Type:	Package
Title:	Goodness-of-Fit Methods for Right-Censored Data
Version:	1.5
Date:	2025-05-29
Description:	Graphical tools and goodness-of-fit tests for right-censored data: 1. Kolmogorov-Smirnov, Cramér-von Mises, and Anderson-Darling tests, which use the empirical distribution function for complete data and are extended for right-censored data. 2. Generalized chi-squared-type test, which is based on the squared differences between observed and expected counts using random cells with right-censored data. 3. A series of graphical tools such as probability or cumulative hazard plots to guide the decision about the most suitable parametric model for the data. These functions share several features as they can handle both complete and right-censored data, and they provide parameter estimates for the distributions under study.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
URL:	https://arnaugarciagrbio.github.io/GofCens/, https://github.com/ArnauGarciaGRBIO/GofCens
Encoding:	UTF-8
Depends:	R (≥ 3.5.0), survival, actuar
Imports:	fitdistrplus, grid, ggplot2, gridExtra, boot, methods
NeedsCompilation:	no
VignetteBuilder:	knitr
Suggests:	knitr, rmarkdown, rms
Packaged:	2025-05-29 18:06:47 UTC; klangohr
Repository:	CRAN
Date/Publication:	2025-05-29 20:20:02 UTC
Author:	Klaus Langohr [aut, cre], Mireia Besalú [aut], Matilde Francisco [aut], Arnau Garcia [aut], Guadalupe Gómez [aut]
Maintainer:	Klaus Langohr <klaus.langohr@upc.edu>

Goodness-of-Fit Methods for Right-Censored Data.

Description

This package implements both graphical tools and goodness-of-fit tests for right-censored data. All functions can handle complete as well as right-censored data. It has implemented:

1. Kolmogorov-Smirnov (KScens), Cramér-von Mises (CvMcens), and Anderson-Darling (ADcens) tests, which use the empirical distribution function for complete data and are extended for right-censored data.

2. Generalized chi-squared-type (chisqcens) test, which is based on the squared differences between observed and expected counts using random cells with right-censored data.

3. A series of graphical tools such as probability (probPlot) or cumulative hazard (cumhazPlot) plots to guide the decision about the most suitable parametric model for the data.

Details

All functions of the GofCens package can be used to check the goodness of fit of the following 8 distributions. The list shows the parametrizations of the survival functions.

1. Exponential Distribution [Exp(\beta)]

   S(t)=e^{-\frac{t}{\beta}}

2. Weibull Distribution [Wei(\alpha,\,\beta)]

   S(t)=e^{-(\frac{t}{\beta})^\alpha}

3. Gumbel Distribution [Gum(\mu,\,\beta)]

   S(t)=1 - e^{-e^{-\frac{t-\mu}{\beta}}}

4. Log-Logistic Distribution [LLogis(\alpha, \beta)]

   S(t)=\frac{1}{1 + \left(\frac{t}{\beta}\right)^\alpha}

5. Logistic Distribution [Logis(\mu,\beta)]

   S(t)=\frac{e^{-\frac{t -\mu}{\beta}}}{1 + e^{-\frac{t - \mu}{\beta}}}

6. Log-Normal Distribution [LN(\mu,\beta)]

   S(t)=\int_{\frac{\log t - \mu}{\beta}}^\infty \!\frac{1}{\sqrt{2 \pi}}

7. Normal Distribution [N(\mu,\beta)]

   S(t)=\int_t^\infty \! \frac{1}{\beta\sqrt{2\pi}}e^{-\frac{(x - \mu)^2}{2 \beta^2}} dx

8. 4-Param. Beta Distribution [Beta(\alpha, \gamma, a, b)]

   S(t)=1 - \frac{B_{(\alpha, \gamma, a, b)}(t)}{B(\alpha, \gamma)}

The list of the parameters of the theoretical distribution can be set manually using the argument params of each function. In that case, the correspondence is: \alpha is the shape value, \gamma is the shape2 value, \mu is the location value and \beta is the scale value.

In addition, the functions KScens, CvMcens, ADcens, and chisqcens can be used with any other distribution for which the corresponding density (dname), distribution (pname), and random generator (rname) functions are defined.

