<div dir="ltr"><div><span style="color:rgb(80,0,80)">Dear Wolfgang and All, </span><br></div><div><div><div><span style="color:rgb(80,0,80)"><div><br></div><div>Thanks so much for your prior help. </div>We have calculated incidence and prevalence rates from a mixed-effects model using the <a href="http://rma.mv/" target="_blank">rma.mv</a> command. We are attaching the results below.</span><div><div><br></div><div><div><table border="0" cellspacing="0" cellpadding="0" width="919" style="width:689.4pt;border-collapse:collapse"><tbody><tr style="height:13.1pt"><td width="369" nowrap valign="bottom" style="width:277.1pt;border-top:1pt solid windowtext;border-left:none;border-bottom:1pt solid windowtext;border-right:none;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">Variable</span></b></p></td><td width="117" nowrap valign="bottom" style="width:87.75pt;border-top:1pt solid windowtext;border-left:none;border-bottom:1pt solid windowtext;border-right:none;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">Cohorts (n)</span></b></p></td><td width="181" nowrap valign="bottom" style="width:135.85pt;border-top:1pt solid windowtext;border-left:none;border-bottom:1pt solid windowtext;border-right:none;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">Incidence Rate/100k</span></b></p></td><td width="127" nowrap valign="bottom" style="width:94.95pt;border-top:1pt solid windowtext;border-left:none;border-bottom:1pt solid windowtext;border-right:none;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">Lower 95% CI</span></b></p></td><td width="125" nowrap valign="bottom" style="width:93.75pt;border-top:1pt solid windowtext;border-left:none;border-bottom:1pt solid windowtext;border-right:none;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">Upper 95% CI</span></b></p></td></tr><tr style="height:13.1pt"><td width="369" nowrap valign="bottom" style="width:277.1pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">World Health Organization Region</span></b></p></td><td width="117" nowrap valign="bottom" style="width:87.75pt;padding:0in 5.4pt;height:13.1pt"></td><td width="181" nowrap valign="bottom" style="width:135.85pt;padding:0in 5.4pt;height:13.1pt"></td><td width="127" nowrap valign="bottom" style="width:94.95pt;padding:0in 5.4pt;height:13.1pt"></td><td width="125" nowrap valign="bottom" style="width:93.75pt;padding:0in 5.4pt;height:13.1pt"></td></tr><tr style="height:13.1pt"><td width="369" nowrap valign="bottom" style="width:277.1pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> Americas</span></p></td><td width="117" nowrap valign="bottom" style="width:87.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">225</span></p></td><td width="181" nowrap valign="bottom" style="width:135.85pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">324.5</span></p></td><td width="127" nowrap valign="bottom" style="width:94.95pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">166.7</span></p></td><td width="125" nowrap valign="bottom" style="width:93.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">482.2</span></p></td></tr><tr style="height:13.1pt"><td width="369" nowrap valign="bottom" style="width:277.1pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> African</span></p></td><td width="117" nowrap valign="bottom" style="width:87.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">30</span></p></td><td width="181" nowrap valign="bottom" style="width:135.85pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1906.4</span></p></td><td width="127" nowrap valign="bottom" style="width:94.95pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1134.1</span></p></td><td width="125" nowrap valign="bottom" style="width:93.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">2678.7</span></p></td></tr><tr style="height:13.1pt"><td width="369" nowrap valign="bottom" style="width:277.1pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> Eastern Mediterranean</span></p></td><td width="117" nowrap valign="bottom" style="width:87.