Computation of the p-Value, Mid-p-Value and p-Value Interval

Description

Methods for computation of the p-value, mid-p-value and p-value interval.

Usage

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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, ...)

Arguments

object

an object from which the p-value, mid-p-value or p-value interval can be computed.

method

a character, the method used for the p-value computation: either "global" (default), "single-step", "step-down" or "unadjusted".

distribution

a character, the distribution used for the computation of adjusted p-values: either "joint" (default) or "marginal".

type

a character, the type of p-value adjustment when the marginal distributions are used: either "Bonferroni" (default) or "Sidak".

q

a numeric, the quantile for which the p-value, mid-p-value or p-value interval is computed.

...

further arguments (currently ignored).

Details

The methods pvalue, midpvalue and pvalue_interval compute the p-value, mid-p-value and p-value interval 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, single-step or free step-down adjusted p-values using max-T procedures are obtained by setting method to "single-step" or "step-down" 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 Šidák-type 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 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".

For 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 pvalue_interval 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 level.

Value

The p-value, mid-p-value or p-value interval computed from object. A numeric vector or matrix.

Note

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 NA.

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.

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 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.

Examples

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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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