Author(s)

Klaus Langohr, Mireia Besalú, Matilde Francisco, Arnau Garcia, Guadalupe Gómez

Maintainer: Klaus Langohr <klaus.langohr@upc.edu>

Anderson-Darling test for complete and right-censored data

Description

Function ADcens computes the Anderson-Darling test statistic and p-value for right-censored data against eight possible predefined or user-specified distributions using bootstrapping. This function also accounts for complete data.

Usage

## Default S3 method:
ADcens(times, cens = rep(1, length(times)),
       distr = c("exponential", "gumbel", "weibull", "normal",
                 "lognormal", "logistic", "loglogistic", "beta"),
       betaLimits = c(0, 1), igumb = c(10, 10), BS = 999,
       params0 = list(shape = NULL, shape2 = NULL,
                       location = NULL, scale = NULL, theta = NULL),
       tol = 1e-04, start = NULL, ...)
## S3 method for class 'formula'
ADcens(formula, data, ...)

Arguments

Details

The parameter estimation is acomplished with the fitdistcens function of the fitdistrplus package.

To avoid long computation times due to bootstrapping, an alternative with complete data is the function ad.test of the goftest package.

The precision of the survival times is important mainly in the data generation step of the bootstrap samples.

Value

ADcens returns an object of class "ADcens".

An object of class "ADcens" is a list containing the following components:

Warning

If the amount of data is large, the execution time of the function can be elevated. The parameter BS can limit the number of random censored samples generated and reduce the execution time.

Author(s)

K. Langohr, M. Besalú, M. Francisco, A. Garcia, G. Gómez.

References

G. Marsaglia and J. Marsaglia. Evaluating the Anderson-Darling Distribution. In: Journal of Statistical Software, Articles, 9 (2) (2004), 1-5. URL: https://doi.org/10.18637/jss.v009.i02

See Also

Function ad.test (Package goftest) for complete data and function gofcens for statistics and p-value of the Kolmogorov-Smirnov, Cramér von-Mises, and Anderson-Darling together for right-censored data.

Examples

# Complete data
set.seed(123)
ADcens(times = rweibull(100, 12, scale = 4), distr = "weibull",
       BS = 199)
summary(ADcens(times = rweibull(100, 12, scale = 4), distr = "exponential",
        BS = 199), outp = "table", print.BIC = FALSE, print.infoBoot = TRUE)

## Not run: 
# Censored data
set.seed(123)
colonsamp <- colon[sample(nrow(colon), 300), ]
ADcens(Surv(time, status) ~ 1, colonsamp, distr = "normal")

## End(Not run)

Cramér-von Mises test for complete and right-censored data

Description

Function CvMcens computes the Cramér-von Mises test statistic and p-value for right-censored data against eight possible predefined or user-specified distributions using bootstrapping. This function also accounts for complete data.

Usage

## Default S3 method:
CvMcens(times, cens = rep(1, length(times)),
        distr = c("exponential", "gumbel", "weibull", "normal",
                  "lognormal", "logistic", "loglogistic", "beta"),
        betaLimits = c(0, 1), igumb = c(10, 10), BS = 999,
        params0 = list(shape = NULL, shape2 = NULL,
                       location = NULL, scale = NULL, theta = NULL),
        tol = 1e-04, start = NULL, ...)
## S3 method for class 'formula'
CvMcens(formula, data, ...)

Arguments

Details

Koziol and Green (1976) proposed a Cramér-von Mises statistic for randomly censored data. This function reproduces this test for a given survival data and a theorical distribution. In presence of ties, different authors provide slightly different definitions of the product-limit estimator, what might provide different values of the test statistic.

The parameter estimation is acomplished with the fitdistcens function of the fitdistrplus package.

To avoid long computation times due to bootstrapping, an alternative with complete data is the function cvm.test of the goftest package.

The precision of the survival times is important mainly in the data generation step of the bootstrap samples.

Value

CvMcens returns an object of class "CvMcens".

An object of class "CvMcens" is a list containing the following components:

Warning

If the amount of data is large, the execution time of the function can be elevated. The parameter BS can limit the number of random censored samples generated and reduce the execution time.