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">24</span></p></td><td width="181" nowrap valign="bottom" style="width:135.85pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">249.1</span></p></td><td width="127" nowrap valign="bottom" style="width:94.95pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">32.3</span></p></td><td width="125" nowrap valign="bottom" style="width:93.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">466.0</span></p></td></tr><tr style="height:13.1pt"><td width="369" nowrap valign="bottom" style="width:277.1pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> European</span></p></td><td width="117" nowrap valign="bottom" style="width:87.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">33</span></p></td><td width="181" nowrap valign="bottom" style="width:135.85pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">767.6</span></p></td><td width="127" nowrap valign="bottom" style="width:94.95pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">407.8</span></p></td><td width="125" nowrap valign="bottom" style="width:93.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1127.3</span></p></td></tr><tr style="height:13.1pt"><td width="369" nowrap valign="bottom" style="width:277.1pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> South-East Asia</span></p></td><td width="117" nowrap valign="bottom" style="width:87.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">48</span></p></td><td width="181" nowrap valign="bottom" style="width:135.85pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1148.7</span></p></td><td width="127" nowrap valign="bottom" style="width:94.95pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">628.6</span></p></td><td width="125" nowrap valign="bottom" style="width:93.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1668.9</span></p></td></tr><tr style="height:13.1pt"><td width="369" nowrap valign="bottom" style="width:277.1pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> Western Pacific</span></p></td><td width="117" nowrap valign="bottom" style="width:87.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">30</span></p></td><td width="181" nowrap valign="bottom" style="width:135.85pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">560.0</span></p></td><td width="127" nowrap valign="bottom" style="width:94.95pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt;background-color:rgb(255,255,0)">-131.7</span></p></td><td width="125" nowrap valign="bottom" style="width:93.75pt;padding:0in 5.4pt;height:13.1pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1251.7</span></p></td></tr></tbody></table></div><div><br></div><div><table border="0" cellspacing="0" cellpadding="0" width="848" style="width:636.2pt;border-collapse:collapse"><tbody><tr style="height:12.6pt"><td width="342" nowrap valign="bottom" style="width:256.5pt;border-top:1pt solid windowtext;border-left:none;border-bottom:1pt solid windowtext;border-right:none;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> </span></p></td><td width="102" nowrap valign="bottom" style="width:76.5pt;border-top:1pt solid windowtext;border-left:none;border-bottom:1pt solid windowtext;border-right:none;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">Cohorts (n)</span></b></p></td><td width="130" nowrap valign="bottom" style="width:97.65pt;border-top:1pt solid windowtext;border-left:none;border-bottom:1pt solid windowtext;border-right:none;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">Prevalence Rate</span></b></p></td><td width="138" nowrap valign="bottom" style="width:103.45pt;border-top:1pt solid windowtext;border-left:none;border-bottom:1pt solid windowtext;border-right:none;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">Lower 95% CI</span></b></p></td><td width="136" nowrap valign="bottom" style="width:102.1pt;border-top:1pt solid windowtext;border-left:none;border-bottom:1pt solid windowtext;border-right:none;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">Upper 95% CI</span></b></p></td></tr><tr style="height:12.6pt"><td width="342" nowrap valign="bottom" style="width:256.