-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathmr2019-okezie-poster.html
714 lines (689 loc) · 19 KB
/
mr2019-okezie-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
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
<!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>Investigating a pseudo-horseshoe prior for the MR-Egger model</title>
<script src="mr2019-okezie-poster_files/htmlwidgets-1.3/htmlwidgets.js"></script>
<script src="mr2019-okezie-poster_files/viz-0.3/viz.js"></script>
<link href="mr2019-okezie-poster_files/DiagrammeR-styles-0.2/styles.css" rel="stylesheet" />
<script src="mr2019-okezie-poster_files/grViz-binding-1.0.1/grViz.js"></script>
<script src="mr2019-okezie-poster_files/kePrint-0.0.1/kePrint.js"></script>
<!--
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: 122pt;
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">Investigating a pseudo-horseshoe prior for the MR-Egger model</h1><br>
<h3 id="author" class="author">
<strong>Okezie Uche Ikonne</strong><sup> 1, <a class="orcid" href="https://orcid.org/0000-0002-8461-8099"><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">
Frank Dondelinger<sup>2</sup>
Tom Palmer<sup>1</sup>
</h5>
<p id="affiliation" class="affiliation">
<sup>1</sup> Department of Mathematics and Statistics, Lancaster University<br> <sup>2</sup> Faculty of Health and Medicine, Lancaster University
</p>
</div>
<div id="introduction" class="section level1">
<h1>Introduction</h1>
<!-- Mendelian randomization (MR) uses genetic variants as instruments to estimate causal effects in epidemiology (@davey2003mendelian). -->
<ul>
<li>The MR-Egger model can consistently estimate the causal effect in the presence of pleiotropy given the InSIDE assumption holds (<span class="citation">Bowden, Davey Smith, and Burgess (2015)</span>).</li>
<li><span class="citation">Schmidt and Dudbridge (2017)</span> used weakly informative priors for the MR-Egger model. Other authors have investigated alternative prior distributions in MR analyses (<span class="citation">Berzuini et al. (2018)</span>).</li>
</ul>
<!--
<div class="figure" style="text-align: center">
<div id="htmlwidget-95e7d781744f1de0f8da" style="width:672px;height:350px;" class="grViz html-widget"></div>
<script type="application/json" data-for="htmlwidget-95e7d781744f1de0f8da">{"x":{"diagram":"\n digraph mrdag {\n\n graph [rankdir=TB, layout=neato]\n\n node [shape=box,eight=0.3, width=0.3]\n U [label=\"Confounders\", pos=\"3,1!\"]\n \n node [shape=box, height=0.3, width=0.3]\n Y1 [label=\"Outcome\", pos=\"4,0!\"]\n\n node [shape=box, height=0.3, width=0.3]\n G2 [label=\"SNPs\", pos=\"0,0!\"]\n X1 [label=\"Exposure\", pos=\"2,0!\"]\n \n { rank = same; Y1 }\n\n \n\t G2 -> {X1}\n U -> X1\n U -> Y1\n X1 -> {Y1}\n \n }\n ","config":{"engine":"dot","options":null}},"evals":[],"jsHooks":[]}</script>
<p class="caption">(\#fig:dag)DAG representing MR analysis</p>
</div>
-->
<p>The objectives of this research work are to:</p>
<ul>
<li>implement Bayesian estimation of IVW and MR-Egger models for a range of prior distributions in an R package.</li>
<li>investigate model performance for a range of simulated pleiotropic scenarios and a range of priors.</li>
</ul>
</div>
<div id="methods" class="section level1">
<h1>Methods</h1>
<ul>
<li><p>We implemented Bayesian estimation of the IVW and MR-Egger models in an R package <code>mrbayes</code> which automates fitting these models in the JAGS software.</p></li>
<li><p>We provide the user with a choice of priors or let them specify their own.</p></li>
<li><p>The MR-Egger model is written as;
<span class="math display">\[\frac{\Gamma_j}{\sigma_{y_j}^2} = \frac{\alpha}{{\sigma_{y_j}^2}} + \frac{\beta\gamma_j}{{\sigma_{y_j}^2}} + \varepsilon_j,\quad \varepsilon_j \sim N(0,\sigma^2)\]</span></p></li>
<li><p>Uninformative Prior <span class="math display">\[p(\alpha) \sim N(0,1000),\ p(\beta) \sim N(0,1000),\ p(\sigma) \sim U(10,10)\]</span></p></li>
<li><p>Weakly Informative Prior <span class="math display">\[p(\alpha) \sim N(0,1),\ p(\beta) \sim N(0,1),\ p(\sigma) \sim U(10,10)\]</span></p></li>
</ul>
<!-- The posterior of the models is decomposed as; $$P(\mu_\beta,\sigma_\beta|\Gamma_j) \propto P(\Gamma_j|\mu_\beta,\sigma_\beta)P(\mu_\beta,\sigma_\beta)$$ -->
<ul>
<li><p>Pseudo-Horseshoe Prior<span class="math display">\[p(\alpha) \sim N(0,1),\ p(\beta) \sim C(0,1),\ p(\sigma) \sim IG(0.5,0.5)\]</span></p></li>
<li><p>Figure <a href="#fig:dens">1</a> shows the densities of the priors.</p></li>
</ul>
<!--
$$
\begin{aligned}
p(\alpha) &\sim N(0,1)\\
p(\beta) &\sim C(0,1)\\
p(\sigma) &\sim IG(0.5,0.5).
