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Question

3.1.7 Delta Method for Log Odds Ratio*
Standard errors for the log odds ratio and the log relative risk result from a multiparameter version of the delta method. Suppose that have a multinomial distribution. The sample proportion has mean and variance
and (3.7)
In Section 14.1.4 we show that for , and have covariance
(3.8)
The sample proportions have a large-sample multivariate normal distribution. For functions of them ,the delta method implies the following result, proved in Section 14.1.4:
Let denote a differentiable function of , with sample value for a multinomial sample. Let

Then as , the distribution of converges to standard normal, where
. (3.9)
The asymptotic variance depends on and the partial derivatives of the measure with respect to .In practice, replacing and in (3.9) by their sample values yields an ML estimate of .Then is an estimated standard error for .A large-sample Wald confidence interval for is
.
With the substitution of for in (3.9), the limiting distribution is still standard normal, bur convergence is slower. The equivalence in the large-sample distribution is justified as follows: The sample proportions converge in probability to ,by the weak law of large numbers. Since is a continuous function of the sample proportions, it converges in proportions to ,and converges in probability to 1. Now

The first term on the right-hand side converges in distribution to standard normal, by (3.9),and the second term converges in probability to 1.Thus , their product also has a limiting standard normal distribution
We now apply the delta method to the log odds ratio, taking .since

. The asymptotic standard error of log for a multinomial sample is .
Since ,the estimated standard error is (3.1).
The delta method also applies directly with to obtain and a Wald confidence interval .This is not recommended; converges more slowly than log to normality, this interval could contain negative values, and it doer not give results equivalent to those obtained with the Wald interval using and its standard error.

Answer 1

3.1.7 三角洲方法为日志可能性比率* 标准误差为日志可能性比率和日志相对风险起因于三角洲方法的multiparameter 版本。假设有多项式发行。样品比例有手段和变化 并且(3.7) 在部分14.1.4 我们显示那为, 和有协变性 (3.8) 样品比例有大样品多维分布的正常分配。为作用的他们, 三角洲方法暗示以下结果, 被证明在部分14.1.4: 让表示一个能区分的作用, 以样品价值为一个多项式样品。让 然后和, 发行聚合对标准法线, ．(3.9) 渐进变化依靠并且措施的部份衍生物谈到In 实践, 替换和(3.9) 由他们的样品价值出产量Then 的机器语言估计是一个估计的标准误差为A大样品Wald 信心间隔时间为是 ． 以代替为(3.9), 限制的发行是标准法线, bur 汇合是更慢的。相等在大样品发行被辩解如下: 样品比例聚合在可能性对,by 大数字微弱的法律。从是样品比例的一个连续函数, 它聚合在比例对, 和聚合在可能性到1 。现在 第一期限在右边聚合在发行对标准法线,(3.9), 并且第二个期限聚合在可能性对1.Thus, 他们的产品并且有限制的标准正常分配 我们向日志可能性比率现在运用三角洲方法, 采取since ．日志渐进标准误差为一个多项式样品是。 因为, 估计的标准误差是(3.1) 。 三角洲方法并且申请直接地与获得并且Wald 信心间隔时间This 不被推荐; 比日志对正常性, 这间隔时间能包含消极价值, 和它实行家不是授予结果等效与那些被获得以Wald 间隔时间使用和它的标准误差慢慢地聚合。