83  Bootstrap Plot (Central Tendency)

The Bootstrap method (Efron 1979) is a random sampling technique which treats the sample as if it were a population. The underlying idea is that one can obtain an empirical distribution of (almost) any statistic by repetitively drawing a sample with replacement.

The Bootstrap Plot (for Central Tendency) can be applied to univariate data series which do not have serial correlation. In practice, this means that the standard i.i.d. bootstrap version is not appropriate for time series; they require a special type of bootstrapping (see Chapter 64, Blocked Bootstrap Plot).

83.1 Definition

The Bootstrap Plot for any given data series \(x\) with \(n\) observations and pre-specified parameter \(k \in \mathbb{N}_0\), is computed according to the following steps:

• generate \(k\) samples \(z_j\) (for \(j=1,2,…,k\)) of size \(n\) (use sampling with replacement)
• compute the following measures of Central Tendency: Arithmetic Mean, Median, Midrange, Harmonic Mean, and the Geometric Mean
• compute a series of pre-specified Quantiles such as \(Quantile(q)_j\) for \(q\) = 0.005, 0.025, 0.25, 0.50, 0.75, 0.975, 0.995 and \(j = 1, 2, …,k\)
• plot the simulated statistics
• plot the Gaussian Kernel Density Plots for each simulated statistic
• plot a summary based on Notched Boxplots for each simulated statistic

83.2 Horizontal axis for Simulated Statistics

The horizontal axis represents the value of the simulated statistic.

83.3 Vertical axis for Simulated Statistics

The vertical axis corresponds to the value of the Central Tendency measure that is drawn from the sample.

83.4 Horizontal axis for Kernel Density of Simulated Statistics

The horizontal axis represents the value of the simulated statistic.

83.5 Vertical axis for Kernel Density of Simulated Statistics

The vertical axis corresponds to the estimated density value.

83.6 Horizontal axis for Notched Boxplots of Simulated Statistics

The horizontal axis represents the measures of Central Tendency (listed in arbitrary order).

83.7 Vertical axis for Notched Boxplots of Simulated Statistics

The vertical axis corresponds to the values of the Central Tendency measures.

83.8 R Module

83.8.1 Public website

The Bootstrap Plot is available on the public website:

• https://compute.wessa.net/rwasp_bootstrapplot1.wasp

83.8.2 RFC

When using the default profile, the Bootstrap Plot can be found under the “Descriptive / Bootstrap Plot” menu item.

To compute the Bootstrap Plot on your local machine, the following script can be used in the R console:

library(boot)
library(psych)

boot.stat <- function(s,i) {
  s.mean <- mean(s[i])
  s.median <- median(s[i])
  s.midrange <- (max(s[i]) + min(s[i])) / 2
  s.hmean <- harmonic.mean(s[i])
  s.gmean <- geometric.mean(s[i])
  c(s.mean, s.median, s.midrange, s.hmean, s.gmean)
}

x <- runif(n = 150, min = 10, max = 50)
par1 <- 200 # number of simulations
par2 <- 5 # significant digits

r <- boot(x, boot.stat, R = par1, stype = "i")
z <- data.frame(cbind(r$t[,1],r$t[,2],r$t[,3],r$t[,4],r$t[,5]))
colnames(z) <- list("mean","median","midrange","harmonic","geometric")

