| Title: | Testing One Hypothesis Twice in Observational Studies |
|---|---|
| Description: | Tests one hypothesis with several test statistics, correcting for multiple testing. The central function in the package is testtwice(). In a sensitivity analysis, the method has the largest design sensitivity of its component tests. The package implements the method and examples in Rosenbaum, P. R. (2012) <doi:10.1093/biomet/ass032> Testing one hypothesis twice in observational studies. Biometrika, 99(4), 763-774. |
| Authors: | Paul R. Rosenbaum |
| Maintainer: | Paul R. Rosenbaum <[email protected]> |
| License: | GPL-2 |
| Version: | 1.0.3 |
| Built: | 2024-12-05 06:46:37 UTC |
| Source: | CRAN |
Tests one hypothesis with several test statistics, correcting for multiple testing. The central function in the package is testtwice(). In a sensitivity analysis, the method has the largest design sensitivity of its component tests. The package implements the method and examples in Rosenbaum, P. R. (2012) <doi:10.1093/biomet/ass032> Testing one hypothesis twice in observational studies. Biometrika, 99(4), 763-774.
The DESCRIPTION file:
| Package: | testtwice |
| Type: | Package |
| Title: | Testing One Hypothesis Twice in Observational Studies |
| Version: | 1.0.3 |
| Author: | Paul R. Rosenbaum |
| Maintainer: | Paul R. Rosenbaum <[email protected]> |
| Description: | Tests one hypothesis with several test statistics, correcting for multiple testing. The central function in the package is testtwice(). In a sensitivity analysis, the method has the largest design sensitivity of its component tests. The package implements the method and examples in Rosenbaum, P. R. (2012) <doi:10.1093/biomet/ass032> Testing one hypothesis twice in observational studies. Biometrika, 99(4), 763-774. |
| License: | GPL-2 |
| Encoding: | UTF-8 |
| LazyData: | true |
| Imports: | stats, mvtnorm |
| NeedsCompilation: | no |
| Packaged: | 2020-09-08 00:35:42 UTC; Paul Rosenbaum |
| Repository: | CRAN |
| Date/Publication: | 2020-09-15 10:00:02 UTC |
Index of help topics:
bmhranks One-step and Two-step Signed Ranks.
multrnk A family of Signed Ranks for Matched Pair
Differences.
smokerlead Pair Differences in Blood Lead Levels in 679
Matched Pairs of One Smoker and One Control.
testtwice Computes the P-value and Sensitivity Bound for
Testing Twice.
testtwice-package Testing One Hypothesis Twice in Observational
Studies
tt Computes the P-value and Sensitivity Bound for
Testing Twice From a User Supplied Matrix.
Paul R. Rosenbaum
Maintainer: Paul R. Rosenbaum <[email protected]>
Rosenbaum, P. R. (2012) <doi:10.1093/biomet/ass032> Testing one hypothesis twice in observational studies. Biometrika, 99(4), 763-774.
data(smokerlead) attach(smokerlead) # The following example reproduces parts of the first column of Table 3 in Rosenbaum (2012). w<-rank(abs(lead)) wd<-rank(abs(lead))*dose wl<-rank(abs(llead)) H<-cbind(w,wd,wl) tt(lead,H,gamma=2.8) tt(lead,H,gamma=3.2) rm(w,wd,wl,H) detach(smokerlead)data(smokerlead) attach(smokerlead) # The following example reproduces parts of the first column of Table 3 in Rosenbaum (2012). w<-rank(abs(lead)) wd<-rank(abs(lead))*dose wl<-rank(abs(llead)) H<-cbind(w,wd,wl) tt(lead,H,gamma=2.8) tt(lead,H,gamma=3.2) rm(w,wd,wl,H) detach(smokerlead)
Calculates either one-step or two-step signed ranks as proposed by Noether (1973), Brown (1981) and Markowski and Hettmansperger (1982).
bmhranks(y, q1 = 1/3, q2 = 2/3)bmhranks(y, q1 = 1/3, q2 = 2/3)
y |
A vector of matched pair differences. |
q1 |
Quantile at which step-ranks rise from 0 to 1. |
q2 |
Quantile at which step-ranks rise from 1 to 2. If q1=q2, then there is only one step, from 0 to 1. |
A vector with the same length as y containing the step-ranks.
The use of step-ranks in observational studies is discussed in Rosenbaum (1999, 2012, 2015). They can have larger design sensitivity than Wilcoxon ranks and higher Bahadur efficiency in a sensitivity analysis.
Paul R. Rosenbaum
Brown, B. M. (1981) <doi:10.1093/biomet/68.1.235> Symmetric quantile averages and related estimators. Biometrika, 68(1), 235-242.