Author(s)

K. Langohr, M. Besalú, M. Francisco, A. Garcia, G. Gómez.

References

J. A. Koziol and S. B. Green. A Cramér-von Mises statistic for randomly censored data. In: Biometrika, 63 (3) (1976), 465-474. URL: https://doi.org/10.1093/biomet/63.3.465

A. N. Pettitt and M. A. Stephens. Modified Cramér-von Mises statistics for censored data. In: Biometrika, 63 (2) (1976), 291-298. URL: https://doi.org/10.1093/biomet/63.2.291

See Also

Function cvm.test (Package goftest) for complete data and gofcens for statistics and p-value of Kolmogorov-Smirnov, Cramér von-Mises and Anderson-Darling together for right-censored data.

Examples

# Complete data
set.seed(123)
CvMcens(times = rweibull(100, 12, scale = 4), distr = "weibull",
        BS = 199)
summary(CvMcens(times = rweibull(100, 12, scale = 4), distr = "normal",
        BS = 99), degs = 4, print.AIC = FALSE, print.BIC = FALSE)

## Not run: 
# Censored data
set.seed(123)
colonsamp <- colon[sample(nrow(colon), 300), ]
CvMcens(Surv(time, status) ~ 1, colonsamp, distr = "normal")

## End(Not run)

Kolmogorov-Smirnov test for complete and right-censored data

Description

Function KScens computes the Kolmogorov-Smirnov statistic and p-value for right-censored data against eight possible predefined or user-specified distributions using either bootstrapping or a modified test. This function also accounts for complete data.

Usage

## Default S3 method:
KScens(times, cens = rep(1, length(times)),
       distr = c("exponential", "gumbel", "weibull", "normal",
                 "lognormal", "logistic", "loglogistic", "beta"),
       betaLimits = c(0, 1), igumb = c(10, 10), BS = 999,
       params0 = list(shape = NULL, shape2 = NULL, location = NULL,
                      scale = NULL, theta = NULL),
       tol = 1e-04, boot = TRUE, start = NULL, ...)
## S3 method for class 'formula'
KScens(formula, data, ...)

Arguments

Details

By default the p-value is computed via bootstrapping methods.

The parameter estimation is acomplished with the fitdistcens function of the fitdistrplus package.

To avoid long computation times due to bootstrapping, an alternative with complete data is the function ks.test of the stats package.

The precision of the survival times is important mainly in the data generation step of the bootstrap samples.

If boot = FALSE a modified test is used to compute the p-value. Fleming et al. (1980) proposed a modified Kolmogorov-Smirnov test to use with right-censored data. This function reproduces this test for a given survival data and a theorical distribution. The approximation for the p-value is acceptable when it is smaller than 0.8 and excellent when it is smaller than 0.2. The output of the function follows the notation of Fleming et al. (1980).

In presence of ties, different authors provide slightly different definitions of \widehat{F}_n(t), with which other values of the test statistic might be obtained.

Value

KScens returns an object of class "KScens".

An object of class "KScens" is a list containing the following components:

Author(s)

K. Langohr, M. Besalú, M. Francisco, A. Garcia, G. Gómez.

References

T. R. Fleming et al. Modified Kolmogorov-Smirnov test procedure with application to arbitrarily right-censored data. In: Biometrics 36 (1980), 607-625. URL: https://doi.org/10.2307/2556114

See Also

Function ks.test (Package stats) for complete data and gofcens for statistics and p-value of Kolmogorov-Smirnov, Cramér von-Mises and Anderson-Darling together for right-censored data.

Examples

# Complete data with bootstrapping
set.seed(123)
KScens(times = rweibull(100, 12, scale = 4), distr = "weibull", BS = 99)

# Censored data with bootstrapping
KScens(Surv(time, status) ~ 1, colon, distr = "norm", BS = 99)

# Censored data using the modified test
KScens(Surv(time, status) ~ 1, colon, distr = "norm", boot = FALSE)

data(nba)
summary(KScens(Surv(survtime, cens) ~ 1, nba, "logis", boot = FALSE), degs = 2)
KScens(Surv(survtime, cens) ~ 1, nba, "beta", betaLimits = c(0, 80),
       boot = FALSE)

Methods for ADcens, chisqcens, CvMcens, gofcens and KScens objects.

