Goodness-of-Fit Tests using the Probability Plot Correlation Coefficient

Description

The function ppccTest performs the Probability Plot Correlation Coefficient test for various continuous distribution functions.

Plotting Point Positions

Description

Calculates plotting point positions according to different authors

Usage

ppPositions(
  n,
  method = c("Gringorton", "Cunane", "Filliben", "Blom", "Weibull", "ppoints")
)

Arguments

Details

The following methods can by selected:

"Gringorton" the plotting point positions are calculated as

m_i = \left(i - 0.44\right) / \left(n + 0.12\right)

"Cunane" the plotting point positions are calculated as

m_i = \left(i - 0.4\right) / \left(n + 0.2\right)

"Blom" the plotting point positions are calculated as

m_i = \left(i - 0.3175\right) / \left(n + 0.25\right)

"Filliben" the order statistic medians are calculated as:

m_i = \left\{ \begin{array}{l l} 1 - 0.5^{1/n} & i = 1 \\ \left(i - 0.3175\right)/\left(n + 0.365\right) & i = 2,\ldots, n - 1 \\ 0.5^{1/n} & i = n \\ \end{array}\right.

"ppoints" R core's default plotting point positions are calculated (see ppoints).

Value

a vector of class numeric that contains the plotting positions

Probability Plot Correlation Coefficient Test

Description

Performs the Probability Plot Correlation Coeffient Test of Goodness-of-Fit

Usage

ppccTest(
  x,
  qfn = c("qnorm", "qlnorm", "qunif", "qexp", "qcauchy", "qlogis", "qgumbel",
    "qweibull", "qpearson3", "qgev", "qkappa2", "qrayleigh", "qglogis"),
  shape = NULL,
  ppos = NULL,
  mc = 10000,
  ...
)

Arguments

Details

Filliben (1975) suggested a probability plot correlation coeffient test to test a sample for normality. The ppcc is defined as the product moment correlation coefficient between the ordered data x_{(i)} and the order statistic medians M_{i},

r = \frac{\sum_{i = 1}^n \left(x_{(i)} - \bar{x} \right)~ \left(M_i - \bar{M}\right)} {\sqrt{\sum_{i=1}^n \left(x_{(i)} - \bar{x}\right)^2 ~ \sum_{j = 1}^n \left(M_j - \bar{M} \right)^2}},

whereas the ordered statistic medians are related to the quantile function of the standard normal distribution, M_{i} = \phi^{-1} (m_i). The values of m_i are estimated by plotting-point position procedures (see ppPositions).

In this function the test is performed by Monte-Carlo simulation:

1. Calculate quantile-quantile \hat{r} for the ordered sample data x and the specified qfn distribution (with shape, if applicable) and given ppos.

2. Draw n (pseudo) random deviates from the specified qfn distribution, where n is the sample size of x.

3. Calculate quantile-quantile r_i for the random deviates and the specified qfn distribution with given ppos.

4. Repeat step 2 and 3 for i = \left\{1, 2, \ldots, mc\right\}.

5. Calculate S = \sum_{i=1}^n \mathrm{sgn}(\hat{r} - r_i) with sgn the sign-function.

6. The estimated p-value is p = S / mc.

The probability plot correlation coeffient is invariant for location and scale. Therefore, the null hypothesis is a composite hypothesis, e.g. H0: X \in N(\mu, \sigma), ~~ \mu \in R,~~ \sigma \in R_{>0}. Furthermore, distributions with one (additional) specified shape parameter can be tested.

The magnitude of \hat{r} depends on the selected method for plotting-point positions (see ppPositions) and the sample size. Several authors extended Filliben's method to assess the goodness-of-fit to other distributions, whereas theoretical quantiles were used as opposed to Filliben's medians.

The default plotting positions (see ppPositions) depend on the selected qfn.

Distributions with none or one single scale parameter that can be tested:

Distributions with location and scale parameters that can be tested:

If Blom's plotting position is used for qnorm, than the ppcc-test is related to the Shapiro-Francia normality test (Royston 1993), where W' = r^2. See sf.test and example(ppccTest).

Distributions with additional shape parameters that can be tested:

If qfn = qpearson3 and shape = 0 is selected, the qnorm distribution is used. If qfn = qgev and shape = 0, the qgumbel distribution is used. If qfn = qglogis and shape = 0 is selected, the qglogis distribution is used.

Value

a list with class 'htest'

Note

As the pvalue is estimated through a Monte-Carlo simulation, the results depend on the selected seed (see set.seed) and the total number of replicates (mc).

The default of mc = 10000 re-runs is sufficient for testing the composite hypothesis on levels of \alpha = [0.1, 0.05]. If a level of \alpha = 0.01 is desired, than larger sizes of re-runs (e.g. mc = 100000) might be required.

