英语翻译3.1.7 Delta Method for Log Odds Ratio*Standard errors fo
英语翻译
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.