I would like to tie off this thread (?!?!) by thanking Jeff Newmiller,
Rui Barradas, Avi Gross and Bill Dunlap for their advice and insight.

I have attached the code that I finally put together, on the basis of
the aforementioned advice, in the file ciPlot.txt.  I have also
attached the necessary data set in the file egDat.txt.

Just in case anyone is interested or in case someone else might benefit
from seeing this code.

cheers,

Rolf Turner
-- 
Honorary Research Fellow
Department of Statistics
University of Auckland
Phone: +64-9-373-7599 ext. 88276
#
# Script ciPlot.txt for doing the confidence interval plots that
# I wanted.
#

library(ggplot2)
cidf   <- dget("egDat.txt")
cidf.a <- cidf[cidf$param=="alpha",]
cidf.b <- cidf[cidf$param=="beta",]
Ylab.a <- expression(paste("Bias in ",alpha))
Ylab.b <- expression(paste("Bias in ",beta))
Part.a <- ggplot(cidf.a, aes(Ndat, estimate)) +
          geom_errorbar(aes(ymin = lower, ymax = upper), width = 50) +
          geom_point(size = 1) +
          geom_hline(yintercept = 0,col="red") +
          labs(x="",y=Ylab.a) +
          theme_bw() +
          theme(axis.title.x=element_text(size=14),
                axis.title.y=element_text(size=14)) +
          theme(axis.text.x=element_text(size=14),
                axis.text.y=element_text(size=14)) +
          theme(panel.border=element_rect(linetype="solid",fill=NA),
                panel.grid.minor=element_blank(),
                panel.grid.major=element_blank())
Part.b <- ggplot(cidf.b, aes(Ndat, estimate)) +
          geom_errorbar(aes(ymin = lower, ymax = upper), width = 50) +
          geom_point(size = 1) +
          geom_hline(yintercept = 0,col="red") +
          labs(x="Sample size",y=Ylab.b) +
          theme_bw() +
          theme(axis.title.x=element_text(size=14),
                axis.title.y=element_text(size=14)) +
          theme(axis.text.x=element_text(size=14),
                axis.text.y=element_text(size=14)) +
          theme(panel.border=element_rect(linetype="solid",fill=NA),
                panel.grid.minor=element_blank(),
                panel.grid.major=element_blank())
ciPlot <- cowplot::plot_grid(Part.a,Part.b,nrow=2)
print(ciPlot)

structure(list(Ndat = c(100, 500, 1000, 1500, 2000, 100, 500, 
1000, 1500, 2000), lower = c(0.209325514321019, 0.0403237235334606, 
0.0145196529159657, 0.0216209917402008, -0.00989822805983288, 
1.67646857307238, 0.293606879421865, 0.126166692157642, 0.135454331658732, 
-0.0506914315997755), estimate = c(0.328510135047226, 0.0804793047005517, 
0.0401822279422254, 0.0422410687789209, 0.00959659685206644, 
2.42401777661119, 0.512738636066594, 0.262346592121412, 0.245742220593958, 
0.0555399174577091), upper = c(0.447694755773433, 0.120634885867643, 
0.0658448029684851, 0.0628611458176411, 0.0290914217639658, 3.17156698015001, 
0.731870392711323, 0.398526492085181, 0.356030109529183, 0.161771266515194
), param = structure(c(1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L
), .Label = c("alpha", "beta"), class = "factor")), class = "data.frame", 
row.names = c(NA, 
-10L))
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