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<h1 class="title toc-ignore">Posterior Inference with Gaussian Derivative Likelihood: Model and Result</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-06-14</em></h4>
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<p><strong>Last updated:</strong> 2018-05-12</p>
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<div id="gd-ash-model" class="section level2">
<h2><code>GD-ASH</code> Model</h2>
<p>Recall the typical <code>GD-ASH</code> model is</p>
<p><span class="math display">\[
\begin{array}{l}
\beta_j \sim \sum\pi_k N\left(0, \sigma_k^2\right) \ ;\\
\hat\beta_j = \beta_j + \hat s_j z_j \ ;\\
z_j \sim N\left(0, 1\right), \text{ correlated} \ .
\end{array}
\]</span> Then we are fitting the empirical distribution of <span class="math inline">\(z\)</span> with Gaussian derivatives</p>
<p><span class="math display">\[
f(z) = \sum w_l\frac{1}{\sqrt{l!}}\varphi^{(l)}(z) \ .
\]</span> Therefore, in essence, we are changing the likelihood of <span class="math inline">\(\hat\beta_j | \hat s_j, \beta_j\)</span> from correlated <span class="math inline">\(N\left(\beta_j, \hat s_j^2\right)\)</span> to independent <span class="math inline">\(\frac{1}{\hat s_j}f\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right)\)</span>, which using Gaussian derivatives is</p>
<p><span class="math display">\[
\frac{1}{\hat s_j}\sum w_l \frac{1}{\sqrt{l!}} \varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right) \ .
\]</span> Note that when <span class="math inline">\(f = \varphi\)</span> it becomes the independent <span class="math inline">\(N\left(\beta_j, \hat s_j^2\right)\)</span> case.</p>
</div>
<div id="gd-lik-model" class="section level2">
<h2><code>GD-Lik</code> Model</h2>
<p>Therefore, if we use Gaussian derivatives instead of Gaussian as the likelihood, the posterior distribution of <span class="math inline">\(\beta_j | \hat s_j, \hat\beta_j\)</span> should be</p>
<p><span class="math display">\[
\begin{array}{rcl}
f\left(\beta_j \mid \hat s_j, \hat\beta_j\right)
&=&
\frac{
\displaystyle g\left(\beta_j\right)
f\left(\hat\beta_j \mid \hat s_j, \beta_j \right)
}{
\displaystyle\int g\left(\beta_j\right) f\left(\hat\beta_j \mid \hat s_j, \beta_j \right) d\beta_j
}\\
&=&
\frac{
\displaystyle
\sum\pi_k\sum w_l
\frac{1}{\sigma_k}
\varphi\left(\frac{\beta_j - \mu_k}{\sigma_k}\right)
\frac{1}{\hat s_j}
\frac{1}{\sqrt{l!}}
\varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right)
}{
\displaystyle
\sum\pi_k\sum w_l \int
\frac{1}{\sigma_k}
\varphi\left(\frac{\beta_j - \mu_k}{\sigma_k}\right)
\frac{1}{\hat s_j}
\frac{1}{\sqrt{l!}}
\varphi^{(l)}\left(\frac{\hat\beta_j - \beta_j}{\hat s_j}\right)
d\beta_j
} \ .
\end{array}
\]</span> The denominator <a href="ash_gd.html#normal_mixture_prior">readily has an analytic form</a> which is</p>
<p><span class="math display">\[
\displaystyle
\sum\pi_k \sum w_l
\frac{\hat s_j^l}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+1}}
\frac{1}{\sqrt{l!}}
\varphi^{(l)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right)
:=
\sum \pi_k \sum w_l f_{jkl}
\ .
