Description

Perform a goodness-of-fit test to determine whether a data set appears to come from a specified probability distribution or if two data sets appear to come from the same distribution.

Usage

Arguments

Details

• Shapiro-Wilk Goodness-of-Fit Test (test="sw").

  The Shapiro-Wilk goodness-of-fit test (Shapiro and Wilk, 1965; Royston, 1992a) is one of the most commonly used goodness-of-fit tests for normality. You can use it to test the following hypothesized distributions: Normal, Lognormal, Three-Parameter Lognormal, Zero-Modified Normal, or Zero-Modified Lognormal (Delta). In addition, you can also use it to test the null hypothesis of any continuous distribution that is available (see the help file for Distribution.df, and see explanation below).
  

  Shapiro-Wilk W-Statistic and P-Value for Testing Normality
  Let X denote a random variable with cumulative distribution function (cdf) F. Suppose we want to test the null hypothesis that F is the cdf of a normal (Gaussian) distribution with some arbitrary mean μ and standard deviation σ against the alternative hypothesis that F is the cdf of some other distribution. The table below shows the random variable for which F is the assumed cdf, given the value of the argument distribution.

  Value of		Random Variable for
  distribution	Distribution Name	which F is the cdf
  "norm"	Normal	X
  "lnorm"	Lognormal (Log-space)	log(X)
  "lnormAlt"	Lognormal (Untransformed)	log(X)
  "lnorm3"	Three-Parameter Lognormal	log(X-γ)
  "zmnorm"	Zero-Modified Normal	X | X > 0
  "zmlnorm"	Zero-Modified Lognormal (Log-space)	log(X) | X > 0
  "zmlnormAlt"	Zero-Modified Lognormal (Untransformed)	log(X) | X > 0

  Note that for the three-parameter lognormal distribution, the symbol γ denotes the threshold parameter.

  Let \underline{x} = (x_1, x_2, …, x_n) denote the vector of n ordered observations assumed to come from a normal distribution.
  

  The Shapiro-Wilk W-Statistic
  Shapiro and Wilk (1965) introduced the following statistic to test the null hypothesis that F is the cdf of a normal distribution:

  W = \frac{(∑_{i=1}^n a_i x_i)^2}{∑_{i=1}^n (x_i - \bar{x})^2} \;\;\;\;\;\; (1)

  where the quantity a_i is the i'th element of the vector \underline{a} defined by:

  \underline{a} = \frac{\underline{m}^T V^{-1}}{[\underline{m}^T V^{-1} V^{-1} \underline{m}]^{1/2}} \;\;\;\;\;\; (2)

  where T denotes the transpose operator, and \underline{m} is the vector of expected values and V is the variance-covariance matrix of the order statistics of a random sample of size n from a standard normal distribution. That is, the values of \underline{a} are the expected values of the standard normal order statistics weighted by their variance-covariance matrix, and normalized so that

  \underline{a}^T \underline{a} = 1 \;\;\;\;\;\; (3)

  It can be shown that the coefficients \underline{a} are antisymmetric, that is,

  a_i = -a_{n-i+1} \;\;\;\;\;\; (4)

  and for odd n,

  a_{(n+1)/2} = 0 \;\;\;\;\;\; (5)

  Now because

  \bar{a} = \frac{1}{n} ∑_{i=1}^n a_i = 0 \;\;\;\;\;\ (6)

  and

  ∑_{i=1}^n (a_i - \bar{a})^2 = ∑_{i=1}^n a_i^2 = \underline{a}^T \underline{a} = 1 \;\;\;\;\;\; (7)

  the W-statistic in Equation (1) is the same as the square of the sample product-moment correlation between the vectors \underline{a} and \underline{x}:

  W = r(\underline{a}, \underline{x})^2 \;\;\;\;\;\; (8)

  where

  r(\underline{x}, \underline{y}) = \frac{∑_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{[∑_{i=1}^n (x_i - \bar{x})^2 ∑_{i=1}^n (y_i - \bar{y})^2]^{1/2}} \;\;\;\;\;\;\; (9)

  (see the R help file for cor).