Markowski, E. P. and Hettmansperger, T. P. (1982) <doi:10.2307/2287325> Inference based on simple rank step score statistics for the location model. Journal of the American Statistical Association, 77(380), 901-907.
Noether, G. E. (1973) <doi:10.2307/2284805> Some simple distribution-free confidence intervals for the center of a symmetric distribution. Journal of the American Statistical Association, 68(343), 716-719.
Rosenbaum, P. R. (1999) <doi:10.1111/1467-9876.00140> Using quantile averages in matched observational studies. Journal of the Royal Statistical Society: Series C (Applied Statistics), 48(1), 63-78.
Rosenbaum, P. R. (2012) <doi:10.1214/11-AOAS508> An exact adaptive test with superior design sensitivity in an observational study of treatments for ovarian cancer. The Annals of Applied Statistics, 6(1), 83-105.
Rosenbaum, P. R. (2015) <doi:10.1080/01621459.2014.960968> Bahadur efficiency of sensitivity analyses in observational studies. Journal of the American Statistical Association, 110(509), 205-217.
The function multrnk() computes an alternative family of signed ranks.
data(smokerlead) attach(smokerlead) w<-rank(abs(llead)) # Wilcoxon ranks o<-order(w) brn<-bmhranks(llead) # Brown (1981) ranks plot(w[o],brn[o],type="n",xlab="Wilcoxon ranks",ylab="Step Ranks", main="Comparison of 3 Step Ranks") lines(w[o],brn[o],col="black") # The following two-step ranks were best for Normal data in # Table 2 of Markowski-Hettmansperger (1982). mhn<-bmhranks(llead,q1=.4,q2=.8) lines(w[o],mhn[o],col="blue") # Noether (1973) ranks take a single step. The case of a step # at q1=q2=2/3 was evaluated in Rosenbaum (2015, Table 2). noe<-bmhranks(llead,q1=2/3,q2=2/3) lines(w[o],noe[o],col="red") legend(20,1.75,c("Brown","MH","Noether"), lty=c(1,1,1),col=c("black","blue","red")) # Adaptive choice of Brown or Noether ranks was considered in # Rosenbaum (2012). In this case, an exact distribution # is available, but function tt() uses a limiting Normal # distribution instead. H<-cbind(brn,noe) tt(llead,H,gamma=1) tt(llead,H,gamma=3.7) rm(w,brn,mhn,noe,H) detach(smokerlead)data(smokerlead) attach(smokerlead) w<-rank(abs(llead)) # Wilcoxon ranks o<-order(w) brn<-bmhranks(llead) # Brown (1981) ranks plot(w[o],brn[o],type="n",xlab="Wilcoxon ranks",ylab="Step Ranks", main="Comparison of 3 Step Ranks") lines(w[o],brn[o],col="black") # The following two-step ranks were best for Normal data in # Table 2 of Markowski-Hettmansperger (1982). mhn<-bmhranks(llead,q1=.4,q2=.8) lines(w[o],mhn[o],col="blue") # Noether (1973) ranks take a single step. The case of a step # at q1=q2=2/3 was evaluated in Rosenbaum (2015, Table 2). noe<-bmhranks(llead,q1=2/3,q2=2/3) lines(w[o],noe[o],col="red") legend(20,1.75,c("Brown","MH","Noether"), lty=c(1,1,1),col=c("black","blue","red")) # Adaptive choice of Brown or Noether ranks was considered in # Rosenbaum (2012). In this case, an exact distribution # is available, but function tt() uses a limiting Normal # distribution instead. H<-cbind(brn,noe) tt(llead,H,gamma=1) tt(llead,H,gamma=3.7) rm(w,brn,mhn,noe,H) detach(smokerlead)
Computes a family of signed ranks that includes the ranks of Wilcoxon, Stephenson (1981) and Rosenbaum (2011).
multrnk(y, m1 = 2, m2 = 2, m = 2, exact=FALSE)multrnk(y, m1 = 2, m2 = 2, m = 2, exact=FALSE)
y |
A vector of matched pair differences. |
m1 |
See m. |
m2 |
See m. |
m |
The three integers (m,m1,m2) with 1<=m1<=m2<=m define a U-statistic and an associated signed rank statistic. See Details. |
exact |
In large samples, it is appropriate to set exact=FALSE, and this is the default. When testing hypotheses, it is reasonable to set exact=FALSE for all sample sizes. If exact=FALSE, then the rank scores are approximated using expression (9) in Rosenbaum (2011). If exact=TRUE, then ranks are proportional to (8) in Rosenbaum (2011). The exact ranks involve combinatorial coefficients that grow very quickly with the sample size, whereas the approximate ranks do not have this property. The confidence intervals for attributable effects in Rosenbaum (2003; 2007; 2011, Appendix) are based on the exact ranks. |
Setting m1=2, m2=2, and m=2 yields the U-statistic that is nearly the same as Wilcoxon's signed rank statistic. See Lehmann (1975, Appendix, Example 6) or Pratt and Gibbons (1981, Section 3.5).
