Description

Perform a goodness-of-fit test to determine whether a data set appears to come from a normal distribution, lognormal distribution, or lognormal distribution (alternative parameterization) based on a sample of data that has been subjected to Type I or Type II censoring.

Usage

Arguments

Details

Let \underline{x} = c(x_1, x_2, …, x_N) denote a vector of N observations from from some distribution with 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.

Assume n (0 < n < N) of these observations are known and c (c=N-n) of these observations are all censored below (left-censored) or all censored above (right-censored) at k fixed censoring levels

T_1, T_2, …, T_k; \; k ≥ 1 \;\;\;\;\;\; (1)

For the case when k ≥ 2, the data are said to be Type I multiply censored. For the case when k=1, set T = T_1. If the data are left-censored and all n known observations are greater than or equal to T, or if the data are right-censored and all n known observations are less than or equal to T, then the data are said to be Type I singly censored (Nelson, 1982, p.7), otherwise they are considered to be Type I multiply censored.

Let c_j denote the number of observations censored below or above censoring level T_j for j = 1, 2, …, k, so that

∑_{i=1}^k c_j = c \;\;\;\;\;\; (2)

Let x_{(1)}, x_{(2)}, …, x_{(N)} denote the “ordered” observations, where now “observation” means either the actual observation (for uncensored observations) or the censoring level (for censored observations). For right-censored data, if a censored observation has the same value as an uncensored one, the uncensored observation should be placed first. For left-censored data, if a censored observation has the same value as an uncensored one, the censored observation should be placed first.

Note that in this case the quantity x_{(i)} does not necessarily represent the i'th “largest” observation from the (unknown) complete sample.

Note that for singly left-censored data:

x_{(1)} = x_{(2)} = \cdots = x_{(c)} = T \;\;\;\;\;\; (3)

and for singly right-censored data:

x_{(n+1)} = x_{(n+2)} = \cdots = x_{(N)} = T \;\;\;\;\;\; (4)

Finally, let Ω (omega) denote the set of n subscripts in the “ordered” sample that correspond to uncensored observations.

Shapiro-Wilk Goodness-of-Fit Test for Singly Censored Data (test="sw")
Equation (8) in the help file for gofTest shows that for the case of complete ordered data \underline{x}, the Shapiro-Wilk W-statistic 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 \;\;\;\;\;\; (5)

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}} \;\;\;\;\;\;\; (6)

and \underline{a} is defined by:

\underline{a} = \frac{\underline{m}^T V^{-1}}{[\underline{m}^T V^{-1} V^{-1} \underline{m}]^{1/2}} \;\;\;\;\;\; (7)

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 \;\;\;\;\;\; (8)

Computing Shapiro-Wilk W-Statistic for Singly Censored Data
For the case of singly censored data, following Smith and Bain (1976) and Verrill and Johnson (1988), Royston (1993) generalizes the Shapiro-Wilk W-statistic to:

W = r(\underline{a}_{Δ}, \underline{x}_{Δ})^2 \;\;\;\;\;\; (9)

where for left singly-censored data:

\underline{a}_{Δ} = (a_{c+1}, a_{c+2}, …, a_{N}) \;\;\;\;\;\; (10)

\underline{x}_{Δ} = (x_{(c+1)}, x_{(c+2)}, …, x_{(N)}) \;\;\;\;\;\; (11)

and for right singly-censored data:

\underline{a}_{Δ} = (a_1, a_2, …, a_n) \;\;\;\;\;\; (12)

\underline{x}_{Δ} = (x_{(1)}, x_{(2)}, …, x_{(n)}) \;\;\;\;\;\; (13)

Just like the function gofTest, when test="sw", the function gofTestCensored uses Royston's (1992a) approximation for the coefficients \underline{a} (see the help file for gofTest).

Computing P-Values for the Shapiro-Wilk Test
Verrill and Johnson (1988) show that the asymptotic distribution of the statistic in Equation (9) above is normal, but the rate of convergence is “surprisingly slow” even for complete samples. They provide a table of empirical percentiles of the distribution for the W-statistic shown in Equation (9) above for several sample sizes and percentages of censoring.

Based on the tables given in Verrill and Johnson (1988), Royston (1993) approximated the 90'th, 95'th, and 99'th percentiles of the distribution of the z-statistic computed from the W-statistic. (The distribution of this z-statistic is assumed to be normal, but not necessarily a standard normal.) Denote these percentiles by Z_{0.90}, Z_{0.95}, and Z_{0.99}. The true mean and standard deviation of the z-statistic are estimated by the intercept and slope, respectively, from the linear regression of Z_{α} on Φ^{-1}(α) for α = 0.9, 0.95, and 0.99, where Φ denotes the cumulative distribution function of the standard normal distribution. The p-value associated with this test is then computed as:

p = 1 - Φ(\frac{z - μ_z}{σ_z}) \;\;\;\;\;\; (14)

Note: Verrill and Johnson (1988) produced their tables based on Type II censoring. Royston's (1993) approximation to the p-value of these tests, however, should be fairly accurate for Type I censored data as well.