Description

Print, summary and print summary methods for ADcens, chisqcens, CvMcens, gofcens and KScens objects.

Usage

## S3 method for class 'ADcens'
print(x, ...)

## S3 method for class 'ADcens'
summary(object, outp = c("list", "table"),
        print.AIC = TRUE, print.BIC = TRUE, print.infoBoot = FALSE, ...)

## S3 method for class 'ADcens'
print.summary(x, degs = 3, ...)

## S3 method for class 'chisqcens'
print(x, ...)

## S3 method for class 'chisqcens'
summary(object, outp = c("list", "table"),
        print.AIC = TRUE, print.BIC = TRUE, print.infoBoot = FALSE, ...)

## S3 method for class 'chisqcens'
print.summary(x, degs = 3, ...)

## S3 method for class 'CvMcens'
print(x, ...)

## S3 method for class 'CvMcens'
summary(object, outp = c("list", "table"),
        print.AIC = TRUE, print.BIC = TRUE, print.infoBoot = FALSE, ...)

## S3 method for class 'CvMcens'
print.summary(x, degs = 3, ...)

## S3 method for class 'gofcens'
print(x, ...)

## S3 method for class 'gofcens'
summary(object, outp = c("list", "table"),
        print.AIC = TRUE, print.BIC = TRUE, print.infoBoot = FALSE, ...)

## S3 method for class 'gofcens'
print.summary(x, degs = 3, ...)

## S3 method for class 'KScens'
print(x, ...)

## S3 method for class 'KScens'
summary(object, outp = c("list", "table"),
        print.AIC = TRUE, print.BIC = TRUE, print.infoBoot = FALSE, ...)

## S3 method for class 'KScens'
print.summary(x, degs = 3, ...)

Arguments

Details

print()

Prints only essential information of the object.

summary()

Produces result summaries, printing information in a more extensive form and with different formats available.

print.summary()

Prints the summary output.

Value

print()

Basic information is returned on the screen.

summary()

A list with the elements:

• Distribution: Null distribution.

• Hypothesis: Parameters under the null hypothesis (if params0 is provided).

• Test: Vector containing the value of the Anderson-Darling statistic (AD) and the estimated p-value (p-value).

• Estimates: Vector with the maximum likelihood estimates of the parameters of the distribution under study.

• StdErrors: Vector containing the estimated standard errors.

• aic: The Akaike information criterion.

• bic: The so-called BIC or SBC (Schwarz Bayesian criterion).

• BS: The number of bootstrap samples used.

print.summary()

x, with the invisible flag set to prevent printing.

Author(s)

K. Langohr, M. Besalú, M. Francisco, A. Garcia, G. Gómez.

Examples

# Generating data
set.seed(123)
survt <- round(rlnorm(300, 2, 1), 2)
censt <- round(rexp(300, 1 / 20), 2)
times <- pmin(survt, censt)
cens <- as.numeric(survt <= censt)

# Print method
set.seed(123)
CvMcens(times, cens, distr = "weibull", BS = 99)

# List output from summary method
set.seed(123)
summary(ADcens(times = rweibull(100, 12, scale = 4), distr = "weibull",
        BS = 149))

# Table output from summary method
set.seed(123)
summary(ADcens(times = rweibull(100, 12, scale = 4), distr = "weibull",
               BS = 99), outp = "table")

General chi-squared statistics for right-censored data.

Description

Function chisqcens computes the general chi-squared test statistic for right-censored data introduced by Kim (1993) and the respective p-value using bootstrapping.

Usage

## Default S3 method:
chisqcens(times, cens = rep(1, length(times)), M,
          distr = c("exponential", "gumbel", "weibull", "normal",
                    "lognormal", "logistic", "loglogistic", "beta"),
          betaLimits=c(0, 1), igumb = c(10, 10), BS = 999,
          params0 = list(shape = NULL, shape2 = NULL,
                       location = NULL, scale = NULL, theta = NULL),
          tol = 1e-04, start = NULL, ...)
## S3 method for class 'formula'
chisqcens(formula, data, ...)

