Computation of the p-Value, Mid-p-Value and p-Value Interval
Methods for computation of the p-value, mid-p-value and p-value interval.
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## S4 method for signature 'IndependenceTest' pvalue(object, ...) ## S4 method for signature 'MaxTypeIndependenceTest' pvalue(object, method = c("global", "single-step", "step-down", "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, ...)
an object from which the p-value, mid-p-value or p-value interval can be computed.
a character, the method used for the p-value computation: either
a character, the distribution used for the computation of adjusted
a character, the type of p-value adjustment when the marginal
distributions are used: either
a numeric, the quantile for which the p-value, mid-p-value or p-value interval is computed.
further arguments (currently ignored).
the p-value, mid-p-value and p-value interval respectively.
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, single-step or free step-down
adjusted p-values using max-T procedures are obtained by setting
"step-down" respectively. In
both cases, the
distribution argument specifies whether the adjustment
is based on the joint distribution (
"joint") or the marginal
"marginal") of the test statistics. For procedures
based on the marginal distributions, Bonferroni- or Šidák-type
adjustment can be specified through the
type argument by setting it to
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 distribution-based procedures are typically more powerful. Details regarding the single-step and free step-down 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 distributions-based single-step and free step-down 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".
midpvalue, the global mid-p-value is given with an associated
99 % mid-p confidence interval when resampling is used to determine the
null distribution. The two-sided mid-p-value is computed according to
the minimum likelihood method (Hirji et al., 1991).
The p-value interval (p_0, p_1] obtained by
was proposed by Berger (2000, 2001), where the upper endpoint p_1 is the
conventional p-value and the mid-point, i.e., p_0.5, is
the mid-p-value. The lower endpoint p_0 is the smallest
p-value obtainable 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 data-dependent
measure of conservatism that is completely independent of the significance
The p-value, mid-p-value or p-value interval computed from
object. A numeric vector or matrix.
The mid-p-value and p-value interval of asymptotic permutation
distributions for maximum-type tests or exact permutation distributions
obtained by the split-up algoritm is reported as
In versions of coin prior to 1.1-0, a min-P procedure computing
Šidák single-step adjusted p-values 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 max-T 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.
Berger, V. W. (2001). The p-value 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 quasi-exact 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). Resampling-Based Multiple Testing: Examples and Methods for p-Value Adjustment. New York: John Wiley & Sons.
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## Two-sample problem dta <- data.frame( y = rnorm(20), x = gl(2, 10) ) ## Exact Ansari-Bradley test (at <- ansari_test(y ~ x, data = dta, distribution = "exact")) pvalue(at) midpvalue(at) pvalue_interval(at) ## Bivariate two-sample problem dta2 <- data.frame( y1 = rnorm(20) + rep(0:1, each = 10), y2 = rnorm(20), x = gl(2, 10) ) ## Approximative (Monte Carlo) bivariate Fisher-Pitman test (it <- independence_test(y1 + y2 ~ x, data = dta2, distribution = approximate(B = 10000))) ## Global p-value pvalue(it) ## Joint distribution single-step p-values pvalue(it, method = "single-step") ## Joint distribution step-down p-values pvalue(it, method = "step-down") ## Sidak step-down p-values pvalue(it, method = "step-down", distribution = "marginal", type = "Sidak") ## Unadjusted p-values 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) Fisher-Pitman 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 step-down p-values pvalue(it, method = "step-down") # subset pivotality is violated
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