  The Shapiro-Wilk W-statistic is also simply the ratio of two estimators of variance, and can be rewritten as

  W = \frac{\hat{σ}_{BLUE}^2}{\hat{σ}_{MVUE}^2} \;\;\;\;\;\; (10)

  where the numerator is the square of the best linear unbiased estimate (BLUE) of the standard deviation, and the denominator is the minimum variance unbiased estimator (MVUE) of the variance:

  \hat{σ}_{BLUE} = \frac{∑_{i=1}^n a_i x_i}{√{n-1}} \;\;\;\;\;\; (11)

  \hat{σ}_{MVUE}^2 = \frac{∑_{i=1}^n (x_i - \bar{x})^2}{n-1} \;\;\;\;\;\; (12)

  Small values of W indicate the null hypothesis is probably not true. Shapiro and Wilk (1965) computed the values of the coefficients \underline{a} and the percentage points for W (based on smoothing the empirical null distribution of W) for sample sizes up to 50. Computation of the W-statistic for larger sample sizes can be cumbersome, since computation of the coefficients \underline{a} requires storage of at least n + [n(n+1)/2] reals followed by n \times n matrix inversion (Royston, 1992a).
  

  The Shapiro-Francia W'-Statistic
  Shapiro and Francia (1972) introduced a modification of the W-test that depends only on the expected values of the order statistics (\underline{m}) and not on the variance-covariance matrix (V):

  W' = \frac{(∑_{i=1}^n b_i x_i)^2}{∑_{i=1}^n (x_i - \bar{x})^2} \;\;\;\;\;\; (13)

  where the quantity b_i is the i'th element of the vector \underline{b} defined by:

  \underline{b} = \frac{\underline{m}}{[\underline{m}^T \underline{m}]^{1/2}} \;\;\;\;\;\; (14)

  Several authors, including Ryan and Joiner (1973), Filliben (1975), and Weisberg and Bingham (1975), note that the W'-statistic is intuitively appealing because it is the squared Pearson correlation coefficient associated with a normal probability plot. That is, it is the squared correlation between the ordered sample values \underline{x} and the expected normal order statistics \underline{m}:

  W' = r(\underline{b}, \underline{x})^2 = r(\underline{m}, \underline{x})^2 \;\;\;\;\;\; (15)

  Shapiro and Francia (1972) present a table of empirical percentage points for W' based on a Monte Carlo simulation. It can be shown that the asymptotic null distributions of W and W' are identical, but convergence is very slow (Verrill and Johnson, 1988).
  

  The Weisberg-Bingham Approximation to the W'-Statistic
  Weisberg and Bingham (1975) introduced an approximation of the Shapiro-Francia W'-statistic that is easier to compute. They suggested using Blom scores (Blom, 1958, pp.68–75) to approximate the element of \underline{m}:

  \tilde{W}' = \frac{(∑_{i=1}^n c_i x_i)^2}{∑_{i=1}^n (x_i - \bar{x})^2} \;\;\;\;\;\; (16)

  where the quantity c_i is the i'th element of the vector \underline{c} defined by:

  \underline{c} = \frac{\underline{\tilde{m}}}{[\underline{\tilde{m}}^T \underline{\tilde{m}}]^{1/2}} \;\;\;\;\;\; (17)

  and

  \tilde{m}_i = Φ^{-1}[\frac{i - (3/8)}{n + (1/4)}] \;\;\;\;\;\; (18)

  and Φ denotes the standard normal cdf. That is, the values of the elements of \underline{m} in Equation (14) are replaced with their estimates based on the usual plotting positions for a normal distribution.
  

  Royston's Approximation to the Shapiro-Wilk W-Test
  Royston (1992a) presents an approximation for the coefficients \underline{a} necessary to compute the Shapiro-Wilk W-statistic, and also a transformation of the W-statistic that has approximately a standard normal distribution under the null hypothesis.

  Noting that, up to a constant, the components of \underline{b} in Equation (14) and \underline{c} in Equation (17) differ from those of \underline{a} in Equation (2) mainly in the first and last two components, Royston (1992a) used the approximation \underline{c} as the basis for approximating \underline{a} using polynomial (quintic) regression analysis. For 4 ≤ n ≤ 1000, the approximation gave the following equations for the last two (and hence first two) components of \underline{a}:

  \tilde{a}_n = c_n + 0.221157 y - 0.147981 y^2 - 2.071190 y^3 + 4.434685 y^4 - 2.706056 y^5 \;\;\;\;\;\; (19)

  \tilde{a}_{n-1} = c_{n-1} + 0.042981 y - 0.293762 y^2 - 1.752461 y^3 + 5.682633 y^4 - 3.582633 y^5 \;\;\;\;\;\; (20)

  where

  y = √{n} \;\;\;\;\;\; (21)

  The other components are computed as:

  \tilde{a}_i = \frac{\tilde{m}_i}{√{η}} \;\;\;\;\;\; (22)

  for i = 2, … , n-1 if n ≤ 5, or i = 3, …, n-2 if n > 5, where

  η = \frac{\underline{\tilde{m}}^T \underline{\tilde{m}} - 2 \tilde{m}_n^2}{1 - 2 \tilde{a}_n^2} \;\;\;\;\;\; (23)

  if n ≤ 5, and

  η = \frac{\underline{\tilde{m}}^T \underline{\tilde{m}} - 2 \tilde{m}_n^2 - 2 \tilde{m}_{n-1}^2}{1 - 2 \tilde{a}_n^2 - 2 \tilde{a}_{n-1}^2} \;\;\;\;\;\; (24)

  if n > 5.