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><b><span style="font-size:11pt">World Health Organization Region</span></b></p></td><td width="102" nowrap valign="bottom" style="width:76.5pt;padding:0in 5.4pt;height:12.6pt"></td><td width="130" nowrap valign="bottom" style="width:97.65pt;padding:0in 5.4pt;height:12.6pt"></td><td width="138" nowrap valign="bottom" style="width:103.45pt;padding:0in 5.4pt;height:12.6pt"></td><td width="136" nowrap valign="bottom" style="width:102.1pt;padding:0in 5.4pt;height:12.6pt"></td></tr><tr style="height:12.6pt"><td width="342" nowrap valign="bottom" style="width:256.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> Americas</span></p></td><td width="102" nowrap valign="bottom" style="width:76.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">33</span></p></td><td width="130" nowrap valign="bottom" style="width:97.65pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1.7</span></p></td><td width="138" nowrap valign="bottom" style="width:103.45pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">0.9</span></p></td><td width="136" nowrap valign="bottom" style="width:102.1pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">2.5</span></p></td></tr><tr style="height:12.6pt"><td width="342" nowrap valign="bottom" style="width:256.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> African</span></p></td><td width="102" nowrap valign="bottom" style="width:76.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">31</span></p></td><td width="130" nowrap valign="bottom" style="width:97.65pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">2.8</span></p></td><td width="138" nowrap valign="bottom" style="width:103.45pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1.4</span></p></td><td width="136" nowrap valign="bottom" style="width:102.1pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">4.1</span></p></td></tr><tr style="height:12.6pt"><td width="342" nowrap valign="bottom" style="width:256.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> Eastern Mediterranean</span></p></td><td width="102" nowrap valign="bottom" style="width:76.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">7</span></p></td><td width="130" nowrap valign="bottom" style="width:97.65pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1.9</span></p></td><td width="138" nowrap valign="bottom" style="width:103.45pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">0.5</span></p></td><td width="136" nowrap valign="bottom" style="width:102.1pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">3.4</span></p></td></tr><tr style="height:12.6pt"><td width="342" nowrap valign="bottom" style="width:256.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> European</span></p></td><td width="102" nowrap valign="bottom" style="width:76.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">21</span></p></td><td width="130" nowrap valign="bottom" style="width:97.65pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1.9</span></p></td><td width="138" nowrap valign="bottom" style="width:103.45pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">0.8</span></p></td><td width="136" nowrap valign="bottom" style="width:102.1pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">3.0</span></p></td></tr><tr style="height:12.6pt"><td width="342" nowrap valign="bottom" style="width:256.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> South-East Asian</span></p></td><td width="102" nowrap valign="bottom" style="width:76.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">10</span></p></td><td width="130" nowrap valign="bottom" style="width:97.65pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">2.4</span></p></td><td width="138" nowrap valign="bottom" style="width:103.45pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt;background-color:rgb(255,255,0)">-1.1</span></p></td><td width="136" nowrap valign="bottom" style="width:102.1pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">6.0</span></p></td></tr><tr style="height:12.6pt"><td width="342" nowrap valign="bottom" style="width:256.