\]</span></p>
</div>
<div id="posterior-mean" class="section level2">
<h2>Posterior mean</h2>
<p>After algebra, the posterior mean is given by</p>
<p><span class="math display">\[
E\left[\beta_j \mid \hat s_j, \hat \beta_j \right]
=
\int \beta_j
f\left(\beta_j \mid \hat s_j, \hat\beta_j\right) d\beta_j
=
\displaystyle
\frac{
\sum \pi_k \sum w_l m_{jkl}
}{
\sum \pi_k \sum w_l f_{jkl}
} \ ,
\]</span> where <span class="math inline">\(f_{jkl}\)</span> is defined as above and <span class="math display">\[
m_{jkl} =
-
\frac{\hat s_j^l \sigma_k^2}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+2}}
\frac{1}{\sqrt{l!}}
\varphi^{(l+1)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right)
+
\frac{\hat s_j^l\mu_k}{\left(\sqrt{\sigma_k^2 + \hat s_j^2}\right)^{l+1}}
\frac{1}{\sqrt{l!}}
\varphi^{(l)}\left(\frac{\hat\beta_j - \mu_k}{\sqrt{\sigma_k^2 + \hat s_j^2}}\right) \ .
\]</span></p>
</div>
<div id="local-fdr" class="section level2">
<h2>Local FDR</h2>
<p>Assuming <span class="math inline">\(\mu_k \equiv 0\)</span>, the <code>lfdr</code> is given by <span class="math display">\[
p\left(\beta_j = 0\mid \hat s_j, \hat \beta_j\right)
=
\frac{
\pi_0
\sum w_l
\frac{1}{\hat s_j}
\frac{1}{\sqrt{l!}}
\varphi^{(l)}\left(\frac{\hat\beta_j}{\hat s_j}\right)
}{
\sum \pi_k
\sum w_l
f_{jkl}
} \ .
\]</span></p>
</div>
<div id="local-fsr" class="section level2">
<h2>Local FSR</h2>
<p>Right now the analytic form of <code>lfsr</code> using Gaussian derivatives is unavailable.</p>
</div>
<div id="simulation" class="section level2">
<h2>Simulation</h2>
<p>The correlated <span class="math inline">\(N\left(0, 1\right)\)</span> <span class="math inline">\(z\)</span> scores are simulated from the GTEx/Liver data by the <a href="nullpipeline.html">null pipeline</a>. In order to get a better sense of the effectiveness of <code>GD-ASH</code> and <code>GD-Lik</code>, we are using data sets more distorted by correlation in the simulation. In particular, we are using an “inflation” batch, defined as the standard error of the correlated <span class="math inline">\(z\)</span> no less than <span class="math inline">\(1.2\)</span>, and a “deflation” batch, defined as that no greater than <span class="math inline">\(0.8\)</span>. Out of <span class="math inline">\(1000\)</span> simulated data sets, there are <span class="math inline">\(109\)</span> inflation ones and <span class="math inline">\(99\)</span> deflation ones.</p>
<p>In order to create realistic heterskedastic estimated standard error, <span class="math inline">\(\hat s_j\)</span>’s are also simulated from the same <a href="nullpipeline.html">null pipeline</a>. Let <span class="math inline">\(\sigma^2 = \frac1n \sum\limits_{j = 1}^n \hat s_j^2\)</span> be the average strength of the heteroskedastic noise, and the true effects <span class="math inline">\(\beta_j\)</span>’s are simulated from <span class="math display">\[
0.6\delta_0 + 0.3N\left(0, \sigma^2\right) + 0.1N\left(0, \left(2\sigma\right)^2\right) \ .
