-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathmr2019-tom-palmer-poster.html
568 lines (544 loc) · 20.8 KB
/
mr2019-tom-palmer-poster.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta charset="utf-8" />
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1">
<title>bpbounds: R package and web app</title>
<style type="text/css">code.sourceCode > span { display: inline-block; line-height: 1.25; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode { white-space: pre; position: relative; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
code.sourceCode { white-space: pre-wrap; }
code.sourceCode > span { text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
background-color: #2a211c;
color: #bdae9d;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #bdae9d; padding-left: 4px; }
div.sourceCode
{ color: #bdae9d; background-color: #2a211c; }
@media screen {
code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { color: #ffff00; } /* Alert */
code span.an { color: #0066ff; font-weight: bold; font-style: italic; } /* Annotation */
code span.at { } /* Attribute */
code span.bn { color: #44aa43; } /* BaseN */
code span.bu { } /* BuiltIn */
code span.cf { color: #43a8ed; font-weight: bold; } /* ControlFlow */
code span.ch { color: #049b0a; } /* Char */
code span.cn { } /* Constant */
code span.co { color: #0066ff; font-weight: bold; font-style: italic; } /* Comment */
code span.do { color: #0066ff; font-style: italic; } /* Documentation */
code span.dt { text-decoration: underline; } /* DataType */
code span.dv { color: #44aa43; } /* DecVal */
code span.er { color: #ffff00; font-weight: bold; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #44aa43; } /* Float */
code span.fu { color: #ff9358; font-weight: bold; } /* Function */
code span.im { } /* Import */
code span.in { color: #0066ff; font-weight: bold; font-style: italic; } /* Information */
code span.kw { color: #43a8ed; font-weight: bold; } /* Keyword */
code span.op { } /* Operator */
code span.pp { font-weight: bold; } /* Preprocessor */
code span.sc { color: #049b0a; } /* SpecialChar */
code span.ss { color: #049b0a; } /* SpecialString */
code span.st { color: #049b0a; } /* String */
code span.va { } /* Variable */
code span.vs { color: #049b0a; } /* VerbatimString */
code span.wa { color: #ffff00; font-weight: bold; } /* Warning */</style>
<!--
Font-awesome icons ie github or twitter
-->
<link rel="stylesheet" href="https://use.fontawesome.com/releases/v5.8.1/css/all.css" integrity="sha384-50oBUHEmvpQ+1lW4y57PTFmhCaXp0ML5d60M1M7uH2+nqUivzIebhndOJK28anvf" crossorigin="anonymous">
<link rel="stylesheet" href="https://use.fontawesome.com/releases/v5.8.1/css/brands.css" integrity="sha384-n9+6/aSqa9lBidZMRCQHTHKJscPq6NW4pCQBiMmHdUCvPN8ZOg2zJJTkC7WIezWv" crossorigin="anonymous">
<!--
Google fonts api stuff
-->
<link href='https://fonts.googleapis.com/css?family=Special Elite' rel='stylesheet'>
<link href='https://fonts.googleapis.com/css?family=Rasa' rel='stylesheet'>
<!--
Here are the required style attributes for css to make this poster work :)
-->
<style>
@page {
size: 33.1in 46.8in;
margin: 0;
padding: 0;
}
body {
margin: 0px;
padding: 0px;
width: 33.1in;
height: 46.8in;
text-align: justify;
font-size: 45px;
line-height: 1.05;
}
/* RMarkdown Class Styles */
/* center align leaflet map,
from https://stackoverflow.com/questions/52112119/center-leaflet-in-a-rmarkdown-document */
.html-widget {
margin: auto;
position: sticky;
margin-top: 2cm;
margin-bottom: 2cm;
}
.leaflet.html-widget.html-widget-static-bound.leaflet-container.leaflet-touch.leaflet-fade-anim.leaflet-grab.leaflet-touch-drag.leaflet-touch-zoom {
position: sticky;
width: 100%;
}
pre.sourceCode.r {
background-color: #dddddd40;
border-radius: 4mm;
padding: 4mm;
width: 75%;
margin: auto;
margin-top: 1em;
margin-bottom: 1em;
/* align-items: center; */
}
code.sourceCode.r{
background-color: transparent;
font-size: 20pt;
/* border-radius: 2mm;
padding: 2mm; */
}
code {
font-size: 20pt;
font-family: monospace;
