# fieller: Confidence Limits for Lethal Dose Estimate From Dose-response... In qra: Quantal Response Analysis for Dose-Mortality Data

## Description

This uses Fieller's formula to calculate a confidence interval for a specified mortality proportion, commonly 0.50, or 0.90, or 0.99. Here "dose" is a generic term for any measure of intensity of a treatment that is designed to induce insect death.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 fieller( phat, b, vv, df.t = Inf, offset = 0, logscale = FALSE, link = "logit", eps = 0, type = c("Fieller", "Delta"), maxg = 0.99 ) fieller2( phat, b, vv, df.t = Inf, offset = 0, logscale = FALSE, link = "fpower", lambda = 0, eps = 0, type = c("Fieller", "Delta"), maxg = 0.99 ) 

## Arguments

 phat Mortality proportion b Length 2 vector of intercept and slope vv Variance-covariance matrix for intercept and slope df.t Degrees of freedom for variance-covariance matrix offset Offset to be added to intercept. This can be of length 2, in order to return values on the original scale, in the case where b and vv are for values that have been scaled by subtracting offset[1] and dividing by offset[2]. logscale Should confidence limits be back transformed from log scale? link Link function that transforms expected mortalities to the scale of the linear predictor eps If eps>0 phat is replaced by \frac{p+ε}{1+2*ε} before applying the transformation. type The default is to use Fieller's formula. The Delta (type="Delta") method, which relies on a first order Taylor series approximation to the variance, is provided so that it can be used for comparative purposes. It can be reliably used only in cases where the interval has been shown to be essentially the same as given by type="Fieller"! maxg Maximum value of g for which a confidence interval will be calculated. Must be < 1. lambda The power λ, when using the link="fpower". (This applies to fieller2 only.)

## Details

See the internal code for details of the value g. The calculation gives increasing wide confidence intervals as g approaches 1. If g>=1, there are no limits. The default value for df.t is a rough guess at what might be reasonable. For models fitted using lme4::lmer(), abilities in the lmerTest package can be used to determine a suitable degrees of freedom approximation — this does not extend to use with glmer() or glmmTMB.

## Value

A vector, with elements

 est Estimate var Variance, calculated using the Delta method lwr Lower bound of confidence interval upr upper bound of confidence interval g If g is close to 0 (perhaps g < 0.05), confidence intervals will be similar to those calculated using the Delta method, and the variance can reasonably be used for normal theory inference.

## References

Joe Hirschberg & Jenny Lye (2010) A Geometric Comparison of the Delta and Fieller Confidence Intervals, The American Statistician, 64:3, 234-241, DOI: 10.1198/ tast.2010.08130

E C Fieller (1944). A Fundamental Formula in the Statistics of Biological Assay, and Some Applications. Quarterly Journal of Pharmacy and Pharmacology, 17, 117-123.

David J Finney (1978). Statistical Method in Biological Assay (3rd ed.), London, Charles Griffin and Company.

varRatio
 1 2 3 4 5 redDel <- subset(qra::codling1988, Cultivar=="Red Delicious") redDel.glm <- glm(cbind(dead,total-dead)~ct, data=redDel, family=quasibinomial(link='cloglog')) vv <- summary(redDel.glm)\$cov.scaled fieller(0.99, b=coef(redDel.glm), vv=vv, link='cloglog')