-
Notifications
You must be signed in to change notification settings - Fork 3
Expand file tree
/
Copy pathPvalues.html
More file actions
416 lines (343 loc) · 14.7 KB
/
Pvalues.html
File metadata and controls
416 lines (343 loc) · 14.7 KB
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
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta http-equiv="X-UA-Compatible" content="IE=EDGE" />
<title>The aggregate consequences of publishing only statistically significant treatment effects</title>
<script src="site_libs/header-attrs-2.14/header-attrs.js"></script>
<script src="site_libs/jquery-3.6.0/jquery-3.6.0.min.js"></script>
<meta name="viewport" content="width=device-width, initial-scale=1" />
<link href="site_libs/bootstrap-3.3.5/css/flatly.min.css" rel="stylesheet" />
<script src="site_libs/bootstrap-3.3.5/js/bootstrap.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/html5shiv.min.js"></script>
<script src="site_libs/bootstrap-3.3.5/shim/respond.min.js"></script>
<style>h1 {font-size: 34px;}
h1.title {font-size: 38px;}
h2 {font-size: 30px;}
h3 {font-size: 24px;}
h4 {font-size: 18px;}
h5 {font-size: 16px;}
h6 {font-size: 12px;}
code {color: inherit; background-color: rgba(0, 0, 0, 0.04);}
pre:not([class]) { background-color: white }</style>
<script src="site_libs/navigation-1.1/tabsets.js"></script>
<link href="site_libs/highlightjs-9.12.0/default.css" rel="stylesheet" />
<script src="site_libs/highlightjs-9.12.0/highlight.js"></script>
<style type="text/css">
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
span.underline{text-decoration: underline;}
div.column{display: inline-block; vertical-align: top; width: 50%;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
ul.task-list{list-style: none;}
</style>
<style type="text/css">code{white-space: pre;}</style>
<script type="text/javascript">
if (window.hljs) {
hljs.configure({languages: []});
hljs.initHighlightingOnLoad();
if (document.readyState && document.readyState === "complete") {
window.setTimeout(function() { hljs.initHighlighting(); }, 0);
}
}
</script>
<style type = "text/css">
.main-container {
max-width: 940px;
margin-left: auto;
margin-right: auto;
}
img {
max-width:100%;
}
.tabbed-pane {
padding-top: 12px;
}
.html-widget {
margin-bottom: 20px;
}
button.code-folding-btn:focus {
outline: none;
}
summary {
display: list-item;
}
details > summary > p:only-child {
display: inline;
}
pre code {
padding: 0;
}
</style>
<style type="text/css">
.dropdown-submenu {
position: relative;
}
.dropdown-submenu>.dropdown-menu {
top: 0;
left: 100%;
margin-top: -6px;
margin-left: -1px;
border-radius: 0 6px 6px 6px;
}
.dropdown-submenu:hover>.dropdown-menu {
display: block;
}
.dropdown-submenu>a:after {
display: block;
content: " ";
float: right;
width: 0;
height: 0;
border-color: transparent;
border-style: solid;
border-width: 5px 0 5px 5px;
border-left-color: #cccccc;
margin-top: 5px;
margin-right: -10px;
}
.dropdown-submenu:hover>a:after {
border-left-color: #adb5bd;
}
.dropdown-submenu.pull-left {
float: none;
}
.dropdown-submenu.pull-left>.dropdown-menu {
left: -100%;
margin-left: 10px;
border-radius: 6px 0 6px 6px;
}
</style>
<script type="text/javascript">
// manage active state of menu based on current page
$(document).ready(function () {
// active menu anchor
href = window.location.pathname
href = href.substr(href.lastIndexOf('/') + 1)
if (href === "")
href = "index.html";
var menuAnchor = $('a[href="' + href + '"]');
// mark it active
menuAnchor.tab('show');
// if it's got a parent navbar menu mark it active as well
menuAnchor.closest('li.dropdown').addClass('active');
// Navbar adjustments
var navHeight = $(".navbar").first().height() + 15;
var style = document.createElement('style');
var pt = "padding-top: " + navHeight + "px; ";
var mt = "margin-top: -" + navHeight + "px; ";
var css = "";
// offset scroll position for anchor links (for fixed navbar)
for (var i = 1; i <= 6; i++) {
css += ".section h" + i + "{ " + pt + mt + "}\n";
}
style.innerHTML = "body {" + pt + "padding-bottom: 40px; }\n" + css;
document.head.appendChild(style);
});
</script>
<!-- tabsets -->
<style type="text/css">
.tabset-dropdown > .nav-tabs {
display: inline-table;
max-height: 500px;
min-height: 44px;
overflow-y: auto;
border: 1px solid #ddd;
border-radius: 4px;
}
.tabset-dropdown > .nav-tabs > li.active:before {
content: "";
font-family: 'Glyphicons Halflings';
display: inline-block;
padding: 10px;
border-right: 1px solid #ddd;
}
.tabset-dropdown > .nav-tabs.nav-tabs-open > li.active:before {
content: "";
border: none;
}
.tabset-dropdown > .nav-tabs.nav-tabs-open:before {
content: "";
font-family: 'Glyphicons Halflings';
display: inline-block;
padding: 10px;
