Utility functions for Integrated Regression Goodness of Fit

Description

Core functions for the computation of the Integrated Regression Goodness of Fit

Usage

compIntRegProc(y, xord, weig = rep(1, length(y)))
compBootSamp(obj, datLT, B = 999, LINMOD = FALSE)
plotIntRegProc(y, x, weig = rep(1, length(y)), ADD = FALSE, ...)
getModelFrame(obj)
getResiduals(obj,type)

Arguments

Details

...TODO: Each of them computes what in which way

Note

Surely they can better implemented.

Author(s)

Jorge Luis Ojeda Cabrera (jojeda@unizar.es).

Utility functions for Integrated Regression Goodness of Fit

Description

Functions that are basic or/and useful for the computation of the Integrated Regression Goodness of Fit

Usage

  getLessThan(x, d)
  mvCumSum(x, ord)
  mvPartOrd(x1, x2)
  getContVar(df, vars = NULL)
  getModelCovars(obj)
  getModelWeights(obj)
  rWildBoot(n)

Arguments

Details

...TODO: Each of them computes what in which way

Note

getLessThan can be ceitainly better implemented.

Author(s)

Jorge Luis Ojeda Cabrera (jojeda@unizar.es).

Integrated Regression Goodness of Fit

Description

Integrated Regression Goodness of Fit to test the adequacy of different model to represent the regression function for a given data.

Usage

  anovarIntReg(objH0, ..., covars = NULL, B = 499, 
    LINMOD = FALSE, INCREMENTAL = FALSE)
  ## S3 method for class 'anovarIntReg'
print(x,...)

Arguments

.

Details

This function implements the test

H_0:m\in M_0 \ \textrm{vs} \ H_1:m\in M_1

for two different models M_0, M_1 using the Integrated Regression Goodness of Fit as os done in intRegGOF, but instead of the accumulation of the residual of a givem model, in this case, the accumuation of the difference in the fits is considered:

R^w_n(x)=n^{-1/2}\sum^n_{i=1}(\hat y_{0i}-\hat y_{1i})I(x_i\le x).

The test statistics considered are $K_n$ and $W^2_n$.

If objH0 and objH1 are lm, glm or nls fits for the models in classes M_0 and M_1 respectively, then anovarIntReg(objH0,objH1) computes test H_0:m\in M_0 vs H_1:m\notin M_1. When anovarIntReg(objH0,objH1,...,objHk) is executed (notice that by default INCREMENTAL=FALSE) we obtain a table with the statistics K_n and W^2_n and its associated p-values for each of the tests H_0:m\in M_0 vs H_i:m\notin M_i being i=1,\dots,k. On the other hand, if the parameter INCREMENTAL is set to TRUE, the command returns the results for the tests H_i:m\in M_i vs H_{i+1}:m\notin M_{i+1} being i=1,\dots,k-1.

Value

This function returns an object of class anovarIntReg, a matrix like structure whose rows refers to models and columns to statistics and its p-values. It also has an attribute heading to support printing the object.

Note

This method requires more testing, and careful study of the effect of factors (discrete random variables) when fitting the model.

Author(s)

Jorge Luis Ojeda Cabrera (jojeda@unizar.es).

See Also

lm, glm, nls, and intRegGOF.

Examples

  n <- 50
  d <- data.frame( X1=runif(n),X2=runif(n))
  d$Y <- 1 - 2*d$X1 - 5*d$X2 + rnorm(n,sd=.125)
  a0 <- lm(Y~1,d) 
  a1 <- lm(Y~X1,d) 
  a2 <- lm(Y~X1+X2,d) 
  anovarIntReg(a0,a1,a2,B=50) 
  anovarIntReg(a0,a1,a2,B=50,INCREMENTAL=TRUE) 

Integrated Regression Goodness of Fit

Description

Integrated Regression Goodness of Fit to test if a given model is suitable to represent the regression function for a given data.

Usage

  intRegGOF(obj, covars = NULL, B = 499, LINMOD = FALSE)
  ## S3 method for class 'intRegGOF'
print(x,...)

