pvaluemethods  R Documentation 
p
Value, Midp
Value, p
Value
Interval and Test SizeMethods for computation of the p
value, midp
value, p
value
interval and test size.
## S4 method for signature 'PValue'
pvalue(object, q, ...)
## S4 method for signature 'NullDistribution'
pvalue(object, q, ...)
## S4 method for signature 'ApproxNullDistribution'
pvalue(object, q, ...)
## S4 method for signature 'IndependenceTest'
pvalue(object, ...)
## S4 method for signature 'MaxTypeIndependenceTest'
pvalue(object, method = c("global", "singlestep",
"stepdown", "unadjusted"),
distribution = c("joint", "marginal"),
type = c("Bonferroni", "Sidak"), ...)
## S4 method for signature 'NullDistribution'
midpvalue(object, q, ...)
## S4 method for signature 'ApproxNullDistribution'
midpvalue(object, q, ...)
## S4 method for signature 'IndependenceTest'
midpvalue(object, ...)
## S4 method for signature 'NullDistribution'
pvalue_interval(object, q, ...)
## S4 method for signature 'IndependenceTest'
pvalue_interval(object, ...)
## S4 method for signature 'NullDistribution'
size(object, alpha, type = c("pvalue", "midpvalue"), ...)
## S4 method for signature 'IndependenceTest'
size(object, alpha, type = c("pvalue", "midpvalue"), ...)
object 
an object from which the 
q 
a numeric, the quantile for which the 
method 
a character, the method used for the 
distribution 
a character, the distribution used for the computation of adjusted

type 

alpha 
a numeric, the nominal significance level 
... 
further arguments (currently ignored). 
The methods pvalue
, midpvalue
, pvalue_interval
and
size
compute the p
value, midp
value, p
value
interval and test size, respectively.
For pvalue()
, the global p
value (method = "global"
) is
returned by default and is given with an associated 99% confidence interval
when resampling is used to determine the null distribution (which for maximum
statistics may be true even in the asymptotic case).
The familywise error rate (FWER) is always controlled under the global null
hypothesis, i.e., in the weak sense, implying that the smallest
adjusted p
value is valid without further assumptions. Control of the
FWER under any partial configuration of the null hypotheses, i.e., in the
strong sense, as is typically desired for multiple tests and
comparisons, requires that the subset pivotality condition holds
(Westfall and Young, 1993, pp. 42–43; Bretz, Hothorn and Westfall, 2011,
pp. 136–137). In addition, for methods based on the joint distribution of
the test statistics, failure of the joint exchangeability assumption
(Westfall and Troendle, 2008; Bretz, Hothorn and Westfall, 2011, pp. 129–130)
may cause excess Type I errors.
Assuming subset pivotality, singlestep or free stepdown
adjusted p
values using maxT
procedures are obtained by setting
method
to "singlestep"
or "stepdown"
, respectively. In
both cases, the distribution
argument specifies whether the adjustment
is based on the joint distribution ("joint"
) or the marginal
distributions ("marginal"
) of the test statistics. For procedures
based on the marginal distributions, Bonferroni or Šidáktype
adjustment can be specified through the type
argument by setting it to
"Bonferroni"
or "Sidak"
, respectively.
The p
value adjustment procedures based on the joint distribution of the
test statistics fully utilizes distributional characteristics, such as
discreteness and dependence structure, whereas procedures using the marginal
distributions only incorporate discreteness. Hence, the joint
distributionbased procedures are typically more powerful. Details regarding
the singlestep and free stepdown procedures based on the joint
distribution can be found in Westfall and Young (1993); in particular, this
implementation uses Equation 2.8 with Algorithm 2.5 and 2.8, respectively.
Westfall and Wolfinger (1997) provide details of the marginal
distributionsbased singlestep and free stepdown procedures. The
generalization of Westfall and Wolfinger (1997) to arbitrary test statistics,
as implemented here, is given by Westfall and Troendle (2008).
Unadjusted p
values are obtained using method = "unadjusted"
.
For midpvalue()
, the global midp
value is given with an
associated 99% midp
confidence interval when resampling is used to
determine the null distribution. The twosided midp
value is computed
according to the minimum likelihood method (Hirji et al., 1991).
The p
value interval (p_0, p_1]
obtained by
pvalue_interval()
was proposed by Berger (2000, 2001), where the upper
endpoint p_1
is the conventional p
value and the midpoint, i.e.,
p_{0.5}
, is the midp
value. The lower endpoint p_0
is the smallest p
value attainable if no conservatism attributable to
the discreteness of the null distribution is present. The length of the
p
value interval is the null probability of the observed outcome and
provides a datadependent measure of conservatism that is completely
independent of the nominal significance level.