\]</span></p>
<p>Then let <span class="math inline">\(\hat\beta_j = \beta_j + \hat s_j z_j\)</span>. We are using <span class="math inline">\(\hat\beta_j\)</span>, <span class="math inline">\(\hat s_j\)</span>, along with <span class="math inline">\(\hat z_j = \hat\beta_j / \hat s_j\)</span>, <span class="math inline">\(\hat p_j= 2\left(1 - \Phi\left(\left|\hat z_j\right|\right)\right)\)</span>, as the summary statistics fed to <code>GD-ASH</code> and <code>GD-Lik</code>, as well as into <code>BH</code>, <code>qvalue</code>, <code>locfdr</code>, <code>ASH</code> for a comparison.</p>
<pre class="r"><code>source("../code/gdash_lik.R")</code></pre>
<pre class="r"><code>z.mat = readRDS("../output/z_null_liver_777.rds")
se.mat = readRDS("../output/sebetahat_null_liver_777.rds")</code></pre>
<pre class="r"><code>z.sd = apply(z.mat, 1, sd)
inflation.index = (z.sd >= 1.2)
deflation.index = (z.sd <= 0.8)
z.inflation = z.mat[inflation.index, ]
se.inflation = se.mat[inflation.index, ]
## Number of inflation data sets
nrow(z.inflation)</code></pre>
<pre><code>[1] 109</code></pre>
<pre class="r"><code>z.deflation = z.mat[deflation.index, ]
se.deflation = se.mat[deflation.index, ]
## Number of deflation data sets
nrow(z.deflation)</code></pre>
<pre><code>[1] 99</code></pre>
</div>
<div id="inflation-data-sets" class="section level2">
<h2>Inflation data sets</h2>
<div id="some-examples-of-inflated-correlated-null-z-scores" class="section level3">
<h3>Some examples of inflated correlated null <span class="math inline">\(z\)</span> scores</h3>
<p><img src="figure/gd_lik_2.rmd/inflation%20examples-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of inflation examples-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/inflation examples-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/853a484bfacf347e109f6c8fb3ffaab5f4d6cc02/docs/figure/gd_lik_2.rmd/inflation examples-1.png" target="_blank">853a484</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-11-07
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/4f032ada03363cc112612f17b5856e75c479feb8/docs/figure/gd_lik_2.rmd/inflation examples-1.png" target="_blank">4f032ad</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-11-05
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/a2512ac60d3081797fad7bea49dc0f4e7e17837b/docs/figure/gd_lik_2.rmd/inflation examples-1.png" target="_blank">a2512ac</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-07-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/b7c3984b48103ee6d72ca9a3be90ce473cbfefa1/docs/figure/gd_lik_2.rmd/inflation examples-1.png" target="_blank">b7c3984</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-24
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/e6d5a6488d6ee53e08cc396a42e9538adc087dd2/docs/figure/gd_lik_2.rmd/inflation examples-1.png" target="_blank">e6d5a64</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/inflation examples-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="hat-pi_0" class="section level3">
<h3><span class="math inline">\(\hat \pi_0\)</span></h3>
<p><code>locfdr</code> overestimates, <code>ASH</code> underestimates, <code>GD-ASH</code> on target, <code>qvalue</code> surprisingly good.</p>
<p><img src="figure/gd_lik_2.rmd/inflation%20pihat0-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of inflation pihat0-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/inflation pihat0-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/inflation pihat0-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="mse" class="section level3">
<h3>MSE</h3>
<p><code>GD-Lik</code> clearly improves the posterior estimates of <code>GD-ASH</code>.</p>
<p><img src="figure/gd_lik_2.rmd/inflation%20mse-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of inflation mse-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/inflation mse-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/inflation mse-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="auc" class="section level3">
<h3>AUC</h3>
<p>For almost all methods, <code>.q</code> means using q values, and <code>.l</code> using local FDRs. <code>GD-Lik</code> is the best, yet even the vanilla <span class="math inline">\(p\)</span> values are not much worse. It indicates that all the methods based on summary statistics indeed make few changes to the order of original <span class="math inline">\(p\)</span> values. Worth noting is that <code>locfdr</code> doesn’t perform well, and <code>lfdr</code>’s give a drastically different result than q values do, probably due to some artifacts like <a href="auc_pvalue.html">ties</a>.</p>
<p><img src="figure/gd_lik_2.rmd/inflation%20auc-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of inflation auc-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/inflation auc-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/inflation auc-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="q-values-and-positive-fdr-pfdr-calibration" class="section level3">