  Royston (1992a) found his approximation to \underline{a} to be accurate to at least \pm 1 in the third decimal place over all values of i and selected values of n, and also found that critical percentage points of W based on his approximation agreed closely with the exact critical percentage points calculated by Verrill and Johnson (1988).
  

  Transformation of the Null Distribution of W to Normality
  In order to compute a p-value associated with a particular value of W, Royston (1992a) approximated the distribution of (1-W) by a three-parameter lognormal distribution for 4 ≤ n ≤ 11, and the upper half of the distribution of (1-W) by a two-parameter lognormal distribution for 12 ≤ n ≤ 2000. Setting

  z = \frac{w - μ}{σ} \;\;\;\;\;\; (25)

  the p-value associated with W is given by:

  p = 1 - Φ(z) \;\;\;\;\;\; (26)

  For 4 ≤ n ≤ 11, the quantities necessary to compute z are given by:

  w = -log[γ - log(1 - W)] \;\;\;\;\;\; (27)

  γ = -2.273 + 0.459 n \;\;\;\;\;\; (28)

  μ = 0.5440 - 0.39978 n + 0.025054 n^2 - 0.000671 n^3 \;\;\;\;\;\; (29)

  σ = exp(1.3822 - 0.77857 n + 0.062767 n^2 - 0.0020322 n^3) \;\;\;\;\;\; (30)

  For 12 ≤ n ≤ 2000, the quantities necessary to compute z are given by:

  w = log(1 - W) \;\;\;\;\;\; (31)

  γ = log(n) \;\;\;\;\;\; (32)

  μ = -1.5861 - 0.31082 y - 0.083751 y^2 + 0.00038915 y^3 \;\;\;\;\;\; (33)

  σ = exp(-0.4803 - 0.082676 y + 0.0030302 y^2) \;\;\;\;\;\; (34)

  For the last approximation when 12 ≤ n ≤ 2000, Royston (1992a) claims this approximation is actually valid for sample sizes up to n = 5000.
  

  Modification for the Three-Parameter Lognormal Distribution
  When distribution="lnorm3", the function gofTest assumes the vector \underline{x} is a random sample from a three-parameter lognormal distribution. It estimates the threshold parameter via the zero-skewness method (see elnorm3), and then performs the Shapiro-Wilk goodness-of-fit test for normality on log(x-\hat{γ}) where \hat{γ} is the estimated threshold parmater. Because the threshold parameter has to be estimated, however, the p-value associated with the computed z-statistic will tend to be conservative (larger than it should be under the null hypothesis). Royston (1992b) proposed the following transformation of the z-statistic:

  z' = \frac{z - μ_z}{σ_z} \;\;\;\;\;\; (35)

  where for 5 ≤ n ≤ 11,

  μ_z = -3.8267 + 2.8242 u - 0.63673 u^2 - 0.020815 v \;\;\;\;\;\; (36)

  σ_z = -4.9914 + 8.6724 u - 4.27905 u^2 + 0.70350 u^3 - 0.013431 v \;\;\;\;\;\; (37)

  and for 12 ≤ n ≤ 2000,

  μ_z = -3.7796 + 2.4038 u - 0.6675 u^2 - 0.082863 u^3 - 0.0037935 u^4 - 0.027027 v - 0.0019887 vu \;\;\;\;\;\; (38)

  σ_z = 2.1924 - 1.0957 u + 0.33737 u^2 - 0.043201 u^3 + 0.0019974 u^4 - 0.0053312 vu \;\;\;\;\;\; (39)

  where

  u = log(n) \;\;\;\;\;\; (40)

  v = u (\hat{σ} - \hat{σ}^2) \;\;\;\;\;\; (41)

  \hat{σ}^2 = \frac{1}{n-1} ∑_{i=1}^n (y_i - \bar{y})^2 \;\;\;\;\;\; (42)

  y_i = log(x_i - \hat{γ}) \;\;\;\;\;\; (43)

  and γ denotes the threshold parameter. The p-value associated with this test is then given by:

  p = 1 - Φ(z') \;\;\;\;\;\; (44)

