twoSampleLinearRankTestCensored: Two-Sample Linear Rank Test to Detect a Difference Between Two Distributions Based on Censored Data

Description

Two-sample linear rank test to detect a difference (usually a shift) between two distributions based on censored data.

Usage

twoSampleLinearRankTestCensored(x, x.censored, y, y.censored, 
    censoring.side = "left", location.shift.null = 0, scale.shift.null = 1, 
    alternative = "two.sided", test = "logrank", variance = "hypergeometric", 
    surv.est = "prentice", shift.type = "location")

Arguments

numeric vector of values for the first sample. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are allowed but will be removed.

x.censored

numeric or logical vector indicating which values of x are censored. This must be the same length as x. If the mode of x.censored is "logical", TRUE values correspond to elements of x that are censored, and FALSE values correspond to elements of x that are not censored. If the mode of x.censored is "numeric", it must contain only 1's and 0's; 1 corresponds to TRUE and 0 corresponds to FALSE. Missing (NA) values are allowed but will be removed.

numeric vector of values for the second sample. Missing (NA), undefined (NaN), and infinite (Inf, -Inf) values are allowed but will be removed.

y.censored

numeric or logical vector indicating which values of y are censored. This must be the same length as y. If the mode of y.censored is "logical", TRUE values correspond to elements of y that are censored, and FALSE values correspond to elements of y that are not censored. If the mode of y.censored is "numeric", it must contain only 1's and 0's; 1 corresponds to TRUE and 0 corresponds to FALSE. Missing (NA) values are allowed but will be removed.

censoring.side

character string indicating on which side the censoring occurs for the data in x and y. The possible values are "left" (the default) and "right".

location.shift.null

numeric scalar indicating the hypothesized value of $\Delta$, the location shift between the two distributions, under the null hypothesis. The default value is location.shift.null=0. This argument is ignored if shift.type="scale".

scale.shift.null

numeric scalar indicating the hypothesized value of $\tau$, the scale shift between the two distributions, under the null hypothesis. The default value is scale.shift.null=1. This argument is ignored if shift.type="location".

alternative

character string indicating the kind of alternative hypothesis. The possible values are "two.sided" (the default), "less", and "greater". See the DETAILS section below for more information.

test

character string indicating which linear rank test to use. The possible values are: "logrank" (the default), "tarone-ware", "gehan", "peto-peto", "normal.scores.1", "normal.scores.2", and "generalized.sign". See the DETAILS section below for more information.

variance

character string indicating which kind of variance to compute for the test. The possible values are: "hypergeometric" (the default), "permutation", and "asymptotic". See the DETAILS section below for more information.

surv.est

character string indicating what method to use to estimate the survival function. The possible values are "prentice" (the default), "kaplan-meier", "peto-peto", and "altshuler". When test="logrank" the argument surv.est is automatically set to "altshuler" and cannot be changed by the user. See the DETAILS section below for more information.

shift.type

character string indicating which kind of shift is being tested. The possible values are "location" (the default) and "scale".

Value

a list of class "htestCensored" containing the results of the hypothesis test. See the help file for htestCensored.object for details.

Details

The function twoSampleLinearRankTestCensored allows you to compare two samples containing censored observations using a linear rank test to determine whether the two samples came from the same distribution. The help file for twoSampleLinearRankTest explains linear rank tests for complete data (i.e., no censored observations are present), and here we assume you are familiar with that material The sections below explain how linear rank tests can be extended to the case of censored data.

Notation Several authors have proposed extensions of the MWW test to the case of censored data, mainly in the context of survival analysis (e.g., Breslow, 1970; Cox, 1972; Gehan, 1965; Mantel, 1966; Peto and Peto, 1972; Prentice, 1978). Prentice (1978) showed how all of these proposed tests are extensions of a linear rank test to the case of censored observations.

Survival analysis usually deals with right-censored data, whereas environmental data is rarely right-censored but often left-censored (some observations are reported as less than some detection limit). Fortunately, all of the methods developed for right-censored data can be applied to left-censored data as well. (See the sub-section Left-Censored Data below.)