<h3>q values and positive FDR (pFDR) calibration</h3>
<p>Dashed lines are <span class="math inline">\(y = x\)</span> and <span class="math inline">\(y = 2x\)</span>. <code>ASH</code> and <code>qvalue</code> are too liberal, and <code>locfdr</code> is too conservative. <code>GD-ASH</code> and <code>BH</code> give very similar results and not far off. <code>BH</code>’s calibrates pFDR relatively well, even though it’s only guaranteed to control FDR under independence. <code>GD-Lik</code> calibrates pFDR almost precisely.</p>
<div id="model-and-data-generation" class="section level4">
<h4>Model and data generation</h4>
<p><span class="math display">\[
\begin{array}{l}
\hat\beta_j = \beta_j + \sigma_j z_j \\
z_j \sim N(0, 1), \text{ correlated, simulated from real data}\\
\sigma_j : \text{ heteroskedastic, simulated from real data}\\
\beta_j \sim 0.6\delta_0 + 0.3N(0, \sigma^2) + 0.1N\left(0, \left(2\sigma\right)^2\right)
\end{array}
\]</span></p>
<p><img src="figure/gd_lik_2.rmd/inflation%20qvalue%20calibration%20plotting-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of inflation qvalue calibration plotting-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/inflation qvalue calibration plotting-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/b7c3984b48103ee6d72ca9a3be90ce473cbfefa1/docs/figure/gd_lik_2.rmd/inflation qvalue calibration plotting-1.png" target="_blank">b7c3984</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-24
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/inflation qvalue calibration plotting-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/gd_lik_2.rmd/inflation%20qvalue%20calibration%20plotting-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of inflation qvalue calibration plotting-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/inflation qvalue calibration plotting-2.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/87c16f961e270fa9688b960346a01683287bb0d3/docs/figure/gd_lik_2.rmd/inflation qvalue calibration plotting-2.png" target="_blank">87c16f9</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-07-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/a2512ac60d3081797fad7bea49dc0f4e7e17837b/docs/figure/gd_lik_2.rmd/inflation qvalue calibration plotting-2.png" target="_blank">a2512ac</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-07-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/inflation qvalue calibration plotting-2.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="fdr-calibration" class="section level3">
<h3>FDR calibration</h3>
<p><img src="figure/gd_lik_2.rmd/inflation%20FDR%20calibration%20plotting-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of inflation FDR calibration plotting-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/inflation FDR calibration plotting-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/a2512ac60d3081797fad7bea49dc0f4e7e17837b/docs/figure/gd_lik_2.rmd/inflation FDR calibration plotting-1.png" target="_blank">a2512ac</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-07-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/gd_lik_2.rmd/inflation%20FDR%20calibration%20plotting-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of inflation FDR calibration plotting-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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</th>
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</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/inflation FDR calibration plotting-2.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/a2512ac60d3081797fad7bea49dc0f4e7e17837b/docs/figure/gd_lik_2.rmd/inflation FDR calibration plotting-2.png" target="_blank">a2512ac</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-07-03
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="s-values-and-positive-fsr-pfsr-calibration" class="section level3">
<h3>s values and positive FSR (pFSR) calibration</h3>
<p>Both <code>ASH</code> and <code>GD-ASH</code> are too liberal, although <code>GD-ASH</code> is not too far off.</p>
<p><img src="figure/gd_lik_2.rmd/inflation%20svalue%20calibration%20plotting-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of inflation svalue calibration plotting-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<th style="text-align:left;">
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</th>
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</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/inflation svalue calibration plotting-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/b7c3984b48103ee6d72ca9a3be90ce473cbfefa1/docs/figure/gd_lik_2.rmd/inflation svalue calibration plotting-1.png" target="_blank">b7c3984</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-24
