Rootograms for Assessing Goodness of Fit of Probability Models

Rootograms graphically compare (square roots) of empirical frequencies with expected (fitted) frequencies from a probabilistic model. If plot = TRUE, the resulting object of class "rootogram" is plotted by plot.rootogram or autoplot.rootogram before it is returned, depending on whether the package ggplot2 is loaded.

rootogram(object, ...)

# S3 method for default
rootogram(
  object,
  newdata = NULL,
  plot = TRUE,
  class = NULL,
  breaks = NULL,
  width = NULL,
  style = c("hanging", "standing", "suspended"),
  scale = c("sqrt", "raw"),
  expected = TRUE,
  confint = TRUE,
  ref = TRUE,
  xlab = NULL,
  ylab = NULL,
  main = NULL,
  ...
)

Arguments

object: an object from which an rootogram can be extracted with procast.
...: further graphical parameters passed to the plotting function.
newdata: an optional data frame in which to look for variables with which to predict. If omitted, the original observations are used.
plot: logical or character. Should the plot or autoplot method be called to draw the computed extended reliability diagram? Logical FALSE will suppress plotting, TRUE (default) will choose the type of plot conditional if the package ggplot2 is loaded. Alternatively "base" or "ggplot2" can be specified to explicitly choose the type of plot.
class: should the invisible return value be either a data.frame or a tbl_df. Can be set to "data.frame" or "tibble" to explicitly specify the return class, or to NULL (default) in which case the return class is conditional on whether the package "tibble" is loaded.
breaks: NULL (default) or numeric to manually specify the breaks for the rootogram intervals. A single numeric (larger 0) specifies the number of breaks to be automatically chosen, multiple numeric values are interpreted as manually specified breaks.
width: NULL (default) or single positive numeric. Width of the histogram bars. Will be ignored for non-discrete distributions.
style: character specifying the syle of rootogram (see 'Details').
scale: character specifying whether "raw" frequencies or their square roots ("sqrt"; default) should be drawn.
expected: logical or character. Should the expected (fitted) frequencies be plotted? Can be set to "both" (same as TRUE; default), "line", "point", or FALSE which will suppress plotting.
confint: logical, defaults to TRUE. Should confident intervals be drawn?
ref: logical, defaults to TRUE. Should a reference line be plotted?
xlab, ylab, main: graphical parameters forwarded to plot.rootogram or autoplot.rootogram.

Value

An object of class "rootogram" inheriting from "data.frame" or "tibble" conditional on the argument class

with the following variables:

observed: observed frequencies,
expected: expected (fitted) frequencies,
mid: histogram interval midpoints on the x-axis,
width: widths of the histogram bars,
confint_lwr, confint_upr: lower and upper confidence interval bound.

Additionally, style, scale, expected, confint, ref, xlab, ylab, amd main are stored as attributes.

Details

Rootograms graphically compare frequencies of empirical distributions and expected (fitted) probability models. For the observed distribution the histogram is drawn on a square root scale (hence the name) and superimposed with a line for the expected frequencies. The histogram can be "hanging" from the expected curve (default), "standing" on the (like bars in barplot), or drawn as a "suspended" histogram of deviations.

The function rootogram leverages the procast generic in order to compute all necessary coordinates based on observed and expected (fitted) frequencies.

In addition to the plot and autoplot method for rootogram objects, it is also possible to combine two (or more) rootograms by c/rbind, which creates a set of rootograms that can then be plotted in one go.

Note

Note that there is also a rootogram function in the vcd package that is similar to the numeric method provided here. However, it is much more limited in scope, hence a function has been created here.

References

Friendly M (2000), Visualizing Categorical Data. SAS Institute, Cary, ISBN 1580256600.

Kleiber C, Zeileis A (2016). “Visualizing Count Data Regressions Using Rootograms.” The American Statistician, 70(3), 296--303. doi:10.1080/00031305.2016.1173590 .

Tukey JW (1977). Exploratory Data Analysis. Addison-Wesley, Reading, ISBN 0201076160.

Examples

## plots and output

## number of deaths by horsekicks in Prussian army (Von Bortkiewicz 1898)
deaths <- rep(0:4, c(109, 65, 22, 3, 1))

## fit glm model
m1_pois <- glm(deaths ~ 1, family = poisson)
rootogram(m1_pois)


## inspect output (without plotting)
r1 <- rootogram(m1_pois, plot = FALSE)
r1
#> A `rootogram` object with `scale = 'sqrt'` and `style = 'hanging'`
#> 
#>    observed   expected mid width
#> 1 10.440307 10.4244987   0   0.9
#> 2  8.062258  8.1417938   1   0.9
#> 3  4.690416  4.4964526   2   0.9
#> 4  1.732051  2.0275628   3   0.9
#> 5  1.000000  0.7917886   4   0.9

## combine plots
plot(c(r1, r1), col = c(1, 2), expected_col = c(1, 2))



#-------------------------------------------------------------------------------
## different styles

## artificial data from negative binomial (mu = 3, theta = 2)
## and Poisson (mu = 3) distribution
set.seed(1090)
y <- rnbinom(100, mu = 3, size = 2)
x <- rpois(100, lambda = 3)

## glm method: fitted values via glm()
m2_pois <- glm(y ~ x, family = poisson)

## correctly specified Poisson model fit
par(mfrow = c(1, 3))
r1 <- rootogram(m2_pois, style = "standing", ylim = c(-2.2, 4.8), main = "Standing")

r2 <- rootogram(m2_pois, style = "hanging", ylim = c(-2.2, 4.8), main = "Hanging")

r3 <- rootogram(m2_pois, style = "suspended", ylim = c(-2.2, 4.8), main = "Suspended")

par(mfrow = c(1, 1))

#-------------------------------------------------------------------------------
## linear regression with normal/Gaussian response: anorexia data

data("anorexia", package = "MASS")

m3_gauss <- glm(Postwt ~ Prewt + Treat + offset(Prewt), family = gaussian, data = anorexia)

## plot rootogram as "ggplot2" graphic
rootogram(m3_gauss, plot = "ggplot2")