background-color: #b5121b24;
color: #b5121b;
padding: 1.2mm;
line-height: 1;
border-radius: 2mm;
}
caption {
margin-bottom: 10px;
font-size: 20pt;
font-style: italic;
}
tbody tr:nth-child(odd) {
background-color: #b5121b20;
}
.table>thead>tr>th, .table>tbody>tr>th, .table>tfoot>tr>th, .table>thead>tr>td, .table>tbody>tr>td, .table>tfoot>tr>td{
border-spacing: 0;
font-size: 40%;
border-style: none;
padding-top: 15px;
padding-bottom: 15px;
padding-right: 1em;
padding-left: 1em;
line-height: 1em;
}
table {
margin: auto;
}
th {
padding-left: 5mm;
padding-right: 5mm;
}
.caption {
font-size: 20pt;
font-style: italic;
padding-top: 0;
}
.references {
font-size: 20px;
line-height: 90%;
}
/* Create three unequal columns that floats next to each other */
.column {
float: left;
padding: 0px;
}
.outer {
width: 33.1in;
height: calc(46.8in * 0.6325 );
-webkit-column-count: 3; /* Chrome, Safari, Opera */
-moz-column-count: 3; /* Firefox */
column-count: 3;
-webkit-column-fill: auto;
-moz-column-fill: auto;
column-fill: auto;
column-gap: 0;
padding-left: 0cm;
padding-right: 0cm;
/* -webkit-column-rule-width: 50%;
-moz-column-rule-width: 50%;
column-rule-width: 50%; */
-webkit-column-rule-style: none;
-moz-column-rule-style: none;
column-rule-style: none;
-webkit-column-rule-color: black;
-moz-column-rule-color: black;
column-rule-color: black;
background-color: #ffffff;
font-family: Rasa;
margin-top: calc(46.8in * 0.25 );
padding-top: 1em;
padding-bottom: 1em;
}
span.citation {
color: #b5121b;
font-weight: bold;
}
a {
text-decoration: none;
color: #b5121b;
}
#title {
font-size: 125pt;
text-align: left;
margin: 0;
line-height: 98%;
border-bottom: 0;
font-weight: normal;
}
#author {
color: #b5121b;
margin: 0;
line-height: 85%;
font-size: 1.17em;
}
#affiliation {
padding-top: 0.1em;
color: #00000080;
font-style: italic;
font-size: 25px;
margin: 0;
}
sup {
color: #b5121b;
}
.affiliation sup {
font-size: 20px;
}
.author {
text-align: left;
}
.author sup {
font-size: 30px;
}
.author_extra {
color: #b5121b;
margin: 0;
line-height: 85%;
font-size: 35px;
text-align: left;
}
.outer h1, h2, h3, h4, h5, h6 {
text-align: center;
margin: 0;
font-weight: bold;
}
.section h1 {
text-align:center;
padding-bottom:5px;
background:
linear-gradient(
to left,
#ffffff 1%,
#ffffff 20%,
#b5121b75 33%,
#b5121b 50%,
#b5121b75 66%,
#ffffff 80%,
#ffffff 99%
)
left
bottom
#ffffff
no-repeat;
background-size:100% 5px ;
margin-top: 0.5em;
margin-bottom: 0.5em;
}
.outer h2 {
text-align: center;
}
.outer p, .level2 {
color: #000000;
}
.outer ol {
padding-left: 8%;
padding-right: 8%;
text-align: left;
}
.main {
width: 33.1in;
height: calc(46.8in * 0.25);
position: absolute;
background-color: #b5121b;
color: #ffffff90;
font-family: Special Elite;
background-image: linear-gradient(#b5121b 50%,#b5121b);
}
.main strong {
color: #ffffff;
}
.main strong > sup {
color: #ffffff;
}
.main sup {
color: #ffffff90;
}
#main-img-left {
width: 10%;
left: 0.5in;
bottom: 0.2in;
position: absolute;
opacity: 1
}
#main-img-center {
width: 10%;
left: calc(33.1in * 0.45);
bottom: 0.5in;
position: absolute;
opacity: 1
}
#main-img-right {
width: 10%;
right: 0.5in;
bottom: 0.2in;
position: absolute;
opacity: 1
}
.main p {
font-size: 150px;
text-align: left;
margin: 0;
position: absolute;
top: 50%;
-ms-transform: translateY(-50%);
transform: translateY(-50%);
margin-left: 1em;
}
.fab {
color: #00000030;
font-size: 25px;
}
.twitter, i {
color: #00000030;
font-size: 35px;
text-decoration: none;
}
a.email {
text-decoration: none;
color: #00000030;
font-size: 35px;
}
.envelope {
color: #00000030;
font-size: 5px;
text-decoration: none;
}
.poster_wrap {
width: 33.1in;
height: 46.8in;
padding: 0cm;
}
.main_bottom {
width: 33.1in;
height: calc(46.8in * 0.1);
margin-top: calc(46.8in * 0.9);
position: absolute;
background-color: #b5121b;
background-image: linear-gradient(#b5121b 10%, #b5121b);
}
.section {
padding-left: 10mm;
padding-right: 10mm;
}
span > #tab:mytable {
font-weight: bold;
}
.orcid img {
width: 3%;