Arguments

Details

The function implements the test introduced by Kim (1993) and returns the value of the test statistic.

The cell boundaries of the test are obtained via the quantiles, which are based on the Kaplan-Meier estimate of the distribution function. In the presence of right-censored data, it is possible that not all quantiles are estimated, and in this case, the value of M provided by the user is reduced.

The parameter estimation is acomplished with the fitdistcens function of the fitdistrplus package.

The precision of the survival times is important mainly in the data generation step of the bootstrap samples.

Value

chisqcens returns an object of class "chisqcens".

An object of class "chisqcens" is a list containing the following components:

Author(s)

K. Langohr, M. Besalú, M. Francisco, A. Garcia, G. Gómez.

References

J. H. Kim. Chi-Square Goodness-of-Fit Tests for Randomly Censored Data. In: The Annals of Statistics, 21 (3) (1993), 1621-1639. URL: https://doi.org/10.1214/aos/1176349275

Examples

# Complete data
set.seed(123)
chisqcens(times = rgumbel(100, 12, scale = 4), M = 8, distr = "gumbel", BS = 99)
summary(chisqcens(times = rlogis(100, 20, scale = 3), M = 8, distr = "loglogistic",
                  BS = 105), print.AIC = FALSE, print.infoBoot = TRUE)

## Not run: 
# Censored data
set.seed(123)
colonsamp <- colon[sample(nrow(colon), 300), ]
chisqcens(Surv(time, status) ~ 1, colonsamp, M = 6, distr = "normal")

## End(Not run)

Cumulative hazard plots to check the goodness of fit of parametric models

Description

Function cumhazPlot uses the cumulative hazard plot to check if a certain distribution is an appropiate choice for the data.

Usage

## Default S3 method:
cumhazPlot(times, cens = rep(1, length(times)), distr = "all6", colour = 1,
           betaLimits = c(0, 1), igumb = c(10, 10), ggp = FALSE, m = NULL,
           prnt = FALSE, degs = 3, print.AIC = TRUE, print.BIC = TRUE,
           outp = c("list", "table"), ...)
## S3 method for class 'formula'
cumhazPlot(formula, data, ...)

Arguments

Details

The cumulative hazard plot is based on transforming the cumulative hazard function \Lambda in such a way that it becomes linear in t or \log(t). This transformation is specific for each distribution. The function uses the data to compute the Nelson-Aalen estimator of the cumulative hazard function, \widehat{\Lambda}, and the maximum likelihood estimators of the parameters of the theoretical distribution under study. If the distribution fits the data, the plot is expected to be a straight line.

The parameter estimation is acomplished with the fitdistcens function of the fitdistrplus package.

Value

If prnt = TRUE, the following output is returned:

In addition, a list with the same contents is returned invisibly.

Author(s)

K. Langohr, M. Besalú, M. Francisco, A. Garcia, G. Gómez.

Examples

# Complete data and default distributions
set.seed(123)
x <- rlogis(1000, 50, 5)
cumhazPlot(x, lwd = 2)

# Censored data comparing three distributions
data(nba)
cumhazPlot(Surv(survtime, cens) ~ 1, nba, distr = c("expo", "normal", "gumbel"),
           outp = "table")

Kolmogorov-Smirnov, Cramér-von Mises, and Anderson-Darling tests for complete and right-censored data

Description

Function gofcens computes the Kolmogorov-Smirnov, Cramér-von Mises, and Anderson-Darling statistics and p-values for right-censored data against eight possible predefined or user-specified distributions using bootstrapping. This function also accounts for complete data.

Usage

## Default S3 method:
gofcens(times, cens = rep(1, length(times)),
        distr = c("exponential", "gumbel", "weibull", "normal",
                  "lognormal", "logistic", "loglogistic", "beta"),
        betaLimits = c(0, 1), igumb = c(10, 10), BS = 999,
        params0 = list(shape = NULL, shape2 = NULL,
                       location = NULL, scale = NULL, theta = NULL),
        tol = 1e-04, start = NULL, ...)
## S3 method for class 'formula'
gofcens(formula, data, ...)

Arguments

Details

All p-values are calculated via bootstrapping methods. For the three hypothesis tests, the same data generated with the bootstrapping method are used.

