# gofstat: Goodness-of-fit statistics In fitdistrplus: Help to Fit of a Parametric Distribution to Non-Censored or Censored Data

## Description

Computes goodness-of-fit statistics for parametric distributions fitted to a same non-censored data set.

## Usage

 ```1 2 3 4 5``` ```gofstat(f, chisqbreaks, meancount, discrete, fitnames=NULL) ## S3 method for class 'gofstat.fitdist' print(x, ...) ```

## Arguments

 `f` An object of class `"fitdist"`, output of the function `fitdist`, or a list of `"fitdist"` objects. `chisqbreaks` A numeric vector defining the breaks of the cells used to compute the chi-squared statistic. If omitted, these breaks are automatically computed from the data in order to reach roughly the same number of observations per cell, roughly equal to the argument `meancount`, or sligthly more if there are some ties. `meancount` The mean number of observations per cell expected for the definition of the breaks of the cells used to compute the chi-squared statistic. This argument will not be taken into account if the breaks are directly defined in the argument `chisqbreaks`. If `chisqbreaks` and `meancount` are both omitted, `meancount` is fixed in order to obtain roughly (4n)^{2/5} cells with n the length of the dataset. `discrete` If `TRUE`, only the Chi-squared statistic and information criteria are computed. If missing, `discrete` is passed from the first object of class `"fitdist"` of the list `f`. `fitnames` A vector defining the names of the fits. `x` An object of class `"fitdist"`. `...` Further arguments to be passed to generic functions.

## Details

Goodness-of-fit statistics are computed. The Chi-squared statistic is computed using cells defined by the argument `chisqbreaks` or cells automatically defined from data, in order to reach roughly the same number of observations per cell, roughly equal to the argument `meancount`, or sligthly more if there are some ties. The choice to define cells from the empirical distribution (data), and not from the theoretical distribution, was done to enable the comparison of Chi-squared values obtained with different distributions fitted on a same data set. If `chisqbreaks` and `meancount` are both omitted, `meancount` is fixed in order to obtain roughly (4n)^{2/5} cells, with n the length of the data set (Vose, 2000). The Chi-squared statistic is not computed if the program fails to define enough cells due to a too small dataset. When the Chi-squared statistic is computed, and if the degree of freedom (nb of cells - nb of parameters - 1) of the corresponding distribution is strictly positive, the p-value of the Chi-squared test is returned.

For continuous distributions, Kolmogorov-Smirnov, Cramer-von Mises and Anderson-Darling and statistics are also computed, as defined by Stephens (1986).

An approximate Kolmogorov-Smirnov test is performed by assuming the distribution parameters known. The critical value defined by Stephens (1986) for a completely specified distribution is used to reject or not the distribution at the significance level 0.05. Because of this approximation, the result of the test (decision of rejection of the distribution or not) is returned only for data sets with more than 30 observations. Note that this approximate test may be too conservative.

For data sets with more than 5 observations and for distributions for which the test is described by Stephens (1986) for maximum likelihood estimations (`"exp"`, `"cauchy"`, `"gamma"` and `"weibull"`), the Cramer-von Mises and Anderson-darling tests are performed as described by Stephens (1986). Those tests take into account the fact that the parameters are not known but estimated from the data by maximum likelihood. The result is the decision to reject or not the distribution at the significance level 0.05. Those tests are available only for maximum likelihood estimations.

Only recommended statistics are automatically printed, i.e. Cramer-von Mises, Anderson-Darling and Kolmogorov statistics for continuous distributions and Chi-squared statistics for discrete ones ( `"binom"`, `"nbinom"`, `"geom"`, `"hyper"` and `"pois"` ).

Results of the tests are not printed but stored in the output of the function.

## Value

`gof.stat` returns an object of class `"gofstat.fitdist"` with following components,

 ` chisq ` a named vector with the Chi-squared statistics or `NULL` if not computed ` chisqbreaks ` common breaks used to define cells in the Chi-squared statistic ` chisqpvalue ` a named vector with the p-values of the Chi-squared statistic or `NULL` if not computed ` chisqdf ` a named vector with the degrees of freedom of the Chi-squared distribution or `NULL` if not computed ` chisqtable ` a table with observed and theoretical counts used for the Chi-squared calculations ` cvm ` a named vector of the Cramer-von Mises statistics or `"not computed"` if not computed ` cvmtest ` a named vector of the decisions of the Cramer-von Mises test or `"not computed"` if not computed ` ad ` a named vector with the Anderson-Darling statistics or `"not computed"` if not computed ` adtest ` a named vector with the decisions of the Anderson-Darling test or `"not computed"` if not computed ` ks ` a named vector with the Kolmogorov-Smirnov statistic or `"not computed"` if not computed ` kstest ` a named vector with the decisions of the Kolmogorov-Smirnov test or `"not computed"` if not computed `aic` a named vector with the values of the Akaike's Information Criterion. `bic` a named vector with the values of the Bayesian Information Criterion. `discrete` the input argument or the automatic definition by the function from the first object of class `"fitdist"` of the list in input. `nbfit` Number of fits in argument.

## Author(s)

Marie-Laure Delignette-Muller and Christophe Dutang.

## References

Cullen AC and Frey HC (1999), Probabilistic techniques in exposure assessment. Plenum Press, USA, pp. 81-155.

Stephens MA (1986), Tests based on edf statistics. In Goodness-of-fit techniques (D'Agostino RB and Stephens MA, eds), Marcel Dekker, New York, pp. 97-194.

Venables WN and Ripley BD (2002), Modern applied statistics with S. Springer, New York, pp. 435-446.

Vose D (2000), Risk analysis, a quantitative guide. John Wiley & Sons Ltd, Chischester, England, pp. 99-143.

Delignette-Muller ML and Dutang C (2015), fitdistrplus: An R Package for Fitting Distributions. Journal of Statistical Software, 64(4), 1-34.

`fitdist`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45``` ```# (1) fit of two distributions to the serving size data # by maximum likelihood estimation # and comparison of goodness-of-fit statistics # data(groundbeef) serving <- groundbeef\$serving (fitg <- fitdist(serving, "gamma")) gofstat(fitg) (fitln <- fitdist(serving, "lnorm")) gofstat(fitln) gofstat(list(fitg, fitln)) # (2) fit of two discrete distributions to toxocara data # and comparison of goodness-of-fit statistics # data(toxocara) number <- toxocara\$number fitp <- fitdist(number,"pois") summary(fitp) plot(fitp) fitnb <- fitdist(number,"nbinom") summary(fitnb) plot(fitnb) gofstat(list(fitp, fitnb),fitnames = c("Poisson","negbin")) # (3) Use of Chi-squared results in addition to # recommended statistics for continuous distributions # set.seed(1234) x4 <- rweibull(n=1000,shape=2,scale=1) # fit of the good distribution f4 <- fitdist(x4,"weibull") # fit of a bad distribution f4b <- fitdist(x4,"cauchy") gofstat(list(f4,f4b),fitnames=c("Weibull", "Cauchy")) ```