}
.section h4 {
break-before: column;
}
</style>
</head>
<body>
<div class="poster_wrap">
<div class="column outer">
<div class="section">
<h1 id="title"><strong>bpbounds</strong>: R package and web app</h1><br>
<h3 id="author" class="author">
<strong>Tom Palmer</strong><sup> 1, <a class="orcid" href="https://orcid.org/0000-0003-4655-4511"><img src="https://raw.githubusercontent.com/brentthorne/posterdown/master/images/orcid.jpg"></a></sup><br>
<a class='envelope'><i class="fas fa-envelope"></i></a> <a href="mailto:[email protected]" class="email">[email protected]</a> <br>
</h3>
<h5 id="author_extra", class="author_extra">
Roland Ramsahai<sup></sup>
Vanessa Didelez<sup>2</sup>
Nuala Sheehan<sup>3</sup>
</h5>
<p id="affiliation" class="affiliation">
<sup>1</sup> Department of Mathematics and Statistics, Lancaster University<br> <sup>2</sup> Leibniz BIPS, Bremen, Germany<br> <sup>3</sup> Department of Health Sciences, University of Leicester
</p>
</div>
<div id="introduction" class="section level1">
<h1>Introduction</h1>
<ul>
<li>We present our bpbounds R package and Shiny web app for the nonparametric bounds for the average causal effect (ACE) due to Balke and Pearl <span class="citation">(Palmer et al. 2018)</span>.</li>
<li>This is an R implementation of our Stata programs <span class="citation">(Palmer et al. 2011)</span>.</li>
<li>The package can be installed from CRAN as follows:</li>
</ul>
<pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1"></a><span class="kw">install.packages</span>(<span class="st">"bpbounds"</span>)</span></code></pre>
<ul>
<li>Code development is on the GitHub repository:
<a href="https://github.com/remlapmot/bpbounds" class="uri">https://github.com/remlapmot/bpbounds</a></li>
</ul>
</div>
<div id="methods" class="section level1">
<h1>Methods</h1>
<ul>
<li>Under the instrumental variable assumptions alone, without additional parametric model assumptions, the ACE is not identified.</li>
<li><span class="citation">Balke and Pearl (1997)</span> showed it is possible to derive bounds for the ACE.</li>
<li>The bounds have the following interpretation:</li>
</ul>
<blockquote>
<p>There is some joint distribution of the unobserved confounders and the observed variables that yields a true ACE as small as the lower bound, while another choice produces an ACE as large as the upper bounds (the bounds are tight).</p>
</blockquote>
<ul>
<li>There are at least two ways to implement the Balke-Pearl bounds:</li>
</ul>
<ol style="list-style-type: lower-roman">
<li>using conditional probabilities calculated from contingency tables;</li>
<li>the polytope method due to <span class="citation">Dawid (2003)</span>.</li>
</ol>
<ul>
<li>We implemented the polytope method since it is generalisable for identified IV models with exposures, outcomes, and instruments with more than 2 categories.</li>
<li>Currently, we allow for a binary or 3 category instrument, and binary exposure and outcome.</li>
</ul>
</div>
<div id="example-mendelian-randomization-analysis" class="section level1">
<h1>Example Mendelian randomization analysis</h1>
<ul>
<li>We extract an example from <span class="citation">Meleady et al. (2003)</span>.</li>
<li>We have a 3 category instrument and binary exposure and outcome.</li>
<li>We use the 677CT polymorphism (rs1801133) in the MTHFR gene, involved in folate metabolism, as an instrumental variable to investigate the causal effect of homocysteine on the risk of cardiovascular disease.</li>
<li>The code is shown on the right.</li>
<li>The ACE lies between a risk difference of -9% to 74% increase in absolute risk.</li>
<li>Additionally, we see that the monotonicity inequality is not satisfied.</li>
</ul>
</div>
<div id="conclusion" class="section level1">
<h1>Conclusion</h1>
<ul>
<li>Use of bounds in instrumental variable analyses is regaining interest <span class="citation">(Swanson et al. 2018; Labrecque and Swanson 2018)</span>.</li>
