Simultaneous Inference
Simultaneous Inference
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DOI：
10.1002/0470011815.b2a15148
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发表时间：
2020
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通讯作者：
Peter C. O'Brien
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作者：
Nancy L. Geller;Peter C. O'Brien
Bonferroni联合置信区间将单个参数的置信系数调整为较高的1 \（\alpha \），以便参数集合的置信系数必须至少为1 \（\alpha \）。这是基于以下不等式：定理（Bonferroni不等式）\[P（\beta_0\cap\beta_1）\geq1-P（\beta_0^c）-P（\beta_1^c）\label{Bonferroni}\]对于任何两个事件\（\beta_0 \）和\（\beta_1 \），其中\（\beta_0^c \）和\（\beta_1^c \）分别是事件\（\beta_0 \）和\（\beta_1 \）的补充。我们采取，（\beta_0 =\）事件的置信区间（\beta_0 \）覆盖\（\beta_0 \）;并且，\（\beta_1 =\）事件的置信区间（\beta_1 \）覆盖\（\beta_1 \）;所以，如果\（P（\beta_0）= 1-\alpha_1\），且（P（\beta_1）= 1-\alpha_2 \），然后\（P（\beta_0\cap\beta_1）\geq1-\alpha_1-\alpha_2 \），通过Bonferroni不等式（等式\ref{Bonferroni}）。请注意，\（\beta_0\cap\beta_1\）是两个参数的置信区间覆盖各自参数的事件。因此，我们取\（\alpha_1 = \alpha_2 = \alpha/2 \），得到置信系数至少为\（1 \alpha \）的联合置信区间，\（\beta_0\）和\（\beta_1\）分别为\（b_0 \pm t（1-\alpha/4; n-2）s（b_0）\）和\（b_1 \pm t（1-\alpha/4;n-2）s（b_1）\）。
Bonferroni Joint Confidence Intervals The confidence coefficients for individual parameters are adjusted to the higher 1 \( \alpha \) so that the confidence coefficient for the collection of parameters must be at least 1 \( \alpha \). This is based on the following inequality: Theorem (Bonferroni's Inequality) \[P( \beta_0\cap\beta_1)\geq1-P(\beta_0^c)-P(\beta_1^c) \label{Bonferroni}\] for any two events \(\beta_0 \) and \( \beta_1 \), where \(\beta_0^c \) and \(\beta_1^c \) are complements of events \( \beta_0 \) and \( \beta_1 \), respectively. We take, \( \beta_0 =\) the event that confidence interval for \(\beta_0 \) covers \(\beta_0 \); and, \( \beta_1 =\) the event that confidence interval for \( \beta_1 \) covers \( \beta_1 \); So, if \(P(\beta_0) = 1-\alpha_1 \), and \(P(\beta_1) = 1-\alpha_2 \), then \(P(\beta_0\cap\beta_1)\geq1-\alpha_1-\alpha_2 \), by Bonferroni's inequality (Equation \ref{Bonferroni}). Note that \(\beta_0\cap\beta_1\) is the event that confidence intervals for both the parameters cover the respective parameters. Therefore we take \(\alpha_1 = \alpha_2 = \alpha/2 \) to get joint confidence intervals with confidence coefficient at least \(1 \alpha \), \(b_0 \pm t(1-\alpha/4;n-2) s(b_0) \) and \(b_1 \pm t(1-\alpha/4;n-2) s(b_1) \) for \(\beta_0\) and \(\beta_1\), respectively.