Description Usage Arguments Details Value Methods Note Author(s) References See Also Examples
Tests the goodness of fit of a regression model against a specified alternative using the Global Test.
Three main functions are provided: gtPS
uses Penalized Splines, gtKS
uses Kernel Smoothers and gtLI
uses Linear Interactions. The other functions are for external use in combination with gt
.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22  gtPS(response, null, data,
model = c("linear", "logistic", "cox", "poisson", "multinomial"),
..., covs, bdeg = 3, nint= 10, pord = 2, interact = FALSE, robust = FALSE,
termlabels = FALSE, returnZ = FALSE)
gtKS(response, null, data,
model = c("linear", "logistic", "cox", "poisson", "multinomial"),
..., covs, quant = .25, metric = c("euclidean", "pearson"),
kernel=c("uniform", "exponential", "triangular", "neighbours", "gauss"),
robust = FALSE, scale = TRUE, termlabels = FALSE, returnZ = FALSE)
gtLI(response, null, data, ..., covs, iorder=2, termlabels = FALSE, standardize = FALSE)
bbase(x, bdeg, nint)
btensor(xs, bdeg, nint, pord, returnU=FALSE)
reparamZ(Z, pord, K=NULL, tol = 1e10, returnU=FALSE)
reweighZ(Z, null.fit)
sterms(object, ...)

response 
The response vector of the regression model. May be
supplied as a vector, as a 
null 
The null design matrix. May be given as a matrix or as a half 
data 
Only used when 
model 
The type of regression model to be tested. If omitted, the function will try to determine the model from the class and values of the 
... 
Any other arguments are also passed on to 
covs 
A variable or a vector of variables that are the covariates the smooth terms are function of. 
bdeg 
A vector or a list of vectors which specifies the degree of the Bspline basis, with default 
nint 
A vector or a list of vectors which specifies the number of intervals determined by equallyspaced knots, with default 
pord 
A vector or a list of vectors which specifies the order of the differences indicating the type of the penalty imposed to the coefficients, with default 
interact 

termlabels 

robust 

returnZ 

quant 
The smoothing bandwidth to be used, expressed as the percentile of the distribution of distance between observations, with default the 25th percentile. To investigate the sensitivity to different choices, 
metric 
A character string specifying the metric to be used. The available options are "euclidean" (the default), "pearson" and "mixed" (to be implemented). "mixed" distance is chosen automatically if some of the selected covariates are not numeric. 
kernel 
A character string giving the smoothing kernel to be used. This must be one of "uniform", "exponential", "triangular", "neighbours", or "Gauss", with default "uniform". 
scale 