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" style="margin:0in 0in 0.0001pt;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt"> Western Pacific</span></p></td><td width="102" nowrap valign="bottom" style="width:76.5pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">54</span></p></td><td width="130" nowrap valign="bottom" style="width:97.65pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">1.2</span></p></td><td width="138" nowrap valign="bottom" style="width:103.45pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt;background-color:rgb(255,255,0)">-0.1</span></p></td><td width="136" nowrap valign="bottom" style="width:102.1pt;padding:0in 5.4pt;height:12.6pt"><p class="MsoNormal" align="center" style="margin:0in 0in 0.0001pt;text-align:center;font-size:12pt;font-family:Arial,sans-serif;color:black"><span style="font-size:11pt">2.4</span></p></td></tr></tbody></table></div><span style="color:rgb(80,0,80)"><div><div><br></div></div><div><br></div><div>Unfortunately we have some negative confidence intervals for some of our incidence and prevalence estimates. We would like to not have any negative confidence intervals and therefore would like to switch the models that we are using.<br></div><div><br></div></span><div>Is there a way to keep our code (which we have put below for both incidence and prevalence) and run a poisson model for incidence and a binary or beta model for prevalence so that we no longer have a negative confidence interval for some of our variables? We noticed that when we run the model using log transformed incident and prevalence rates, the confidence intervals are positive. We are also wondering what the difference is between using PR/IR versus PLN/IRLN for fitting the model, and why the latter would result in all positive confidence intervals.</div><span style="color:rgb(80,0,80)"><div><br></div><div>Thank you again for all your help! </div><div><br></div><div>Best</div><div>Olivia and Leo</div><div><br></div></span></div></div></div><div><br></div><div><div><u>Code for incidence rates: </u></div><div>#data subsetted by WHO region</div><div><span style="color:rgb(80,0,80)"><br>pd_ec <- escalc(measure = 'IR', xi = data_sub$inc_positive,ti = data_sub$inc_person_years, append = TRUE, <br> data = data_sub) </span></div><div><span style="color:rgb(80,0,80)"><br>m0 <- <a href="http://rma.mv/" rel="noreferrer" target="_blank">rma.mv</a>(yi, vi, method='REML', mods = formula,<br> random= ~ 1 | study_id/cohort_id,<br> tdist=TRUE,<br> data=pd_ec)</span></div><div><br></div><div><u>Code for prevalence rates: </u></div><div>#data subsetted by WHO region</div><div><br><div></div><div>pd_ec <- escalc(<br> measure = 'PR', xi = data_sub$prev_positive,ni = data_sub$prev_total_n, append = TRUE,<br> data = data_sub)<br></div></div></div><div><br></div><div> m0 <- <a href="http://rma.mv/" target="_blank">rma.mv</a>(yi, vi, method='REML', mods = formula,<br> random= ~ 1 | study_id/cohort_id,<br> tdist=TRUE,<br> data=pd_ec)</div></div></div><div><br></div><div>Thank you for any help you can provide!</div><div><br></div><div>Best</div><div>Leo and Olivia</div><div><div><div><div><br><div><br><div><br clear="all"><div><div dir="ltr" data-smartmail="gmail_signature"><div dir="ltr"><div><div dir="ltr"><span style="font-size:12.8px">Leonardo Martinez, PhD, MPH</span><br style="font-size:12.8px"><span style="font-size:12.8px">Stanford University School of Medicine</span><br style="font-size:12.8px"><span style="font-size:12.8px">Division of Infectious Diseases and Geographic Medicine</span><br style="font-size:12.8px"><span style="font-size:12.8px">300 Pasteur Drive, Lane Building, Stanford, CA 94305</span><br style="font-size:12.8px"><span style="font-size:12.8px">Phone: +1.202.769.8090 </span><br style="font-size:12.8px"><span style="font-size:12.8px">Email: <a href="mailto:leomarti@stanford.edu" target="_blank">leomarti@stanford.edu</a>; <a href="mailto:chopotin@gmail.com" target="_blank">chopotin@gmail.com</a></span><br style="font-size:12.8px"><a href="https://profiles.stanford.edu/leonardo-martinez-pantoja" style="font-size:12.8px" target="_blank">Website</a><br></div></div></div></div></div><br></div></div></div></div></div></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Wed, Sep 4, 2019 at 10:57 AM Viechtbauer, Wolfgang (SP) <<a href="mailto:wolfgang.viechtbauer@maastrichtuniversity.nl" target="_blank">wolfgang.viechtbauer@maastrichtuniversity.nl</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">Forgot to cc the mailing list, so resending this.<br>
