Fitting a Zeta distribution to a P survey—number of groups data

suppressWarnings(suppressPackageStartupMessages({library(fitPS)}))

In this example we will learn how to use fitPS to fit a Zeta distribution to some data from a survey where the number of groups of glass found is recorded. The data in this example comes from Roux et al. (2001) who surveyed the footwear of 776 individuals in south-eastern Australia, and is summarised in the table below.

This data set is built into the package and can be accessed from the Psurveys object. That is, we can type:

roux = Psurveys$roux

The data is stored as an object of class psData. This probably will not be of importance to most users. If you are interested in the details, then these can be found in the Value section of the help page for readData. There is an S3 print method for objects of time, meaning that if we print the object—either by typing its name at the command prompt, or by explicitly calling print—then we will get formatted printing of the information contained within the object. Specifically:

• the values of \(n\) and \(r_n\) will be printed out in tabular format (where \(n\) is the number of groups or the size of the groups, and \(r_n\) is the number of times \(n\) has been observed in the survey),
• the type of survey will be printed—either “Number of Groups” or “Group Size”,
• and if the object has a reference or notes attached to it, then these will be printed as well

For example

roux
#> 
#> 
#> Number of Groups
#> 
#>   n    rn
#> ---  ----
#>   0   754
#>   1     9
#>   2     8
#>   3     4
#>   4     1
#> Roux C, Kirk R, Benson S, Van Haren T, Petterd C (2001). "Glass
#> particles in footwear of members of the public in south-eastern
#> Australia-a survey." _Forensic Science International_, *116*(2),
#> 149-156. doi:10.1016/S0379-0738(00)00355-8
#> <https://doi.org/10.1016/S0379-0738%2800%2900355-8>.

It is very simple to fit a Zeta distribution to this data set. We do this using the fitDist function.

fit = fitDist(roux)

The function returns an object of class psFit, the details of which can be found in the help page for fitDist. There are both S3 print and S3 plot methods for objects of this class. The print method method displays an estimate of the shape parameter \(s\), an estimate of the standard deviation—the standard error—of the estimate of \(s\) (\(\widehat{\mathrm{sd}}(\hat{s})=\mathrm{se}(\hat{s})\)). NOTE: it is important to understand that the value of the shape parameter that is displayed, and the value that is stored in the fitted object differ by 1. That is, \(s\) is shown,m and \(s^\prime = s - 1\) is stored. This is done because the package which assists in the fitting, VGAM, is parameterised in terms of \(s^\prime\). This difference only has consequences if the fitted value is being used in conjuction with other VGAM functions. The print method also displays the first 10 fitted probabilities from the model by default. The number of probabilities shown can be altered by changing the value of nterms. We will do this here to only display the first six probabilities (from \(P_0\) to \(P_5\)).

print(fit, nterms = 6)
#> The estimated shape parameter is  4.9544 
#> The standard error of shape parameter is  0.2366 
#> ------
#> NOTE: The shape parameter is reported so that it is consistent with Coulson et al.
#> However, the value returned is actually s' = shape - 1 to be consistent with the 
#> VGAM parameterisation, which is used for computation. This has flow on effects, for
#> example in confInt. This will be changed at some point.
#> ------
#> 
#> The first  6 fitted values are:
#>           P0           P1           P2           P3           P4           P5 
#> 0.9631546669 0.0310644678 0.0041670821 0.0010019171 0.0003316637 0.0001344002

The package provides a confint method for the fitted value. The method returns both a Wald confidence interval and profile likelihood interval. The Wald interval takes the usual the form where the lower and upper bound are given by \(\hat{s} \pm z^*(1-\alpha/2)\times se(\hat{s}))\). The profile likelihood interval finds the end-points of the interval that satisfies \[ -2\left[\ell(\hat{s};\mathbf{x})-\ell(s;\mathbf{x})\right] \le \chi^2_1(\alpha) \] where \(\ell(s;\mathbf{x})\) is value of the log-likelihood given shape parameter \(s\). The two intervals are returned as elements of a list named wald and prof respectively.

ci = confint(fit)
ci$wald
#>     2.5%    97.5% 
#> 3.490761 4.418099
ci$prof
#>     2.5%    97.5% 
#> 3.520495 4.451277

You will notice that neither of these intervals contain the value shown in the previous output. However, this simply because they are confidence intervals on \(s^\prime\) and not \(s\). This can be remedied by adding one to each interval:

ci$wald + 1
#>     2.5%    97.5% 
#> 4.490761 5.418099
ci$prof + 1
#>     2.5%    97.5% 
#> 4.520495 5.451277

The reason for not correcting these intervals is that the method mostly exists to feed into other parts of the package, especially the plot method.