Setting m1=m2=m>=2 yields the method of Stephenson (1981). The ranks used by Stephenson (1981) are closely related to ranks proposed by Conover and Salsburg (1988) who aimed for higher power when only a subset of treated individuals are affected by the treatment; see Rosenbaum (2007).
Stephenson (1981) looks at m differences in y, finds the one pair difference with the largest absolute value, and score a 1 if the difference is positive or a zero if it is negative. He sums this score over all subsets of m distinct differences.
The general case, 1<=m1<=m2<=m is discussed in Rosenbaum (2011), with further evaluation of performance in Rosenbaum (2015). The statistic in Rosenbaum (2011) may be inverted to obtain a confidence interval for an attributable effect; see also Rosenbaum (2003) for the special case of m1=m2=m=2 analogous to Wilcoxon's test.
For instance, the statistic (8,5,8) has Pitman efficiency of .97 compared to Wilcoxon for Normal or logistic errors in a randomized experiment, but has higher design sensitivity when used in a sensitivity analysis in an observational study; see Rosenbaum (2011, Tables 1 and 3).
A vector with the same length as y containing the ranks.
The performance of various ranks when used in sensitivity analyses is discussed in Rosenbaum (2015).
Paul R. Rosenbaum
Conover, W. J. and Salsburg, D. S. (1988) <doi:10.2307/2531906> Locally most powerful tests for detecting treatment effects when only a subset of patients can be expected to respond to treatment. Biometrics, 189-196.
Lehmann, E. L. (1975). Nonparametrics. San Francisco: Holden-Day.
Pratt, J. W. and Gibbons, J. D. (1981) <doi:10.1007/978-1-4612-5931-2> Concepts of Nonparametric Theory. New York: Springer.
Rosenbaum, P. R. (2003) <doi:10.1198/0003130031405> Exact confidence intervals for nonconstant effects by inverting the signed rank test. American Statistician, 57(2), 132-138.
Rosenbaum, P. R. (2007) <doi:10.1111/j.1541-0420.2007.00783.x> Confidence intervals for uncommon but dramatic responses to treatment. Biometrics, 63(4), 1164-1171.
Rosenbaum, P. R. (2011) <doi:10.1111/j.1541-0420.2010.01535.x> A New U-Statistic with Superior Design Sensitivity in Matched Observational Studies. Biometrics, 67(3), 1017-1027.
Rosenbaum, P. R. (2012) <doi:10.1093/biomet/ass032> Testing one hypothesis twice in observational studies. Biometrika, 99(4), 763-774.
Rosenbaum, P. R. (2015) <doi:10.1080/01621459.2014.960968> Bahadur efficiency of sensitivity analyses in observational studies. Journal of the American Statistical Association, 110(509), 205-217.
Stephenson, W. R. (1981) <doi:10.1080/01621459.1981.10477749> A general class of one-sample nonparametric test statistics based on subsamples. Journal of the American Statistical Association, 76(376), 960-966.
An alternative family of signed ranks is calculated by the function bmhranks().
# The following example reproduces part of column 3 of Table 3 of Rosenbaum (2012). data("smokerlead") attach(smokerlead) u878<-multrnk(lead,m1=7,m2=8,m=8) u867<-multrnk(lead,m1=6,m2=7,m=8) u878l<-multrnk(llead,m1=7,m2=8,m=8) u867l<-multrnk(llead,m1=6,m2=7,m=8) H<-cbind(u878,u867,u878l,u867l) tt(lead,H,gamma=3) tt(lead,H,gamma=3.4) rm(u878,u867,u878l,u867l,H) detach(smokerlead) # # ---------------------- # The following examples are intended to aid understanding of # some of the technical details. # Exact and approximate ranks # Exact and approximate ranks are highly correlated. a<-multrnk(1:50,m1=4,m2=5,m=5,exact=FALSE) b<-multrnk(1:50,m1=4,m2=5,m=5,exact=TRUE) cor(a,b) # Compare the following with Section 3.5 in Pratt and Gibbons (1981) multrnk(1:10,exact=TRUE) # Stephenson (1981) ranks for m=5 with 10 pair differences. a<-multrnk(1:10,m1=5,m2=5,m=5,exact=TRUE) sum(a) choose(10,5) a # There are 252 ways to pick 5 differences from the 10 differences. # In 70/252 subsets of size 5, the pair with absolute rank 9 # has the largest absolute pair difference, choose(9-1,5-1) = 70, # and determines the sign.