In order to explain Prentice's (1978) generalization of linear rank tests to censored data, we will use the following notation that closely follows Prentice (1978), Prentice and Marek (1979), and Latta (1981). Let $X$ denote a random variable representing measurements from group 1 with cumulative distribution function (cdf): $$F_1(t) = Pr(X \le t) \;\;\;\;\;\; (1)$$ and let $x_1, x_2, \ldots, x_m$ denote $m$ independent observations from this distribution. Let $Y$ denote a random variable from group 2 with cdf: $$F_2(t) = Pr(Y \le t) \;\;\;\;\;\; (2)$$ and let $y_1, y_2, \ldots, y_n$ denote $n$ independent observations from this distribution. Set $N = m + n$, the total number of observations. Assume the data are right-censored so that some observations are only recorded as greater than some censoring level, with possibly several different censoring levels. Let $t_1, t_2, \ldots, t_k$ denote the $k$ ordered, unique, uncensored observations for the combined samples (in the context of survival data, $t$ usually stands for “time of death”). For $i = 1, 2, \ldots, k$, let $d_{1i}$ denote the number of observations from sample 1 (the $X$ observations) that are equal to $t_i$, and let $d_{2i}$ denote the observations from sample 2 (the $Y$ observations) equal to this value. Set $$d_i = d_{1i} + d_{2i} \;\;\;\;\;\; (3)$$ the total number of observations equal to $t_i$. If there are no tied uncensored observations, then $t_i = 1$ for $i = 1, 2, \ldots, k$, otherwise it is greater than 1 for at least on value of $i$.

For $i = 1, 2, \ldots, k$, let $e_{1i}$ denote the number of censored observations from sample 1 (the $X$ observations) with censoring levels that fall into the interval $[t_i, t_{i+1})$ where $t_{k+1} = \infty$ by definition, and let $e_{2i}$ denote the number of censored observations from sample 2 (the $Y$ observations) with censoring levels that fall into this interval. Set $$e_i = e_{1i} + e_{2i} \;\;\;\;\;\; (4)$$ the total number of censoring levels that fall into this interval.

Finally, set $n_{1i}$ equal to the number of observations from sample 1 (uncensored and censored) known to be greater than or equal to $t_i$, i.e., that lie in the interval $[t_i, \infty)$, set $n_{2i}$ equal to the number of observations from sample 2 (uncensored and censored) that lie in this interval, and set $$n_i = n_{1i} + n_{2i} \;\;\;\;\;\; (5)$$ In survival analysis jargon, $n_{1i}$ denotes the number of people from sample 1 who are “at risk” at time $t_i$, that is, these people are known to still be alive at this time. Similarly, $n_{2i}$ denotes the number of people from sample 2 who are at risk at time $t_i$, and $n_i$ denotes the total number of people at risk at time $t_i$.

Score Statistics for Multiply Censored Data Prentice's (1978) generalization of the two-sample score (linear rank) statistic is given by: $$\nu = \sum_{i=1}^k (d_{1i} c_i + e_{1i} C_i) \;\;\;\;\;\; (6)$$ where $c_i$ and $C_i$ denote the scores associated with the uncensored and censored observations, respectively. As for complete data, the form of the scores depends upon the assumed underlying distribution. Table 1 below shows scores for various assumed distributions as presented in Prentice (1978) and Latta (1981) (also see Table 5 of Millard and Deverel, 1988, p.2091). The last column shows what these tests reduce to in the case of complete data (no censored observations).

Table 1. Scores Associated with Various Censored Data Rank Tests

	Uncensored	Censored		Uncensored
Distribution	Score ($c_i$)	Score ($C_i$)	Test Name	Analogue
Logistic	$2\hat{F}_i - 1$	$\hat{F}_i$	Peto-Peto	Wilcoxon Rank Sum

"	$i - n_i$	$i$	Gehan or Breslow	"

"	$i - \sqrt{n_i}$	$i$	Tarone-Ware	"


Normal,	$\Phi^{-1}(\hat{F}_i)$	$\phi(c_i)/\hat{S}_i$	Normal Scores 1	Normal Scores
Lognormal
"	"	$\frac{n_i C_{i-1} - c_i}{n_i-1}$	Normal Scores 2	"


Double	$sign(\hat{F}_i - 0.5)$	$\frac{\hat{F}_i}{1-\hat{F}_i}, if \hat{F_i} < 0.5$	Generalized	Mood's Median
Exponential		$1, if \hat{F}_i \ge 0.5$	Sign


Exponential,	$-log(\tilde{S}_i) - 1$	$-log(\tilde{S}_i)$	Logrank	Savage Scores

In Table 1 above, $\Phi$ denotes the cumulative distribution function of the standard normal distribution, $\phi$ denotes the probability density function of the standard normal distribution, and $sign$ denotes the sign function. Also, the quantities $\hat{F}_i$ and $\hat{S}_i$ denote the estimates of the cumulative distribution function (cdf) and survival function, respectively, at time $t_i$ for the combined sample. The estimated cdf is related to the estimated survival function by: $$\hat{F}_i = 1 - \hat{S}_i \;\;\;\;\;\; (7)$$ The quantity $\tilde{S}_i$ denotes the Altshuler (1970) estimate of the survival function at time $t_i$ for the combined sample (see below).

