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<h1 class="title toc-ignore">Variability of <code>Knockoff</code>’s FDR Control</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2018-04-03</em></h4>
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<p><strong>Last updated:</strong> 2018-05-15</p>
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<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>We hypothesize that <code>Knockoff</code> might have a potential problem: the variability of the actual FDP could be large, although the mean of FDP is theoretically and empirically controlled at FDR.</p>
<p>The reason is that <code>Knockoff</code> relies on the <strong>flip sign property</strong> of the test statistics <span class="math inline">\(W_j\)</span> for all null variables. Actually the events <span class="math inline">\(\{W_j \geq t_j\}\)</span> or <span class="math inline">\(\{W_j \leq -t_j\}\)</span> are independent for all null variables. However, they are <strong>not identically distributed</strong>.</p>
<p>Therefore, if every null variable is “unique,” in the sense that its test statistic has a distinct distribution, for example, if it tends to be larger or in a different region than anyone else, then this test statistic could randomly be positive or negative, and affect the estimate of FDP for different random samples.</p>
<p>In common high-dimensional settings it’s usually not a big problem since in those simulations no null variables are unique, meaning that we would expect to see <span class="math inline">\(W_i\)</span> and <span class="math inline">\(W_j\)</span> showing up in regions symmetric at zero.</p>
</div>
<div id="a-simple-illustration" class="section level2">
<h2>A simple illustration</h2>
<p><span class="math inline">\(X\)</span> is <span class="math inline">\(30 \times 4\)</span>, <span class="math inline">\(X_1\)</span> and <span class="math inline">\(X_3\)</span> are “true” variables, and <span class="math inline">\(X_2\)</span> and <span class="math inline">\(X_4\)</span> are nulls. <span class="math inline">\(X_2\)</span> is highly correlated with <span class="math inline">\(X_1\)</span>, <span class="math inline">\(\rho = 0.99\)</span>, whereas <span class="math inline">\(X_4\)</span> is almost uncorrelated with <span class="math inline">\(X_3\)</span>, <span class="math inline">\(\rho = 0.01\)</span>.</p>
<p>Below is a plot of <span class="math inline">\(W_2\)</span> vs <span class="math inline">\(W_4\)</span>. As we can see, <span class="math inline">\(|W_2|\)</span> is usually much greater than <span class="math inline">\(|W_4|\)</span>, and in many samples, only <span class="math inline">\(W_2\)</span> will show up in the rejection region <span class="math inline">\(W \geq t\)</span> or its symmetric null estimation region <span class="math inline">\(W \le -t\)</span>, which brings variability of <span class="math inline">\(\hat{FDP}\)</span> as the null statistic <span class="math inline">\(W_2\)</span> shows up in one region or the other.</p>
<p><img src="figure/knockoff_var.rmd/simple_var-1.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="example" class="section level2">
<h2>Example</h2>
<p><span class="math inline">\(X\)</span> is <span class="math inline">\(n \times 20\)</span>. There are <span class="math inline">\(10\)</span> “true” variables: <span class="math inline">\(X_1, X_3, X_5, \ldots, X_{17}, X_{19}\)</span> and <span class="math inline">\(10\)</span> null variables: <span class="math inline">\(X_2, X_4, X_6, \ldots, X_{18}, X_{20}\)</span>. The correlation between the true and null variables are decreasing such that <span class="math inline">\(\rho(X_1, X_2) = 0.9, \rho(X_3, X_4) = 0.8, \rho(X_5, X_6) = 0.7, \ldots, \rho(X_{17}, X_{18}) = 0.2, \rho(X_{19}, X_{20}) = 0.1\)</span>, and all other pairwise correlations are zero.</p>
<div id="n-30" class="section level3">
<h3><span class="math inline">\(n = 30\)</span></h3>
<p>The variability in FDP given by <code>Knockoff</code> is obvious.</p>