<br>
-----Original Message-----<br>
From: Viechtbauer, Wolfgang (SP) <br>
Sent: Wednesday, 04 September, 2019 19:36<br>
To: 'Leo Martinez'; 'Olivia Cords'<br>
Subject: RE: [R-meta] metafor package in R - Risk ratios using <a href="http://rma.mv" rel="noreferrer" target="_blank">rma.mv</a>()<br>
<br>
Dear Leo, Dear Olivia,<br>
<br>
Late response (to Olivia), but I was out of the office the entire August.<br>
<br>
Q2) Yes, one can estimate rate ratios this way.<br>
<br>
They are different because the log transformation is non-linear. Also, the normal approximation of the sampling distributions doesn't work in the same way on the raw and on the log scale. To illustrate:<br>
<br>
Let's say we observe x=5 cases in t=100 person years, so IR = 5/100. For IR values, the normal approximation is IR ~ N(theta, theta/t), where theta is the true rate per person year (this follows from assuming that x is Poisson distributed with rate t*theta), so we estimate the sampling variance with v = IR/t. Hence, a 95% CI for theta is given by:<br>
<br>
x <- 5<br>
t <- 100<br>
IR <- x/t<br>
IR + c(-1,1) * qnorm(.975) * sqrt(IR/t)<br>
<br>
which yields<br>
<br>
0.006173873 0.093826127<br>
<br>
For log(IR), the normal approximation (after using the delta method) is log(IR) ~ N(log(theta), 1/(t*theta)), so we estimate the sampling variance with v = 1/x. Hence, a 95% CI for theta is given by:<br>
<br>
exp(log(IR) + c(-1,1) * qnorm(.975) * sqrt(1/x))<br>
<br>
which yields<br>
<br>
0.02081139 0.12012652<br>
<br>
As you can see, these results are not the same. And this doesn't yet get into the additional complexities involved when fitting the model you are fitting (where we estimate additional variance components, which in turn also has implications for how the estimates are weighted and combined).<br>
<br>
Q3) Just exponentiate the CIs for the model coefficients for the model fitted with measure = "IRLN". So, exp(3.1798) is the first rate ratio with (approximate) 95% CI exp(1.9348) and exp(4.4248).<br>
<br>
There is also a technical issue here that is relevant whenever we analyze outcomes on some transformed scale where the transformation is non-linear. exp(3.1798) is actually not the estimated *average* incidence rate for the African region. To be precise, the correct interpretation is that exp(3.1798) is the estimated *median* incidence rate for the African region. The problem is that f(E(X)) != E(f(X)) whenever f() is non-linear (Jensen's inequality). However, f(M(X)) = M(f(X)) when M() is the median.<br>
<br>
So, if we have the estimated average log incidence rate (which, under the normality assumptions of the model, is equal to the estimated median log incidence rate), the back-transformation gives us the estimated median incidence rate (and not the estimated average incidence rate). So, this is another reason why results are different when you analyze raw or log transformed incidence rates.<br>
<br>
Essentially everybody ignores this issue when analyzing transformed outcomes. This also applies to correlations, where there was a lot of debate in the literature around the question whether we should analyze raw or r-to-z transformed correlations (Adam Hafdahl eventually pointed out this issue in this context).<br>
<br>
Best,<br>
Wolfgang<br>
<br>
-----Original Message-----<br>
From: R-sig-meta-analysis [mailto:<a href="mailto:r-sig-meta-analysis-bounces@r-project.org" target="_blank">r-sig-meta-analysis-bounces@r-project.org</a>] On Behalf Of Leo Martinez<br>
Sent: Wednesday, 04 September, 2019 18:10<br>
To: <a href="mailto:r-sig-meta-analysis@r-project.org" target="_blank">r-sig-meta-analysis@r-project.org</a><br>
Subject: Re: [R-meta] metafor package in R - Risk ratios using <a href="http://rma.mv" rel="noreferrer" target="_blank">rma.mv</a>()<br>
<br>
Dear All,<br>
<br>
Thanks for your previous help on this thread. I just wanted to follow up on<br>
this topic with a few additional questions regarding incident rate ratios<br>
and confidence intervals using the metafor package and the <a href="http://rma.mv" rel="noreferrer" target="_blank">rma.mv</a>()<br>
command.<br>
<br>
*Incidence Rates*I calculated the incidence rates (measure = "IR", ti =<br>