The argument surv.est determines what method to use estimate the survival function. When surv.est="prentice" (the default), the survival function is estimated as: $$\hat{S}_{i,P} = \prod_{j=1}^{i} \frac{n_j - d_j + 1}{n_j + 1} \;\;\;\;\;\; (8)$$ (Prentice, 1978). When surv.est="kaplan-meier", the survival function is estimated as: $$\hat{S}_{i,KM} = \prod_{j=1}^{i} \frac{n_j - d_j}{n_j} \;\;\;\;\;\; (9)$$ (Kaplan and Meier, 1958), and when surv.est="peto-peto", the survival function is estimated as: $$\hat{S}_{i,PP} = \frac{1}{2} (\hat{S}_{i,KM} + \hat{S}_{i-1, KM}) \;\;\;\;\;\; (10)$$ where $\hat{S}_{0,KM} = 0$ (Peto and Peto, 1972). All three of these estimators of the survival function should produce very similar results. When surv.est="altshuler", the survival function is estimated as: $$\tilde{S}_i = exp(-\sum_{j=1}^i \frac{d_j}{n_j}) \;\;\;\;\;\; (11)$$ (Altshuler, 1970). The scores for the logrank test use this estimator of survival.

Lee and Wang (2003, p. 116) present a slightly different version of the Peto-Peto test. They use the Peto-Peto estimate of the survival function for $c_i$, but use the Kaplan-Meier estimate of the survival function for $C_i$.

The scores for the “Normal Scores 1” test shown in Table 1 above are based on the approximation (30) of Prentice (1978). The scores for the “Normal Scores 2” test are based on equation (7) of Prentice and Marek (1979). For the “Normal Scores 2” test, the following rules are used to construct the scores for the censored observations: $C_0 = 0$, and $C_k = 0$ if $n_k = 1$.

The Distribution of the Score Statistic Under the null hypothesis that the two distributions are the same, the expected value of the score statistic $\nu$ in Equation (6) is 0. The variance of $\nu$ can be computed in at least three different ways. If the censoring mechanism is the same for both groups, the permutation variance is appropriate (variance="permutation") and its estimate is given by: $$\hat{\sigma}_{\nu}^2 = \frac{mn}{N(N-1)} \sum_{i=1}^k (d_i c_i^2 + e_i C_i^2) \;\;\;\;\;\; (12)$$ Often, however, it is not clear whether this assumption is valid, and both Prentice (1978) and Prentice and Marek (1979) caution against using the permuation variance (Prentice and Marek, 1979, state it can lead to inflated estimates of variance).

If the censoring mechanisms for the two groups are not necessarily the same, a more general estimator of the variance is based on a conditional permutation approach. In this case, the statistic $\nu$ in Equation (6) is re-written as: $$\nu = \sum_{i=1}^k w_i [d_{1i} - d_i \frac{n_{1i}}{n_i}] \;\;\;\;\;\; (13)$$ where $$w_i = c_i - C_i \;\;\;\;\;\; (14)$$ $c_i$ and $C_i$ are given above in Table 1, and the conditional permutation or hypergeometric estimate (variance="hypergeometric") is given by: $$\hat{\sigma}_{\nu}^2 = \sum_{i=1}^k d_i w_i^2 (\frac{n_{1i}}{n_i}) (1 - \frac{n_{1i}}{n_i}) (\frac{n_i - d_i}{n_i - 1}) \;\;\;\;\;\; (15)$$ (Prentice and Marek, 1979; Latta, 1981; Millard and Deverel, 1988). Note that Equation (13) can be thought of as the sum of weighed values of observed minus expected observations.

