class: center, middle, inverse, title-slide .title[ # 逻辑斯蒂回归 ] .subtitle[ ## 社会统计学——第11讲 ] .author[ ### 李代 ] .institute[ ### 中国政法大学社会学院 ] .date[ ### 2025-05-06 ] --- class: center, middle, inverse <!-- background-image: url("images/cool.png") --> # 逻辑斯蒂回归 ## 比数比 ## 逻辑斯蒂回归 ## 注意 --- # 比数比 ## 线性回归 线性回归模型，是对变量之间相关关系的一种设想。它认为，变量之间存在“线性”的关系。也就是说，一条直线可以描述自变量与因变量之间的关系。 $$ y = \beta_0 + \beta_1 x_1 + \epsilon$$ 该模型认为，x每增加1单位，y增加 `\(\beta_1\)` 单位。当 `\(x=0\)` 时， `\(y = \beta_0\)`。 --- # 比数比 ## 线性回归模型的使用条件 1. 因变量是连续变量（定距、定比）。 1. `\(x\)` 和 `\(y\)` 之间的关系是线性的（能写成一次方程的形式）。 1. 独立：误差（ `\(\epsilon_i\)`）之间不相关。（Durbin-Watson检验） 1. 同分布：误差 `\(\epsilon_i\)`的分布相同，均值 `\(E(\epsilon_i) = 0\)`， 方差 `\(Var(\epsilon_i) = \sigma^2\)`。 1. 正态分布：误差 `\(\epsilon_i\)`符合正态分布 `\(\epsilon_i \sim N(0, \sigma^2)\)`。（Q-Q图） --- # 比数比 ## 线性回归不适用怎么办？ 如果因变量不是连续变量，应该怎么办？ 例如： 二分类变量：性别、好恶、中外…… 多分类变量：爱好、品牌、国籍…… 定序变量：教育程度、满意程度…… --- # 比数比 ## 比数比 发生比、比数：odds What are the **odds**? [国足仍有世界杯夺冠赔率 1赔3501倒数第二压泰国](https://www.163.com/sports/article/BJG7MT7700051C8M.html) 用1元钱赌中国队夺冠，中国队如果真的夺冠了，拿回来3501元钱。 odds: 1:3501 --- # 比数比 ## 升学率 假设有n个学生考大学，考上大学的人数是x。 升学率，是一个百分比，proportion。 `\(Proportion = \frac{x}{n}\)`. 概率，probability。概率不可观测，但可以通过统计推断推算。 `\(p = \frac{x}{n}\)`. 比数，也就是事件发生的概率比事件不发生的概率： `\(odds = \frac{p}{1-p}\)`. --- # 比数比 ## 升学率 .pull-left[ <!-- --> ] .pull-left[ <!-- --> ] --- # 比数比 ## 谁更占优势？ 假设有m个男学生考大学，考上大学的男生人数是x。 假设有n个女学生考大学，考上大学的女生人数是y。 假如男生的升学率比女生大，是否说明男生更占优势？ $$ p_1 - p_2 = \frac{x}{m} - \frac{y}{n} $$ $$ \frac{p_1}{p_2} = \frac{\frac{x}{m}}{\frac{y}{n}} $$ --- # 比数比 ## 谁更占优势？ <!-- --> --- # 比数比 ## 比数比 odds ratio 男生升学的发生比是： `\(\frac{p_1}{1 - p_1}\)` 女生升学的发生比是： `\(\frac{p_2}{1 - p_2}\)` 比数比是： `\(\frac{\frac{p_1}{1 - p_1}}{\frac{p_2}{1 - p_2}}\)` --- # 比数比 ## 比数比 <!-- --> --- # 比数比 ## 比数比的对数 (log odds ratio) <!-- --> --- class: center, middle, inverse <!-- background-image: url("images/cool.png") --> # 逻辑斯蒂回归 --- # 逻辑斯蒂回归 ## 泰坦尼克号 <!-- --> --- # 逻辑斯蒂回归 ## 泰坦尼克号 <div class="tabwid"><style>.cl-8996b098{}.cl-898f2db4{font-family:'Arial';font-size:11pt;font-weight:normal;font-style:normal;text-decoration:none;color:rgba(0, 0, 0, 1.00);background-color:transparent;}.cl-89922d8e{margin:0;text-align:left;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-89922da2{margin:0;text-align:right;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);padding-bottom:5pt;padding-top:5pt;padding-left:5pt;padding-right:5pt;line-height: 1;background-color:transparent;}.cl-89924f30{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-89924f3a{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 1.5pt solid rgba(102, 102, 102, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-89924f3b{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-89924f44{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 0 solid rgba(0, 0, 0, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-89924f45{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}.cl-89924f4e{width:0.75in;background-color:transparent;vertical-align: middle;border-bottom: 1.5pt solid rgba(102, 102, 102, 1.00);border-top: 0 solid rgba(0, 0, 0, 1.00);border-left: 0 solid rgba(0, 0, 0, 1.00);border-right: 0 solid rgba(0, 0, 0, 1.00);margin-bottom:0;margin-top:0;margin-left:0;margin-right:0;}</style><table data-quarto-disable-processing='true' class='cl-8996b098'><thead><tr style="overflow-wrap:break-word;"><th class="cl-89924f30"><p class="cl-89922d8e"><span class="cl-898f2db4">class</span></p></th><th class="cl-89924f3a"><p class="cl-89922da2"><span class="cl-898f2db4">No</span></p></th><th class="cl-89924f3a"><p class="cl-89922da2"><span class="cl-898f2db4">Yes</span></p></th><th class="cl-89924f3a"><p class="cl-89922da2"><span class="cl-898f2db4">p</span></p></th><th class="cl-89924f3a"><p class="cl-89922da2"><span class="cl-898f2db4">1-p</span></p></th><th class="cl-89924f3a"><p