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<summary><em>Expand here to see past versions of unnamed-chunk-4-3.png:</em></summary>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/21e639d9fabe7fb1e54a45ab6bd470a68097924a/docs/figure/knockoff_var.rmd/unnamed-chunk-4-3.png" target="_blank">21e639d</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-05
</td>
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</tbody>
</table>
</details>
<p><img src="figure/knockoff_var.rmd/unnamed-chunk-4-4.png" width="672" style="display: block; margin: auto;" /></p>
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<a href="https://github.com/LSun/truncash/blob/21e639d9fabe7fb1e54a45ab6bd470a68097924a/docs/figure/knockoff_var.rmd/unnamed-chunk-4-4.png" target="_blank">21e639d</a>
</td>
<td style="text-align:left;">
LSun
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2018-04-05
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</tbody>
</table>
</details>
</div>
<div id="n-10k" class="section level3">
<h3><span class="math inline">\(n = 10K\)</span></h3>
<p>The variability in FDP given by <code>Knockoff</code> is less than when <span class="math inline">\(n = 30\)</span>, but still bigger than that given by <code>BH</code>.</p>
<p><img src="figure/knockoff_var.rmd/unnamed-chunk-5-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-5-1.png:</em></summary>
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<a href="https://github.com/LSun/truncash/blob/21e639d9fabe7fb1e54a45ab6bd470a68097924a/docs/figure/knockoff_var.rmd/unnamed-chunk-5-1.png" target="_blank">21e639d</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-05
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_var.rmd/unnamed-chunk-5-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-5-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<a href="https://github.com/LSun/truncash/blob/21e639d9fabe7fb1e54a45ab6bd470a68097924a/docs/figure/knockoff_var.rmd/unnamed-chunk-5-2.png" target="_blank">21e639d</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-05
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_var.rmd/unnamed-chunk-5-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-5-3.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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<a href="https://github.com/LSun/truncash/blob/21e639d9fabe7fb1e54a45ab6bd470a68097924a/docs/figure/knockoff_var.rmd/unnamed-chunk-5-3.png" target="_blank">21e639d</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-05
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_var.rmd/unnamed-chunk-5-4.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-5-4.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
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</th>
<th style="text-align:left;">
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/21e639d9fabe7fb1e54a45ab6bd470a68097924a/docs/figure/knockoff_var.rmd/unnamed-chunk-5-4.png" target="_blank">21e639d</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-04-05
</td>
</tr>
</tbody>
</table>
</details>
</div>
</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] forcats_0.2.0 stringr_1.3.0 dplyr_0.7.4 purrr_0.2.4
[5] readr_1.1.1 tidyr_0.7.2 tibble_1.4.1 ggplot2_2.2.1
[9] tidyverse_1.2.1 Matrix_1.2-12 knockoff_0.3.0
loaded via a namespace (and not attached):
[1] reshape2_1.4.3 haven_1.1.0 lattice_0.20-35
[4] colorspace_1.3-2 htmltools_0.3.6 yaml_2.1.18
[7] rlang_0.1.6 R.oo_1.21.0 pillar_1.0.1
[10] foreign_0.8-69 glue_1.2.0 R.utils_2.6.0
[13] readxl_1.0.0 modelr_0.1.1 bindrcpp_0.2
[16] bindr_0.1 plyr_1.8.4 cellranger_1.1.0
[19] munsell_0.4.3 gtable_0.2.0 workflowr_1.0.1
[22] rvest_0.3.2 R.methodsS3_1.7.1 psych_1.7.8
[25] evaluate_0.10.1 labeling_0.3 knitr_1.20
[28] parallel_3.4.3 broom_0.4.3 Rcpp_0.12.16
[31] backports_1.1.2 scales_0.5.0 jsonlite_1.5
[34] mnormt_1.5-5 hms_0.4.0 digest_0.6.15
[37] stringi_1.1.6 grid_3.4.3 rprojroot_1.3-2
[40] cli_1.0.0 tools_3.4.3 magrittr_1.5
[43] lazyeval_0.2.1 crayon_1.3.4 whisker_0.3-2
[46] pkgconfig_2.0.1 xml2_1.1.1 lubridate_1.7.1
[49] rstudioapi_0.7 assertthat_0.2.0 rmarkdown_1.9
[52] httr_1.3.1 R6_2.2.2 nlme_3.1-131
[55] git2r_0.21.0 compiler_3.4.3 </code></pre>
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