\end{aligned}
$$
-->
<!--
The posterior distribution for inference is given by,
$$
P(\alpha,\beta,\sigma|\Gamma_j) \propto P(\Gamma_j|\alpha,\beta,\sigma)P(\alpha,\beta,\sigma).
$$
-->
<!--
Implementing the horseshoe prior on a summary-level dataset would lead to overshrinkage reducing the effect of strong instruments.
-->
</div>
<div id="results" class="section level1">
<h1>Results</h1>
<div id="simulations" class="section level2">
<h2>Simulations</h2>
<ul>
<li>We simulated two-sample summary-level data under directional pleiotropy with a true value of the causal effect of 0.05. The performance of the model was assessed using coverage and power. Results in table <a href="#tab:simtab">1</a> and figure <a href="#fig:simplot">2</a>.</li>
</ul>
<!-- Comments about results -->
</div>
<div id="example" class="section level2">
<h2>Example</h2>
<ul>
<li>We fitted summary data models to a dataset investigating the effect of body mass index on insulin resistance <span class="citation">(Richmond et al. 2017)</span>.</li>
<li>We compared Bayesian MR-Egger model estimates from models including horseshoe priors from the horseshoe package <span class="citation">(van der Pas et al. 2016)</span>.</li>
<li>Results are presented in table <a href="#tab:restab">2</a> and figure <a href="#fig:plot">3</a>.</li>
</ul>
</div>
</div>
<div id="conclusion" class="section level1">
<h1>Conclusion</h1>
<ul>
<li>We present Bayesian estimation of the IVW and MR-Egger models in our <code>mrbayes</code> package.</li>
<li>In future work we could implement Bayesian estimation of MVMR models and perform estimation using other programs such as Stan.</li>
</ul>
</div>
<div id="references" class="section level1">
<h1>References</h1>
<div id="refs" class="references">
<div id="ref-berzuini2018bayesian">
<p>Berzuini, Carlo, Hui Guo, Stephen Burgess, and Luisa Bernardinelli. 2018. “A Bayesian Approach to Mendelian Randomization with Multiple Pleiotropic Variants.” <em>Biostatistics</em>.</p>
</div>
<div id="ref-bowden-ije-2015">
<p>Bowden, Jack, George Davey Smith, and Stephen Burgess. 2015. “Mendelian randomization with invalid instruments: effect estimation and bias detection through Egger regression.” <em>International Journal of Epidemiology</em> 44 (2): 512–25. <a href="https://dx.doi.org/10.1093/ije/dyv080">https://dx.doi.org/10.1093/ije/dyv080</a>.</p>
</div>
<div id="ref-richmond2017investigating">
<p>Richmond, Rebecca, Kaitlin Wade, Laura Corbin, Jack Bowden, Gibran Hemani, Nicholas Timpson, and George Davey Smith. 2017. “Investigating the role of insulin in increased adiposity: Bi-directional Mendelian randomization study.” <em>bioRxiv</em>, 155739. <a href="https://doi.org/10.1101/155739">https://doi.org/10.1101/155739</a>.</p>
</div>
<div id="ref-schmidt-ije-2017">
<p>Schmidt, A F, and F Dudbridge. 2017. “Mendelian randomization with Egger pleiotropy correction and weakly informative Bayesian priors.” <em>International Journal of Epidemiology</em> 47 (4): 1217–28. <a href="https://dx.doi.org/10.1093/ije/dyx254">https://dx.doi.org/10.1093/ije/dyx254</a>.</p>
</div>
<div id="ref-horsepackage">
<p>van der Pas, Stephanie, James Scott, Antik Chakraborty, and Anirban Bhattacharya. 2016. <em>Horseshoe: Implementation of the Horseshoe Prior</em>. <a href="https://CRAN.R-project.org/package=horseshoe">https://CRAN.R-project.org/package=horseshoe</a>.</p>
</div>
</div>
<div id="section" class="section level4">
<h4></h4>
</div>
</div>
<div id="figures-and-tables" class="section level1">
<h1>Figures and Tables</h1>
<div class="figure" style="text-align: center"><span id="fig:dens"></span>
<img src="mr2019-okezie-poster_files/figure-html/dens-1.png" alt="Density of alternative prior distributions implemented in our package." width="672" />
<p class="caption">
Figure 1: Density of alternative prior distributions implemented in our package.
</p>
</div>
<table class="table table-striped" style="margin-left: auto; margin-right: auto;">
<caption>
<span id="tab:simtab">Table 1: </span>Model performance under directional pleiotropy.