Prentice (1978) derived an asymptotic estimator of the variance of the score statistic $\nu$ given in Equation (6) above based on the log likelihood of the rank vector (variance="asymptotic"). This estimator is the same as the hypergeometric variance estimator for the logrank and Gehan tests (assuming no tied uncensored observations), but for the Peto-Peto test, this estimator is given by: $$\hat{\sigma}_{\nu}^2 = \sum_{i=1}^k \{ \hat{S}_i (1 - a_i) b_i - (a_i - \hat{S}_i) b_i [\hat{S}_i b_i + 2 \sum_{j=i+1}^k \hat{S}_j b_j ]\} \;\;\;\;\;\; (16)$$ where $$a_i = \prod_{j=1}^i \frac{n_j + 1}{n_j + 2} \;\;\;\;\;\; (17)$$ $$b_i = 2 d_{1i} + e_{1i} \;\;\;\;\;\; (18)$$ (Prentice, 1978; Latta, 1981; Millard and Deverel, 1988). Note that equation (14) of Millard and Deverel (1988) contains a typographical error.

The Treatment of Ties If the hypergeometric estimator of variance is being used, no modifications need to be made for ties; Equations (13)-(15) already account for ties. For the case of the permutation or asymptotic variance estimators, Equations (6), (12), and (16) all assume no ties in the uncensored observations. If ties exist in the uncensored observations, Prentice (1978) suggests computing the scores shown in Table 1 above as if there were no ties, and then assigning average scores to the tied observations. (This modification also applies to the quantities $a_i$ and $\hat{S}_i$ in Equation (16) above.) For this algorithm, the statistic in Equation (6) is not in general the same as the one in Equation (13).

Computing a Test Statistic Under the null hypothesis that the two distributions are the same, the statistic $$z = \frac{\nu}{\hat{\sigma}_{\nu}} \;\;\;\;\;\; (19)$$ is approximately distributed as a standard normal random variable for “large” sample sizes. This statistic will tend to be large if the observations in group 1 (the $X$ observations) tend to be larger than the observations in group 2 (the $Y$ observations).

Left-Censored Data Most of the time, if censored observations occur in environmental data, they are left-censored (e.g., observations are reported as less than one or more detection limits). For the two-sample test of differences between groups, the methods that apply to right-censored data are easily adapted to left-censored data: simply multiply the observations by -1, compute the z-statistic shown in Equation (20), then reverse the sign of this statistic before computing the p-value.

References

Altshuler, B. (1970). Theory for the Measurement of Competing Risks in Animal Experiments. Mathematical Biosciences 6, 1--11.

Breslow, N.E. (1970). A Generalized Kruskal-Wallis Test for Comparing K Samples Subject to Unequal Patterns of Censorship. Biometrika 57, 579--594.

Conover, W.J. (1980). Practical Nonparametric Statistics. Second Edition. John Wiley and Sons, New York, Chapter 4.

Cox, D.R. (1972). Regression Models and Life Tables (with Discussion). Journal of the Royal Statistical Society of London, Series B 34, 187--220.

Divine, G., H.J. Norton, R. Hunt, and J. Dinemann. (2013). A Review of Analysis and Sample Size Calculation Considerations for Wilcoxon Tests. Anesthesia \& Analgesia 117, 699--710.

Fleming, T.R., and D.P. Harrington. (1981). A Class of Hypothesis Tests for One and Two Sample Censored Survival Data. Communications in Statistics -- Theory and Methods A10(8), 763--794.

Fleming, T.R., and D.P. Harrington. (1991). Counting Processes \& Survival Analysis. John Wiley and Sons, New York, Chapter 7.

Gehan, E.A. (1965). A Generalized Wilcoxon Test for Comparing Arbitrarily Singly-Censored Samples. Biometrika 52, 203--223.

Harrington, D.P., and T.R. Fleming. (1982). A Class of Rank Test Procedures for Censored Survival Data. Biometrika 69(3), 553--566.

Heller, G., and E. S. Venkatraman. (1996). Resampling Procedures to Compare Two Survival Distributions in the Presence of Right-Censored Data. Biometrics 52, 1204--1213.

Hettmansperger, T.P. (1984). Statistical Inference Based on Ranks. John Wiley and Sons, New York, 323pp.

Hollander, M., and D.A. Wolfe. (1999). Nonparametric Statistical Methods, Second Edition. John Wiley and Sons, New York.

Kaplan, E.L., and P. Meier. (1958). Nonparametric Estimation From Incomplete Observations. Journal of the American Statistical Association 53, 457--481.

Latta, R.B. (1981). A Monte Carlo Study of Some Two-Sample Rank Tests with Censored Data. Journal of the American Statistical Association 76(375), 713--719.

Mantel, N. (1966). Evaluation of Survival Data and Two New Rank Order Statistics Arising in its Consideration. Cancer Chemotherapy Reports 50, 163-170.