Arguments

Details

The Integrated Regression Goodness of Fit technique is introduce in Stute(1997). The main idea is to study the process that results from the cumulation of the residuals up to a given value of the covariates. Once this process is built, different functional over it can be considered to measure the discrepany between the true regression function and its estimation.

The tests that implements this function is

H_0:m\in M \ \textrm{vs} \ H_1:m\notin M

being m the regression function, and M a given class of functions. The statistics considered are

K_n=\sup_{x\in R^d}|R^w_n(x)|

W^2_n=\int_{R^d}R^w_n(z)^2 \,dF(z).

where R^w_n(z) is the cumulated residual process:

R^w_n(x)=n^{-1/2}\sum^n_{i=1}(y_i-\hat y_i)I(x_i\le x).

As the stochastic behaviour of this cumulated residual process is quite complex, the implementation of the technique is based on resampling techniques. In particular the chosen implementation is based on Wild Bootstrap methods.

The method also handles selection biased data by means of compensation, by means of the weights used to fit the resgression function when computing the cumulated residual process.

At the moment only 'response' type of residuals are considered, jointly with wild bootstrap resampling technique and the result for discrete responses might no be proper.

Value

This function returns an object of class intRegGOF, a list which cointains following objects:

Note

This method requires more testing, and careful study of the effect of factors (discrete random variables) when fitting the model.

Author(s)

Jorge Luis Ojeda Cabrera (jojeda@unizar.es).

References

Stute, W. (1997). Nonparametric model checks for regression. Ann. Statist., 25(2), pp. 613–641.

Ojeda, J. L., W. González-Manteiga W. and Cristóbal, J. A A bootstrap based Model Checking for Selection–Biased data Reports in Statistics and Operations Research, U. de Santiago de Compostela. Report 07-05 http://eio.usc.es/eipc1/BASE/BASEMASTER/FORMULARIOS-PHP-DPTO/REPORTS/447report07_05.pdf

Ojeda, J. L., Cristóbal, J. A., and Alcalá, J. T. (2008). A bootstrap approach to model checking for linear models under length-biased data. Ann. Inst. Statist. Math., 60(3), pp. 519–543.

See Also

lm, glm, nls and its methods summary, print, plot, etc...

Examples

n <- 50
d <- data.frame( X1=runif(n),X2=runif(n))
d$Y <- 1 + 2*d$X1 + rnorm(n,sd=.125)
plot( d ) 
intRegGOF(lm(Y~X1+X2,d),B=99)
intRegGOF(a <- lm(Y~X1-1,d),B=99) 
intRegGOF(a,c("X1","X2"),B=99) 
intRegGOF(a,~X2+X1,B=99) 

Integrated Regression Goodness of Fit graphical output

Description

Methods to develop model validation and visualization of Integrated Regression Goodness of Fit technique.

Usage

  plotAsIntRegGOF(obj, covar = 1, ADD = FALSE, ...)
  pointsAsIntRegGOF(obj,covar=1,...)
  linesAsIntRegGOF(obj,covar=1,...)

Arguments

Details

Currently, the implementation computes the accumulated residual process against a single covariate (covar). When the value of covar is set to 0, the response is used as the variable whose residual are accumulated against.

Notice that if covar is a vector its lenght should be equal to the number of residuals.

Note

lm objects that does not have a data parameter set when the call is executed does not work presently when the covar parameter is different than 0.

Author(s)

Jorge Luis Ojeda Cabrera (jojeda@unizar.es).

See Also

lm, glm, nls its associated plot method and intRegGOF.

Examples

  n <- 50
  d <- data.frame( X1=runif(n),X2=runif(n))
  d$Y <- 1 + 2*d$X1 + rnorm(n,sd=.125)
  par(ask=TRUE)
  plot( d ) 
  plotAsIntRegGOF(lm(Y~X1+X2,d),covar="X1") 
  plotAsIntRegGOF(a <- lm(Y~X1-1,d)) 
  plotAsIntRegGOF(a,c("X1")) 
  plotAsIntRegGOF(a,0) 
  plotAsIntRegGOF(a,fitted(a)) 
  par(ask=FALSE)