grcmle function

Maximum likelihood estimation of Poisson or ZIP parameters at the aggregate level.

This function infers Poisson or zero-inflated Poisson (ZIP) parameters from grouped and right-censored count data, and conducts a chi-squared goodness-of-fit test. A grouped and right-censored scheme may look like

0, 1, 2--4, 5--8, 9+.

For grouped and right-censored count data collected in a survey, such as frequency of alcohol drinking, number of births or occurrence of crimes, the response category designed as the example above means never, once, 2 to 4 times, 5 to 8 times, 9 times and more. The frequency distribution from a sample corresponding to the example above may look like

3, 15, 168, 155, 15.

grcmle(counts, scheme, method = c("Poisson", "ZIP"),
  do.plot = T, init.guess = NULL,
  optimizing.algorithm.index = 2, lambda.extend.ratio = 3,
  conf.level = 0.95)

Arguments

• counts: specifies the frequency distribution of the grouped and right-censored count data. For the example above, one may input

  counts = c(3, 15, 168, 155, 15).

• scheme: specifies the grouping scheme. It should be a vector of integers containing the starting point (or the lowest integer) of each group. For example, to input the scheme above

  0, 1, 2--4, 5--8, 9+,

  one may use

  scheme = c(0, 1, 2, 5, 9).

• method: a string parameter specifies which statistical model to use. Currently there are two options "Poisson" and "ZIP". The default value is "Poisson". It can be abbreviated.

• do.plot: a logical variable indicating whether or not to plot the log likelihood. The default is T.

• init.guess: the initial value used for the optimization procedure of the likelihood estimation. The default value is NULL, which instructs the function grcmle to select the initial value automatically.

• optimizing.algorithm.index: defines which optimization algorithm to use. Currently the possible values are 1,2,3,4,5,6,7 and 8, representing the following algorithms, respectively:

  NLOPT_GN_DIRECT_L

  NLOPT_GN_DIRECT

  NLOPT_GN_DIRECT_L_RAND

  NLOPT_GN_DIRECT_NOSCAL

  NLOPT_GN_DIRECT_L_NOSCAL

  NLOPT_GN_DIRECT_L_RAND_NOSCAL

  NLOPT_GN_ORIG_DIRECT

  NLOPT_GN_ORIG_DIRECT_L

  For details of these algorithms, please see the manual of the package "nloptr". The default value is 2.

• lambda.extend.ratio: specifies the searching interval of possible $\lambda$ as $[0, nr]$, where $n$ is the left end (i.e., the lowest integer) of the last right-censored group, and $r$ is lambda.extend.ratio. By default, we set

  lambda.extend.ratio=3.

• conf.level: confidence level of the confidence interval(s) for the parameter(s) inferred

Details

Maximum likelihood estimation is used for the inference.

Returns

The returned value is a list containing - mle: the parameter(s) inferred. For Poisson model, it is the estimate of $\lambda$. For ZIP model, it shows a vector of length 2: the first element is the estimate of $p$ and the second element is the estimate of $\lambda$.

• p.value: the p-value of the chi-squared test of goodness-of-fit.

• df: the degree(s) of freedom of the chi-squared test of goodness-of-fit.

• CI.lambda: the confidence interval of $\lambda$ obtained by normal approximation

• CI.p: the confidence interval of $p$ obtained by normal approximation

• conf.level: the confidence level

• std.err: the standard error of $\lambda$

  or the standard errors of $(p, \lambda)$, if a ZIP model is specified

Author(s)

Authors: Xin Guo x.guo@polyu.edu.hk, Qiang Fu qiang.fu@ubc.ca

Examples

grcmle(counts=c(6, 15, 168, 155, 15), scheme = c(0, 1, 2, 5, 9))

grcmle(counts=c(6, 15, 168, 155, 15), scheme = c(0, 1, 2, 5, 9), method = "ZIP")