For size()
, the test size, i.e., the actual significance level, at the
nominal significance level \alpha
is computed using either the rejection
region corresponding to the p
value (type = "pvalue"
, default)
or the midp
value (type = "midpvalue"
). The test size is, in
contrast to the p
value interval, a dataindependent measure of
conservatism that depends on the nominal significance level. A test size
smaller or larger than the nominal significance level indicates that the test
procedure is conservative or anticonservative, respectively, at that
particular nominal significance level. However, as pointed out by Berger
(2001), even when the actual and nominal significance levels are identical,
conservatism may still affect the p
value.
The p
value, midp
value, p
value interval or test size
computed from object
. A numeric vector or matrix.
The midp
value, p
value interval and test size of asymptotic
permutation distributions or exact permutation distributions obtained by the
splitup algorithm is reported as NA
.
In versions of coin prior to 1.10, a minP
procedure computing
Šidák singlestep adjusted p
values accounting for
discreteness was available when specifying method = "discrete"
. This
was made defunct in version 1.20 due to the introduction of a more
general maxT
version of the same algorithm.
Berger, V. W. (2000). Pros and cons of permutation tests in clinical trials. Statistics in Medicine 19(10), 1319–1328. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/(SICI)10970258(20000530)19:10<1319::AIDSIM490>3.0.CO;20")}
Berger, V. W. (2001). The p
value interval as an inferential tool.
The Statistician 50(1), 79–85. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1111/14679884.00262")}
Bretz, F., Hothorn, T. and Westfall, P. (2011). Multiple Comparisons Using R. Boca Raton: CRC Press.
Hirji, K. F., Tan, S.J. and Elashoff, R. M. (1991). A quasiexact test for comparing two binomial proportions. Statistics in Medicine 10(7), 1137–1153. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/sim.4780100713")}
Westfall, P. H. and Troendle, J. F. (2008). Multiple testing with minimal assumptions. Biometrical Journal 50(5), 745–755. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1002/bimj.200710456")}
Westfall, P. H. and Wolfinger, R. D. (1997). Multiple tests with discrete distributions. The American Statistician 51(1), 3–8. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00031305.1997.10473577")}
Westfall, P. H. and Young, S. S. (1993). ResamplingBased Multiple
Testing: Examples and Methods for p
Value Adjustment. New York: John
Wiley & Sons.
## Twosample problem
dta < data.frame(
y = rnorm(20),
x = gl(2, 10)
)
## Exact AnsariBradley test
(at < ansari_test(y ~ x, data = dta, distribution = "exact"))
pvalue(at)
midpvalue(at)
pvalue_interval(at)
size(at, alpha = 0.05)
size(at, alpha = 0.05, type = "midpvalue")
## Bivariate twosample problem
dta2 < data.frame(
y1 = rnorm(20) + rep(0:1, each = 10),
y2 = rnorm(20),
x = gl(2, 10)
)
## Approximative (Monte Carlo) bivariate FisherPitman test
(it < independence_test(y1 + y2 ~ x, data = dta2,
distribution = approximate(nresample = 10000)))
## Global pvalue
pvalue(it)
## Joint distribution singlestep pvalues
pvalue(it, method = "singlestep")
## Joint distribution stepdown pvalues
pvalue(it, method = "stepdown")
## Sidak stepdown pvalues
pvalue(it, method = "stepdown", distribution = "marginal", type = "Sidak")
## Unadjusted pvalues
pvalue(it, method = "unadjusted")
## Length of YOY Gizzard Shad (Hollander and Wolfe, 1999, p. 200, Tab. 6.3)
yoy < data.frame(
length = c(46, 28, 46, 37, 32, 41, 42, 45, 38, 44,
42, 60, 32, 42, 45, 58, 27, 51, 42, 52,
38, 33, 26, 25, 28, 28, 26, 27, 27, 27,
31, 30, 27, 29, 30, 25, 25, 24, 27, 30),
site = gl(4, 10, labels = as.roman(1:4))
)
## Approximative (Monte Carlo) FisherPitman test with contrasts
## Note: all pairwise comparisons
(it < independence_test(length ~ site, data = yoy,
distribution = approximate(nresample = 10000),
xtrafo = mcp_trafo(site = "Tukey")))
## Joint distribution stepdown pvalues
pvalue(it, method = "stepdown") # subset pivotality is violated
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