http://sonorouschocolate.com/covid19/index.php/Special:RecentChanges Track the most recent changes to the wiki in this feed. en-GB MediaWiki 1.43.11 Sat, 24 Oct 2026 15:02:11 GMT http://sonorouschocolate.com/covid19/index.php?title=Growth_in_Variant_of_Concern&diff=475&oldid=266 http://sonorouschocolate.com/covid19/index.php?title=Growth_in_Variant_of_Concern&diff=475&oldid=266 <p><span class="autocomment">Estimating transmission advantage using variant counts only</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en-GB"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:50, 21 October 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l99">Line 99:</td> <td colspan="2" class="diff-lineno">Line 99:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A simpler version of this is to combine the estimates weighted by their inverse variance, and this tends to give more-or-less similar answers to the above likelihood method (effectively it&#039;s a special case of the likelihood method where we&#039;ve replaced likelihoods by normal approximations). Under usual Poisson assumptions, the variance of $$\log(AD/(BC))$$ is approximately $$A^{-1}+B^{-1}+C^{-1}+D^{-1}$$, which leads to this estimate for the overall log cross ratio:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>A simpler version of this is to combine the estimates weighted by their inverse variance, and this tends to give more-or-less similar answers to the above likelihood method (effectively it&#039;s a special case of the likelihood method where we&#039;ve replaced likelihoods by normal approximations). Under usual Poisson assumptions, the variance of $$\log(AD/(BC))$$ is approximately $$A^{-1}+B^{-1}+C^{-1}+D^{-1}$$, which leads to this estimate for the overall log cross ratio:</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\[</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\[\frac{\sum(A_i^{-1}+B_i^{-1}+C_i^{-1}+D_i^{-1})^{-1}\log((A_iD_i)/(B_iC_i))}{\sum(A_i^{-1}+B_i^{-1}+C_i^{-1}+D_i^{-1})^{-1}},\]</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">  </del>\frac{\sum(A_i^{-1}+B_i^{-1}+C_i^{-1}+D_i^{-1})^{-1}\log((A_iD_i)/(B_iC_i))}{\sum(A_i^{-1}+B_i^{-1}+C_i^{-1}+D_i^{-1})^{-1}},</div></td><td colspan="2" class="diff-side-added"></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\]</div></td><td colspan="2" class="diff-side-added"></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where $$i$$ ranges over different subgroups (typically geographic). Note that such a decomposition would fix the problem with the example above: if $$(A_1,B_1,C_1,D_1)=(100,1,100,2)$$ and $$(A_2,B_2,C_2,D_2)=(1,100,5,1000)$$ then the overall estimate for the cross ratio will be exactly $$2$$ simply because each component is equal to $$2$$.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where $$i$$ ranges over different subgroups (typically geographic). Note that such a decomposition would fix the problem with the example above: if $$(A_1,B_1,C_1,D_1)=(100,1,100,2)$$ and $$(A_2,B_2,C_2,D_2)=(1,100,5,1000)$$ then the overall estimate for the cross ratio will be exactly $$2$$ simply because each component is equal to $$2$$.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table> Wed, 21 Oct 2026 22:50:57 GMT Alex http://sonorouschocolate.com/covid19/index.php/Talk:Growth_in_Variant_of_Concern