Millard, S.P., and S.J. Deverel. (1988). Nonparametric Statistical Methods for Comparing Two Sites Based on Data With Multiple Nondetect Limits. Water Resources Research, 24(12), 2087--2098.

Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton, FL, pp.432--435.

Peto, R., and J. Peto. (1972). Asymptotically Efficient Rank Invariant Test Procedures (with Discussion). Journal of the Royal Statistical Society of London, Series A 135, 185--206.

Prentice, R.L. (1978). Linear Rank Tests with Right Censored Data. Biometrika 65, 167--179.

Prentice, R.L. (1985). Linear Rank Tests. In Kotz, S., and N.L. Johnson, eds. Encyclopedia of Statistical Science. John Wiley and Sons, New York. Volume 5, pp.51--58.

Prentice, R.L., and P. Marek. (1979). A Qualitative Discrepancy Between Censored Data Rank Tests. Biometrics 35, 861--867.

Tarone, R.E., and J. Ware. (1977). On Distribution-Free Tests for Equality of Survival Distributions. Biometrika 64(1), 156--160.

USEPA. (2009). Statistical Analysis of Groundwater Monitoring Data at RCRA Facilities, Unified Guidance. EPA 530/R-09-007, March 2009. Office of Resource Conservation and Recovery Program Implementation and Information Division. U.S. Environmental Protection Agency, Washington, D.C.

USEPA. (2010). Errata Sheet - March 2009 Unified Guidance. EPA 530/R-09-007a, August 9, 2010. Office of Resource Conservation and Recovery, Program Information and Implementation Division. U.S. Environmental Protection Agency, Washington, D.C.

Examples

Run this code

# NOT RUN {
  # The last part of the EXAMPLES section in the help file for 
  # cdfCompareCensored compares the empirical distribution of copper and zinc 
  # between two sites:  Alluvial Fan and Basin-Trough (Millard and Deverel, 1988).  
  # The data for this example are stored in Millard.Deverel.88.df.  Perform a 
  # test to determine if there is a significant difference between these two 
  # sites (perform a separate test for the copper and the zinc).

  Millard.Deverel.88.df
  #    Cu.orig Cu Cu.censored Zn.orig  Zn Zn.censored         Zone Location
  #1       < 1  1        TRUE     <10  10        TRUE Alluvial.Fan        1
  #2       < 1  1        TRUE       9   9       FALSE Alluvial.Fan        2
  #3         3  3       FALSE      NA  NA       FALSE Alluvial.Fan        3
  #.
  #.
  #.
  #116       5  5       FALSE      50  50       FALSE Basin.Trough       48
  #117      14 14       FALSE      90  90       FALSE Basin.Trough       49
  #118       4  4       FALSE      20  20       FALSE Basin.Trough       50


  #------------------------------
  # First look at the copper data 
  #------------------------------

  Cu.AF <- with(Millard.Deverel.88.df, 
    Cu[Zone == "Alluvial.Fan"])

  Cu.AF.cen <- with(Millard.Deverel.88.df, 
    Cu.censored[Zone == "Alluvial.Fan"]) 

  Cu.BT <- with(Millard.Deverel.88.df, 
    Cu[Zone == "Basin.Trough"])

  Cu.BT.cen <- with(Millard.Deverel.88.df, 
    Cu.censored[Zone == "Basin.Trough"])

  # Note the large number of tied observations in the copper data
  #-------------------------------------------------------------- 

  table(Cu.AF[!Cu.AF.cen]) 
  # 1  2  3  4  5  7  8  9 10 11 12 16 20 
  # 5 21  6  3  3  3  1  1  1  1  1  1  1

  table(Cu.BT[!Cu.BT.cen]) 
  # 1  2  3  4  5  6  8  9 12 14 15 17 23 
  # 7  4  8  5  1  2  1  2  1  1  1  1  1


  # Logrank test with hypergeometric variance:
  #-------------------------------------------
  twoSampleLinearRankTestCensored(x = Cu.AF, x.censored = Cu.AF.cen, 
    y = Cu.BT, y.censored = Cu.BT.cen)