The precision of the survival times is important mainly in the data generation step of the bootstrap samples.

When dealing with complete data, we recommend the use of functions ks.test of the stats package and cvm.test and ad.test of the goftest package.

Value

gofcens returns an object of class "gofcens".

An object of class "gofcens" is a list containing the following components:

Warning

If the amount of data is large, the execution time of the function can be elevated. The parameter BS can limit the number of random censored samples generated and reduce the execution time.

Author(s)

K. Langohr, M. Besalú, M. Francisco, A. Garcia, G. Gómez.

References

J. A. Koziol and S. B. Green. A Cramér-von Mises statistic for randomly censored data. In: Biometrika, 63 (3) (1976), 465-474. URL: https://doi.org/10.1093/biomet/63.3.465

A. N. Pettitt and M. A. Stephens. Modified Cramér-von Mises statistics for censored data. In: Biometrika, 63 (2) (1976), 291-298. URL: https://doi.org/10.1093/biomet/63.2.291

See Also

ks.test (Package stats), cvm.test (Package goftest), and ad.test (Package goftest) for complete data, and KScens for the Kolmogorov-Smirnov test for right-censored data, which returns the p-value.

Examples

## Not run: 
# Complete data
set.seed(123)
gofcens(times = rweibull(100, 12, scale = 4), distr = "weibull", BS = 499)
summary(gofcens(times = rweibull(100, 12, scale = 4), distr = "exponential"),
        outp = "table", print.infoBoot = TRUE)

# Censored data
data(colon)
set.seed(123)
colonsamp <- colon[sample(nrow(colon), 300), ]
gofcens(Surv(time, status) ~ 1, colonsamp, distr = "normal")

## End(Not run)

Plot of the Kaplan-Meier and parametric estimations

Description

Function kmPlot is a function that generates a plot that combines a Kaplan-Meier survival curve and a parametric survival curve in the same graph. It is useful for comparing non-parametric survival estimates with the fitted parametric survival model.

Usage

## Default S3 method:
kmPlot(times, cens = rep(1, length(times)), distr = "all6",
       colour = c("black", "blue", "cornflowerblue"),
       betaLimits = c(0, 1), igumb = c(10, 10), ggp = FALSE, m = NULL,
       prnt = FALSE, degs = 3, print.AIC = TRUE, print.BIC = TRUE, ...)
## S3 method for class 'formula'
kmPlot(formula, data, ...)

Arguments

Details

The parameter estimation is acomplished with the fitdistcens function of the fitdistrplus package.

Value

If prnt = TRUE, the following output is returned:

In addition, a list with the same contents is returned invisibly.

Author(s)

K. Langohr, M. Besalú, M. Francisco, A. Garcia, G. Gómez.

References

Peterson Jr, Arthur V. Expressing the Kaplan-Meier estimator as a function of empirical subsurvival functions. In: Journal of the American Statistical Association 72.360a (1977): 854-858. URL: https://doi.org/10.1080/01621459.1977.10479970

Examples

# Plots for complete data and default distributions
set.seed(123)
x <- rexp(1000, 0.1)
kmPlot(x)

# Plots for censored data using ggplot2
kmPlot(Surv(time, status) ~ 1, colon, distr= "lognormal", ggp = TRUE)

# Plots for censored data from three distributions
data(nba)
kmPlot(Surv(survtime, cens) ~ 1, nba, distr = c("normal", "weibull", "lognormal"),
       prnt = FALSE)

Survival times of former NBA players.

Description

Survival times of former NBA players after their NBA career.

Usage

data("nba")

Format

A data frame with 3962 observations on the following 3 variables.

id

Player ID

survtime

Time (in years) from end of NBA career until either death or July 31, 2019.

cens

Death indicator (1, exact survival time; 0, right-censored survival time).

Details

The survival times of former NBA players were analyzed by Martínez et al. (2022).