  Testing Goodness-of-Fit for Any Continuous Distribution
  The function gofTest extends the Shapiro-Wilk test to test for goodness-of-fit for any continuous distribution by using the idea of Chen and Balakrishnan (1995), who proposed a general purpose approximate goodness-of-fit test based on the Cramer-von Mises or Anderson-Darling goodness-of-fit tests for normality. The function gofTest modifies the approach of Chen and Balakrishnan (1995) by using the same first 2 steps, and then applying the Shapiro-Wilk test:

  1. Let \underline{x} = x_1, x_2, …, x_n denote the vector of n ordered observations. Compute cumulative probabilities for each x_i based on the cumulative distribution function for the hypothesized distribution. That is, compute p_i = F(x_i, \hat{θ}) where F(x, θ) denotes the hypothesized cumulative distribution function with parameter(s) θ, and \hat{θ} denotes the estimated parameter(s).

  2. Compute standard normal deviates based on the computed cumulative probabilities:
     y_i = Φ^{-1}(p_i)

  3. Perform the Shapiro-Wilk goodness-of-fit test on the y_i's.

• Shapiro-Francia Goodness-of-Fit Test (test="sf").

  The Shapiro-Francia goodness-of-fit test (Shapiro and Francia, 1972; Weisberg and Bingham, 1975; Royston, 1992c) is also one of the most commonly used goodness-of-fit tests for normality. You can use it to test the following hypothesized distributions: Normal, Lognormal, Zero-Modified Normal, or Zero-Modified Lognormal (Delta). In addition, you can also use it to test the null hypothesis of any continuous distribution that is available (see the help file for Distribution.df). See the section Testing Goodness-of-Fit for Any Continuous Distribution above for an explanation of how this is done.
  

  Royston's Transformation of the Shapiro-Francia W'-Statistic to Normality
  Equation (13) above gives the formula for the Shapiro-Francia W'-statistic, and Equation (16) above gives the formula for Weisberg-Bingham approximation to the W'-statistic (denoted \tilde{W}'). Royston (1992c) presents an algorithm to transform the \tilde{W}'-statistic so that its null distribution is approximately a standard normal. For 5 ≤ n ≤ 5000, Royston (1992c) approximates the distribution of (1-\tilde{W}') by a lognormal distribution. Setting

  z = \frac{w-μ}{σ} \;\;\;\;\;\; (45)

  the p-value associated with \tilde{W}' is given by:

  p = 1 - Φ(z) \;\;\;\;\;\; (46)

  The quantities necessary to compute z are given by:

  w = log(1 - \tilde{W}') \;\;\;\;\;\; (47)

  ν = log(n) \;\;\;\;\;\; (48)

  u = log(ν) - ν \;\;\;\;\;\; (49)

  μ = -1.2725 + 1.0521 u \;\;\;\;\;\; (50)

  v = log(ν) + \frac{2}{ν} \;\;\;\;\;\; (51)

  σ = 1.0308 - 0.26758 v \;\;\;\;\;\; (52)

  Testing Goodness-of-Fit for Any Continuous Distribution
  The function gofTest extends the Shapiro-Francia test to test for goodness-of-fit for any continuous distribution by using the idea of Chen and Balakrishnan (1995), who proposed a general purpose approximate goodness-of-fit test based on the Cramer-von Mises or Anderson-Darling goodness-of-fit tests for normality. The function gofTest modifies the approach of Chen and Balakrishnan (1995) by using the same first 2 steps, and then applying the Shapiro-Francia test:

  1. Let \underline{x} = x_1, x_2, …, x_n denote the vector of n ordered observations. Compute cumulative probabilities for each x_i based on the cumulative distribution function for the hypothesized distribution. That is, compute p_i = F(x_i, \hat{θ}) where F(x, θ) denotes the hypothesized cumulative distribution function with parameter(s) θ, and \hat{θ} denotes the estimated parameter(s).

  2. Compute standard normal deviates based on the computed cumulative probabilities:
     y_i = Φ^{-1}(p_i)

  3. Perform the Shapiro-Francia goodness-of-fit test on the y_i's.

• Probability Plot Correlation Coefficient (PPCC) Goodness-of-Fit Test (test="ppcc").

  The PPPCC goodness-of-fit test (Filliben, 1975; Looney and Gulledge, 1985) can be used to test the following hypothesized distributions: Normal, Lognormal, Zero-Modified Normal, or Zero-Modified Lognormal (Delta). In addition, you can also use it to test the null hypothesis of any continuous distribution that is available (see the help file for Distribution.df). The function gofTest computes the PPCC test statistic using Blom plotting positions.