  #Results of Hypothesis Test
  #Based on Censored Data
  #--------------------------
  #
  #Null Hypothesis:                 Fy(t) = Fx(t)
  #
  #Alternative Hypothesis:          Fy(t) != Fx(t) for at least one t
  #
  #Test Name:                       Two-Sample Linear Rank Test:
  #                                 Logrank Test
  #                                 with Hypergeometric Variance
  #
  #Censoring Side:                  left
  #
  #Censoring Level(s):              x =  1  5 10 20 
  #                                 y =  1  2  5 10 15 
  #
  #Data:                            x = Cu.AF
  #                                 y = Cu.BT
  #
  #Censoring Variable:              x = Cu.AF.cen
  #                                 y = Cu.BT.cen
  #
  #Number NA/NaN/Inf's Removed:     x = 3
  #                                 y = 1
  #
  #Sample Sizes:                    nx = 65
  #                                 ny = 49
  #
  #Percent Censored:                x = 26.2%
  #                                 y = 28.6%
  #
  #Test Statistics:                 nu     = -1.8791355
  #                                 var.nu = 13.6533490
  #                                 z      = -0.5085557
  #
  #P-value:                         0.6110637


  # Compare the p-values produced by the Normal Scores 2 test
  # using the hypergeomtric vs. permutation variance estimates.
  # Note how much larger the estimated variance is based on
  # the permuation variance estimate:
  #-----------------------------------------------------------

  twoSampleLinearRankTestCensored(x = Cu.AF, x.censored = Cu.AF.cen, 
    y = Cu.BT, y.censored = Cu.BT.cen, 
    test = "normal.scores.2")$p.value
  #[1] 0.2008913

  twoSampleLinearRankTestCensored(x = Cu.AF, x.censored = Cu.AF.cen, 
    y = Cu.BT, y.censored = Cu.BT.cen, 
    test = "normal.scores.2", variance = "permutation")$p.value
  #[1] [1] 0.657001


  #--------------------------
  # Now look at the zinc data
  #--------------------------

  Zn.AF <- with(Millard.Deverel.88.df, 
    Zn[Zone == "Alluvial.Fan"])

  Zn.AF.cen <- with(Millard.Deverel.88.df, 
    Zn.censored[Zone == "Alluvial.Fan"])

  Zn.BT <- with(Millard.Deverel.88.df, 
    Zn[Zone == "Basin.Trough"])

  Zn.BT.cen <- with(Millard.Deverel.88.df, 
    Zn.censored[Zone == "Basin.Trough"])

  # Note the moderate number of tied observations in the zinc data, 
  # and the "outlier" of 620 in the Alluvial Fan data.
  #--------------------------------------------------------------- 

  table(Zn.AF[!Zn.AF.cen]) 
  #  5   7   8   9  10  11  12  17  18  19  20  23  29  30  33  40  50 620 
  #  1   1   1   1  20   2   1   1   1   1  14   1   1   1   1   1   1   1 

  table(Zn.BT[!Zn.BT.cen]) 
  # 3  4  5  6  8 10 11 12 13 14 15 17 20 25 30 40 50 60 70 90 
  # 2  2  2  1  1  5  1  2  1  1  1  2 11  1  4  3  2  2  1  1


  # Logrank test with hypergeometric variance:
  #-------------------------------------------
  twoSampleLinearRankTestCensored(x = Zn.AF, x.censored = Zn.AF.cen, 
    y = Zn.BT, y.censored = Zn.BT.cen)

  #Results of Hypothesis Test
  #Based on Censored Data
  #--------------------------
  #
  #Null Hypothesis:                 Fy(t) = Fx(t)
  #
  #Alternative Hypothesis:          Fy(t) != Fx(t) for at least one t
  #
  #Test Name:                       Two-Sample Linear Rank Test:
  #                                 Logrank Test
  #                                 with Hypergeometric Variance
  #
  #Censoring Side:                  left
  #
  #Censoring Level(s):              x =  3 10 
  #                                 y =  3 10 
  #
  #Data:                            x = Zn.AF
  #                                 y = Zn.BT
  #
  #Censoring Variable:              x = Zn.AF.cen
  #                                 y = Zn.BT.cen
  #
  #Number NA/NaN/Inf's Removed:     x = 1
  #                                 y = 0
  #
  #Sample Sizes:                    nx = 67
  #                                 ny = 50
  #
  #Percent Censored:                x = 23.9%
  #                                 y =  8.0%
  #
  #Test Statistics:                 nu     = -6.992999
  #                                 var.nu = 17.203227
  #                                 z      = -1.686004
  #
  #P-value:                         0.09179512

