summary.scaleboot {scaleboot}

P-value Calculation for Multiscale Bootstrap

Description

summary method for class "scaleboot" and "scalebootv".

Usage

## S3 method for class 'scaleboot'
summary(object,models=names(object$fi),k=3,sk=k,s=1,sp=-1,
              hypothesis=c("auto","null","alternative"),
              type=c("Frequentist","Bayesian"),...)

## S3 method for class 'scalebootv'
summary(object,models=attr(object,"models"),k=3,sk=k,
              hypothesis="auto",type="Frequentist", select="average",...)

## S3 method for class 'summary.scaleboot'
print(x,sort.by=c("aic","none"),verbose=FALSE,...)

## S3 method for class 'summary.scalebootv'
print(x,...)

Arguments

object

an object used to select a method.

models

character vector of model names. If numeric, names(object$fi)[models] is used for each "scaleboot" object.

k

numeric vector of k for calculating p-values.

sk

numeric vector of k for calculating selective inference p-values.

s

\sigma_0^2

sp

\sigma_p^2

hypothesis

specifies type of selective infernece. "null" takes the region as null hypothesis, and "alternative" takes the region as alternative hypothesis. "auto" determins it by the sign of beta0. The selectice pvalues (sk.1, sk.2, ...) are selective pvalues when "null", and they are one minus selective pvalues when "alternative".

type

If numeric, it is passed to sbpsi functions as lambda to specify p-value type. If "Frequentist" or "Bayesian", then equivalent to specifying lambda = 1 or 0, respectively.

select

character of model name (such as "poly.3") or one of "average" and "best". If "average" or "best", then the averaging by Akaike weights or the best model is used, respectively.

x

object.

sort.by

sort key.

verbose

logical.

...

further arguments passed to and from other methods.

Details

For each model, a class of approximately unbiased p-values, indexed by k=1,2,..., is calculaed. The p-values are named k.1, k.2, ..., where k=1 (k.1) corresponds to the ordinary bootstrap probability, and k=2 (k.2) corresponds to the third-order accurate p-value of Shimodaira (2002). As the k value increases, the bias of testing decreases, although the p-value becomes less stable numerically and the monotonicity of rejection regions becomes worse. Typically, k=3 provides a reasonable compromise. The sbpval method is available to extract p-values from the "summary.scaleboot" object.

The p-value is defined as

p_k = 1 - \Phi\left( \sum_{j=0}^{k-1} \frac{(\sigma_p^2-\sigma_0^2)^j}{j!} \frac{d^j \psi(x|\beta)}{d x^j}\Bigr|_{\sigma_0^2} \right),

where \psi(\sigma^2|\beta) is the model specification function, \sigma_0^2 is the evaluation point for the Taylor series, and \sigma_p^2 is an additional parameter. Typically, we do not change the default values \sigma_0^2=1 and \sigma_p^2=-1.

The p-values are justified only for good fitting models. By default, the model which minimizes the AIC value is selected. We can modify the AIC value by using the sbaic function. We also diagnose the fitting by using the plot method.

Now includes selective inference p-values. The method is described in Terada and Shimodaira (2017; arXiv:1711.00949) "Selective inference for the problem of regions via multiscale bootstrap".

Value

summary.scaleboot returns an object of the class "summary.scaleboot", which is inherited from the class "scaleboot". It is a list containing all the components of class "scaleboot" and the following components:

pv

matrix of p-values of size length(models) * length(k) with elements p_k.

pe

matrix of standard errors of p-values.

spv

matrix of selective inference p-values of size length(models) * length(sk) with elements sp_k.

spe

matrix of standard errors of selective inference p-values.

betapar

list array containing (beta0, beta1) and its covariance matrix for each model. They are obtained by linear extrapolation. This will be used for interpreting the fitting in terms of signed distance and curvature.

best

a list consisting of components model for the best fitting model name, aic for its AIC value, pv and spv for vector of p-values, and pe and spe for vectors of standard errors. Also includes betapar for the best model.

average

a list of results for the average model computed by Akaike weight.

parex

a list of components k, s, and sp.

Author(s)

Hidetoshi Shimodaira

See Also

sbfit, sbpsi, sbpval, sbaic.

Examples

data(mam15)
## For a single hypothesis
a <- mam15.relltest[["t4"]] # an object of class "scaleboot"
summary(a) # calculate and print p-values (k=3)
summary(a,k=2) # calculate and print p-values (k=2)
summary(a,k=1:4) # up to "k.4" p-value.

## For multiple hypotheses
b <- mam15.relltest[1:15] # an object of class "scalebootv"
summary(b) # calculate and print p-values (k=3)
summary(b,k=1:4) # up to "k.4" p-value.