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<h1 class="title toc-ignore">Factor Model for <span class="math inline">\(\hat\beta\)</span> and Column Randomization for <code>Knockoff</code></h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2018-02-20</em></h4>
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<p><strong>Last updated:</strong> 2018-05-12</p>
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<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p><code>Knockoff</code> has 3 steps.</p>
<ol style="list-style-type: decimal">
<li><p>Generate knockoff variables, which keep the same correlation structure as original variables but has no effect on the response.</p></li>
<li><p>Generate test statistics such that these statistics tend to be large positive number for non-null variables but iid positive or negative for null variables.</p></li>
<li><p>Find a cutoff threshold for those test statistics to control the estimated FDR under <span class="math inline">\(q\)</span>.</p></li>
</ol>
<p>The default <code>knockoff::knockoff.filter</code> function uses <code>SDP</code> construction in step 1 and <code>LASSO</code>-related statistics in step 2. However, <a href="knockoff_5.html#scenario_3:_(hatbeta)_from_a_factor_model7">we’ve found</a> that <code>Knockoff</code> coded in this way failed to control FDR in simulations when variables are generated such that <span class="math inline">\(\hat\beta\)</span> has heavy average absolute pairwise correlation, which seems to contradict Theorem 2 in the <code>Knockoff</code> paper. Now we take a closer look to see what went wrong.</p>
<p>In step 1, we use two construction methods: <code>equi</code> and <code>sdp</code>. <code>sdp</code> is believed to be more powerful. In step 2, we use two statistics: <code>marginal</code> and <code>lasso</code>-related. <code>lasso</code>-related is believed to be more powerful.</p>
<pre class="r"><code>n <- 3e3
p <- 1e3
k <- 50
d <- 7
q <- 0.1</code></pre>
</div>
<div id="fixed-x-knockoffs-1000-simulation-trials" class="section level2">
<h2>Fixed <span class="math inline">\(X\)</span> Knockoffs: 1000 simulation trials</h2>
<p><img src="figure/knockoff_7.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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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/3fcf8155d94e5fee6d389fbfd027dfcfafedc2be/docs/figure/knockoff_7.rmd/unnamed-chunk-5-1.png" target="_blank">3fcf815</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-23
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_7.rmd/unnamed-chunk-6-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-6-1.png:</em></summary>
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<a href="https://github.com/LSun/truncash/blob/3fcf8155d94e5fee6d389fbfd027dfcfafedc2be/docs/figure/knockoff_7.rmd/unnamed-chunk-6-1.png" target="_blank">3fcf815</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-23
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_7.rmd/unnamed-chunk-6-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-6-2.png:</em></summary>
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<a href="https://github.com/LSun/truncash/blob/3b0b20d63100fe9db11b7a9e14b60fd8812a4823/docs/figure/knockoff_7.rmd/unnamed-chunk-6-2.png" target="_blank">3b0b20d</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-23
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="model-x-knockoffs-1000-simulation-trials" class="section level2">
<h2>Model <span class="math inline">\(X\)</span> Knockoffs: 1000 simulation trials</h2>
<p><img src="figure/knockoff_7.rmd/unnamed-chunk-9-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-9-1.png:</em></summary>
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<a href="https://github.com/LSun/truncash/blob/1d0f7e0c9da605465a7de32a917146937b7c8972/docs/figure/knockoff_7.rmd/unnamed-chunk-9-1.png" target="_blank">1d0f7e0</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-28
</td>
</tr>
<tr>
<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/3fcf8155d94e5fee6d389fbfd027dfcfafedc2be/docs/figure/knockoff_7.rmd/unnamed-chunk-9-1.png" target="_blank">3fcf815</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-23
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_7.rmd/unnamed-chunk-10-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-10-1.png:</em></summary>
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<a href="https://github.com/LSun/truncash/blob/1d0f7e0c9da605465a7de32a917146937b7c8972/docs/figure/knockoff_7.rmd/unnamed-chunk-10-1.png" target="_blank">1d0f7e0</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-28
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/3fcf8155d94e5fee6d389fbfd027dfcfafedc2be/docs/figure/knockoff_7.rmd/unnamed-chunk-10-1.png" target="_blank">3fcf815</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-23
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_7.rmd/unnamed-chunk-11-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-11-1.png:</em></summary>