References

J. U. Chowdhury, J. R. Stedinger, L.-H. Lu (1991), Goodness-of-Fit Tests for Regional Generalized Extreme Value Flood Distributions, Water Resources Research 27, 1765–1776.

J. J. Filliben (1975), The Probability Plot Correlation Coefficient Test for Normality, Technometrics 17, 111–117.

S. Kim, H. Shin, T. Kim, J.-H. Heo (2010), Derivation of the Probability Plot Correlation Coefficient Test Statistics for the Generalized Logistic Distribution. Intern. Workshop Adv. in Stat. Hydrol., May 23 - 25, 2010 Taormina.

S. W. Looney, T. R. Gulledge (1985), Use of Correlation Coefficient with Normal Probability Plots, The American Statistician 39, 75–79.

P. W. Mielke (1973), Another family of distributions for describing and analyzing precipitation data. Journal of Applied Meteorology 12, 275–280.

P. Royston, P. (1993), A pocket-calculator algorithm for the Shapiro-Francia test for non-normality: an application to medicine. Statistics in Medicine 12, 181-184.

R. M. Vogel (1986), The Probability Plot Correlation Coefficient Test for the Normal, Lognormal, and Gumbel Distributional Hypotheses, Water Resources Research 22, 587–590.

R. M. Vogel, C. N. Kroll (1989), Low-flow frequency analysis using probability-plot correlation coefficients, Journal of Water Resources Planning and Management 115, 338–357.

R. M. Vogel, D. E. McMartin (1991), Probability Plot Goodness-of-Fit and Skewness Estimation Procedures for the Pearson Type 3 Distribution, Water Resources Research 27, 3149–3158.

See Also

qqplot, qqnorm, ppoints, ppPositions, Normal, Lognormal, Uniform, Exponential, Cauchy, Logistic, qgumbel, Weibull, qgev.

Examples

## Filliben (1975, p.116)
## Note: Filliben's result was 0.98538
## decimal accuracy in 1975 is assumed to be less than in 2017
x <- c(6, 1, -4, 8, -2, 5, 0)
set.seed(100)
ppccTest(x, "qnorm", ppos="Filliben")
## p between .75 and .9
## see Table 1 of Filliben (1975, p.113)
##
set.seed(100)
## Note: default plotting position for
## qnorm is ppos ="Blom"
ppccTest(x, "qnorm")
## p between .75 and .9
## see Table 2 of Looney and Gulledge (1985, p.78)
##
## 
set.seed(300)
x <- rnorm(30)
qn <- ppccTest(x, "qnorm")
qn
## p between .5 and .75
## see Table 2 for n = 30 of Looney and Gulledge (1985, p.78)
##
## Compare with Shapiro-Francia test
if(require(nortest)){
   sn <- sf.test(x)
   print(sn)
   W <- sn$statistic
   rr <- qn$statistic^2
   names(W) <- NULL
   names(rr) <- NULL
   print(all.equal(W, rr))
}
ppccTest(x, "qunif")
ppccTest(x, "qlnorm")
old <- par()
par(mfrow=c(1,3))
xlab <- "Theoretical Quantiles"
ylab <- "Empirical Quantiles"
qqplot(x = qnorm(ppPositions(30, "Blom")),
       y = x, xlab=xlab, ylab=ylab, main = "Normal q-q-plot")
qqplot(x = qunif(ppPositions(30, "Weibull")),
       y = x, xlab=xlab, ylab=ylab, main = "Uniform q-q-plot")
qqplot(x = qlnorm(ppPositions(30, "Blom")),
       y = x, xlab=xlab, ylab=ylab, main = "log-Normal q-q-plot")
par(old)
##
if (require(VGAM)){
set.seed(300)
x <- rgumbel(30)
gu <- ppccTest(x, "qgumbel")
print(gu)
1000 * (1 -  gu$statistic)
}
##
## see Table 2 for n = 30 of Vogel (1986, p.589) 
## for n = 30 and Si = 0.5, the critical value is 16.9 
##
set.seed(200)
x <- runif(30)
un <- ppccTest(x, "qunif")
print(un)
1000 * (1 - un$statistic)
##
## see Table 1 for n = 30 of Vogel and Kroll (1989, p.343)
## for n = 30 and Si = 0.5, the critical value is 10.5
##
set.seed(200)
x <- rweibull(30, shape = 2.5)
ppccTest(x, "qweibull", shape=2.5)
ppccTest(x, "qweibull", shape=1.5)
##
if (require(VGAM)){
set.seed(200)
x <- rgev(30, shape = -0.2)
ev <- ppccTest(x, "qgev", shape=-0.2)
print(ev)
1000 * (1 - ev$statistic)
##
## see Table 3 for n = 30 and shape = -0.2
## of Chowdhury et al. (1991, p.1770)
## The tabulated critical value is 80.
}