range([-2, 2], {step: 0.01, value: 0, label: tex`\\text{Mean: }(\\mu)`}),\n sd: Inputs.range([0.1, 2], {step: 0.1, value: 1, label: tex`\\text{SD: }(\\sigma)`}),\n side: Inputs.radio([\"left\", \"right\", \"both\"], {label: \"1-side/2-sides\", value: 'both'})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":216,"value":"$31","pinned":false,"hidden":false,"mode":"ojs"},{"id":529,"value":"$32","pinned":false,"hidden":false,"mode":"md"},{"id":545,"value":"

Type I error and Type II error

\n>

${tex`H_0: \\mu \\leq \\mu_0`}

${tex`H_1: \\mu > \\mu_0`}

${tex`\\alpha = P(z > z_a | \\mu = \\mu_0)`}

${tex`\\beta = P(z < z_a | \\mu > \\mu_0)`}

\n
","pinned":false,"hidden":false,"mode":"md"},{"id":339,"value":"viewof pars1_3 = Inputs.form({\n sig: Inputs.range([0.01, 1], {step: 0.01, value: 0.05, label: tex`\\alpha`}),\n mean0: Inputs.range([3, 7], {step: 0.1, value: 5, label: tex`H_0 \\text{ Mean: }(\\mu_0)`}),\n mean1: Inputs.range([6, 10], {step: 0.1, value: 8, label: tex`H_1 \\text{ Mean: }(\\mu)`}),\n sd: Inputs.range([0.1, 2], {step: 0.1, value: 1.2, label: tex`\\text{SD: }(\\sigma)`})\n})","pinned":false,"hidden":false,"mode":"ojs"},{"id":323,"value":"$33","pinned":true,"hidden":false,"mode":"ojs"},{"id":563,"value":"

Survival Analysis

","pinned":false,"hidden":false,"mode":"md"},{"id":818,"value":"**Kaplan-Meier (KM) Curve**\n\n* Sample size: 5\n* Subject 1: event free during the follow-up\n* Subject 2: hospitalized at Month 8\n* Subject 3: died at Month 3\n* Subject 4: withdrew at Month 14\n* Subject 5: hospitalized at Month 16","pinned":false,"hidden":false,"mode":"md"},{"id":812,"value":"viewof month = Inputs.radio([0, 3, 8, 14, 16, 24], {label: \"Month\", value: 0})","pinned":false,"hidden":false,"mode":"ojs"},{"id":808,"value":"plot6_comb = html`\n

\n ${[plot6_1, plot6_2]}\n

\nd.time == month ? 'orange' : 'black'}\n ),\n Plot.dot(data2.plot1, \n {\n x: \"time\", y: \"id\", \n fill: d=> d.censor ? 'white': d.time == month ? 'orange' : 'black', \n stroke: d=>d.time == month ? 'orange' : 'black', \n r: 4}\n ),\n // Plot.ruleY([]),\n Plot.ruleX([0, 24]),\n Plot.text(data2.plot1, \n {x: \"time\", y: \"id\", \n text: d=>d.event, \n fill: d=>d.time == month ? 'orange' : 'black', \n dy: -10, \n dx: -3}\n )\n ]\n})","pinned":true,"hidden":false,"mode":"ojs"},{"id":712,"value":"// viewof month = Inputs.range([0, 24], {step: 1, value: 0, label: 'Month'})","pinned":true,"hidden":false,"mode":"ojs"},{"id":647,"value":"$34","pinned":true,"hidden":false,"mode":"ojs"},{"id":824,"value":"

Adaptive Design

","pinned":false,"hidden":false,"mode":"md"},{"id":834,"value":"viewof interim = Inputs.radio([1, 2, 3], {label: \"Interim\", value: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":829,"value":"$35","pinned":false,"hidden":false,"mode":"ojs"},{"id":846,"value":"$36","pinned":false,"hidden":false,"mode":"ojs"},{"id":959,"value":"

How Prevalence affects Clinical Utility

","pinned":false,"hidden":false,"mode":"md"},{"id":961,"value":"viewof scenario = Inputs.radio([1, 2, 3], {label: \"Scenario\", value: 1})","pinned":false,"hidden":false,"mode":"ojs"},{"id":965,"value":"$37","pinned":false,"hidden":false,"mode":"ojs"},{"id":6,"value":"

\n## Appendix","pinned":false,"hidden":false,"mode":"md"},{"id":842,"value":"data3 = [\n {i:1, event:22, b:-4.05, z:-2.02, p:0.0434},\n {i:2, event:44, b:-2.86, z:-2.38, p:0.0174},\n {i:3, event:66, b:-2.34, z:-2.92, p:0.0036},\n {i:4, event:88, b:-2.04, z:null, p:null},\n {i:5, event:88, b:2.04, z:null, p:null},\n {i:6, event:66, b:2.34, z:null, p:null},\n {i:7, event:44, b:2.86, z:null, p:null},\n {i:8, event:22, b:4.05, z:null, p:null},\n]","pinned":true,"hidden":false,"mode":"ojs"},{"id":568,"value":"$38","pinned":true,"hidden":false,"mode":"ojs"},{"id":29,"value":"data1_1 = {\n let x = d3.range(-2, 2, 4/pars1_1.n);\n let y = x.map(d => jstat.normal.pdf(d, pars1_1.mean, pars1_1.sd));\n let z = x.map(d => jstat.normal.pdf(d, pars1_1.mean, pars1_1.sd / Math.sqrt(pars1_1.n) ));\n return x.map((d,i) => ({'x': d, 'sd': y[i], 'sem': z[i] }))\n}","pinned":false,"hidden":false,"mode":"ojs"},{"id":15,"value":"jstat = require('jstat')","pinned":false,"hidden":false,"mode":"ojs"},{"id":7,"value":"import { plotter } from '50dadfdec01c15a8'","pinned":false,"hidden":false,"mode":"ojs"},{"id":18,"value":"import {Plot, d3} from \"@observablehq/plot\"","pinned":false,"hidden":false,"mode":"ojs"}],"files":"$W39","stdlib":"1","titling":"auto"}