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<a href="https://github.com/LSun/truncash/blob/1d0f7e0c9da605465a7de32a917146937b7c8972/docs/figure/knockoff_7.rmd/unnamed-chunk-11-1.png" target="_blank">1d0f7e0</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-28
</td>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/3b0b20d63100fe9db11b7a9e14b60fd8812a4823/docs/figure/knockoff_7.rmd/unnamed-chunk-11-1.png" target="_blank">3b0b20d</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-02-23
</td>
</tr>
</tbody>
</table>
</details>
</div>
<div id="observation" class="section level2">
<h2>Observation</h2>
<p>A set of well-bahaving <code>Knockoff</code> variables <span class="math inline">\(X^k\)</span> should have the property that <span class="math display">\[
\begin{array}{c}
cor(X^k_i, X^k_j) = cor(X_i, X_j)\\
cor(X_i, X^k_j) = cor(X_i, X_j)
\end{array}
\]</span> while <span class="math inline">\(cor(X_i, X^k_i)\)</span> should be as small as possible. It turns out it’s just not that easy to generate these well-behaving <code>Knockoff</code> variables when columns in <span class="math inline">\(X\)</span> are correlated in a certain way. Especially when using <code>SDP</code> optimization, it could generate a lot of knockoffs that are exactly the same as the originals.</p>
<p><img src="figure/knockoff_7.rmd/unnamed-chunk-12-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-12-1.png:</em></summary>
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<a href="https://github.com/LSun/truncash/blob/f4c819abc133850e8df68cbbdb52a3f4c249fccc/docs/figure/knockoff_7.rmd/unnamed-chunk-12-1.png" target="_blank">f4c819a</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-01
</td>
</tr>
</tbody>
</table>
</details>
<p>If an original variable and its knockoff are too similar, it essentially makes little difference which one is included in the model, from a goodness of fit point of view.</p>
<p>The problem becomes more severe when we fit models like LASSO using methods like coordinate descent. The result depends in large part on the sequence of variables getting into the model. So if we feed LASSO with <code>cbind(X, Xk)</code>, for every iteration, it’s always <code>X[j]</code> being optimized before <code>Xk[j]</code>. That’s a major reason we see asymmetric test statistics as above, and why <code>Knockoff</code> loses FDR control in these circumstances.</p>
<p>One way to fix that is to randomize the order of variables in <code>cbind(X, Xk)</code> before feeding them to LASSO. The following is a simulation.</p>
<p><img src="figure/knockoff_7.rmd/unnamed-chunk-14-1.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-14-1.png:</em></summary>
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<a href="https://github.com/LSun/truncash/blob/f4c819abc133850e8df68cbbdb52a3f4c249fccc/docs/figure/knockoff_7.rmd/unnamed-chunk-14-1.png" target="_blank">f4c819a</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-01
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_7.rmd/unnamed-chunk-14-2.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-14-2.png:</em></summary>
<table style = "border-collapse:separate; border-spacing:5px;">
<thead>
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<a href="https://github.com/LSun/truncash/blob/f4c819abc133850e8df68cbbdb52a3f4c249fccc/docs/figure/knockoff_7.rmd/unnamed-chunk-14-2.png" target="_blank">f4c819a</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-01
</td>
</tr>
</tbody>
</table>
</details>
<p><img src="figure/knockoff_7.rmd/unnamed-chunk-14-3.png" width="672" style="display: block; margin: auto;" /></p>
<details>
<summary><em>Expand here to see past versions of unnamed-chunk-14-3.png:</em></summary>
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<td style="text-align:left;">
<a href="https://github.com/LSun/truncash/blob/f4c819abc133850e8df68cbbdb52a3f4c249fccc/docs/figure/knockoff_7.rmd/unnamed-chunk-14-3.png" target="_blank">f4c819a</a>
</td>
<td style="text-align:left;">
LSun
</td>
<td style="text-align:left;">
2018-03-01
</td>
</tr>
</tbody>
</table>
</details>
<p>After column randomization, the test statistcs for null variables are back to normal and <code>Knockoff</code> controls FDR again. The low power is another issue.</p>
</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] parallel stats graphics grDevices utils datasets methods
[8] base
other attached packages:
[1] lattice_0.20-35 doMC_1.3.5 iterators_1.0.9 foreach_1.4.4
[5] ggplot2_2.2.1 reshape2_1.4.3 knockoff_0.3.0
loaded via a namespace (and not attached):
[1] Rcpp_0.12.16 RSpectra_0.12-0 compiler_3.4.3
[4] pillar_1.0.1 git2r_0.21.0 plyr_1.8.4
[7] workflowr_1.0.1 R.methodsS3_1.7.1 R.utils_2.6.0
[10] tools_3.4.3 digest_0.6.15 evaluate_0.10.1
[13] tibble_1.4.1 gtable_0.2.0 rlang_0.1.6
[16] Matrix_1.2-12 yaml_2.1.18 stringr_1.3.0
[19] knitr_1.20 rprojroot_1.3-2 grid_3.4.3
[22] rmarkdown_1.9 Rdsdp_1.0.4-2 magrittr_1.5
[25] whisker_0.3-2 backports_1.1.2 scales_0.5.0
[28] codetools_0.2-15 htmltools_0.3.6 colorspace_1.3-2
[31] labeling_0.3 stringi_1.1.6 lazyeval_0.2.1
[34] munsell_0.4.3 R.oo_1.21.0 </code></pre>
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