morrisCI: Analytical confidence intervals using the Morris (1988)...

In cir: Centered Isotonic Regression and Dose-Response Utilities

Analytical confidence intervals using the Morris (1988) algorithm

Description

Analytical confidence intervals for CIR and IR, using the recursive algorithm by Morris (1988), equation (4.3), for ordered-binomial point estimates. Optionally, the intervals are narrowed further using a backup pointwise interval estimate.

Usage

morrisCI(
  y,
  n,
  phat = y/n,
  conf = 0.9,
  narrower = TRUE,
  alternate = wilsonCI,
  ...
)

Arguments

y	integer or numeric vector, the pointwise Binomial counts
n	integer or numeric vector, the pointwise sample sizes
phat	numeric vector, the point estimates. Defaults to y/n, but when called by isotInterval is overridden by the actual CIR/IR point estimate.
conf	numeric, the interval's confidence level as a fraction in (0,1). Default 0.9.
narrower	logical, if the alternate-produced interval is narrower at any point, should it replace the Morris result? Also, can we enforce straightforward monotonocity to narrow the bounds? Default TRUE.
alternate	function to use for alternate pointwise interval. Default wilconCI.
...	parameters passed on to alternate.

Details

The default for backup is Wilson's (wilconCI). Also available are Jeffrys' (jeffCI) and Agresti-Coull (agcouCI).

Value

A two-column matrix with the same number of rows as length(phat), containing the calculated lower and upper bounds, respectively.

Note

This function found and corrected a typo in equation (4.3), namely the use of G_(j+1) in the recursion. The recursion cannot start in this way. Rather, it is the use of theta_(j+1) that delivers information from adjacent doses. Or perhaps in other words, there is only one G function rather than a different one for each dose. The correction has been verified by reproducing the numbers in the Morris (1988) example (Table 1), and also approved by the original author.

Author(s)

Assaf P. Oron <assaf.oron.at.gmail.com>

References

Morris, M., 1988. Small-sample confidence limits for parameters under inequality constraints with application to quantal bioassay. Biometrics 44, 1083-1092.

See Also

isotInterval

Examples

# Interesting run (#664) from a simulated up-and-down ensemble:
# (x will be auto-generated as dose levels 1:5)
dat=doseResponse(y=c(1/7,1/8,1/2,1/4,4/17),wt=c(7,24,20,12,17))
# The experiment's goal is to find the 30th percentile
slow1=cirPAVA(dat,full=TRUE)
# Default interval (Morris+Wilson); same as you get by directly calling 'quickIsotone'
int1=isotInterval(slow1)
# Morris without Wilson; the 'narrower=FALSE' argument is passed on to 'morrisCI'
int1_0=isotInterval(slow1,narrower=FALSE)
# Wilson without Morris
int2=isotInterval(slow1,intfun=wilsonCI)
# Agresti=Coull (the often-used "plus 2")
int3=isotInterval(slow1,intfun=agcouCI)
# Jeffrys (Bayesian-inspired) is also available
int4=isotInterval(slow1,intfun=jeffCI)

### Showing the data and the intervals
par(mar=c(3,3,4,1),mgp=c(2,.5,0),tcl=-0.25)
plot(dat,ylim=c(0,0.65),refsize=4,las=1,main="Forward-Estimation CIs") # uses plot.doseResponse()

# The true response function; true target is where it crosses the y=0.3 line
lines(seq(0,7,0.1),pweibull(seq(0,7,0.1),shape=1.1615,scale=8.4839),col=4)

lines(int1$ciLow,lty=2,col=2,lwd=2) 
lines(int1$ciHigh,lty=2,col=2,lwd=2) 

lines(int1_0$ciLow,lty=2) 
lines(int1_0$ciHigh,lty=2) 

lines(int2$ciLow,lty=2,col=3) 
lines(int2$ciHigh,lty=2,col=3) 
# Plotting the remaining 2 is skipped, as they are very similar to Wilson.

# Note how the default (red) boundaries take the tighter of the two options everywhere, 
# except for one place (dose 1 upper bound) where they go even tighter thanks to monotonicity 
# enforcement. This can often happen when sample size is uneven; since bounds tend to be 
# conservative it is rather safe to do.

legend('topleft',pch=c(NA,'X',NA,NA,NA),lty=c(1,NA,2,2,2),col=c(4,1,2,1,3),lwd=c(1,1,2,1,1),legend
=c('True Curve','Observations','Morris+Wilson (default)','Morris only','Wilson only'),bty='n')

cir documentation built on April 27, 2023, 9:05 a.m.