# The following example reproduces part of column 3 of Table 3 of Rosenbaum (2012). data("smokerlead") attach(smokerlead) u878<-multrnk(lead,m1=7,m2=8,m=8) u867<-multrnk(lead,m1=6,m2=7,m=8) u878l<-multrnk(llead,m1=7,m2=8,m=8) u867l<-multrnk(llead,m1=6,m2=7,m=8) H<-cbind(u878,u867,u878l,u867l) tt(lead,H,gamma=3) tt(lead,H,gamma=3.4) rm(u878,u867,u878l,u867l,H) detach(smokerlead) # # ---------------------- # The following examples are intended to aid understanding of # some of the technical details. # Exact and approximate ranks # Exact and approximate ranks are highly correlated. a<-multrnk(1:50,m1=4,m2=5,m=5,exact=FALSE) b<-multrnk(1:50,m1=4,m2=5,m=5,exact=TRUE) cor(a,b) # Compare the following with Section 3.5 in Pratt and Gibbons (1981) multrnk(1:10,exact=TRUE) # Stephenson (1981) ranks for m=5 with 10 pair differences. a<-multrnk(1:10,m1=5,m2=5,m=5,exact=TRUE) sum(a) choose(10,5) a # There are 252 ways to pick 5 differences from the 10 differences. # In 70/252 subsets of size 5, the pair with absolute rank 9 # has the largest absolute pair difference, choose(9-1,5-1) = 70, # and determines the sign.
Data from the 2007-2008 US National Health and Nutrition Examination Survey. There are 679 matched pair differences in blood lead levels in ug per dl, comparing a daily smoker and a nonsmoking control. A daily smoker smoked every day for the last 30 days and smoked at least 10 cigarettes per day. Nonsmokers smoked fewer than 100 cigarettes in their lives. The data are described further in Rosenbaum (2012).
data("smokerlead")data("smokerlead")
A data frame with 679 observations on the following 3 variables.
leadSmoker-minus-control pair differences in blood lead levels, ug per dl.
lleadSmoker-minus-control pair differences in logs of blood lead levels.
doseNumber of cigarettes smoked per day for the smoker in the matched pair.
Pairs were matched for age, gender, education, income and ethnicity.
The data are originally from the 2007-2008 US National Health and Nutrition Examination Survey, but the current example appeared in Rosenbaum (2012).
Rosenbaum, P. R. (2012) <doi:10.1093/biomet/ass032> Testing one hypothesis twice in observational studies. Biometrika, 99(4), 763-774.
US National Health and Nutrition Examination Survey.
data(smokerlead) attach(smokerlead) # Compare with Table 1 in Rosenbaum (2012). quantile(lead,c(0,1/16,1/8,1/4,1/2,3/4,7/8,15/16,1)) quantile(dose,c(0,1/16,1/8,1/4,1/2,3/4,7/8,15/16,1)) oldpar<-par(mfrow=c(1,2)) boxplot(llead,ylab="Difference in Logs of Lead Levels",xlab="Smoker-Control") abline(h=0,col="red") plot(dose,llead,ylab="Difference in Logs of Lead Levels",xlab="Cigarettes Per Day") lines(lowess(dose,llead),col="red") detach(smokerlead) par(oldpar)data(smokerlead) attach(smokerlead) # Compare with Table 1 in Rosenbaum (2012). quantile(lead,c(0,1/16,1/8,1/4,1/2,3/4,7/8,15/16,1)) quantile(dose,c(0,1/16,1/8,1/4,1/2,3/4,7/8,15/16,1)) oldpar<-par(mfrow=c(1,2)) boxplot(llead,ylab="Difference in Logs of Lead Levels",xlab="Smoker-Control") abline(h=0,col="red") plot(dose,llead,ylab="Difference in Logs of Lead Levels",xlab="Cigarettes Per Day") lines(lowess(dose,llead),col="red") detach(smokerlead) par(oldpar)
The function testtwice() is a convenient way to call the function tt(). Conversely, the function tt() is a less convenient but more flexible way to test twice. The function tt() requires you to build a matrix of signed ranks, but testtwice() builds that matrix for you.
The function tt() computes the P-value for testing twice from from a vector y of matched pair differences and a matrix H of ranks of the absolute values of y. In contrast, testwice() follows your instructions and builds H according to those instructions. Alternatively, by setting do.test=FALSE, testtwice() will assist in constructing some columns of H for use by tt(). See Details.