  #----------

  # Compare the p-values produced by the Logrank, Gehan, Peto-Peto, 
  # and Tarone-Ware tests using the hypergeometric variance.
  #-----------------------------------------------------------

  twoSampleLinearRankTestCensored(x = Zn.AF, x.censored = Zn.AF.cen, 
    y = Zn.BT, y.censored = Zn.BT.cen, 
    test = "logrank")$p.value
  #[1] 0.09179512

  twoSampleLinearRankTestCensored(x = Zn.AF, x.censored = Zn.AF.cen, 
    y = Zn.BT, y.censored = Zn.BT.cen, 
    test = "gehan")$p.value
  #[1] 0.0185445

  twoSampleLinearRankTestCensored(x = Zn.AF, x.censored = Zn.AF.cen, 
    y = Zn.BT, y.censored = Zn.BT.cen, 
    test = "peto-peto")$p.value
  #[1] 0.009704529

  twoSampleLinearRankTestCensored(x = Zn.AF, x.censored = Zn.AF.cen, 
    y = Zn.BT, y.censored = Zn.BT.cen, 
    test = "tarone-ware")$p.value
  #[1] 0.03457803

  #----------

  # Clean up
  #---------

  rm(Cu.AF, Cu.AF.cen, Cu.BT, Cu.BT.cen, 
     Zn.AF, Zn.AF.cen, Zn.BT, Zn.BT.cen)


  #==========

  # Example 16.5 on pages 16-22 to 16.23 of USEPA (2009) shows how to perform 
  # the Tarone-Ware two sample linear rank test based on censored data using 
  # observations on tetrachloroethylene (PCE) (ppb) collected at one background 
  # and one compliance well.  The data for this example are stored in 
  # EPA.09.Ex.16.5.PCE.df.

  EPA.09.Ex.16.5.PCE.df

  #    Well.type PCE.Orig.ppb PCE.ppb Censored
  #1  Background           <4     4.0     TRUE
  #2  Background          1.5     1.5    FALSE
  #3  Background           <2     2.0     TRUE
  #4  Background          8.7     8.7    FALSE
  #5  Background          5.1     5.1    FALSE
  #6  Background           <5     5.0     TRUE
  #7  Compliance          6.4     6.4    FALSE
  #8  Compliance         10.9    10.9    FALSE
  #9  Compliance            7     7.0    FALSE
  #10 Compliance         14.3    14.3    FALSE
  #11 Compliance          1.9     1.9    FALSE
  #12 Compliance           10    10.0    FALSE
  #13 Compliance          6.8     6.8    FALSE
  #14 Compliance           <5     5.0     TRUE

  with(EPA.09.Ex.16.5.PCE.df, 
    twoSampleLinearRankTestCensored(
      x = PCE.ppb[Well.type == "Compliance"], 
      x.censored = Censored[Well.type == "Compliance"], 
      y = PCE.ppb[Well.type == "Background"], 
      y.censored = Censored[Well.type == "Background"], 
      test = "tarone-ware", alternative = "greater"))

  #Results of Hypothesis Test
  #Based on Censored Data
  #--------------------------
  #
  #Null Hypothesis:                 Fy(t) = Fx(t)
  #
  #Alternative Hypothesis:          Fy(t) > Fx(t) for at least one t
  #
  #Test Name:                       Two-Sample Linear Rank Test:
  #                                 Tarone-Ware Test
  #                                 with Hypergeometric Variance
  #
  #Censoring Side:                  left
  #
  #Censoring Level(s):              x = 5 
  #                                 y = 2 4 5 
  #
  #Data:                            x = PCE.ppb[Well.type == "Compliance"]
  #                                 y = PCE.ppb[Well.type == "Background"]
  #
  #Censoring Variable:              x = Censored[Well.type == "Compliance"]
  #                                 y = Censored[Well.type == "Background"]
  #
  #Sample Sizes:                    nx = 8
  #                                 ny = 6
  #
  #Percent Censored:                x = 12.5%
  #                                 y = 50.0%
  #
  #Test Statistics:                 nu     =  8.458912
  #                                 var.nu = 20.912407
  #                                 z      =  1.849748
  #
  #P-value:                         0.03217495