  Filliben (1975) proposed using the correlation coefficient r from a normal probability plot to perform a goodness-of-fit test for normality, and he provided a table of critical values for r under the for samples sizes between 3 and 100. Vogel (1986) provided an additional table for sample sizes between 100 and 10,000.

  Looney and Gulledge (1985) investigated the characteristics of Filliben's probability plot correlation coefficient (PPCC) test using the plotting position formulas given in Filliben (1975), as well as three other plotting position formulas: Hazen plotting positions, Weibull plotting positions, and Blom plotting positions (see the help file for qqPlot for an explanation of these plotting positions). They concluded that the PPCC test based on Blom plotting positions performs slightly better than tests based on other plotting positions, and they provide a table of empirical percentage points for the distribution of r based on Blom plotting positions.

  The function gofTest computes the PPCC test statistic r using Blom plotting positions. It can be shown that the square of this statistic is equivalent to the Weisberg-Bingham Approximation to the Shapiro-Francia W'-Test (Weisberg and Bingham, 1975; Royston, 1993). Thus the PPCC goodness-of-fit test is equivalent to the Shapiro-Francia goodness-of-fit test.
  

• Anderson-Darling Goodness-of-Fit Test (test="ad").

  The Anderson-Darling goodness-of-fit test (Stephens, 1986a; Thode, 2002) can be used to test the following hypothesized distributions: Normal, Lognormal, Zero-Modified Normal, or Zero-Modified Lognormal (Delta).

  When test="ad", the function gofTest calls the function ad.test in the package nortest. Documentation from that package is as follows:

  The Anderson-Darling test is an EDF omnibus test for the composite hypothesis of normality. The test statistic is:

  A = -n - \frac{1}{n} ∑_{i=1}^n [2i - 1][ln(p_{(i)}) + ln(1 - p_{(n-i+1)})]

  where p_{(i)} = Φ([x_{(i)} - \bar{x}]/s). Here, Φ is the cumulative distribution function of the standard normal distribution, and \bar{x} and s are mean and standard deviation of the data values. The p-value is computed from the modified statistic Z = A (1.0 + 0.75/n + 2.25/n^2) according to Table 4.9 in Stephens [(1986a)].
  

• Cramer-von Mises Goodness-of-Fit Test (test="cvm").

  The Cramer-von Mises goodness-of-fit test (Stephens, 1986a; Thode, 2002) can be used to test the following hypothesized distributions: Normal, Lognormal, Zero-Modified Normal, or Zero-Modified Lognormal (Delta).

  When test="cvm", the function gofTest calls the function cvm.test in the package nortest. Documentation from that package is as follows:

  The Cramer-von Mises test is an EDF omnibus test for the composite hypothesis of normality. The test statistic is:

  W = \frac{1}{12n} + ∑_{i=1}^n ≤ft(p_{(i)} - \frac{2i-1}{2n}\right)^2

  where p_{(i)} = Φ([x_{(i)} - \bar{x}]/s). Here, Φ is the cumulative distribution function of the standard normal distribution, and \bar{x} and s are mean and standard deviation of the data values. The p-value is computed from the modified statistic Z = W (1.0 + 0.75/n) according to Table 4.9 in Stephens [(1986a)].
  

• Lilliefors Goodness-of-Fit Test (test="lillie").

  The Lilliefors goodness-of-fit test (Stephens, 1974; Dallal and Wilkinson, 1986; Thode, 2002) can be used to test the following hypothesized distributions: Normal, Lognormal, Zero-Modified Normal, or Zero-Modified Lognormal (Delta).

  When test="lillie", the function gofTest calls the function lillie.test in the package nortest. Documentation from that package is as follows:

  The Lilliefors (Kolmogorov-Smirnov) test is an EDF omnibus test for the composite hypothesis of normality. The test statistic is the maximal absolute difference between empirical and hypothetical cumulative distribution function. It may be computed as D = max\{D^+, D^-\} with

  D^+ = \max_{i = 1, …, n} \{i/n - p_{(i)}\}, \;\; D^- = \max_{i = 1, …, n} \{p_{(i)} - (i-1)/n\}

  where p_{(i)} = Φ([x_{(i)} - \bar{x}]/s). Here, Φ is the cumulative distribution function of the standard normal distribution, and \bar{x} and s are mean and standard deviation of the data values. The p-value is computed from the Dallal-Wilkinson (1986) formula, which is claimed to be only reliable when the p-value is smaller than 0.1. If the Dallal-Wilkinson p-value turns out to be greater than 0.1, then the p-value is computed from the distribution of the modified statistic Z = D (√{n} - 0.01 + 0.85/√{n}), see Stephens (1974), the actual p-value formula being obtained by a simulation and approximation process.
  