data$person_years/1000) for tuberculosis on the data subsetted by World<br>
Health Organization Region and got the following results from the model.<br>
<br>
m0 <- <a href="http://rma.mv" rel="noreferrer" target="_blank">rma.mv</a>(yi, vi, method='REML', mods = formula,<br>
random= ~ 1 | study_id/cohort_id,<br>
tdist=TRUE,<br>
data=pd_ec)<br>
<br>
Americas Region 3.244<br>
African Region 19.06<br>
Eastern Mediterranean Region 2.491<br>
European Region 7.675<br>
South-East Asian Region 11.48<br>
Western Pacific Region 5.600<br>
<br>
Rate Ratios:<br>
<br>
Following your suggestion above for calculating rate ratios for each WHO<br>
Region, I used the measure = "IRLN" and exponentiated the coefficients of<br>
the model. I got the following model output:<br>
<br>
Multivariate Meta-Analysis Model (k = 390; method: REML)<br>
<br>
Variance Components:<br>
<br>
estim sqrt nlvls fixed factor<br>
sigma^2.1 2.3898 1.5459 76 no study_id<br>
sigma^2.2 0.5833 0.7637 390 no study_id/cohort_id<br>
<br>
Test for Residual Heterogeneity:<br>
QE(df = 384) = 118469.9261, p-val < .0001<br>
<br>
Test of Moderators (coefficients 2:6):<br>
F(df1 = 5, df2 = 384) = 7.3370, p-val < .0001<br>
<br>
Model Results:<br>
<br>
estimate se tval pval<br>
intrcpt -7.1326 0.2449 -29.1259 <.0001<br>
who_region.1African Region 3.1798 0.6332 5.0217 <.0001<br>
who_region.1Eastern Mediterranean Region 0.9541 0.9634 0.9904 0.3226<br>
who_region.1European Region 1.8665 0.5612 3.3261 0.0010<br>
who_region.1South-East Asian Region 2.6693 0.9360 2.8518 0.0046<br>
who_region.1Western Pacific Region 1.4697 0.8381 1.7536 0.0803<br>
<a href="http://ci.lb" rel="noreferrer" target="_blank">ci.lb</a> ci.ub<br>
intrcpt -7.6141 -6.6511 ***<br>
who_region.1African Region 1.9348 4.4248 ***<br>
who_region.1Eastern Mediterranean Region -0.9401 2.8483<br>
who_region.1European Region 0.7631 2.9698 ***<br>
who_region.1South-East Asian Region 0.8290 4.5096 **<br>
who_region.1Western Pacific Region -0.1781 3.1176 .<br>
<br>
And exponentiating the coefficients, I got the following rate ratios:<br>
<br>
Intrcpt (Americas) 0.000799<br>
African Region 24.04225<br>
Eastern Mediterranean Region 2.596419<br>
European Region 6.465383<br>
South-East Asian Region 14.42971<br>
Western Pacific Region 4.348128<br>
<br>
Based on the incidence rates produced by using the entire dataset in the<br>
model and first subsetting by region, these Rate Ratios don't seem to be<br>
correct. Simply dividing the incidence rates by a comparator (Region of the<br>
Americas) to produce rate ratios would give the following:<br>
<br>
Region of the Americas<br>
African Region 5.875294245<br>
Eastern Mediterranean Region 0.767788539<br>
European Region 2.365564921<br>
South-East Asian Region 3.540304225<br>
Western Pacific Region 1.725952561<br>
<br>
*Q2) Is exponentiating the coefficients (measure = IRLN) the way to<br>
calculate rate ratios? Why are these results so different?*<br>
<br>
*Confidence Intervals*To calculate the 95% confidence intervals for the<br>
rate ratios, I first calculated the standard deviation (SD[ln(IR)] = (1/A1<br>
+ 1/A2)^0.5, where A1 and A2 are the number of tuberculosis cases in each<br>
region), and then used the following: 95% CI's = exp[ln(IR) ± 1.96(SD)]).<br>
It seems that this does not take into account the nested structure of the<br>
data.<br>
<br>
*Q3) Is there a way to calculate the confidence intervals from the model<br>
(either measure = "IRLN" or measure = "IR") output that takes into account<br>
the nested structure of the data?*<br>
Thank you again for your advice and the creation of this package.<br>
<br>
Best<br>
Leo<br>
<br>
Leonardo Martinez, PhD, MPH<br>
Stanford University School of Medicine<br>
Division of Infectious Diseases and Geographic Medicine<br>
300 Pasteur Drive, Lane Building, Stanford, CA 94305<br>
Phone: +1.202.769.8090<br>
Email: <a href="mailto:leomarti@stanford.edu" target="_blank">leomarti@stanford.edu</a>; <a href="mailto:chopotin@gmail.com" target="_blank">chopotin@gmail.com</a><br>
Website <<a href="https://profiles.stanford.edu/leonardo-martinez-pantoja" rel="noreferrer" target="_blank">https://profiles.stanford.edu/leonardo-martinez-pantoja</a>><br>
</blockquote></div>