<li>The empirical experience that the bounds are often wide is not a bad property of the method, it is a property of the typical data: Mendelian randomization data simply often are uninformative in that sense due to weak instrumental variables.</li>
<li>We recommend using the bounds when the variables are genuinely discrete, but not when the exposure is genuinely continuous <span class="citation">(Sheehan and Didelez 2019)</span>.</li>
<li>Our R package and app provide a convenient interface to the bounds.</li>
</ul>
</div>
<div id="references" class="section level1">
<h1>References</h1>
<div id="refs" class="references">
<div id="ref-balke-jasa-1997">
<p>Balke, A., and J. Pearl. 1997. “Bounds on treatment effects from studies with imperfect compliance.” <em>Journal of the American Statistical Association</em> 92 (439): 1172–6. <a href="https://doi.org/10.1080/01621459.1997.10474074">https://doi.org/10.1080/01621459.1997.10474074</a>.</p>
</div>
<div id="ref-dawid-hsss-2003">
<p>Dawid, A. P. 2003. “Causal Inference Using Influence Diagrams: The Problem of Partial Compliance (with Discusssion).” In <em>Highly Structured Stochastic Systems</em>, edited by P. J. Green, N. L. Hjort, and S. Richardson, 45–65. New York: Oxford University Press.</p>
</div>
<div id="ref-labrecque-cer-2018">
<p>Labrecque, Jeremy, and Sonja A Swanson. 2018. “Understanding the Assumptions Underlying Instrumental Variable Analyses: A Brief Review of Falsification Strategies and Related Tools.” <em>Current Epidemiology Reports</em> 5 (3): 214–20. <a href="https://doi.org/10.1007/s4047">https://doi.org/10.1007/s4047</a>.</p>
</div>
<div id="ref-meleady-ajcn-2003">
<p>Meleady, Raymond, Per M Ueland, Henk Blom, Alexander S Whitehead, Helga Refsum, Leslie E Daly, Stein Emil Vollset, et al. 2003. “Thermolabile Methylenetetrahydrofolate Reductase, Homocysteine, and Cardiovascular Disease Risk: The European Concerted Action Project.” <em>The American Journal of Clinical Nutrition</em> 77 (1): 63–70. <a href="https://doi.org/10.1093/ajcn/77.1.63">https://doi.org/10.1093/ajcn/77.1.63</a>.</p>
</div>
<div id="ref-bpbounds-package">
<p>Palmer, T. M., R. Ramsahai, V. Didelez, and N. A. Sheehan. 2018. <em>bpbounds: R package implementing Balke-Pearl bounds for the average causal effect</em>. <a href="https://CRAN.R-project.org/package=bpbounds">https://CRAN.R-project.org/package=bpbounds</a>.</p>
</div>
<div id="ref-palmer-sj-2011">
<p>Palmer, T. M., R. R. Ramsahai, V. Didelez, and N. A Sheehan. 2011. “Nonparametric Bounds for the Causal Effect in a Binary Instrumental-Variable Model.” <em>Stata Journal</em> 11 (3): 345–67. <a href="http://www.stata-journal.com/article.html?article=st0232">http://www.stata-journal.com/article.html?article=st0232</a>.</p>
</div>
<div id="ref-sheehan-hg-2019">
<p>Sheehan, Nuala A, and Vanessa Didelez. 2019. “Epidemiology, genetic epidemiology and Mendelian randomisation: more need than ever to attend to detail.” <em>Human Genetics</em>, 1–16. <a href="https://doi.org/10.1007/s00439-019-02027-3">https://doi.org/10.1007/s00439-019-02027-3</a>.</p>
</div>
<div id="ref-swanson-jasa-2018">
<p>Swanson, Sonja A., Miguel A. Hernán, Matthew Miller, James M. Robins, and Thomas S. Richardson. 2018. “Partial Identification of the Average Treatment Effect Using Instrumental Variables: Review of Methods for Binary Instruments, Treatments, and Outcomes.” <em>Journal of the American Statistical Association</em> 113 (522): 933–47. <a href="https://doi.org/10.1080/01621459.2018.1434530">https://doi.org/10.1080/01621459.2018.1434530</a>.</p>
</div>
</div>
<div id="section" class="section level4">
<h4></h4>
</div>
</div>
<div id="extra-figures-tables" class="section level1">
<h1>Extra Figures & Tables</h1>
<pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1"></a><span class="kw">library</span>(bpbounds)</span>
<span id="cb2-2"><a href="#cb2-2"></a>mt3 <-<span class="st"> </span><span class="kw">c</span>(.<span class="dv">83</span>, <span class="fl">.05</span>, <span class="fl">.11</span>, <span class="fl">.01</span>, </span>