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/inflation svalue calibration plotting-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/gd_lik_2.rmd/inflation%20svalue%20calibration%20plotting-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of inflation svalue calibration plotting-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<th style="text-align:left;">
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</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/inflation svalue calibration plotting-2.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/inflation svalue calibration plotting-2.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="deflation-data-sets" class="section level2">
<h2>Deflation data sets</h2>
<div id="some-examples-of-deflated-correlated-null-z-scores" class="section level3">
<h3>Some examples of deflated correlated null <span class="math inline">\(z\)</span> scores</h3>
<p><img src="figure/gd_lik_2.rmd/deflation%20examples-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of deflation examples-1.png:</em></summary>
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<th style="text-align:left;">
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</thead>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/deflation examples-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/e6d5a6488d6ee53e08cc396a42e9538adc087dd2/docs/figure/gd_lik_2.rmd/deflation examples-1.png" target="_blank">e6d5a64</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/deflation examples-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="hat-pi_0-1" class="section level3">
<h3><span class="math inline">\(\hat \pi_0\)</span></h3>
<p>Almost all methods except <code>GD-ASH</code> overestimate as expected. <code>GD-ASH</code> occasionally severely underestimates as <a href="simulation_real_se.html#global_null">seen before</a>.</p>
<p><img src="figure/gd_lik_2.rmd/deflation%20pihat0-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of deflation pihat0-1.png:</em></summary>
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</thead>
<tbody>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/deflation pihat0-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/deflation pihat0-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="mse-1" class="section level3">
<h3>MSE</h3>
<p><code>GD-Lik</code> does better than <code>ASH</code> and <code>GD-ASH</code> but not as significantly as in the inflation case.</p>
<p><img src="figure/gd_lik_2.rmd/deflation%20mse-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of deflation mse-1.png:</em></summary>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/deflation mse-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/deflation mse-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="auc-1" class="section level3">
<h3>AUC</h3>
<p>Similar story as in the inflation case, although this time <code>qvalue</code>’s <code>lfdr</code> behaves weirdly.</p>
<p><img src="figure/gd_lik_2.rmd/deflation%20auc-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of deflation auc-1.png:</em></summary>
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</thead>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/deflation auc-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/deflation auc-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="q-values-and-positive-fdr-pfdr-calibration-1" class="section level3">
<h3>q values and positive FDR (pFDR) calibration</h3>
<p>Essentially all methods successfully control pFDR. <code>GD-Lik</code> looks good although off a little. <code>qvalue</code> is the most conservative, followed by <code>ASH</code>, <code>GD-ASH</code>, and <code>locfdr</code>.</p>
<p><img src="figure/gd_lik_2.rmd/deflation%20qvalue%20calibration%20plotting-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of deflation qvalue calibration plotting-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
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</th>
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</thead>
<tbody>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/deflation qvalue calibration plotting-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/b7c3984b48103ee6d72ca9a3be90ce473cbfefa1/docs/figure/gd_lik_2.rmd/deflation qvalue calibration plotting-1.png" target="_blank">b7c3984</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-24
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/deflation qvalue calibration plotting-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/gd_lik_2.rmd/deflation%20qvalue%20calibration%20plotting-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of deflation qvalue calibration plotting-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<th style="text-align:left;">
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</th>