The default is the same as u858 = TRUE and u878 = TRUE. If you do not take the default, then you must set at least two of the statistics to TRUE; otherwise, an error will result.
testtwice(y, dose = NULL, gamma = 1, u858 = FALSE, u888 = FALSE, u878 = FALSE, u868 = FALSE, u867 = FALSE, u222 = FALSE, brown = FALSE, noether = FALSE, tailored = FALSE, alternative="greater", do.test = TRUE)testtwice(y, dose = NULL, gamma = 1, u858 = FALSE, u888 = FALSE, u878 = FALSE, u868 = FALSE, u867 = FALSE, u222 = FALSE, brown = FALSE, noether = FALSE, tailored = FALSE, alternative="greater", do.test = TRUE)
y |
A vector of matched pair differences. |
dose |
If is.null(dose), then there are no doses. Otherwise, dose is a vector with length(dose)=length(y) giving nonnegative doses of treatment for the treated individual in a matched pair, where the control received dose zero. If there are doses, the ranks are multiplied by the doses. An error will result if some doses are negative. |
gamma |
Value of the sensitivity parameter, gamma>=1, with gamma=1 for a randomization test. |
u858 |
If u858 is TRUE, one column of H is created by the function multrnk() as multrnk(y, m1 = 5, m2 = 8, m = 8). See the documentaion for multrnk and Rosenbaum (2011). |
u888 |
If u888 is TRUE, one column of H is created by the function multrnk() as multrnk(y, m1 = 8, m2 = 8, m = 8). See the documentaion for multrnk, Stephenson (1981), and Rosenbaum (2007, 2011). |
u878 |
If u878 is TRUE, one column of H is created by the function multrnk() as multrnk(y, m1 = 7, m2 = 8, m = 8). See the documentaion for multrnk and Rosenbaum (2011). |
u868 |
If u868 is TRUE, one column of H is created by the function multrnk() as multrnk(y, m1 = 6, m2 = 8, m = 8). See the documentaion for multrnk and Rosenbaum (2011). |
u867 |
If u868 is TRUE, one column of H is created by the function multrnk() as multrnk(y, m1 = 6, m2 = 7, m = 8). See the documentaion for multrnk and Rosenbaum (2011). |
u222 |
If u868 is TRUE, one column of H is created by the function multrnk() as multrnk(y, m1 = 2, m2 = 2, m = 2). See the documentaion for multrnk and Rosenbaum (2011). These ranks are nearly the same as Wilcoxon's ranks; see Stephenson (1981) and Pratt and Gibbons (1981, Section 3.5). Specifically, these ranks produce the U-statistic that is nearly identical to Wilcoxon's statistic. |
brown |
If brown is TRUE, one column of H is created by the function bmhranks() as bmhranks(y, q1=1/3, q2=2/3). This yields Brown (1981)'s test. See the documentaion for bmhranks() and Rosenbaum (2012a). These ranks are one special case of the two-step ranks proposed by Markowski and Hettmansperger (1982). |
noether |
If noether is TRUE, one column of H is created by the function bmhranks() as bmhranks(y, q1=2/3, q2=2/3). This yields one version of Noether (1973)'s test and one version of the one-step tests proposed by Markowski and Hettmansperger (1982). See the documentaion for bmhranks and and Rosenbaum (2012a). |
tailored |
If tailored is TRUE, one column of H is contains the tailored ranks in Rosenbaum (2015, Section 4.3 and Table 3). Although somewhat complex in form, these ranks have attractive design sensitivity and Bahadur efficiency compared with Noether's ranks above. |
alternative |
If alternative="greater"", then the null hypothesis of no effect is tested against an alternative of a positive effect. If alternative="less"", then the null hypothesis of no effect is tested against an alternative of a negative effect. For a two-sided test, do both tests, double the smaller of the two one-sided P-values, and replace values above 1 by 1. |
do.test |
If do.test=TRUE, then testtwice calls tt() to perform the test. If do.test=FALSE, then testtwice does not perform the test, but instead returns one or more columns of H. With do.test=FALSE, the user can build H using cbind() to combine several columns built by one or more calls to testtwice, and perhaps several other columns built by the user. |
The function testtwice() is a convenient way to call the fucntion tt() which computes the P-value for testing twice from from a vector y of matched pair differences and a matrix H of ranks of the absolute values of y. The function testtwice() can create the matrix H and call tt(). Alternatively, testtwice() can create one or more columns of H that may be combined using cbind() to create H for use with tt(). The function testtwice() automates the construction of H in some common situations, but the function tt() gives the user total control over the construction of H, albeit with greater effort. Literally, one is testing twice if H has two columns, but H can have more than two columns.
With matched pair differences in an observational study, the functions testtwice() and tt() perform several signed rank tests with different ways of scoring the absolute ranks of the differences, and corrects for multiple testing using the joint limiting Normal distribution of the tests. For gamma>1, the functions perform a sensitivity analysis, reporting an upper bound on the one-sided P-value. The method and example are from Rosenbaum (2012b).