  # Compare the p-value for the Tarone-Ware test with p-values from 
  # the logrank, Gehan, and Peto-Peto tests
  #-----------------------------------------------------------------

  with(EPA.09.Ex.16.5.PCE.df, 
    twoSampleLinearRankTestCensored(
      x = PCE.ppb[Well.type == "Compliance"], 
      x.censored = Censored[Well.type == "Compliance"], 
      y = PCE.ppb[Well.type == "Background"], 
      y.censored = Censored[Well.type == "Background"], 
      test = "tarone-ware", alternative = "greater"))$p.value
  #[1] 0.03217495


  with(EPA.09.Ex.16.5.PCE.df, 
    twoSampleLinearRankTestCensored(
      x = PCE.ppb[Well.type == "Compliance"], 
      x.censored = Censored[Well.type == "Compliance"], 
      y = PCE.ppb[Well.type == "Background"], 
      y.censored = Censored[Well.type == "Background"], 
      test = "logrank", alternative = "greater"))$p.value
  #[1] 0.02752793


  with(EPA.09.Ex.16.5.PCE.df, 
    twoSampleLinearRankTestCensored(
      x = PCE.ppb[Well.type == "Compliance"], 
      x.censored = Censored[Well.type == "Compliance"], 
      y = PCE.ppb[Well.type == "Background"], 
      y.censored = Censored[Well.type == "Background"], 
      test = "gehan", alternative = "greater"))$p.value
  #[1] 0.03656224

  with(EPA.09.Ex.16.5.PCE.df, 
    twoSampleLinearRankTestCensored(
      x = PCE.ppb[Well.type == "Compliance"], 
      x.censored = Censored[Well.type == "Compliance"], 
      y = PCE.ppb[Well.type == "Background"], 
      y.censored = Censored[Well.type == "Background"], 
      test = "peto-peto", alternative = "greater"))$p.value
  #[1] 0.03127296

  #==========

  # The results shown in Table 9 of Millard and Deverel (1988, p.2097) are correct 
  # only for the hypergeometric variance and the modified MWW tests; the other 
  # results were computed as if there were no ties.  Re-compute the correct 
  # z-statistics and p-values for the copper and zinc data. 

  test <- c(rep(c("gehan", "logrank", "peto-peto"), 2), "peto-peto", 
    "normal.scores.1", "normal.scores.2", "normal.scores.2")

  variance <- c(rep("permutation", 3), rep("hypergeometric", 3), 
    "asymptotic", rep("permutation", 2), "hypergeometric")

  stats.mat <- matrix(as.numeric(NA), ncol = 4, nrow = 10)

  for(i in 1:10) {
    dum.list <- with(Millard.Deverel.88.df, 
        twoSampleLinearRankTestCensored(
          x = Cu[Zone == "Basin.Trough"], 
          x.censored = Cu.censored[Zone == "Basin.Trough"],
          y = Cu[Zone == "Alluvial.Fan"], 
          y.censored = Cu.censored[Zone == "Alluvial.Fan"], 
          test = test[i], variance = variance[i]))
    stats.mat[i, 1:2] <- c(dum.list$statistic["z"], dum.list$p.value)

    dum.list <- with(Millard.Deverel.88.df, 
        twoSampleLinearRankTestCensored(
          x = Zn[Zone == "Basin.Trough"], 
          x.censored = Zn.censored[Zone == "Basin.Trough"],
          y = Zn[Zone == "Alluvial.Fan"], 
          y.censored = Zn.censored[Zone == "Alluvial.Fan"], 
          test = test[i], variance = variance[i]))
    stats.mat[i, 3:4] <- c(dum.list$statistic["z"], dum.list$p.value)
  }

  dimnames(stats.mat) <- list(paste(test, variance, sep = "."), 
    c("Cu.Z", "Cu.p.value", "Zn.Z", "Zn.p.value"))

  round(stats.mat, 2)
  #                               Cu.Z Cu.p.value Zn.Z Zn.p.value
  #gehan.permutation              0.87       0.38 2.49       0.01
  #logrank.permutation            0.79       0.43 1.75       0.08
  #peto-peto.permutation          0.92       0.36 2.42       0.02
  #gehan.hypergeometric           0.71       0.48 2.35       0.02
  #logrank.hypergeometric         0.51       0.61 1.69       0.09
  #peto-peto.hypergeometric       1.03       0.30 2.59       0.01
  #peto-peto.asymptotic           0.90       0.37 2.37       0.02
  #normal.scores.1.permutation    0.94       0.34 2.37       0.02
  #normal.scores.2.permutation    0.98       0.33 2.39       0.02
  #normal.scores.2.hypergeometric 1.28       0.20 2.48       0.01

  #----------

  # Clean up
  #---------
  rm(test, variance, stats.mat, i, dum.list)
# }

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