twosidedpval: Computation of Conditional Two-Sided p-Values
In skedastic: Handling Heteroskedasticity in the Linear Regression Model

twosidedpval

R Documentation

Computation of Conditional Two-Sided p-Values

Description

Computes the conditional p-value P_C for a continuous or discrete test statistic, as defined in \insertCiteKulinskaya08;textualskedastic. This provides a method for computing a two-sided p-value from an asymmetric null distribution.

Usage

twosidedpval(
  q,
  CDF,
  continuous,
  method = c("doubled", "kulinskaya", "minlikelihood"),
  locpar,
  supportlim = c(-Inf, Inf),
  ...
)

Arguments

`q`	A double representing the quantile, i.e. the observed value of the test statistic for which a two-sided p-value is to be computed
`CDF`	A function representing the cumulative distribution function of the test statistic under the null hypothesis, i.e. \Pr(T≤ q\|\mathrm{H}_0).
`continuous`	A logical indicating whether the test statistic is a continuous (`TRUE`) or discrete (`FALSE`) random variable. Defaults to `TRUE`.
`method`	A character specifying the method to use to calculate two-sided p-value; one of `"doubled"` (representing doubling of the one-sided p-value), `"kulinskaya"` (representing the method of \insertCiteKulinskaya08;textualskedastic), or `"minlikelihood"` (representing the sum of probabilities for values with probability less than or equal to that of the observed value. Partial matching is used. Note that the `"minlikelihood"` method is available only for discrete distributions.
`locpar`	a double representing a generic location parameter chosen to separate the tails of the distribution. Note that if `locpar` corresponds to the median of `CDF`, there is no difference between the two methods, in the continuous case. If `locpar` is not specified, the function attempts to compute the expectation of `CDF` using numerical integration and, if successful, uses this as `locpar`. However, this may yield unexpected results, especially if `CDF` is not one of the cumulative distribution functions of well-known distributions included in the `stats` package.
`supportlim`	A numeric vector of `length` 2, giving the minimum and maximum values in the support of the distribution whose cumulative distribution function is `CDF`. This argument is only used if the distribution is discrete (i.e. if `continuous` is `FALSE`) and if `method` is `"minlikelihood"` or `locpar` is not specified. If `supportlim` is not supplied, the function assumes that the support is `-1e6:1e6`. Values of `-Inf` and `Inf` may be supplied, but if so, the support is truncated at `-1e6` and `1e6` respectively.
`...`	Optional arguments to pass to `CDF`.

Details

Let T be a statistic that, under the null hypothesis, has cumulative distribution function F and probability density or mass function f. Denote by A a generic location parameter chosen to separate the two tails of the distribution. Particular examples include the mean E(T|\mathrm{H}_0), the mode \arg \sup_{t} f(t), or the median F^{-1}≤ft(\frac{1}{2}\right). Let q be the observed value of T.

In the continuous case, the conditional two-sided p-value centered at A is defined as

P_C^A(q)=\frac{F(q)}{F(A)}1_{q ≤ A} + \frac{1-F(q)}{1-F(A)}1_{q > A}

where 1_{\cdot} is the indicator function. In the discrete case, P_C^A depends on whether A is an attainable value within the support of T. If A is not attainable, the conditional two-sided p-value centred at A is defined as

P_C^{A}(q)=\frac{\Pr(T≤ q)}{\Pr(T<A)}1_{q<A} + \frac{\Pr(T≥ q)}{\Pr(T>A)}1_{q>A}

If A is attainable, the conditional two-sided p-value centred at A is defined as

P_C^{A}(q)=\frac{\Pr(T≤ q)}{\Pr(T≤ A)/≤ft(1+\Pr(T=A)\right)} 1_{q<A} + 1_{q=A}+\frac{\Pr(T≥ q)}{\Pr(T ≥ A)/≤ft(1+\Pr(T=A)\right)} 1_{q>A}

Value

A double.

References

\insertAllCited

Examples

# Computation of two-sided p-value for F test for equality of variances
n1 <- 10
n2 <- 20
set.seed(1234)
x1 <- stats::rnorm(n1, mean = 0, sd = 1)
x2 <- stats::rnorm(n2, mean = 0, sd = 3)
# 'Conventional' two-sided p-value obtained by doubling one-sided p-value:
stats::var.test(x1, x2, alternative = "two.sided")$p.value
# This is replicated in `twosidedpval` by setting `method` argument to `"doubled"`
twosidedpval(q = var(x1) / var(x2), CDF = stats::pf, continuous = TRUE,
 method = "doubled", locpar = 1, df1 = n1 - 1, df2 = n2 - 1)
# Conditional two-sided p-value centered at df (mean of chi-squared r.v.):
twosidedpval(q = var(x1) / var(x2), CDF = stats::pf, continuous = TRUE,
 method = "kulinskaya", locpar = 1, df1 = n1 - 1, df2 = n2 - 1)

skedastic documentation built on Nov. 10, 2022, 5:43 p.m.