IMPORTANT: The default is equivalent to setting u858 = TRUE and u878 = TRUE. That is, if no test statistic is selected, then u858 and u878 are selected.
Setting alternative="less" has the same effect as replacing y by -y with alternative="greater".
For a textbook discussion of adaptive inference by testing twice, see Rosenbaum (2020a, section 19.3). A different approach to adaptive inference in observational studies is discussed in Rosenbaum (2020b).
Use the function senU() in the DOS2 package if you do not wish to test twice, but do wish to do a sensitivity analysis using the U-statistic in Rosenbaum (2011), with confidence intervals and point estimates.
If do.test=TRUE, then a list containing the following items is returned.
pval |
The upper bound on the one-sided P-value from the joint test. If gamma=1, then this is the P-value, not an upper bound on the P-value. |
dev |
The standardized deviates from the joint test, one for each column of H. The test uses the largest standardized deviate, correcting for multiple testing. |
cr |
The correlation matrix of the test statistics under the null hypothesis at the given value of gamma. |
If do.test=FALSE, then a vector or matrix of signed ranks is returned for use in the function tt(). See the examples.
Various signed rank statistics have been proposed by Brown (1981), Markowski and Hettmansperger (1982), Noether (1973), Rosenbaum (2007, 2011) and Stephenson (1981). The function testtwice() uses two or more of these signed ranks to perform the test. See also the documentation for tt().
If y[i]=0, then the ith pair difference does not contribute to the permutation test.
Paul R. Rosenbaum
Berk, R. H. and Jones, D. H. (1978) <doi:10.2307/4615706> Relatively optimal combinations of test statistics. Scandinavian Journal of Statistics, 158-162.
Brown, B. M. (1981) <doi:10.1093/biomet/68.1.235> Symmetric quantile averages and related estimators. Biometrika, 68(1), 235-242.
Conover, W. J. and Salsburg, D. S. (1988) <doi:10.2307/2531906> Locally most powerful tests for detecting treatment effects when only a subset of patients can be expected to respond to treatment. Biometrics, 189-196.
Markowski, E. P. and Hettmansperger, T. P. (1982) <doi:10.2307/2287325> Inference based on simple rank step score statistics for the location model. Journal of the American Statistical Association, 77(380), 901-907.
Noether, G. E. (1973) <doi:10.2307/2284805> Some simple distribution-free confidence intervals for the center of a symmetric distribution. Journal of the American Statistical Association, 68(343), 716-719.
Pratt, J. W. and Gibbons, J. D. (1981) <doi:10.1007/978-1-4612-5931-2> Concepts of Nonparametric Theory. New York: Springer. (Section 3.5)
Rosenbaum, P. R. (1999) <doi:10.1111/1467-9876.00140> Using quantile averages in matched observational studies. Journal of the Royal Statistical Society: Series C (Applied Statistics), 48(1), 63-78.
Rosenbaum, P. R. (2007) <doi:10.1111/j.1541-0420.2007.00783.x> Confidence intervals for uncommon but dramatic responses to treatment. Biometrics, 63(4), 1164-1171.
Rosenbaum, P. R. (2011) <doi:10.1111/j.1541-0420.2010.01535.x> A new U-Statistic with superior design sensitivity in matched observational studies. Biometrics, 67(3), 1017-1027.
Rosenbaum, P. R. (2012a) <doi:10.1214/11-AOAS508> An exact adaptive test with superior design sensitivity in an observational study of treatments for ovarian cancer. The Annals of Applied Statistics, 6(1), 83-105.
Rosenbaum, P. R. (2012b) <doi:10.1093/biomet/ass032> Testing one hypothesis twice in observational studies. Biometrika, 99(4), 763-774.
Rosenbaum, P. R. (2015) <doi:10.1080/01621459.2014.960968> Bahadur efficiency of sensitivity analyses in observational studies. Journal of the American Statistical Association, 110(509), 205-217.
Rosenbaum, P. R. (2020a) <doi:10.1007/978-1-4419-1213-8> Design of Observational Studies (2nd edition). NY: Springer.
Rosenbaum, P. R. (2020b) <doi:10.1093/biomet/asaa032> A conditional test with demonstrated insensitivity to unmeasured bias in matched observational studies. Biometrika, to appear.