• Zero-Skew Goodness-of-Fit Test (test="skew").

  The Zero-skew goodness-of-fit test (D'Agostino, 1970) can be used to test the following hypothesized distributions: Normal, Lognormal, Zero-Modified Normal, or Zero-Modified Lognormal (Delta).

  When test="skew", the function gofTest tests the null hypothesis that the skew of the distribution is 0:

  H_0: √{β}_1 = 0 \;\;\;\;\;\; (53)

  where

  √{β}_1 = \frac{μ_3}{μ_2^{3/2}} \;\;\;\;\;\; (54)

  and the quantity μ_r denotes the r'th moment about the mean (also called the r'th central moment). The quantity √{β_1} is called the coefficient of skewness, and is estimated by:

  √{b}_1 = \frac{m_3}{m_2^{3/2}} \;\;\;\;\;\; (55)

  where

  m_r = \frac{1}{n} ∑_{i=1}^n (x_i - \bar{x})^r \;\;\;\;\;\; (56)

  denotes the r'th sample central moment.

  The possible alternative hypotheses are:

  H_a: √{β}_1 \ne 0 \;\;\;\;\;\; (57)

  H_a: √{β}_1 < 0 \;\;\;\;\;\; (58)

  H_a: √{β}_1 > 0 \;\;\;\;\;\; (59)

  which correspond to alternative="two-sided", alternative="less", and
  alternative="greater", respectively.

  To test the null hypothesis of zero skew, D'Agostino (1970) derived an approximation to the distribution of √{b_1} under the null hypothesis of zero-skew, assuming the observations comprise a random sample from a normal (Gaussian) distribution. Based on D'Agostino's approximation, the statistic Z shown below is assumed to follow a standard normal distribution and is used to compute the p-value associated with the test of H_0:

  Z = δ \;\; log\{ \frac{Y}{α} + [(\frac{Y}{α})^2 + 1]^{1/2} \} \;\;\;\;\;\; (60)

  where

  Y = √{b_1} [\frac{(n+1)(n+3)}{6(n-2)}]^{1/2} \;\;\;\;\;\; (61)

  β_2 = \frac{3(n^2 + 27n - 70)(n+1)(n+3)}{(n-2)(n+5)(n+7)(n+9)} \;\;\;\;\;\; (62)

  W^2 = -1 + √{2β_2 - 2} \;\;\;\;\;\; (63)

  δ = 1 / √{log(W)} \;\;\;\;\;\; (64)

  α = [2 / (W^2 - 1)]^{1/2} \;\;\;\;\;\; (65)

  When the sample size n is at least 150, a simpler approximation may be used in which Y in Equation (61) is assumed to follow a standard normal distribution and is used to compute the p-value associated with the hypothesis test.
  

• Kolmogorov-Smirnov Goodness-of-Fit Test (test="ks").

  When test="ks", the function gofTest calls the R function ks.test to compute the test statistic and p-value. Note that for the one-sample case, the distribution parameters should be pre-specified and not estimated from the data, and if the distribution parameters are estimated from the data you will receive a warning that this test is very conservative (Type I error smaller than assumed; high Type II error) in this case.
  

• ProUCL Kolmogorov-Smirnov Goodness-of-Fit Test for Gamma (test="proucl.ks.gamma").

  When test="proucl.ks.gamma", the function gofTest calls the R function ks.test to compute the Kolmogorov-Smirnov test statistic based on the maximum likelihood estimates of the shape and scale parameters (see egamma). The p-value is computed based on the simulated critical values given in ProUCL.Crit.Vals.for.KS.Test.for.Gamma.array (USEPA, 2015). The sample size must be between 5 and 1000, and the value of the maximum likelihood estimate of the shape parameter must be between 0.025 and 50. The critical value for the test statistic is computed using the simulated critical values and linear interpolation.
  

• ProUCL Anderson-Darling Goodness-of-Fit Test for Gamma (test="proucl.ad.gamma").

  When test="proucl.ad.gamma", the function gofTest computes the Anderson-Darling test statistic (Stephens, 1986a, p.101) based on the maximum likelihood estimates of the shape and scale parameters (see egamma). The p-value is computed based on the simulated critical values given in ProUCL.Crit.Vals.for.AD.Test.for.Gamma.array (USEPA, 2015). The sample size must be between 5 and 1000, and the value of the maximum likelihood estimate of the shape parameter must be between 0.025 and 50. The critical value for the test statistic is computed using the simulated critical values and linear interpolation.
  