<span id="cb2-3"><a href="#cb2-3"></a> <span class="fl">.88</span>, <span class="fl">.06</span>, <span class="fl">.05</span>, <span class="fl">.01</span>, </span>
<span id="cb2-4"><a href="#cb2-4"></a> <span class="fl">.72</span>, <span class="fl">.05</span>, <span class="fl">.20</span>, <span class="fl">.03</span>)</span>
<span id="cb2-5"><a href="#cb2-5"></a>p3 <-<span class="st"> </span><span class="kw">array</span>(mt3, <span class="dt">dim =</span> <span class="kw">c</span>(<span class="dv">2</span>, <span class="dv">2</span>, <span class="dv">3</span>),</span>
<span id="cb2-6"><a href="#cb2-6"></a> <span class="dt">dimnames =</span> <span class="kw">list</span>(<span class="dt">x =</span> <span class="kw">c</span>(<span class="dv">0</span>, <span class="dv">1</span>),</span>
<span id="cb2-7"><a href="#cb2-7"></a> <span class="dt">y =</span> <span class="kw">c</span>(<span class="dv">0</span>, <span class="dv">1</span>),</span>
<span id="cb2-8"><a href="#cb2-8"></a> <span class="dt">z =</span> <span class="kw">c</span>(<span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">2</span>)))</span>
<span id="cb2-9"><a href="#cb2-9"></a>bpres3 <-<span class="st"> </span><span class="kw">bpbounds</span>(<span class="kw">as.table</span>(p3))</span>
<span id="cb2-10"><a href="#cb2-10"></a><span class="kw">summary</span>(bpres3)</span>
<span id="cb2-11"><a href="#cb2-11"></a><span class="co">## </span></span>
<span id="cb2-12"><a href="#cb2-12"></a><span class="co">## Data: trivariate</span></span>
<span id="cb2-13"><a href="#cb2-13"></a><span class="co">## Instrument categories: 3</span></span>
<span id="cb2-14"><a href="#cb2-14"></a><span class="co">## </span></span>
<span id="cb2-15"><a href="#cb2-15"></a><span class="co">## Instrumental inequality: TRUE </span></span>
<span id="cb2-16"><a href="#cb2-16"></a><span class="co">## Causal parameter Lower bound Upper bound</span></span>
<span id="cb2-17"><a href="#cb2-17"></a><span class="co">## ACE -0.09 0.74000</span></span>
<span id="cb2-18"><a href="#cb2-18"></a><span class="co">## P(Y|do(X=0)) 0.06 0.12000</span></span>
<span id="cb2-19"><a href="#cb2-19"></a><span class="co">## P(Y|do(X=1)) 0.03 0.80000</span></span>
<span id="cb2-20"><a href="#cb2-20"></a><span class="co">## CRR 0.25 13.33333</span></span>
<span id="cb2-21"><a href="#cb2-21"></a><span class="co">## </span></span>
<span id="cb2-22"><a href="#cb2-22"></a><span class="co">## Monotonicity inequality: FALSE</span></span></code></pre>
<div class="figure" style="text-align: center"><span id="fig:qrcode-shiny"></span>
<img src="Figures/qr-code-bpbounds-shiny-app.png" alt="Shiny app https://remlapmot.shinyapps.io/bpbounds" width="35%" />
<p class="caption">
Figure 1: Shiny app <a href="https://remlapmot.shinyapps.io/bpbounds" class="uri">https://remlapmot.shinyapps.io/bpbounds</a>
</p>
</div>
<div class="figure" style="text-align: center"><span id="fig:shiny-app-screenshot"></span>
<img src="Figures/bpbounds-shiny-app-screenshot.png" alt="Screenshot of our Shiny app." width="75%" />
<p class="caption">
Figure 2: Screenshot of our Shiny app.
</p>
</div>
<div class="figure" style="text-align: center"><span id="fig:qrcode-pkgdown"></span>
<img src="Figures/qrcode-bpbounds-pkgdown.png" alt="Package website https://remlapmot.github.io/bpbounds/" width="35%" />
<p class="caption">
Figure 3: Package website <a href="https://remlapmot.github.io/bpbounds/" class="uri">https://remlapmot.github.io/bpbounds/</a>
</p>
</div>
</div>
</div>
<div class="main">
<p>What <strong>range</strong> could your <strong>causal effect</strong> lie between if the instrumental variable assumptions held?<br><br> <strong>Find out</strong> with our <strong>bpbounds R package</strong> and <strong>Shiny app</strong>!</p>
</div>
<div class="main_bottom">
<img id="main-img-left" src=Figures/Shield-Twitter.jpg>
<img id="main-img-center" src=Figures/qr-code-poster-github-pages.png>
<img id="main-img-right" src=Figures/Shield-Twitter.jpg>
</div>
</div>
</body>
</html>