# Introduction

Belia, Fidler, Williams, and Cumming (2005) found that researchers in
psychology, behavior neuroscience and medicine are really bad at
interpreting when error bars signify that two means are significantly
different (*p* = 0.05). What they did was to email a bunch of
researchers and invite them to take a web-based test, and they got 473
usable responses. The test consisted of an interactive plot with error
bars for two independent groups, the participants were asked to move the
error bars to a position they believed would represent a significant
*t*-test at *p*=0.05. They did this for error bars based on the 95 % CI
and the group’s standard errors. The participants did on average set the
95 % CI too far apart with their mean placement corresponding to a *p*
value of .009. They did the opposite with the SE error bars, which they
put too close together yielding placements corresponding to *p* = 0.109.
And if you’re wondering they found no difference between the three
disciplines.

# Plots

I wanted to pull my weight, and I have therefore created some various
plots in R that show error bars that are significant at various
*p*-values.

*Figure 1.* Error bars corresponding to a significant difference at p = .05 (equal group sizes and equal variances)

*Figure 2.* Error bars corresponding to a significant difference at p = .01 (equal group sizes and equal variances)

*Figure 3.* Error bars corresponding to a significant difference at p = .001 (equal group sizes and equal variances)

Based on the first plot we see that an overlap of about one third of the
95 % CIs corresponds to *p* = 0.05. For the SE error bars we see that
they are about 1 SE apart when *p* = 0.05.

## R Code

Here's the complete R code used to produce these plots

library(ggplot2) library(ggplot2) library(plyr) m2 <- 100 # initital group size, should be the same as m1 p <- 1 # starting p-value m1 <- 100 # mean group 1 sd1 <- 10 # sd group 1 sd2 <- 10 # sd group 2 n <- 20 # n per group s <- sqrt(0.5 * (sd1^2 + sd2^2)) # pooled sd while(p>0.05) { # loop til p = 0.05 t <- (min(c(m1,m2)) - max(c(m1,m2))) / (s * sqrt(2/n)) # t statistics df <- (n*2)-2 # degress of freedom p <-pt(t, df)*2 # p value m2 <- m2 - (m2/10000) # adjust mean for group 2 } get_CI <- function(x, sd, CI) { # calculate error bars se <- sd/sqrt(n) # standard error lwr <- c(x - qt((1 + CI)/2, n - 1) * se, x - se) # 95 % CI and SE lower limit upr <- c(x + qt((1 + CI)/2, n - 1) * se, x + se) # 95 % CI and SE upper limit data.frame("lwr" = lwr, "upr" = upr, "se" = se) # result } plot_df <- data.frame("mu" = rep(c(m1,m2), each=2)) # means plot_df$group <- gl(2,2, labels=c("group1", "group2")) # group factor plot_df$type <- gl(2,1,4, labels=c("95 % CI", "se errorbars")) # type of errorbar plot_df <- cbind(plot_df, rbind(get_CI(m1, sd1, .95), get_CI(m2, sd2, .95))) # put it all together get_overlap <- function(arg) { # calculate overlap % x <-subset(plot_df, type == arg) # subset for type of errorbar x_range <- abs(mean(x$lwr - x$upr)) # length of error bar x_lwr <- max(x$lwr) # lwr limit for group with highest lwr limit x_upr <- min(x$upr) # upr limit for group with lowest lwr limit overlap <- abs( (x_upr - x_lwr) / x_range) # % overlap data.frame("type"=arg, "range" = x_range, "lwr" = x_lwr, "upr" = x_upr, "overlap" = round(overlap, 2)) # result } overlap <-ldply(levels(plot_df$type), get_overlap) # get overlap and put into dataframe overlap$text <- paste(overlap$overlap * 100, "% of errorbar") # label text overlap$text_y <- c(overlap[1,4], overlap[2,3]) # quick-fix ggplot(plot_df, aes(group, mu, group=group)) + geom_point(size=3) + # point for group mean geom_errorbar(aes(ymax=upr, ymin=lwr), width=0.2) + # error bars for means opts(title=paste("Illustration of errorbars for a significant 2-sample t-test, p =", round(p,3))) + # plot title facet_grid(. ~ type) + # split plot after error bar type geom_errorbar(data=overlap, aes(ymax=upr, ymin=lwr, x=1.5, y=NULL, group=type), width=0.1, color="red") + # add overlap error bar geom_text(data=overlap, aes(label = text, group=type, y=text_y, x=1.5, vjust=-1)) + # annotate overlap ylab(expression(bar(x))) # change y label

Belia
S, Fidler F, Williams J, & Cumming G (2005). Researchers misunderstand
confidence intervals and standard error bars.
Psychological methods, 10 (4),
389-96 PMID: 16392994