Description

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

Usage

Arguments

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 ≤ t) \;\;\;\;\;\; (1)

and let x_1, x_2, …, 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 ≤ t) \;\;\;\;\;\; (2)

and let y_1, y_2, …, 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, …, 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, …, 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, …, k, otherwise it is greater than 1 for at least on value of i.

For i = 1, 2, …, 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} = ∞ 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, ∞), 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:

ν = ∑_{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

In Table 1 above, Φ denotes the cumulative distribution function of the standard normal distribution, φ 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} = ∏_{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} = ∏_{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(-∑_{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 ν in Equation (6) is 0. The variance of ν 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{σ}_{ν}^2 = \frac{mn}{N(N-1)} ∑_{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 ν in Equation (6) is re-written as:

ν = ∑_{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{σ}_{ν}^2 = ∑_{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 ν 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{σ}_{ν}^2 = ∑_{i=1}^k \{ \hat{S}_i (1 - a_i) b_i - (a_i - \hat{S}_i) b_i [\hat{S}_i b_i + 2 ∑_{j=i+1}^k \hat{S}_j b_j ]\} \;\;\;\;\;\; (16)

where

a_i = ∏_{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{ν}{\hat{σ}_{ν}} \;\;\;\;\;\; (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.

Value

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

Note

All of the tests computed by twoSampleLinearRankTestCensored (logrank, Tarone-Ware, Gehan, Peto-Peto, normal scores, and generalized sign) are based on a statistic that is essentially the sum over all uncensored time points of the weighted difference between the observed and expected number of observations at each time point (see Equation (15) above). The tests differ in how they weight the differences between the observed and expected number of observations.

Prentice and Marek (1979) point out that the Gehan test uses weights that depend on the censoring rates within each group and can lead to non-significant outcomes in the case of heavy censoring when in fact a very large difference between the two groups exists.

Latta (1981) performed a Monte Carlo simulation to study the power of the Gehan, logrank, and Peto-Peto tests using all three different estimators of variance (permutation, hypergeometric, and asymptotic). He used lognormal, Weibull, and exponential distributions to generate the observations, and studied two different cases of censoring: uniform censoring for both samples vs. no censoring in the first sample and uniform censoring in the second sample. Latta (1981) used sample sizes of 10 and 50 (both the equal and unequal cases were studied). Latta (1981) found that all three tests maintained the nominal Type I error level (α-level) in the case of equal sample sizes and equal censoring. Also, the Peto-Peto test based on the asymptotic variance appeared to maintain the nominal α-level in all situations, but the other tests were slightly biased in the case of unequal sample sizes and/or unequal censoring. In particular, tests based on the hypergeometric variance are slightly biased for unequal sample sizes. Latta (1981) concludes that if there is no censoring or light censoring, any of the tests may be used (but the hypergeometric variance should not be used if the sample sizes are very different). In the case of heavy censoring where sample sizes are far apart and/or the censoring is very different between samples, the Peto-Peto test based on the asymptotic variance should be used.

Millard and Deverel (1988) also performed a Monte Carlo simulation similar to Latta's (1981) study. They only used the lognormal distribution to generate observations, but also looked at the normal scores test and two ad-hoc modifications of the MWW test. They found the “Normal Scores 2” test shown in Table 1 above to be the best behaved test in terms of maintaining the nominal α-level, but the other tests behaved almost as well. As Latta (1981) found, when sample sizes and censoring are very different between the two groups, the nominal α-level of most of the tests is slightly biased. In the cases where the nominal α-level was maintained, the Peto-Peto test based on the asymptotic variance appeared to be as powerful or more powerful than the normal scores tests.

Neither of the Monte Carlo studies performed by Latta (1981) and Millard and Deverel (1988) looked at the behavior of the two-sample linear rank tests in the presence of several tied uncensored observations (because both studies generated observations from continuous distributions). Note that the results shown in Table 9 of Millard and Deverel (1988, p.2097) are not all correct because they did not allow for tied uncensored values. The last example in the EXAMPLES section below shows the correct values that should appear in that table.

Heller and Venkatraman (1996) performed a Monte Carlo simulation study to compare the behaviors of the Peto-Peto test (using the Prentice, 1978, estimator of survival; they call this the Prentice-Wilcoxon test) and logrank test under varying censoring conditions with sample sizes of 20 and 50 per group based on using the following methods to compute p-values: the asymptotic standard normal approximation, a permutation test approach (this is NOT the same as the permutation variance), and a bootstrap approach. Observed times were generated from Weibull and lognormal survival time distributions with independent uniform censoring. They found that for the Peto-Peto test, "the asymptotic test procedure was the most accurate; resampling procedures did not improve upon its accuracy." For the logrank test, with sample sizes of 20 per group, the usual test based on the asymptotic standard normal approximation tended to have a very slightly higher Type I error rate than assumed (however, for an assumed Type I error rate of 0.05, the largest Type I error rate observed was less than 0.065), whereas the permuation and bootstrap tests performed better; with sample sizes of 50 per group there was no difference in test performance.

Fleming and Harrington (1981) introduced a family of tests (sometimes called G-rho tests) that contain the logrank and Peto-Peto tests as special cases. A single parameter ρ (rho) controls the weights given to the uncensored and censored observations. Positive values of ρ produce tests more sensitive to early differences in the survival function, that is, differences in the cdf at small values. Negative values of ρ produce tests more sensitive to late differences in the survival function, that is, differences in the cdf at large values.

The function survdiff in the R package survival implements the G-rho family of tests suggested by Flemming and Harrington (1981). Calling survdiff with rho=0 (the default) yields the logrank test. Calling survdiff with rho=1 yields the Peto-Peto test based on the Kaplan-Meier estimate of survival. The function survdiff always uses the hypergeometric estimate of variance and the Kaplan-Meier estimate of survival, but it uses the “left-continuous” version of the Kaplan-Meier estimate. The left-continuous K-M estimate of survival is defined as follows: at each death (unique uncensored observation), the estimated survival is equal to the estimated survival based on the ordinary K-M estimate at the prior death time (or 1 for the first death).

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

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.

See Also

twoSampleLinearRankTest, survdiff, wilcox.test, htestCensored.object.

Examples