Stephenson, W. R. (1981) <doi:10.2307/2287596> A general class of one-sample nonparametric test statistics based on subsamples. Journal of the American Statistical Association, 76(376), 960-966.
data(smokerlead) attach(smokerlead) testtwice(lead,gamma=3) testtwice(llead,gamma=3) testtwice(llead,u858=TRUE,u888=TRUE,gamma=3) # Same calculation, done differently. H<-testtwice(llead,u858=TRUE,u888=TRUE,do.test=FALSE) dim(H) tt(llead,H,gamma=3) # The following example reproduces parts of the second # column (Brown) of Table 3 in Rosenbaum (2012). # An example in the documentation for function tt() # does the same calculation in a different way. brn<-testtwice(lead,brown=TRUE,do.test=FALSE) brnd<-testtwice(lead,dose=dose,brown=TRUE,do.test=FALSE) brnl<-testtwice(llead,brown=TRUE,do.test=FALSE) tt(lead,cbind(brn,brnd,brnl),gamma=3.2) # The following example reproduces parts of the third # column (U-statistic) of Table 3 in Rosenbaum (2012). u878<-testtwice(lead,u878=TRUE,do.test=FALSE) u867<-testtwice(lead,u867=TRUE,do.test=FALSE) u878L<-testtwice(llead,u878=TRUE,do.test=FALSE) u867L<-testtwice(llead,u867=TRUE,do.test=FALSE) tt(lead,cbind(u878,u867,u878L,u867L),gamma=3.2) tt(lead,cbind(u878,u867,u878L,u867L),gamma=3.6) # The following example compares noether=TRUE and tailored=TRUE. testtwice(llead,brown=TRUE,noether=TRUE,gamma=3.74) testtwice(llead,brown=TRUE,tailored=TRUE,gamma=3.74) rm(brn,brnd,brnl,u878,u878L,u867,u867L,H) detach(smokerlead)data(smokerlead) attach(smokerlead) testtwice(lead,gamma=3) testtwice(llead,gamma=3) testtwice(llead,u858=TRUE,u888=TRUE,gamma=3) # Same calculation, done differently. H<-testtwice(llead,u858=TRUE,u888=TRUE,do.test=FALSE) dim(H) tt(llead,H,gamma=3) # The following example reproduces parts of the second # column (Brown) of Table 3 in Rosenbaum (2012). # An example in the documentation for function tt() # does the same calculation in a different way. brn<-testtwice(lead,brown=TRUE,do.test=FALSE) brnd<-testtwice(lead,dose=dose,brown=TRUE,do.test=FALSE) brnl<-testtwice(llead,brown=TRUE,do.test=FALSE) tt(lead,cbind(brn,brnd,brnl),gamma=3.2) # The following example reproduces parts of the third # column (U-statistic) of Table 3 in Rosenbaum (2012). u878<-testtwice(lead,u878=TRUE,do.test=FALSE) u867<-testtwice(lead,u867=TRUE,do.test=FALSE) u878L<-testtwice(llead,u878=TRUE,do.test=FALSE) u867L<-testtwice(llead,u867=TRUE,do.test=FALSE) tt(lead,cbind(u878,u867,u878L,u867L),gamma=3.2) tt(lead,cbind(u878,u867,u878L,u867L),gamma=3.6) # The following example compares noether=TRUE and tailored=TRUE. testtwice(llead,brown=TRUE,noether=TRUE,gamma=3.74) testtwice(llead,brown=TRUE,tailored=TRUE,gamma=3.74) rm(brn,brnd,brnl,u878,u878L,u867,u867L,H) detach(smokerlead)
The function tt computes the P-value for testing twice from from a vector y of matched pair differences and a matrix H of ranks of the absolute values of y. Literally, one is testing twice if H has two columns, but H can have more than two columns.
The function testtwice() will create the matrix H and then call tt(); however, tt() allows you to create H according to your own specifications.
tt(y, H, gamma = 1)tt(y, H, gamma = 1)
y |
A vector of matched pair differences. |
H |
A matrix with length(y) rows giving various ways of ranking the absolute y's. Entries in H must be nonnegative. The matrix H must have at least two columns and at most 20. In a typical application, H might have 2 to 4 columns. If the columns of H have names, then these names are used to label the output; so, it is helpful if H has column names. |
gamma |
Value of the sensitivity parameter, gamma>=1, with gamma=1 for a randomization test. |
With matched pair differences in an observational study, the function tt() performs several signed rank tests with different ways of scoring the absolute ranks of the differences, and corrects for multiple testing using the joint limiting Normal distribution of the tests. For gamma>1, the function tt() performs a sensitivity analysis, reporting an upper bound on the one-sided P-value. The method and example are from Rosenbaum (2012b).
pval |
The upper bound on the one-sided P-value from the joint test. If gamma=1, then this is the P-value, not an upper bound on the P-value. |
dev |
The standardized deviates from the joint test, one for each column of H. The test uses the largest standardized deviate, correcting for multiple testing. |
cr |
The correlation matrix of the test statistics under the null hypothesis at the given value of gamma. |
Various signed rank statistics have been proposed by Brown (1981), Markowski and Hettmansperger (1982), Noether (1973), Rosenbaum (2007, 2011), and Stephenson (1981). The function bmhranks() generates ranks proposed by Brown (1981), Markowski and Hettmansperger (1982) and Noether (1973); see also Rosenbaum (1999, 2012a). The function multrnk() generates ranks proposed by Stephenson (1981), Rosenbaum (2007, 2011); see also Conover and Salsburg (1988). So, the columns of H may be built using bmhranks() and multrnk().