Testing Goodness-of-Fit for Any Continuous Distribution
The function gofTestCensored extends the Shapiro-Wilk test that accounts for censoring 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 gofTestCensored modifies the approach of Chen and Balakrishnan (1995) by using the same first 2 steps, and then applying the Shapiro-Wilk test that accounts for censoring:

1. Let \underline{x} = x_1, x_2, …, x_n denote the vector of n ordered observations, ignoring censoring status. 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) using an estimation method that accounts for censoring (e.g., assuming a Gamma distribution with alternative parameterization, call the function link{egammaAltCensored}).

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 (that accounts for censoring) on the y_i's.

Shapiro-Francia Goodness-of-Fit Test (test="sf")
Equation (15) in the help file for gofTest shows that for the complete ordered data \underline{x}, the Shapiro-Francia W'-statistic is the same as the squared Pearson correlation coefficient associated with a normal probability plot.

Computing Shapiro-Francia W'-Statistic for Censored Data
For the case of singly censored data, following Smith and Bain (1976) and Verrill and Johnson (1988), Royston (1993) extends the computation of the Weisberg-Bingham Approximation to the W'-statistic to the case of singly censored data:

\tilde{W}' = r(\underline{c}_{Δ}, \underline{x}_{Δ})^2 \;\;\;\;\;\; (14)

where for left singly-censored data:

\underline{c}_{Δ} = (c_{c+1}, c_{c+2}, …, c_{N}) \;\;\;\;\;\; (15)

\underline{x}_{Δ} = (x_{(c+1)}, x_{(c+2)}, …, x_{(N)}) \;\;\;\;\;\; (16)

and for right singly-censored data:

\underline{a}_{Δ} = (a_1, a_2, …, a_n) \;\;\;\;\;\; (17)

\underline{x}_{Δ} = (x_{(1)}, x_{(2)}, …, x_{(n)}) \;\;\;\;\;\; (18)

and \underline{c} is defined as:

\underline{c} = \frac{\underline{\tilde{m}}}{[\underline{\tilde{m}}' \underline{\tilde{m}}]^{1/2}} \;\;\;\;\;\; (19)

where

\tilde{m}_i = Φ^{-1}(\frac{i - (3/8)}{n + (1/4)}) \;\;\;\;\;\; (20)

and Φ denotes the standard normal cdf. Note: Do not confuse the elements of the vector \underline{c} with the scalar c which denotes the number of censored observations. We use \underline{c} here to be consistent with the notation in the help file for gofTest.

Just like the function gofTest, when test="sf", the function gofTestCensored uses Royston's (1992a) approximation for the coefficients \underline{c} (see the help file for gofTest).

In general, the Shapiro-Francia test statistic can be extended to multiply censored data using Equation (14) with \underline{c}_{Δ} defined as the orderd values of c_i associated with uncensored observations, and \underline{x}_{Δ} defined as the ordered values of x_i associated with uncensored observations:

\underline{c}_{Δ} = \cup_{i \in Ω} \;\; c_{(i)} \;\;\;\;\;\; (21)

\underline{x}_{Δ} = \cup_{i \in Ω} \;\; x_{(i)} \;\;\;\;\;\; (22)

and where the plotting positions in Equation (20) are replaced with any of the plotting positions available in ppointsCensored (see the description for the argument prob.method).

Computing P-Values for the Shapiro-Francia Test
Verrill and Johnson (1988) show that the asymptotic distribution of the statistic in Equation (14) above is normal, but the rate of convergence is “surprisingly slow” even for complete samples. They provide a table of empirical percentiles of the distribution for the \tilde{W}'-statistic shown in Equation (14) above for several sample sizes and percentages of censoring.