border-right: 1px solid #ddd;
}
.tabset-dropdown > .nav-tabs > li.active {
display: block;
}
.tabset-dropdown > .nav-tabs > li > a,
.tabset-dropdown > .nav-tabs > li > a:focus,
.tabset-dropdown > .nav-tabs > li > a:hover {
border: none;
display: inline-block;
border-radius: 4px;
background-color: transparent;
}
.tabset-dropdown > .nav-tabs.nav-tabs-open > li {
display: block;
float: none;
}
.tabset-dropdown > .nav-tabs > li {
display: none;
}
</style>
<!-- code folding -->
</head>
<body>
<div class="container-fluid main-container">
<div class="navbar navbar-default navbar-fixed-top" role="navigation">
<div class="container">
<div class="navbar-header">
<button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-bs-toggle="collapse" data-target="#navbar" data-bs-target="#navbar">
<span class="icon-bar"></span>
<span class="icon-bar"></span>
<span class="icon-bar"></span>
</button>
<a class="navbar-brand" href="index.html">The Social Science Knowledge Accumulation Initiative</a>
</div>
<div id="navbar" class="navbar-collapse collapse">
<ul class="nav navbar-nav">
<li>
<a href="index.html">Home</a>
</li>
<li>
<a href="Interventions.html">Evidence on Interventions</a>
</li>
<li>
<a href="Theories.html">Evidence on Theories</a>
</li>
<li>
<a href="methods.html">Methods</a>
</li>
<li>
<a href="about.html">About</a>
</li>
</ul>
<ul class="nav navbar-nav navbar-right">
</ul>
</div><!--/.nav-collapse -->
</div><!--/.container -->
</div><!--/.navbar -->
<div id="header">
<h1 class="title toc-ignore">The aggregate consequences of publishing only statistically significant treatment effects</h1>
</div>
<p><strong>This article is in process.</strong> <strong>You can fork it and propose new writing and simulations by following the steps <a href="https://chabefer.github.io/SKY/tutoSKY.html">here</a>.</strong> <strong>You can leave a comment on what you think should be addressed <a href="https://economics.stackexchange.com/questions/29648/the-aggregate-consequences-of-publishing-only-statistically-significant-treatmen">here</a>.</strong></p>
<p>There is ample debate recently around the use of p-values and test statistics in science. On the one hand, some argue that only statistically significant results are interesting while others suggest that p-values bias scientific results and that every result should be published. In this paper, I want to put some flesh into this debate by comparing the consequences of using test statistics and p-values to filter scientific results with the practice of publishing all results for a decision-maker faced with the problem of deciding which projects to scale up.</p>
<div id="the-problem" class="section level1">
<h1>The problem</h1>
<p>A decision maker has a portfolio a feasible projects that she can implement at scale. Each project <span class="math inline">\(p\)</span> provides a return <span class="math inline">\(\theta^p\)</span>. For simplicity, <span class="math inline">\(\theta^p\)</span> is distributed normally with mean <span class="math inline">\(\bar{\theta}\)</span> and variance <span class="math inline">\(\bar{\sigma}^2\)</span>. Each project costs <span class="math inline">\(C\)</span> in order to be implemented at scale.</p>
<p>The decision maker has access to a set of results from <span class="math inline">\(J\)</span> RCTs run for each project <span class="math inline">\(p\)</span>. Each RCT reports on an estimate <span class="math inline">\(\hat{\theta}^p_j\)</span> for <span class="math inline">\(\theta^p\)</span>, which has a variance of <span class="math inline">\(\sigma^2\)</span>.</p>
<p>I am going to compare two decision rules for the moment:</p>
<ol style="list-style-type: decimal">
<li>Aggregate all <span class="math inline">\(J\)</span> results into an average <span class="math inline">\(\hat{\theta}^p_J\)</span> and implement the programs for which <span class="math inline">\(\hat{\theta}^p_J>C\)</span>.</li>
<li>Aggregate only the results that are statistically significant into an average <span class="math inline">\(\hat{\theta}^p_{J^S}\)</span> and implement the programs for which <span class="math inline">\(\hat{\theta}^p_{J^S}>C\)</span>.</li>
</ol>
<p>In the future, I plan to model two additional decision procedures:</p>
<ol style="list-style-type: decimal">
<li>Proportion of statistically significant results</li>
<li>Sophisticated/Bayesian decision maker correcting and not correcting for publication bias</li>
</ol>
<p>For each scenario, I’m comparing the total value of the outcome reached: each implemented project brings <span class="math inline">\(\theta^p-C\)</span> while each non implemented project brings <span class="math inline">\(C\)</span>.</p>
</div>
<div id="simulations" class="section level1">