Source

J. A. Martínez, K. Langohr, J. Felipo, L. Consuegra and M. Casals. Data set on mortality of national basketball association (NBA) players. In: Data in Brief, 45 (2022). URL: https://doi.org/10.1016/j.dib.2022.108615

Examples

data(nba)
cumhazPlot(Surv(survtime, cens) ~ 1, nba)

Probability plots to check the goodness of fit of parametric models

Description

probPlot provides four types of probability plots: P-P plot, Q-Q plot, Stabilised probability plot, and Empirically Rescaled plot to check if a certain distribution is an appropiate choice for the data.

Usage

## Default S3 method:
probPlot(times, cens = rep(1, length(times)),
         distr = c("exponential", "gumbel", "weibull", "normal",
                   "lognormal", "logistic", "loglogistic", "beta"),
         plots = c("PP", "QQ", "SP", "ER"),
         colour = c("green4", "deepskyblue4", "yellow3",
                    "mediumvioletred"), mtitle = TRUE, ggp = FALSE,
         m = NULL, betaLimits = c(0, 1), igumb = c(10, 10),
         prnt = FALSE, degs = 3,
         params0 = list(shape = NULL, shape2 = NULL,
                        location = NULL, scale = NULL), print.AIC = TRUE,
                        print.BIC = TRUE, ...)
## S3 method for class 'formula'
probPlot(formula, data, ...)

Arguments

Details

By default, function probPlot draws four plots: P-P plot, SP plot, Q-Q plot, and EP plot. Following, a description is given for each plot.

The Probability-Probability plot (P-P plot) depicts the empirical distribution, \widehat{F}(t), which is obtained with the Kaplan-Meier estimator if data are right-censored, versus the theoretical cumulative distribution function (cdf), \widehat{F_0}(t). If the data come from the chosen distribution, the points of the resulting graph are expected to lie on the identity line.

The Stabilised Probability plot (SP plot), proposed by Michael (1983), is a transformation of the P-P plot. It stabilises the variance of the plotted points. If F_0 = F and the parameters of F_0 are known, \widehat{F_0}(t) corresponds to the cdf of a uniform order statistic, and the arcsin transformation stabilises its variance. If the data come from distribution F_0, the SP plot will resemble the identity line.

The Quartile-Quartile plot (Q-Q plot) is similar to the P-P plot, but it represents the sample quantiles versus the theoretical ones, that is, it plots t versus \widehat{F}_0^{-1}(\widehat{F}(t)). Hence, if F_0 fits the data well, the resulting plot will resemble the identity line.

A drawback of the Q-Q plot is that the plotted points are not evenly spread. Waller and Turnbull (1992) proposed the Empirically Rescaled plot (EP plot), which plots \widehat{F}_u(t) against \widehat{F}_u(\widehat{F}_0^{-1}(\widehat{F}(t))), where \widehat{F}_u(t) is the empirical cdf of the points corresponding to the uncensored observations. Again, if \widehat{F}_0 fits the data well, the ER plot will resemble the identity line.

By default, all four probability plots are drawn and the maximum likelihood estimates of the parameters of the chosen parametric model are returned. The parameter estimation is acomplished with the fitdistcens function of the fitdistrplus package.

Value

If prnt = TRUE, the following output is returned:

In addition, a list with the same contents is returned invisibly.

Author(s)

K. Langohr, M. Besalú, M. Francisco, A. Garcia, G. Gómez.

References

J. R. Michael. The Stabilized Probability Plot. In: Biometrika 70 (1) (1983), 11-17. URL: https://doi.org/10.1093/biomet/70.1.11

L.A. Waller and B.W. Turnbull. Probability Plotting with Censored Data. In: American Statistician 46 (1) (1992), 5-12. URL: https://doi.org/10.1080/00031305.1992.10475837

Examples

# P-P, Q-Q, SP, and EP plots for complete data
set.seed(123)
x <- rlnorm(1000, 3, 2)
probPlot(x)
probPlot(x, distr = "lognormal")

# P-P, Q-Q, SP, and EP plots for censored data using ggplot2
probPlot(Surv(time, status) ~ 1, colon, "weibull", ggp = TRUE)

# P-P, Q-Q and SP plots for censored data and lognormal distribution
data(nba)
probPlot(Surv(survtime, cens) ~ 1, nba, "lognorm", plots = c("PP", "QQ", "SP"),
         ggp = TRUE, m = matrix(1:3, nr = 1))