Computation of the pValue, MidpValue and pValue Interval
Description
Methods for computation of the pvalue, midpvalue and pvalue interval.
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18  ## 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'
pvalue(object, q, ...)
## S4 method for signature 'IndependenceTest'
midpvalue(object, ...)
## S4 method for signature 'NullDistribution'
midpvalue(object, q, ...)
## S4 method for signature 'IndependenceTest'
pvalue_interval(object, ...)
## S4 method for signature 'NullDistribution'
pvalue_interval(object, q, ...)

Arguments
object 
an object from which the pvalue, midpvalue or pvalue interval can be computed. 
method 
a character, the method used for the pvalue computation: either

distribution 
a character, the distribution used for the computation of adjusted
pvalues: either 
type 
a character, the type of pvalue adjustment when the marginal
distributions are used: either 
q 
a numeric, the quantile for which the pvalue, midpvalue or pvalue interval is computed. 
... 
further arguments (currently ignored). 
Details
The methods pvalue
, midpvalue
and pvalue_interval
compute
the pvalue, midpvalue and pvalue interval respectively.
For pvalue
, the global pvalue (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 pvalue 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 pvalues 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 pvalue 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 pvalues are obtained using method = "unadjusted"
.
For midpvalue
, the global midpvalue is given with an associated
99 % midp confidence interval when resampling is used to determine the
null distribution. The twosided midpvalue is computed according to
the minimum likelihood method (Hirji et al., 1991).
The pvalue interval (p_0, p_1] obtained by pvalue_interval
was proposed by Berger (2000, 2001), where the upper endpoint p_1 is the
conventional pvalue and the midpoint, i.e., p_0.5, is
the midpvalue. The lower endpoint p_0 is the smallest
pvalue obtainable if no conservatism attributable to the discreteness
of the null distribution is present. The length of the pvalue interval
is the null probability of the observed outcome and provides a datadependent
measure of conservatism that is completely independent of the significance
level.
Value
The pvalue, midpvalue or pvalue interval computed from
object
. A numeric vector or matrix.
Note
The midpvalue and pvalue interval of asymptotic permutation
distributions for maximumtype tests or exact permutation distributions
obtained by the splitup algoritm is reported as NA
.
In versions of coin prior to 1.10, a minP procedure computing
Šidák singlestep adjusted pvalues accounting for
discreteness was available when specifying method = "discrete"
.
This is now deprecated and will be removed in a future release due to
the introduction of a more general maxT version of the same algorithm.
References
Berger, V. W. (2000). Pros and cons of permutation tests in clinical trials. Statistics in Medicine 19(10), 1319–1328.
Berger, V. W. (2001). The pvalue interval as an inferential tool. The Statistician 50(1), 79–85.
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.
Westfall, P. H. and Troendle, J. F. (2008). Multiple testing with minimal assumptions. Biometrical Journal 50(5), 745–755.
Westfall, P. H. and Wolfinger, R. D. (1997). Multiple tests with discrete distributions. The American Statistician 51(1), 3–8.
Westfall, P. H. and Young, S. S. (1993). ResamplingBased Multiple Testing: Examples and Methods for pValue Adjustment. New York: John Wiley & Sons.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57  ## 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)
## 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(B = 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(B = 10000),
xtrafo = mcp_trafo(site = "Tukey")))
## Joint distribution stepdown pvalues
pvalue(it, method = "stepdown") # subset pivotality is violated
