Fieller’s Method for the LD(50).

Fieller’s method is an alternative to the delta method for obtaining an asymptotic confidence interval for the LD(50), cf. Exercise 11.8.2.

Fieller’s method is thought to be less sensitive to the high correlation that is typically present between ˆα and βˆ. From standard results, one can obtain the estimated asymptotic variance and covariance for ˆα and βˆ; thus, for any fixed but unknown value w, an asymptotic standard error for ˆα + βwˆ is readily available as a function of w. Denote this standard error by ˆσ(w).

If α + βw is some known value Q, a 99% confidence region for w can be obtained from

.99 = Pr 

−2.5758 ≤ (ˆα + βwˆ ) − Q

σˆ(w) ≤ 2.5758

= Pr 

(ˆα + βwˆ − Q)

2 − 2.57582 σˆ2(w) ≤ 0



.

The 99% confidence region consists of all values of w that satisfy

(ˆα + βwˆ − Q)

2 − 2.57582σˆ2(w) ≤ 0.

Show how to use the quadratic formula to find the end points of the region.

Under what conditions is the confidence region an interval? What other possibilities exist? How does this result apply to estimating the LD(50)?

Using the data of Exercise 4.6.11, give a 99% confidence interval for the LD(50).