• Chi-Squared Goodness-of-Fit Test (test="chisq").

  The method used by gofTest is a modification of what is used for chisq.test. If the hypothesized distribution function is completely specified, the degrees of freedom are m-1 where m denotes the number of classes. If any parameters are estimated, the degrees of freedom depend on the method of estimation. The function gofTest follows the convention of computing degrees of freedom as m-1-k, where k is the number of parameters estimated. It can be shown that if the parameters are estimated by maximum likelihood, the degrees of freedom are bounded between m-1 and m-1-k. Therefore, especially when the sample size is small, it is important to compare the test statistic to the chi-squared distribution with both m-1 and m-1-k degrees of freedom. See Kendall and Stuart (1991, Chapter 30) for a more complete discussion.

  The distribution theory of chi-square statistics is a large sample theory. The expected cell counts are assumed to be at least moderately large. As a rule of thumb, each should be at least 5. Although authors have found this rule to be conservative (especially when the class probabilities are not too different from each other), the user should regard p-values with caution when expected cell counts are small.
  

• Wilk-Shapiro Goodness-of-Fit Test for Uniform [0, 1] Distribution (test="ws").

  Wilk and Shapiro (1968) suggested this test in the context of jointly testing several independent samples for normality simultaneously. If p_1, p_2, …, p_n denote the p-values associated with the test for normality of n independent samples, then under the null hypothesis that all n samples come from a normal distribution, the p-values are a random sample of n observations from a Uniform [0,1] distribution, that is a Uniform distribution with minimum 0 and maximum 1. Wilk and Shapiro (1968) suggested two different methods for testing whether the p-values come from a Uniform [0, 1] distribution:

  • Test Based on Normal Scores. Under the null hypothesis, the normal scores

    Φ^{-1}(p_1), Φ^{-1}(p_2), …, Φ^{-1}(p_n)

    are a random sample of n observations from a standard normal distribution. Wilk and Shapiro (1968) denote the i'th normal score by

    G_i = Φ^{-1}(p_i) \;\;\;\;\;\; (66)

    and note that under the null hypothesis, the quantity G defined as

    G = \frac{1}{√{n}} \, ∑^n_{1}{G_i} \;\;\;\;\;\; (67)

    has a standard normal distribution. Wilk and Shapiro (1968) were interested in the alternative hypothesis that some of the n independent samples did not come from a normal distribution and hence would be associated with smaller p-values than expected under the null hypothesis, which translates to the alternative that the cdf for the distribution of the p-values is greater than the cdf of a Uniform [0, 1] distribution (alternative="greater"). In terms of the test statistic G, this alternative hypothesis would tend to make G smaller than expected, so the p-value is given by Φ(G). For the one-sided lower alternative that the cdf for the distribution of p-values is less than the cdf for a Uniform [0, 1] distribution, the p-value is given by

    p = 1 - Φ(G) \;\;\;\;\;\; (68)

    .

  • Test Based on Chi-Square Scores. Under the null hypothesis, the chi-square scores

    -2 \, log(p_1), -2 \, log(p_2), …, -2 \, log(p_n)

    are a random sample of n observations from a chi-square distribution with 2 degrees of freedom (Fisher, 1950). Wilk and Shapiro (1968) denote the i'th chi-square score by

    C_i = -2 \, log(p_i) \;\;\;\;\;\; (69)

    and note that under the null hypothesis, the quantity C defined as

    C = ∑^n_{1}{C_i} \;\;\;\;\;\; (70)

    has a chi-square distribution with 2n degrees of freedom. Wilk and Shapiro (1968) were interested in the alternative hypothesis that some of the n independent samples did not come from a normal distribution and hence would be associated with smaller p-values than expected under the null hypothesis, which translates to the alternative that the cdf for the distribution of the p-values is greater than the cdf of a Uniform [0, 1] distribution (alternative="greater"). In terms of the test statistic C, this alternative hypothesis would tend to make C larger than expected, so the p-value is given by

    p = 1 - F_{2n}(C) \;\;\;\;\;\; (71)

    where F_2n denotes the cumulative distribution function of the chi-square distribution with 2n degrees of freedom. For the one-sided lower alternative that the cdf for the distribution of p-values is less than the cdf for a Uniform [0, 1] distribution, the p-value is given by

    p = F_{2n}(C) \;\;\;\;\;\; (72)

Value

a list of class "gof" containing the results of the goodness-of-fit test, unless the two-sample
Kolmogorov-Smirnov test is used, in which case the value is a list of class "gofTwoSample". Objects of class "gof" and "gofTwoSample" have special printing and plotting methods. See the help files for gof.object and gofTwoSample.object for details.