As for the Shapiro-Wilk test, based on the tables given in Verrill and Johnson (1988), Royston (1993) approximated the 90'th, 95'th, and 99'th percentiles of the distribution of the z-statistic computed from the \tilde{W}'-statistic. (The distribution of this z-statistic is assumed to be normal, but not necessarily a standard normal.) Denote these percentiles by Z_{0.90}, Z_{0.95}, and Z_{0.99}. The true mean and standard deviation of the z-statistic are estimated by the intercept and slope, respectively, from the linear regression of Z_{α} on Φ^{-1}(α) for α = 0.9, 0.95, and 0.99, where Φ denotes the cumulative distribution function of the standard normal distribution. The p-value associated with this test is then computed as:

p = 1 - Φ(\frac{z - μ_z}{σ_z}) \;\;\;\;\;\; (23)

Note: Verrill and Johnson (1988) produced their tables based on Type II censoring. Royston's (1993) approximation to the p-value of these tests, however, should be fairly accurate for Type I censored data as well, although this is an area that requires further investigation.

Testing Goodness-of-Fit for Any Continuous Distribution
The function gofTestCensored extends the Shapiro-Francia test that accounts for censoring 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 gofTestCensored modifies the approach of Chen and Balakrishnan (1995) by using the same first 2 steps, and then applying the Shapiro-Francia test that accounts for censoring:

1. Let \underline{x} = x_1, x_2, …, x_n denote the vector of n ordered observations, ignoring censoring status. 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) using an estimation method that accounts for censoring (e.g., assuming a Gamma distribution with alternative parameterization, call the function link{egammaAltCensored}).

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 (that accounts for censoring) on the y_i's.

Probability Plot Correlation Coefficient (PPCC) Goodness-of-Fit Test (test="ppcc")
The function gofTestCensored computes the PPCC test statistic 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.

Value

a list of class "gofCensored" containing the results of the goodness-of-fit test. See the help files for gofCensored.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 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 qqPlotCensored).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

Birnbaum, Z.W., and F.H. Tingey. (1951). One-Sided Confidence Contours for Probability Distribution Functions. Annals of Mathematical Statistics 22, 592-596.

Blom, G. (1958). Statistical Estimates and Transformed Beta Variables. John Wiley and Sons, New York.

Conover, W.J. (1980). Practical Nonparametric Statistics. Second Edition. John Wiley and Sons, New York.

Dallal, G.E., and L. Wilkinson. (1986). An Analytic Approximation to the Distribution of Lilliefor's Test for Normality. The American Statistician 40, 294-296.

D'Agostino, R.B. (1970). Transformation to Normality of the Null Distribution of g1. Biometrika 57, 679-681.

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.

D'Agostino, R.B., and G.L. Tietjen (1973). Approaches to the Null Distribution of √{b1}. Biometrika 60(1), 169-173.

Fisher, R.A. (1950). Statistical Methods for Research Workers. 11'th Edition. Hafner Publishing Company, New York, pp.99-100.

Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken.

Kendall, M.G., and A. Stuart. (1991). The Advanced Theory of Statistics, Volume 2: Inference and Relationship. Fifth Edition. Oxford University Press, New York.

Royston, J.P. (1992a). Approximating the Shapiro-Wilk W-Test for Non-Normality. Statistics and Computing 2, 117-119.

Royston, J.P. (1992b). Estimation, Reference Ranges and Goodness of Fit for the Three-Parameter Log-Normal Distribution. Statistics in Medicine 11, 897-912.

Royston, J.P. (1992c). A Pocket-Calculator Algorithm for the Shapiro-Francia Test of Non-Normality: An Application to Medicine. Statistics in Medicine 12, 181-184.

Royston, P. (1993). A Toolkit for Testing for Non-Normality in Complete and Censored Samples. The Statistician 42, 37-43.

Ryan, T., and B. Joiner. (1973). Normal Probability Plots and Tests for Normality. Technical Report, Pennsylvannia State University, Department of Statistics.

Shapiro, S.S., and R.S. Francia. (1972). An Approximate Analysis of Variance Test for Normality. Journal of the American Statistical Association 67(337), 215-219.

Shapiro, S.S., and M.B. Wilk. (1965). An Analysis of Variance Test for Normality (Complete Samples). Biometrika 52, 591-611.

Verrill, S., and R.A. Johnson. (1987). The Asymptotic Equivalence of Some Modified Shapiro-Wilk Statistics – Complete and Censored Sample Cases. The Annals of Statistics 15(1), 413-419.

Verrill, S., and R.A. Johnson. (1988). Tables and Large-Sample Distribution Theory for Censored-Data Correlation Statistics for Testing Normality. Journal of the American Statistical Association 83, 1192-1197.

Weisberg, S., and C. Bingham. (1975). An Approximate Analysis of Variance Test for Non-Normality Suitable for Machine Calculation. Technometrics 17, 133-134.

See Also

gofTest, gofCensored.object, print.gofCensored, plot.gofCensored, shapiro.test, Normal, Lognormal, enormCensored, elnormCensored, elnormAltCensored, qqPlotCensored.

Examples