iorder 
Order of the linear interactions, e.g. second order interactions, third order etc., with default 
standardize 
TRUE standardizes all covariates of the alternative to have unit second central moment. This makes sure that the test result is independent of the relative scaling of the covariates. 
x 
A numeric vector of values at which to evaluate the Bspline basis. 
xs 
A matrix or dataframe where the columns correspond to covariates values. 
returnU 
codeTRUE gives back the nonpenalized part. 
Z 
Alternative design matrix. 
K 
Penalty matrix (i.e. the penalty term is the quadratic form of K and the spline coefficients). 
tol 
Eigenvalues smaller than 
null.fit 
Fitted null model. 
object 
A 
These are functions to test for specific types of lack of fit by using the Global Test.
Suppose that we are concerned with the adequacy of some regression model response ~ null
, such as Y ~ X1 + X2
.
The alternative model can be cast into the generic form response ~ null + alternative
, which comprises different models that accomodate to different types of lack of fit. Thus, the specification of alternative
is required. It identifies the type of lack of fit the test is directed against.
By using gtPS
, the alternative is given by a user specified sum of smooth functions of continuous covariates,
e.g. alternative= ~ s(X1)
when covs="X1"
and alternative= ~ s(X1) + s(X2)
when covs=c("X1","X2")
.
Smooth terms are constructed using Psplines as proposed by Eilers and Marx (1996). This approach consists in constructing
a Bspline basis of degree bdeg
with nint + 1
equidistant knots, where a difference penalty of order pord
is applied to the basis coefficients. If interact=TRUE
, the alternative is given by a multidimensional smooth function of covs
, which is represented by a tensor product of marginal Bsplines bases and Kronecker sum of the marginal penalties, e.g. alternative= ~ s(X1,X2)
when covs=c("X1","X2")
and interact=TRUE
.
By using gtKS
the alternative is given by a user specified multidimensional smooth term, e.g. alternative= ~ s(X1, X2)
when covs=c("X1","X2")
.
Multidimensional smooth terms are represented by a kernel smoother defined by a distance measure (metric
), a kernel shape (kernel
) and a bandwidth (quant
).
Because the test is sensitive to the chosen value of quant
, it is possible to specify quant
as a vector of different values in combination with robust=TRUE
.
Distance measures for factor covariates and for the situation that both continuous and factor covariates are present are constructed as in le Cessie and van Houwelingen (1995), e.g. covs=c("X1","X2")
and distance="mixed"
when X1
continuous and X2
factor (to be implemented).
By using gtLI
, the alternative is given by all the possible ithorder linear interactions between covs
, e.g. alternative= ~ X1:X2 + X1:X3 + X2:X3
when covs=c("X1","X2","X3")
and iorder=2
.
The remaining functions are meant for constructing the alternative design matrix that will be used in the alternative
argument of the gt
function.
bbase
constructs the Bspline basis for the covariate x
. This function is based on the functions provided by Eilers and Marx (1996).
btensor
builts a tensor product of Bsplines for the covariates xs
, which is reparameterized according with a Kroneker sum of penalties.
reparamZ
reparameterizes the alternative design matrix (e.g. a spline basis B
) according with the order of differences pord
or via the spectral decomposition of a roughness matrix K
. When several smooth terms are to be combined, reweighZ
assigns equal weight to each component term.
See the vignette for more examples.
The function returns an object of class gt.object
. Several operations and diagnostic plots can be made from this object.
(gt.object): Prints the smooth terms specified by gtPS
, gtKS
or gtLI
.
Currently linear (normal), logistic, multinomial logistic and Poisson regression models with canonical links and Cox's proportional hazards regression model are supported.
Aldo Solari: aldo.solari@unimib.it
Eilers, Marx (1996). Flexible smoothing with Bsplines and penalties. Statistical Science, 11: 89121.
le Cessie, van Houwelingen (1995). Testing the Fit of a Regression Model Via Score Tests in Random Effects Models. Biometrics 51: 600614.
For references related to applications of the test, see the vignette GlobalTest.pdf included with this package.
The gt
function. The gt.object
and useful functions associated with that object.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56  # Random data
set.seed(0)
X1<runif(50)
s1 < function(x) exp(2 * x)
e < rnorm(50)
Y < s1(X1) + e
### gtPS
res<gtPS(Y~X1)
res@result
sterms(res)
# model input
rdata<data.frame(Y,X1)
nullmodel<lm(Y~X1,data=rdata)
gtPS(nullmodel)
# formula input and termlabels
gtPS(Y~exp(2*X1),data=rdata)
gtPS(Y~exp(2*X1),covs="exp(2 * X1)",data=rdata)
sterms(gtPS(Y~exp(2*X1),data=rdata,termlabels=TRUE))
# Psplines arguments
gtPS(Y~X1, bdeg=3, nint=list(a=10, b=30), pord=0)
gtPS(Y~X1, bdeg=3, nint=list(a=10, b=30), pord=0, robust=TRUE)
# Random data: additive model
X2<runif(50)
s2 < function(x) 0.2 * x^11 * (10 * (1  x))^6 + 10 * (10 * x)^3 * (1  x)^10
Y < s1(X1) + s2(X2) + e
gtPS(Y~X1+X2)
gtPS(Y~X1+X2, covs="X2")
sterms(gtPS(Y~X1+X2, nint=list(a=c(10,30), b=20)))
# Random data: smooth surface
s12 < function(a, b, sa = 1, sb = 1) {
(pi^sa * sb) * (1.2 * exp((a  0.2)^2/sa^2  (b  0.3)^2/sb^2) +
0.8 * exp((a  0.7)^2/sa^2  (b  0.8)^2/sb^2))
}
Y < s12(X1,X2) + e
# Tensor product of Psplines
res<gtPS(Y~X1*X2, interact=TRUE)
res@result
sterms(res)
### gtKS
res<gtKS(Y~X1*X2)
res@result
sterms(res)
gtKS(Y~X1*X2, quant=seq(.05,.95,.05), robust=TRUE)
### gtLI
library(MASS)
data(Boston)
gtLI(medv~., data=Boston, standardize=TRUE)

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