summary method for class "scaleboot" and "scalebootv".
# S3 method for scaleboot
summary(object,models=names(object$fi),k=3,sk=k,s=1,sp=-1,
hypothesis=c("auto","null","alternative"),
type=c("Frequentist","Bayesian"),...)# S3 method for scalebootv
summary(object,models=attr(object,"models"),k=3,sk=k,
hypothesis="auto",type="Frequentist", select="average",...)
# S3 method for summary.scaleboot
print(x,sort.by=c("aic","none"),verbose=FALSE,...)
# S3 method for summary.scalebootv
print(x,...)
an object used to select a method.
character vector of model names. If numeric,
names(object$fi)[models] is used for each "scaleboot" object.
numeric vector of \(k\) for calculating p-values.
numeric vector of \(k\) for calculating selective inference p-values.
\(\sigma_0^2\)
\(\sigma_p^2\)
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".
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.
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.
object.
sort key.
logical.
further arguments passed to and from other methods.
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:
matrix of p-values of size length(models) *
length(k) with elements \(p_k\).
matrix of standard errors of p-values.
matrix of selective inference p-values of size length(models) *
length(sk) with elements \(sp_k\).
matrix of standard errors of selective inference p-values.
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.
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.
a list of results for the average model computed by Akaike weight.
a list of components k, s, and sp.
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".
# NOT RUN {
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.
# }
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