class="cl-89922da2"><span class="cl-898f2db4">odds</span></p></th><th class="cl-89924f3a"><p class="cl-89922da2"><span class="cl-898f2db4">or</span></p></th><th class="cl-89924f3a"><p class="cl-89922da2"><span class="cl-898f2db4">log or</span></p></th></tr></thead><tbody><tr style="overflow-wrap:break-word;"><td class="cl-89924f3b"><p class="cl-89922d8e"><span class="cl-898f2db4">1st</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">122</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">203</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">0.625</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">0.375</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">1.664</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">5.282</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">1.664</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-89924f3b"><p class="cl-89922d8e"><span class="cl-898f2db4">2nd</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">167</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">118</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">0.414</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">0.586</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">0.707</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">2.243</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">0.808</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-89924f3b"><p class="cl-89922d8e"><span class="cl-898f2db4">3rd</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">528</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">178</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">0.252</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">0.748</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">0.337</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">1.070</span></p></td><td class="cl-89924f44"><p class="cl-89922da2"><span class="cl-898f2db4">0.068</span></p></td></tr><tr style="overflow-wrap:break-word;"><td class="cl-89924f45"><p class="cl-89922d8e"><span class="cl-898f2db4">Crew</span></p></td><td class="cl-89924f4e"><p class="cl-89922da2"><span class="cl-898f2db4">673</span></p></td><td class="cl-89924f4e"><p class="cl-89922da2"><span class="cl-898f2db4">212</span></p></td><td class="cl-89924f4e"><p class="cl-89922da2"><span class="cl-898f2db4">0.240</span></p></td><td class="cl-89924f4e"><p class="cl-89922da2"><span class="cl-898f2db4">0.760</span></p></td><td class="cl-89924f4e"><p class="cl-89922da2"><span class="cl-898f2db4">0.315</span></p></td><td class="cl-89924f4e"><p class="cl-89922da2"><span class="cl-898f2db4">1.000</span></p></td><td class="cl-89924f4e"><p class="cl-89922da2"><span class="cl-898f2db4">0.000</span></p></td></tr></tbody></table></div> 因为 `\(OR > 1\)` 或者 `\(log or > 0\)`，乘客跟船员比都更容易幸存，一等舱最容易幸存，其次是二等舱，再次是三等舱。 --- # 逻辑斯蒂回归 ## 泰坦尼克号 <div id="fsgvhusryq" 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gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="label"><span class='gt_from_md'><strong>Characteristic</strong></span></th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="estimate"><span class='gt_from_md'><strong>log(OR)</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;">¹</span></th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="std.error"><span class='gt_from_md'><strong>SE</strong></span></th> </tr> </thead> <tbody class="gt_table_body"> <tr><td headers="label" class="gt_row gt_left">(Intercept)</td> <td headers="estimate" class="gt_row gt_center">-1.2***</td> <td headers="std.error" class="gt_row gt_center">0.079</td></tr> <tr><td headers="label" class="gt_row gt_left">Class</td> <td headers="estimate" class="gt_row gt_center"><br /></td> <td headers="std.error" class="gt_row gt_center"><br /></td></tr> <tr><td headers="label" class="gt_row gt_left">    