<h1>Simulations</h1>
<p>In this section, I’m parameterizing the problem in order to provide a simple illustration. Let’s choose the parameter values first:</p>
<pre class="r"><code>theta.bar <- 0.5
sigma.bar <- 0.5
C <- 1
sigma <- 0.5
Np <- 1000
J <- 10</code></pre>
<p>Let’s illustrate now the distribution of possible project returns with the red dotted line showing <span class="math inline">\(C\)</span>, the threshold over which a project is profitable:</p>
<pre class="r"><code>set.seed(1234)
theta.p <- rnorm(Np,theta.bar,sigma.bar)
data <- as.data.frame(theta.p)
ggplot(data, aes(x=theta.p)) +
geom_histogram(colour="black", fill="white") +
geom_vline(aes(xintercept=C),color="red", linetype="dashed", size=1)+
theme_bw()</code></pre>
<p><img src="Pvalues_files/figure-html/returns-1.png" width="672" /></p>
<p>Let’s now draw <span class="math inline">\(J\)</span> samples for each of these projects.</p>
<pre class="r"><code>draw.samples <- function(seed,J,data,sigma){
set.seed(seed)
return(rnorm(J,data[seed,1],sigma))
}
test <- draw.samples(2,J,data,sigma)
samples <- lapply(1:Np,draw.samples,J=J,data=data,sigma=sigma)
samples <- data.frame(unlist(samples))
samples$p <- unlist(lapply(1:Np,rep,times=J))
samples$theta.p <- unlist(lapply(theta.p,rep,times=J))
samples$id <- 1:(Np*J)
colnames(samples) <- c('theta.jp','p','theta.p','id')</code></pre>
<p>Before looking at the results, let’s first compute the two decisions rules: the average outcome of all estimates for each program and the average of the significant treatment effects (the ones with a t-stat larger than 1.96, according to the usual aproach):</p>
<pre class="r"><code>means.samples <- samples %>%
group_by(p) %>%
summarise(theta.p.hat = mean(theta.jp))
samples <- merge(samples,means.samples,by='p')
samples <- samples %>%
mutate(test = theta.jp/sigma) %>%
mutate(theta.jp.s = theta.jp) %>%
mutate(theta.jp.s = ifelse(test<1.96,0,theta.jp.s))
means.samples.s <- samples %>%
group_by(p) %>%
filter(theta.jp.s>0) %>%
summarise(theta.p.hat.s = mean(theta.jp.s))
samples <- merge(samples,means.samples.s,by='p',all.x = TRUE)
samples[is.na(samples)] <- 0</code></pre>
<p>Let’s see what it looks like for the first three programs:</p>
<pre class="r"><code>ggplot(filter(samples,p<=3), aes(x=as.factor(id), y=theta.jp,fill=as.factor(p))) +
geom_bar(position=position_dodge(), stat="identity") +
geom_errorbar(aes(ymin=theta.jp-1.96*sigma, ymax=theta.jp+1.96*sigma), width=.2,position=position_dodge(.9))+
geom_hline(aes(yintercept=theta.p, colour=as.factor(p)), linetype="solid")+
geom_hline(aes(yintercept=theta.p.hat, colour=as.factor(p)), linetype="dashed")+
geom_hline(aes(yintercept=theta.p.hat.s, colour=as.factor(p)), linetype="dotted")+
theme_bw()+
xlab('RCTs')+
ylab('estimates')+
scale_fill_discrete(name="Programs")+
scale_colour_discrete(name="Programs")+
scale_linetype_manual(name="Estimates",
values=c('solid','dashed','dotted'),
labels=c('Theta.p','Theta.p.hat','Theta.p.hat.s'))</code></pre>
<p><img src="Pvalues_files/figure-html/example.samples-1.png" width="672" /></p>
<p>On the plot, the solid lines give the true effect of the program while the dashed lines give the estimate of the effet of the program obtained by aggregating ALL the individual estimates and the dotted lines give the estimates of the effect of the program obtained by aggregating only the statistically significant effects.</p>
</div>
<div id="results" class="section level1">
<h1>Results</h1>
<p>As appears clearly on the plot, aggregating only statistically significant results blurs the decision: program 2 appears more effective than program 3, whereas the reverse is actually true, and both programs get implemented, whereas only program 3 should be (it is the only one to have true returns above <span class="math inline">\(C=1\)</span>).</p>
</div>
</div>
<script>
// add bootstrap table styles to pandoc tables
function bootstrapStylePandocTables() {
$('tr.odd').parent('tbody').parent('table').addClass('table table-condensed');
}
$(document).ready(function () {
bootstrapStylePandocTables();
});
</script>
<!-- tabsets -->
<script>
$(document).ready(function () {
window.buildTabsets("TOC");
});
$(document).ready(function () {
$('.tabset-dropdown > .nav-tabs > li').click(function () {
$(this).parent().toggleClass('nav-tabs-open');
});
});
</script>
<!-- code folding -->
<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
(function () {
var script = document.createElement("script");
script.type = "text/javascript";
script.src = "https://mathjax.rstudio.com/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML";
document.getElementsByTagName("head")[0].appendChild(script);
})();
</script>
</body>
</html>