</caption>
<thead>
<tr>
<th style="text-align:left;">
</th>
<th style="text-align:right;">
IVW
</th>
<th style="text-align:right;">
MR-Egger
</th>
<th style="text-align:right;">
MR-Egger with pseudo-HS prior
</th>
<th style="text-align:right;">
MR-Egger with horseshoe prior
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
Estimate
</td>
<td style="text-align:right;">
0.1607
</td>
<td style="text-align:right;">
0.0293
</td>
<td style="text-align:right;">
0.0302
</td>
<td style="text-align:right;">
0.0374
</td>
</tr>
<tr>
<td style="text-align:left;">
Power
</td>
<td style="text-align:right;">
1.0000
</td>
<td style="text-align:right;">
0.1044
</td>
<td style="text-align:right;">
0.0950
</td>
<td style="text-align:right;">
0.0994
</td>
</tr>
<tr>
<td style="text-align:left;">
Coverage
</td>
<td style="text-align:right;">
0.0036
</td>
<td style="text-align:right;">
0.8946
</td>
<td style="text-align:right;">
0.9046
</td>
<td style="text-align:right;">
0.9044
</td>
</tr>
</tbody>
</table>
<div class="figure" style="text-align: center"><span id="fig:simplot"></span>
<img src="mr2019-okezie-poster_files/figure-html/simplot-1.png" alt="Distribution of causal effect estimates under directional pleiotropy." width="672" />
<p class="caption">
Figure 2: Distribution of causal effect estimates under directional pleiotropy.
</p>
</div>
<div class="figure" style="text-align: center"><span id="fig:plot"></span>
<img src="mr2019-okezie-poster_files/figure-html/plot-1.png" alt="Scatter plot of genotype-disease versus genotype-phenotype estimates for the effect of BMI on insulin resistance." width="672" />
<p class="caption">
Figure 3: Scatter plot of genotype-disease versus genotype-phenotype estimates for the effect of BMI on insulin resistance.
</p>
</div>
<table class="table table-striped" style="margin-left: auto; margin-right: auto;">
<caption>
<span id="tab:restab">Table 2: </span>Estimates of the causal effect of BMI on insulin resistance.
</caption>
<thead>
<tr>
<th style="text-align:left;">
Model
</th>
<th style="text-align:left;">
Coefficient
</th>
<th style="text-align:left;">
Estimate
</th>
<th style="text-align:left;">
95% Confidence/Credible Interval
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
IVW
</td>
<td style="text-align:left;">
Slope
</td>
<td style="text-align:left;">
0.5797
</td>
<td style="text-align:left;">
-0.1985, 1.3579
</td>
</tr>
<tr>
<td style="text-align:left;">
MR-Egger
</td>
<td style="text-align:left;">
Intercept
</td>
<td style="text-align:left;">
-0.0544
</td>
<td style="text-align:left;">
-0.1258, 04
</td>
</tr>
<tr>
<td style="text-align:left;">
MR-Egger
</td>
<td style="text-align:left;">
Slope
</td>
<td style="text-align:left;">
3.7586
</td>
<td style="text-align:left;">
-0.4793, 7.9966
</td>
</tr>
<tr>
<td style="text-align:left;">
MR-Egger with pseudo-HS prior
</td>
<td style="text-align:left;">
Intercept
</td>
<td style="text-align:left;">
-0.0143
</td>
<td style="text-align:left;">
-0.0862, 0.0327
</td>
</tr>
<tr>
<td style="text-align:left;">
MR-Egger with pseudo-HS prior
</td>
<td style="text-align:left;">
Slope
</td>
<td style="text-align:left;">
1.3488
</td>
<td style="text-align:left;">
-1.2967, 5.6133
</td>
</tr>
<tr>
<td style="text-align:left;">
MR-Egger with HS prior
</td>
<td style="text-align:left;">
Intercept
</td>
<td style="text-align:left;">
-0.023
</td>
<td style="text-align:left;">
-0.0997, 0.0248
</td>
</tr>
<tr>
<td style="text-align:left;">
MR-Egger with HS prior
</td>
<td style="text-align:left;">
Slope
</td>
<td style="text-align:left;">
1.8779
</td>
<td style="text-align:left;">
-0.9604, 64
</td>
</tr>
</tbody>
</table>
</div>
</div>
<div class="main">
<p>Bayesian estimation of the <strong>MR-Egger</strong> model using <strong>informative priors</strong> can <strong>reduce bias</strong> in the presence of <strong>pleiotropy</strong>.</p>
</div>
<div class="main_bottom">
<img id="main-img-left" src=Figures/Shield-Twitter.jpg>
<img id="main-img-center" src=>
<img id="main-img-right" src=Figures/Shield-Twitter.jpg>
</div>
</div>
<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
(function () {
var script = document.createElement("script");
script.type = "text/javascript";
var src = "true";
if (src === "" || src === "true") src = "https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-MML-AM_CHTML";
if (location.protocol !== "file:" && /^https?:/.test(src))
src = src.replace(/^https?:/, '');
script.src = src;
document.getElementsByTagName("head")[0].appendChild(script);
})();
</script>
</body>
</html>