Crew</td> <td headers="estimate" class="gt_row gt_center">—</td> <td headers="std.error" class="gt_row gt_center">—</td></tr> <tr><td headers="label" class="gt_row gt_left">    1st</td> <td headers="estimate" class="gt_row gt_center">1.7***</td> <td headers="std.error" class="gt_row gt_center">0.139</td></tr> <tr><td headers="label" class="gt_row gt_left">    2nd</td> <td headers="estimate" class="gt_row gt_center">0.81***</td> <td headers="std.error" class="gt_row gt_center">0.144</td></tr> <tr><td headers="label" class="gt_row gt_left">    3rd</td> <td headers="estimate" class="gt_row gt_center">0.07</td> <td headers="std.error" class="gt_row gt_center">0.117</td></tr> <tr><td headers="label" class="gt_row gt_left" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">AIC</td> <td headers="estimate" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">2,597</td> <td headers="std.error" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;"><br /></td></tr> <tr><td headers="label" class="gt_row gt_left">No. Obs.</td> <td headers="estimate" class="gt_row gt_center">2,201</td> <td headers="std.error" class="gt_row gt_center"><br /></td></tr> </tbody> <tfoot class="gt_sourcenotes"> <tr> <td class="gt_sourcenote" colspan="3"><span class='gt_from_md'>Abbreviations: CI = Confidence Interval, OR = Odds Ratio, SE = Standard Error</span></td> </tr> </tfoot> <tfoot class="gt_footnotes"> <tr> <td class="gt_footnote" colspan="3"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;">¹</span> <span class='gt_from_md'>*p<0.05; **p<0.01; ***p<0.001</span></td> </tr> </tfoot> </table> </div> --- # 逻辑斯蒂回归 ## 星球大战角色的性别与身高 <!-- --> --- # 逻辑斯蒂回归 ## 线性关系可以很好的描述两个变量之间的关系吗？ <!-- --> --- # 逻辑斯蒂回归 连接函数是logit函数的回归： `\(logit(p) = log(\frac{p}{1-p})\)` <!-- --> --- # 逻辑斯蒂回归 <div id="dnvujakhil" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;"> <style>#dnvujakhil table { font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji'; -webkit-font-smoothing: antialiased; -moz-osx-font-smoothing: grayscale; } #dnvujakhil thead, #dnvujakhil tbody, #dnvujakhil tfoot, #dnvujakhil tr, #dnvujakhil td, #dnvujakhil th { border-style: none; } #dnvujakhil p { margin: 0; padding: 0; } #dnvujakhil .gt_table { display: table; border-collapse: collapse; line-height: normal; margin-left: auto; margin-right: auto; color: #333333; font-size: 16px; font-weight: normal; font-style: normal; background-color: #FFFFFF; width: auto; border-top-style: solid; border-top-width: 2px; border-top-color: #A8A8A8; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #A8A8A8; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3; } #dnvujakhil .gt_caption { padding-top: 4px; padding-bottom: 4px; } #dnvujakhil .gt_title { color: #333333; font-size: 125%; font-weight: initial; padding-top: 4px; padding-bottom: 4px; padding-left: 5px; padding-right: 5px; border-bottom-color: #FFFFFF; border-bottom-width: 0; } #dnvujakhil .gt_subtitle { color: #333333; font-size: 85%; font-weight: initial; padding-top: 3px; padding-bottom: 5px; padding-left: 5px; padding-right: 5px; border-top-color: #FFFFFF; border-top-width: 0; } #dnvujakhil .gt_heading { background-color: #FFFFFF; text-align: center; border-bottom-color: #FFFFFF; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; } #dnvujakhil .gt_bottom_border { 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padding-top: 0; padding-bottom: 0; padding-left: 4px; padding-right: 4px; } #dnvujakhil .gt_column_spanner_outer:first-child { padding-left: 0; } #dnvujakhil .gt_column_spanner_outer:last-child { padding-right: 0; } #dnvujakhil .gt_column_spanner { border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 5px; overflow-x: hidden; display: inline-block; width: 100%; } #dnvujakhil .gt_spanner_row { border-bottom-style: hidden; } #dnvujakhil .gt_group_heading { padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: initial; text-transform: inherit; border-top-style: solid; border-top-width: 2px; border-top-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; text-align: left; } #dnvujakhil .gt_empty_group_heading { padding: 0.5px; color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: initial; border-top-style: solid; border-top-width: 2px; border-top-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; vertical-align: middle; } #dnvujakhil .gt_from_md > :first-child { margin-top: 0; } #dnvujakhil .gt_from_md > :last-child { margin-bottom: 0; } #dnvujakhil .gt_row { padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; } #dnvujakhil .gt_stub { color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: initial; text-transform: inherit; border-right-style: solid; border-right-width: 2px; border-right-color: #D3D3D3; padding-left: 5px; padding-right: 5px; } #dnvujakhil .gt_stub_row_group { color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: initial; text-transform: inherit; border-right-style: solid; border-right-width: 2px; border-right-color: #D3D3D3; padding-left: 5px; padding-right: 5px; vertical-align: top; } #dnvujakhil .gt_row_group_first td { border-top-width: 2px; } #dnvujakhil .gt_row_group_first th { border-top-width: 2px; } #dnvujakhil .gt_summary_row { color: #333333; background-color: #FFFFFF; text-transform: inherit; padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; } #dnvujakhil .gt_first_summary_row { border-top-style: solid; border-top-color: #D3D3D3; } #dnvujakhil .gt_first_summary_row.thick { border-top-width: 2px; } #dnvujakhil .gt_last_summary_row { padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; } #dnvujakhil .gt_grand_summary_row { color: #333333; background-color: #FFFFFF; text-transform: inherit; padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; } #dnvujakhil .gt_first_grand_summary_row { padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; border-top-style: double; border-top-width: 6px; border-top-color: #D3D3D3; } #dnvujakhil .gt_last_grand_summary_row_top { padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; border-bottom-style: double; border-bottom-width: 6px; border-bottom-color: #D3D3D3; } #dnvujakhil .gt_striped { background-color: rgba(128, 128, 128, 0.05); } #dnvujakhil .gt_table_body { border-top-style: solid; border-top-width: 2px; border-top-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; } #dnvujakhil .gt_footnotes { color: #333333; background-color: #FFFFFF; border-bottom-style: none; border-bottom-width: 2px; border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; } #dnvujakhil .gt_footnote { margin: 0px; font-size: 90%; padding-top: 4px; padding-bottom: 4px; padding-left: 5px; padding-right: 5px; } #dnvujakhil .gt_sourcenotes { color: #333333; background-color: #FFFFFF; border-bottom-style: none; border-bottom-width: 2px; border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; } #dnvujakhil .gt_sourcenote { font-size: 90%; padding-top: 4px; padding-bottom: 4px; padding-left: 5px; padding-right: 5px; } #dnvujakhil .gt_left { text-align: left; } #dnvujakhil .gt_center { text-align: center; } #dnvujakhil .gt_right { text-align: right; font-variant-numeric: tabular-nums; } #dnvujakhil .gt_font_normal { font-weight: normal; } #dnvujakhil .gt_font_bold { font-weight: bold; } #dnvujakhil .gt_font_italic { font-style: italic; } #dnvujakhil .gt_super { font-size: 65%; } #dnvujakhil .gt_footnote_marks { font-size: 75%; vertical-align: 0.4em; position: initial; } #dnvujakhil .gt_asterisk { font-size: 100%; vertical-align: 0; } #dnvujakhil .gt_indent_1 { text-indent: 5px; } #dnvujakhil .gt_indent_2 { text-indent: 10px; } #dnvujakhil .gt_indent_3 { text-indent: 15px; } #dnvujakhil .gt_indent_4 { text-indent: 20px; } #dnvujakhil .gt_indent_5 { text-indent: 25px; } #dnvujakhil .katex-display { display: inline-flex !important; margin-bottom: 0.75em !important; } #dnvujakhil div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after { height: 0px !important; } </style> <table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false"> <thead> <tr class="gt_col_headings"> <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="label"><span class='gt_from_md'><strong>Characteristic</strong></span></th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="estimate"><span class='gt_from_md'><strong>log(OR)</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;">¹</span></th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="std.error"><span class='gt_from_md'><strong>SE</strong></span></th> </tr> </thead> <tbody class="gt_table_body"> <tr><td headers="label" class="gt_row gt_left">(Intercept)</td> <td headers="estimate" class="gt_row gt_center">-15**</td> <td headers="std.error" class="gt_row gt_center">5.42</td></tr> <tr><td headers="label" class="gt_row gt_left">height</td> <td headers="estimate" class="gt_row gt_center">0.09**</td> <td headers="std.error" class="gt_row gt_center">0.031</td></tr> <tr><td headers="label" class="gt_row gt_left" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">AIC</td> <td headers="estimate" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">57.0</td> <td headers="std.error" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;"><br /></td></tr> <tr><td headers="label" class="gt_row gt_left">No. Obs.</td> <td headers="estimate" class="gt_row gt_center">64</td> <td headers="std.error" class="gt_row gt_center"><br /></td></tr> </tbody> <tfoot class="gt_sourcenotes"> <tr> <td class="gt_sourcenote" colspan="3"><span class='gt_from_md'>Abbreviations: CI = Confidence Interval, OR = Odds Ratio, SE = Standard Error</span></td> </tr> </tfoot> <tfoot class="gt_footnotes"> <tr> <td class="gt_footnote" colspan="3"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;">¹</span> <span class='gt_from_md'>*p<0.05; **p<0.01; ***p<0.001</span></td> </tr> </tfoot> </table> </div> --- # 逻辑斯蒂回归 ## 定类变量vs连续变量 泰坦尼克号的结果是把定类变量作为虚拟变量放入模型得到的。可以把它看成是logit函数的一种特殊情况。 假如自变量 `\(x\)`的取值是 `\(0 = 船员, 1 = 一等舱\)`。 当 `\(x = 0\)`时， `\(log(\frac{p_0}{1-p_0}) = \beta_0\)`。此时对等式两边同时取指数，可得 `\(or_0 = \frac{p_0}{1-p_0} = e^{\beta_0}\)`，这是船员幸存的比数。 当 `\(x = 1\)`时， `\(log(\frac{p_1}{1-p_1}) = \beta_0 + \beta_1\)`。此时对等式两边同时取指数，可得 `\(or_1 = \frac{p_1}{1-p_1} = e^{\beta_0 + \beta_1}\)`，这是一等舱乘客幸存的比数。 两个比数如果相除，会得到什么？ $$ \frac{or_1}{or_0} = \frac{e ^{\beta_0 + \beta_1}}{e^{\beta_0 }} = e^{\beta_1} $$ --- # 逻辑斯蒂回归 ## 定类变量vs连续变量 所以说，通过logit函数作为连接函数得到模型系数，这个系数恰好是**比数比的对数**。在写作中，有时不报告模型系数而是**比数比**，亦可。 <div id="oiwztajmgz" style="padding-left:0px;padding-right:0px;padding-top:10px;padding-bottom:10px;overflow-x:auto;overflow-y:auto;width:auto;height:auto;"> <style>#oiwztajmgz table { font-family: system-ui, 'Segoe UI', Roboto, Helvetica, Arial, sans-serif, 'Apple Color Emoji', 'Segoe UI Emoji', 'Segoe UI Symbol', 'Noto Color Emoji'; -webkit-font-smoothing: antialiased; -moz-osx-font-smoothing: grayscale; } #oiwztajmgz thead, #oiwztajmgz tbody, #oiwztajmgz tfoot, #oiwztajmgz tr, #oiwztajmgz td, #oiwztajmgz th { border-style: none; } #oiwztajmgz p { margin: 0; padding: 0; } #oiwztajmgz .gt_table { display: table; border-collapse: collapse; line-height: normal; margin-left: auto; margin-right: auto; color: #333333; font-size: 16px; font-weight: normal; font-style: normal; background-color: #FFFFFF; width: auto; border-top-style: solid; border-top-width: 2px; border-top-color: #A8A8A8; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #A8A8A8; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3; } #oiwztajmgz .gt_caption { padding-top: 4px; 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border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; } #oiwztajmgz .gt_col_heading { color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 6px; padding-left: 5px; padding-right: 5px; overflow-x: hidden; } #oiwztajmgz .gt_column_spanner_outer { color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: normal; text-transform: inherit; padding-top: 0; padding-bottom: 0; padding-left: 4px; padding-right: 4px; } #oiwztajmgz .gt_column_spanner_outer:first-child { padding-left: 0; } #oiwztajmgz .gt_column_spanner_outer:last-child { padding-right: 0; } #oiwztajmgz .gt_column_spanner { border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; vertical-align: bottom; padding-top: 5px; padding-bottom: 5px; overflow-x: hidden; display: inline-block; width: 100%; } #oiwztajmgz .gt_spanner_row { border-bottom-style: hidden; } #oiwztajmgz .gt_group_heading { padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: initial; text-transform: inherit; border-top-style: solid; border-top-width: 2px; border-top-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; text-align: left; } #oiwztajmgz .gt_empty_group_heading { padding: 0.5px; color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: initial; border-top-style: solid; border-top-width: 2px; border-top-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; vertical-align: middle; } #oiwztajmgz .gt_from_md > :first-child { margin-top: 0; } #oiwztajmgz .gt_from_md > :last-child { margin-bottom: 0; } #oiwztajmgz .gt_row { padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; margin: 10px; border-top-style: solid; border-top-width: 1px; border-top-color: #D3D3D3; border-left-style: none; border-left-width: 1px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 1px; border-right-color: #D3D3D3; vertical-align: middle; overflow-x: hidden; } #oiwztajmgz .gt_stub { color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: initial; text-transform: inherit; border-right-style: solid; border-right-width: 2px; border-right-color: #D3D3D3; padding-left: 5px; padding-right: 5px; } #oiwztajmgz .gt_stub_row_group { color: #333333; background-color: #FFFFFF; font-size: 100%; font-weight: initial; text-transform: inherit; border-right-style: solid; border-right-width: 2px; border-right-color: #D3D3D3; padding-left: 5px; padding-right: 5px; vertical-align: top; } #oiwztajmgz .gt_row_group_first td { border-top-width: 2px; } #oiwztajmgz .gt_row_group_first th { border-top-width: 2px; } #oiwztajmgz .gt_summary_row { color: #333333; background-color: #FFFFFF; text-transform: inherit; padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; } #oiwztajmgz .gt_first_summary_row { border-top-style: solid; border-top-color: #D3D3D3; } #oiwztajmgz .gt_first_summary_row.thick { border-top-width: 2px; } #oiwztajmgz .gt_last_summary_row { padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; } #oiwztajmgz .gt_grand_summary_row { color: #333333; background-color: #FFFFFF; text-transform: inherit; padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; } #oiwztajmgz .gt_first_grand_summary_row { padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; border-top-style: double; border-top-width: 6px; border-top-color: #D3D3D3; } #oiwztajmgz .gt_last_grand_summary_row_top { padding-top: 8px; padding-bottom: 8px; padding-left: 5px; padding-right: 5px; border-bottom-style: double; border-bottom-width: 6px; border-bottom-color: #D3D3D3; } #oiwztajmgz .gt_striped { background-color: rgba(128, 128, 128, 0.05); } #oiwztajmgz .gt_table_body { border-top-style: solid; border-top-width: 2px; border-top-color: #D3D3D3; border-bottom-style: solid; border-bottom-width: 2px; border-bottom-color: #D3D3D3; } #oiwztajmgz .gt_footnotes { color: #333333; background-color: #FFFFFF; border-bottom-style: none; border-bottom-width: 2px; border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; } #oiwztajmgz .gt_footnote { margin: 0px; font-size: 90%; padding-top: 4px; padding-bottom: 4px; padding-left: 5px; padding-right: 5px; } #oiwztajmgz .gt_sourcenotes { color: #333333; background-color: #FFFFFF; border-bottom-style: none; border-bottom-width: 2px; border-bottom-color: #D3D3D3; border-left-style: none; border-left-width: 2px; border-left-color: #D3D3D3; border-right-style: none; border-right-width: 2px; border-right-color: #D3D3D3; } #oiwztajmgz .gt_sourcenote { font-size: 90%; padding-top: 4px; padding-bottom: 4px; padding-left: 5px; padding-right: 5px; } #oiwztajmgz .gt_left { text-align: left; } #oiwztajmgz .gt_center { text-align: center; } #oiwztajmgz .gt_right { text-align: right; font-variant-numeric: tabular-nums; } #oiwztajmgz .gt_font_normal { font-weight: normal; } #oiwztajmgz .gt_font_bold { font-weight: bold; } #oiwztajmgz .gt_font_italic { font-style: italic; } #oiwztajmgz .gt_super { font-size: 65%; } #oiwztajmgz .gt_footnote_marks { font-size: 75%; vertical-align: 0.4em; position: initial; } #oiwztajmgz .gt_asterisk { font-size: 100%; vertical-align: 0; } #oiwztajmgz .gt_indent_1 { text-indent: 5px; } #oiwztajmgz .gt_indent_2 { text-indent: 10px; } #oiwztajmgz .gt_indent_3 { text-indent: 15px; } #oiwztajmgz .gt_indent_4 { text-indent: 20px; } #oiwztajmgz .gt_indent_5 { text-indent: 25px; } #oiwztajmgz .katex-display { display: inline-flex !important; margin-bottom: 0.75em !important; } #oiwztajmgz div.Reactable > div.rt-table > div.rt-thead > div.rt-tr.rt-tr-group-header > div.rt-th-group:after { height: 0px !important; } </style> <table class="gt_table" data-quarto-disable-processing="false" data-quarto-bootstrap="false"> <thead> <tr class="gt_col_headings"> <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1" scope="col" id="label"><span class='gt_from_md'><strong>Characteristic</strong></span></th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="estimate"><span class='gt_from_md'><strong>OR</strong></span><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;">¹</span></th> <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col" id="std.error"><span class='gt_from_md'><strong>SE</strong></span></th> </tr> </thead> <tbody class="gt_table_body"> <tr><td headers="label" class="gt_row gt_left">(Intercept)</td> <td headers="estimate" class="gt_row gt_center">0.00**</td> <td headers="std.error" class="gt_row gt_center">5.42</td></tr> <tr><td headers="label" class="gt_row gt_left">height</td> <td headers="estimate" class="gt_row gt_center">1.10**</td> <td headers="std.error" class="gt_row gt_center">0.031</td></tr> <tr><td headers="label" class="gt_row gt_left" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">AIC</td> <td headers="estimate" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;">57.0</td> <td headers="std.error" class="gt_row gt_center" style="border-top-width: 2px; border-top-style: solid; border-top-color: #D3D3D3;"><br /></td></tr> <tr><td headers="label" class="gt_row gt_left">No. Obs.</td> <td headers="estimate" class="gt_row gt_center">64</td> <td headers="std.error" class="gt_row gt_center"><br /></td></tr> </tbody> <tfoot class="gt_sourcenotes"> <tr> <td class="gt_sourcenote" colspan="3"><span class='gt_from_md'>Abbreviations: CI = Confidence Interval, OR = Odds Ratio, SE = Standard Error</span></td> </tr> </tfoot> <tfoot class="gt_footnotes"> <tr> <td class="gt_footnote" colspan="3"><span class="gt_footnote_marks" style="white-space:nowrap;font-style:italic;font-weight:normal;line-height:0;">¹</span> <span class='gt_from_md'>*p<0.05; **p<0.01; ***p<0.001</span></td> </tr> </tfoot> </table> </div> --- class: center, middle, inverse <!-- background-image: url("images/cool.png") --> # 注意 --- # 注意 比较若干个模型的系数时需要注意。参考下文： [Logistic模型的系数比较问题及解决策略：一个综述](https://www.society.shu.edu.cn/CN/Y2015/V35/I4/220)