Note

The Shapiro-Wilk test (Shapiro and Wilk, 1965) and the Shapiro-Francia test (Shapiro and Francia, 1972) are probably the two most commonly used hypothesis tests to test departures from normality. The Shapiro-Wilk test is most powerful at detecting short-tailed (platykurtic) and skewed distributions, and least powerful against symmetric, moderately long-tailed (leptokurtic) distributions. Conversely, the Shapiro-Francia test is more powerful against symmetric long-tailed distributions and less powerful against short-tailed distributions (Royston, 1992b; 1993). In general, the Shapiro-Wilk and Shapiro-Francia tests outperform the Anderson-Darling test, which in turn outperforms the Cramer-von Mises test, which in turn outperforms the Lilliefors test (Stephens, 1986a; Razali and Wah, 2011; Romao et al., 2010).

The zero-skew goodness-of-fit test for normality is one of several tests that have been proposed to test the assumption of a normal distribution (D'Agostino, 1986b). This test has been included mainly because it is called by elnorm3. Ususally, the Shapiro-Wilk or Shapiro-Francia test is preferred to this test, unless the direction of the alternative to normality (e.g., positive skew) is known (D'Agostino, 1986b, pp. 405–406).

Kolmogorov (1933) introduced a goodness-of-fit test to test the hypothesis that a random sample of n observations x comes from a specific hypothesized distribution with cumulative distribution function H. This test is now usually called the one-sample Kolmogorov-Smirnov goodness-of-fit test. Smirnov (1939) introduced a goodness-of-fit test to test the hypothesis that a random sample of n observations x comes from the same distribution as a random sample of m observations y. This test is now usually called the two-sample Kolmogorov-Smirnov goodness-of-fit test. Both tests are based on the maximum vertical distance between two cumulative distribution functions. For the one-sample problem with a small sample size, the Kolmogorov-Smirnov test may be preferred over the chi-squared goodness-of-fit test since the KS-test is exact, while the chi-squared test is based on an asymptotic approximation.

The chi-squared test, introduced by Pearson in 1900, is the oldest and best known goodness-of-fit test. The idea is to reduce the goodness-of-fit problem to a multinomial setting by comparing the observed cell counts with their expected values under the null hypothesis. Grouping the data sacrifices information, especially if the hypothesized distribution is continuous. On the other hand, chi-squared tests can be be applied to any type of variable: continuous, discrete, or a combination of these.

The Wilk-Shapiro (1968) tests for a Uniform [0, 1] distribution were introduced in the context of testing whether several independent samples all come from normal distributions, with possibly different means and variances. The function gofGroupTest extends this idea to allow you to test whether several independent samples come from the same distribution (e.g., gamma, extreme value, etc.), with possibly different parameters.

In practice, almost any goodness-of-fit test will not reject the null hypothesis if the number of observations is relatively small. Conversely, almost any goodness-of-fit test will reject the null hypothesis if the number of observations is very large, since “real” data are never distributed according to any theoretical distribution (Conover, 1980, p.367). For most cases, however, the distribution of “real” data is close enough to some theoretical distribution that fairly accurate results may be provided by assuming that particular theoretical distribution. One way to asses the goodness of the fit is to use goodness-of-fit tests. Another way is to look at quantile-quantile (Q-Q) plots (see qqPlot).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

Juergen Gross and Uwe Ligges for the Anderson-Darling, Carmer-von Mises, and Lilliefors tests called from the package nortest.

References

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D'Agostino, R.B. (1971). An Omnibus Test of Normality for Moderate and Large Size Samples. Biometrika 58, 341-348.

D'Agostino, R.B. (1986b). Tests for the Normal Distribution. In: D'Agostino, R.B., and M.A. Stephens, eds. Goodness-of Fit Techniques. Marcel Dekker, New York.

D'Agostino, R.B., and E.S. Pearson (1973). Tests for Departures from Normality. Empirical Results for the Distributions of b2 and √{b1}. Biometrika 60(3), 613-622.

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Razali, N.M., and Y.B. Wah. (2011). Power Comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors, and Anderson-Darling Tests. Journal of Statistical Modeling and Analytics 2(1), 21–33.

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See Also

rosnerTest, gof.object, print.gof, plot.gof, shapiro.test, ks.test, chisq.test, Normal, Lognormal, Lognormal3, Zero-Modified Normal, Zero-Modified Lognormal (Delta), enorm, elnorm, elnormAlt, elnorm3, ezmnorm, ezmlnorm, ezmlnormAlt, qqPlot.

Examples