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</thead>
<tbody>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/deflation qvalue calibration plotting-2.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/87c16f961e270fa9688b960346a01683287bb0d3/docs/figure/gd_lik_2.rmd/deflation qvalue calibration plotting-2.png" target="_blank">87c16f9</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-07-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/a2512ac60d3081797fad7bea49dc0f4e7e17837b/docs/figure/gd_lik_2.rmd/deflation qvalue calibration plotting-2.png" target="_blank">a2512ac</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-07-03
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/deflation qvalue calibration plotting-2.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="fdr-calibration-1" class="section level3">
<h3>FDR calibration</h3>
<p><img src="figure/gd_lik_2.rmd/deflation%20FDR%20calibration%20plotting-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of deflation FDR calibration plotting-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/deflation FDR calibration plotting-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/a2512ac60d3081797fad7bea49dc0f4e7e17837b/docs/figure/gd_lik_2.rmd/deflation FDR calibration plotting-1.png" target="_blank">a2512ac</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-07-03
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/gd_lik_2.rmd/deflation%20FDR%20calibration%20plotting-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of deflation FDR calibration plotting-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/deflation FDR calibration plotting-2.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/a2512ac60d3081797fad7bea49dc0f4e7e17837b/docs/figure/gd_lik_2.rmd/deflation FDR calibration plotting-2.png" target="_blank">a2512ac</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-07-03
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="s-values-and-positive-fsr-pfsr-calibration-1" class="section level3">
<h3>s values and positive FSR (pFSR) calibration</h3>
<p>Both <code>ASH</code> and <code>GD-ASH</code> seem too conservative, although <code>GD-ASH</code> is more powerful.</p>
<p><img src="figure/gd_lik_2.rmd/deflation%20svalue%20calibration%20plotting-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of deflation svalue calibration plotting-1.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/deflation svalue calibration plotting-1.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/b7c3984b48103ee6d72ca9a3be90ce473cbfefa1/docs/figure/gd_lik_2.rmd/deflation svalue calibration plotting-1.png" target="_blank">b7c3984</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-24
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/deflation svalue calibration plotting-1.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/gd_lik_2.rmd/deflation%20svalue%20calibration%20plotting-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of deflation svalue calibration plotting-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/0f36d998db26444c5dd01502ea1af7fbd1129b22/docs/figure/gd_lik_2.rmd/deflation svalue calibration plotting-2.png" target="_blank">0f36d99</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-12-21
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/73b4bb7b980ef52f3b6c6cafb8139e33894a9dc0/docs/figure/gd_lik_2.rmd/deflation svalue calibration plotting-2.png" target="_blank">73b4bb7</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2017-06-17
</td>
</tr>
</tbody>
</table>
</details>
</div>
</div>
<div id="remarks" class="section level2">
<h2>Remarks</h2>
<ol style="list-style-type: decimal">
<li>Would be nice to come up with a way to calculate <code>lfsr</code> in <code>GD-Lik</code>.</li>
<li>Many methods are too liberal for inflation cases and too conservative for deflation cases, showing a lack of robustness against correlation. Although, on average they probably seem about right.</li>
</ol>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.4
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] ashr_2.2-2 Rmosek_8.0.69 PolynomF_1.0-1 CVXR_0.95
[5] REBayes_1.2 Matrix_1.2-12 SQUAREM_2017.10-1 EQL_1.0-0
[9] ttutils_1.0-1
loaded via a namespace (and not attached):
[1] gmp_0.5-13.1 Rcpp_0.12.16 compiler_3.4.3
[4] git2r_0.21.0 workflowr_1.0.1 R.methodsS3_1.7.1
[7] R.utils_2.6.0 iterators_1.0.9 tools_3.4.3
[10] digest_0.6.15 bit_1.1-12 evaluate_0.10.1
[13] lattice_0.20-35 foreach_1.4.4 yaml_2.1.18
[16] parallel_3.4.3 Rmpfr_0.6-1 ECOSolveR_0.4
[19] stringr_1.3.0 knitr_1.20 rprojroot_1.3-2
[22] bit64_0.9-7 grid_3.4.3 R6_2.2.2
[25] rmarkdown_1.9 magrittr_1.5 whisker_0.3-2
[28] MASS_7.3-47 backports_1.1.2 codetools_0.2-15
[31] htmltools_0.3.6 scs_1.1-1 stringi_1.1.6
[34] pscl_1.5.2 doParallel_1.0.11 truncnorm_1.0-7
[37] R.oo_1.21.0 </code></pre>
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