myq.1 <- 0.005
myq.2 <- 0.025
myq.3 <- 0.975
myq.4 <- 0.995

df = data.frame(statistic = c("mean",
                              "median",
                              "midrange",
                              "harmonic",
                              "geometric"),
                P0.5 = c(signif(quantile(r$t[,1],myq.1)[[1]], par2),
                       signif(quantile(r$t[,2],myq.1)[[1]], par2),
                       signif(quantile(r$t[,3],myq.1)[[1]], par2),
                       signif(quantile(r$t[,4],myq.1)[[1]], par2),
                       signif(quantile(r$t[,5],myq.1)[[1]], par2)
                       ),
                P2.5 = c(signif(quantile(r$t[,1],myq.2)[[1]], par2),
                       signif(quantile(r$t[,2],myq.2)[[1]], par2),
                       signif(quantile(r$t[,3],myq.2)[[1]], par2),
                       signif(quantile(r$t[,4],myq.2)[[1]], par2),
                       signif(quantile(r$t[,5],myq.2)[[1]], par2)
                       ),
                Q1 = c(signif(quantile(r$t[,1],0.25)[[1]], par2),
                       signif(quantile(r$t[,2],0.25)[[1]], par2),
                       signif(quantile(r$t[,3],0.25)[[1]], par2),
                       signif(quantile(r$t[,4],0.25)[[1]], par2),
                       signif(quantile(r$t[,5],0.25)[[1]], par2)
                       ),
                Estimate = c(signif(r$t0[1], par2),
                             signif(r$t0[2], par2),
                             signif(r$t0[3], par2),
                             signif(r$t0[4], par2),
                             signif(r$t0[5], par2)
                             ),
                Q3 = c(signif(quantile(r$t[,1],0.75)[[1]], par2),
                       signif(quantile(r$t[,2],0.75)[[1]], par2),
                       signif(quantile(r$t[,3],0.75)[[1]], par2),
                       signif(quantile(r$t[,4],0.75)[[1]], par2),
                       signif(quantile(r$t[,5],0.75)[[1]], par2)
                       ),
                P97.5 = c(signif(quantile(r$t[,1],myq.3)[[1]], par2),
                        signif(quantile(r$t[,2],myq.3)[[1]], par2),
                        signif(quantile(r$t[,3],myq.3)[[1]], par2),
                        signif(quantile(r$t[,4],myq.3)[[1]], par2),
                        signif(quantile(r$t[,5],myq.3)[[1]], par2)
                        ),
                P99.5 = c(signif(quantile(r$t[,1],myq.4)[[1]], par2),
                        signif(quantile(r$t[,2],myq.4)[[1]], par2),
                        signif(quantile(r$t[,3],myq.4)[[1]], par2),
                        signif(quantile(r$t[,4],myq.4)[[1]], par2),
                        signif(quantile(r$t[,5],myq.4)[[1]], par2)
                        ),
                SD = c(signif(sd(r$t[,1]), par2),
                       signif(sd(r$t[,2]), par2),
                       signif(sd(r$t[,3]), par2),
                       signif(sd(r$t[,4]), par2),
                       signif(sd(r$t[,5]), par2)
                       ),
                IQR = c(signif(quantile(r$t[,1],0.75)[[1]] - quantile(r$t[,1],0.25)[[1]], par2),
                        signif(quantile(r$t[,2],0.75)[[1]] - quantile(r$t[,2],0.25)[[1]], par2),
                        signif(quantile(r$t[,3],0.75)[[1]] - quantile(r$t[,3],0.25)[[1]], par2),
                        signif(quantile(r$t[,4],0.75)[[1]] - quantile(r$t[,4],0.25)[[1]], par2),
                        signif(quantile(r$t[,5],0.75)[[1]] - quantile(r$t[,5],0.25)[[1]], par2)
                        )
                ) 
print(df)

op <- par(mfrow=c(2,3))
plot(density(r$t[,1]),main="Density Plot",xlab="mean")
plot(density(r$t[,2]),main="Density Plot",xlab="median")
plot(density(r$t[,3]),main="Density Plot",xlab="midrange")
plot(density(r$t[,4]),main="Density Plot",xlab="harmonic mean")
plot(density(r$t[,5]),main="Density Plot",xlab="geometric mean")
colnames(z) = c("mean", "median", "midrange", "harmonic", "geometric")
boxplot(z,notch=TRUE,ylab="simulated values",main="Bootstrap Simulation - Central Tendency")
grid()

par(op)

  statistic   P0.5   P2.5     Q1 Estimate     Q3  P97.5  P99.5      SD     IQR
1      mean 29.495 29.806 31.054   31.663 32.335 33.285 33.936 0.91422 1.28140
2    median 29.100 29.825 31.608   32.603 33.610 35.287 37.107 1.52210 2.00180
3  midrange 29.414 29.523 29.731   30.087 30.111 30.157 30.190 0.20982 0.38022
4  harmonic 24.567 24.765 25.880   26.630 27.292 28.894 29.517 1.05070 1.41180
5 geometric 27.259 27.376 28.621   29.310 29.920 31.310 31.697 0.98695 1.29890

To compute the Bootstrap Plot, the R code uses the libraries boot, and psych. The boot.stat function was created to define all the measures of Central Tendency that are included in the analysis.

83.9 Purpose

The Bootstrap Plot is used when it is important to compute and compare important Central Tendency measures with empirical confidence intervals. Some statistical measures do not have a theoretical distribution (e.g. the median): therefore, the bootstrap approach can be used to obtain confidence intervals.

83.10 Pros & Cons

83.10.1 Pros

The Bootstrap Plot has the following advantages:

• It allows one to obtain confidence intervals for Central Tendency measures without the need to know the underlying distribution.
• It provides a lot of useful information about the empirical distribution of the Central Tendency measures.

83.10.2 Cons

The Bootstrap Plot has the following disadvantages:

• Most readers are not familiar with this type of analysis.
• It cannot be computed with many statistical software packages.

83.11 Example

Interactive Shiny app (click to load).

Open in new tab

We generated a series of 100 random numbers based on the rand function (i.e. a “pseudo-random number generator”) in a spreadsheet. We know that the rand function generates numbers between 0 and 1 with an average of 0.5 (at least this is what we would expect).

The output shows the Quantiles for the various measures of Central Tendency. It is not surprising that the estimated Midrange is closest to the true value. Also observe how all other measures of Central Tendency have an uncertainty which is substantially larger than the one of the Midrange (which is has an extremely small Inter Quartile Range -- see Section 66.17).

The Bootstrap Plot shows the Notched Boxplots for each of the simulated measures of Central Tendency. The Midrange seems to achieve the best results because it estimates the right value (0.5) and has a low variability.

Efron, Bradley. 1979. “Bootstrap Methods: Another Look at the Jackknife.” The Annals of Statistics 7 (1): 1–26. https://doi.org/10.1214/aos/1176344552.