If y[i]=0, then the ith pair difference does not contribute to the permutation test.
The P-value is one-sided, upper-tailed. To obtain a one sided, lower-tailed P-value, replace y by -y. See the documentation for the testtwice() function.
Technical note: The function tt() calls the pmvnorm() function in the mvtnorm package. In this call, tt() sets the algorithm option to Miwa(steps=512). This choice of algorithm avoids the default algorithm in pmvnorm(), namely GenzBretz, which is a randomized algorithm, returning slightly different P-values each time it is called.
Paul R. Rosenbaum
Berk, R. H. and Jones, D. H. (1978) <doi:10.2307/4615706> Relatively optimal combinations of test statistics. Scandinavian Journal of Statistics, 158-162.
Brown, B. M. (1981) <doi:10.1093/biomet/68.1.235> Symmetric quantile averages and related estimators. Biometrika, 68(1), 235-242.
Conover, W. J. and Salsburg, D. S. (1988) <doi:10.2307/2531906> Locally most powerful tests for detecting treatment effects when only a subset of patients can be expected to respond to treatment. Biometrics, 189-196.
Markowski, E. P. and Hettmansperger, T. P. (1982) <doi:10.2307/2287325> Inference based on simple rank step score statistics for the location model. Journal of the American Statistical Association, 77(380), 901-907.
Noether, G. E. (1973) <doi:10.2307/2284805> Some simple distribution-free confidence intervals for the center of a symmetric distribution. Journal of the American Statistical Association, 68(343), 716-719.
Rosenbaum, P. R. (1999) <doi:10.1111/1467-9876.00140> Using quantile averages in matched observational studies. Journal of the Royal Statistical Society: Series C (Applied Statistics), 48(1), 63-78.
Rosenbaum, P. R. (2007) <doi:10.1111/j.1541-0420.2007.00783.x> Confidence intervals for uncommon but dramatic responses to treatment. Biometrics, 63(4), 1164-1171.
Rosenbaum, P. R. (2011) <doi:10.1111/j.1541-0420.2010.01535.x> A new U-statistic with superior design sensitivity in matched observational studies. Biometrics, 67(3), 1017-1027.m
Rosenbaum, P. R. (2012a) <doi:10.1214/11-AOAS508> An exact adaptive test with superior design sensitivity in an observational study of treatments for ovarian cancer. The Annals of Applied Statistics, 6(1), 83-105.
Rosenbaum, P. R. (2012b) <doi:10.1093/biomet/ass032> Testing one hypothesis twice in observational studies. Biometrika, 99(4), 763-774.
Stephenson, W. R. (1981) <doi:10.2307/2287596> A general class of one-sample nonparametric test statistics based on subsamples. Journal of the American Statistical Association, 76(376), 960-966.
data(smokerlead) attach(smokerlead) # The following example reproduces parts of the first column of Table 3 in Rosenbaum (2012). w<-rank(abs(lead)) wd<-rank(abs(lead))*dose wl<-rank(abs(llead)) H<-cbind(w,wd,wl) tt(lead,H,gamma=2.8) tt(lead,H,gamma=3.2) # The following example reproduces parts of the second column of Table 3 in Rosenbaum (2012). brn<-bmhranks(lead) brnd<-brn*dose brnl<-bmhranks(llead) H2<-cbind(brn,brnd,brnl) tt(lead,H2,gamma=3.2) tt(lead,H2,gamma=3.4) rm(w,wd,wl,H,brn,brnd,brnl,H2) detach(smokerlead)data(smokerlead) attach(smokerlead) # The following example reproduces parts of the first column of Table 3 in Rosenbaum (2012). w<-rank(abs(lead)) wd<-rank(abs(lead))*dose wl<-rank(abs(llead)) H<-cbind(w,wd,wl) tt(lead,H,gamma=2.8) tt(lead,H,gamma=3.2) # The following example reproduces parts of the second column of Table 3 in Rosenbaum (2012). brn<-bmhranks(lead) brnd<-brn*dose brnl<-bmhranks(llead) H2<-cbind(brn,brnd,brnl) tt(lead,H2,gamma=3.2) tt(lead,H2,gamma=3.4) rm(w,wd,wl,H,brn,brnd,brnl,H2) detach(smokerlead)