sbpsi() R function from [scaleboot]

Model Specification Functions

sbpsi.poly and sbpsi.sing are $\psi$ functions to specify a polynomial model and a singular model, respectively.


sbpsi.poly(beta,s=1,k=1,sp=-1,lambda=NULL,aux=NULL,check=FALSE)

sbpsi.sing(beta,s=1,k=1,sp=-1,lambda=NULL,aux=NULL,check=FALSE)

sbpsi.sphe(beta,s=1,k=1,sp=-1,lambda=NULL,aux=NULL,check=FALSE)

sbpsi.generic(beta,s=1,k=1,sp=-1,lambda=NULL,aux=NULL,check=FALSE,zfun,eps=0.01)

sbmodelnames(m=1:3,one.sided=TRUE,two.sided=FALSE,rev.sided=FALSE,
  poly,sing,poa,pob,poc,pod,sia,sib,sic,sid,sphe,pom,sim)

Arguments

beta: numeric vector of parameters; $\beta_0$ =beta[1], $\beta_1$ =beta[2],... $\beta_{m-1}$ =beta[m], where $m$ is the number of parameters.
s: $\sigma_0^2$ .
k: numeric to specify the order of derivatives.
sp: $\sigma_p^2$ .
lambda: a numeric of specifying the type of p-values; Bayesian (lambda=0) Frequentist (lambda=1).
aux: auxiliary parameter. Currently not used.
check: logical for boundary check.
zfun: z-value function with (s,beta) as parameters.
eps: delta for numerical computation of derivatives.
m: numeric vector to specify the numbers of parameters.
one.sided: logical to include poly and sing models.
two.sided: logical to include poa and sia models.
rev.sided: logical to include pob and sib models.
poly: maximum number of parameters in poly models.
sing: maximum number of parameters in sing models.
sphe: maximum number of parameters in sphe models.
poa: maximum number of parameters in poa models.
pob: maximum number of parameters in pob models.
poc: maximum number of parameters in poc models.
pod: maximum number of parameters in pod models.
sia: maximum number of parameters in sia models.
sib: maximum number of parameters in sib models.
sic: maximum number of parameters in sic models.
sid: maximum number of parameters in sid models.
pom: maximum number of parameters in pom models.
sim: maximum number of parameters in sim models.

Details

For $k=1$ , the sbpsi functions return their $\psi$ function values at $\sigma^2=\sigma_0^2$ . Currently, four types of sbpsi functions are implemented. sbpsi.poly defines the polynomial model;

\psi(\sigma^2 | \beta) =\sum_{j=0}^{m-1} \beta_j \sigma^{2j}

for $m\ge1$ . sbpsi.sing defines the singular model;

\psi(\sigma^2 | \beta) = \beta_0 +\sum_{j=1}^{m-2} \frac{\beta_j \sigma^{2j}}{1 + \beta_{m-1}(\sigma-1)}

for $m\ge3$ and $0\le\beta_{m-1}\le1$ . sbpsi.sphe defines the spherical model; currently the number of parameters must be $m=3$ . sbpsi.generic is a generic sbpsi function for specified zfun.

For $k>1$ , the sbpsi functions return values extrapolated at $\sigma^2=\sigma_p^2$ using derivatives up to order $k-1$

evaluated at $\sigma^2=\sigma_0^2$ ;

q_k = \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},

which reduces to $\psi(\sigma_0^2|\beta)$ for $k=1$ . In the summary.scaleboot, the AU p-values are defined by $p_k = 1-\Phi(q_k)$ for $k\ge1$ .

Returns

sbpsi.poly and sbpsi.sing are examples of a sbpsi function; users can develop their own sbpsi functions for better model fitting by preparing sbpsi.foo and sbini.foo

functions for model foo. If check=FALSE, a sbpsi function returns the $\psi$ function value or the extrapolation value. If check=TRUE, a sbpsi function returns NULL when all the elements of beta are included in the their valid intervals. Otherwise, a sbpsi function returns a list with components beta for the parameter value being modified to be on a boundary of the interval and mask, a logical vector indicating which elements are not on the boundary.

sbmodelnames returns a character vector of model names.

Author(s)

Hidetoshi Shimodaira

sbpsi function

Model Specification Functions

Arguments

Details

Returns

Author(s)

See Also