Wald statistic for variable selection in generalized linear models
Computes the value of Wald's statistic, testing the significance of the excluded variables, in the context of variable subset selection in generalized linear models
wald.coef(mat, H, indices, tolval=10*.Machine$double.eps, tolsym=1000*.Machine$double.eps)
mat |
An estimate (FI) of Fisher's information matrix for the full model variable-coefficient estimates |
H |
A matrix product of the form |
indices |
a numerical vector, matrix or 3-d array of integers giving the indices of the variables in the subset. If a matrix is specified, each row is taken to represent a different k-variable subset. If a 3-d array is given, it is assumed that the third dimension corresponds to different cardinalities. |
tolval |
the tolerance level to be used in checks for
ill-conditioning and positive-definiteness of the Fisher Information and
the auxiliar (H) matrices. Values smaller than |
tolsym |
the tolerance level for symmetry of the Fisher Information and the auxiliar (H) matrices. If corresponding matrix entries differ by more than this value, the input matrices will be considered asymmetric and execution will be aborted. If corresponding entries are different, but by less than this value, the input matrix will be replaced by its symmetric part, i.e., input matrix A becomes (A+t(A))/2. |
Variable selection in the context of generalized linear models is typically based on the minimization of statistics that test the significance of excluded variables. In particular, the likelihood ratio, Wald's, Rao's and some adaptations of such statistics, are often proposed as comparison criteria for variable subsets of the same dimensionality. All these statistics are assympotically equivalent and can be converted into information criteria, like the AIC, that are also able to compare subsets of different dimensionalities (see references [1] and [2] for further details).
Among these criteria, Wald's statistic has some computational advantages because it can
always be derived from the same (concerning the full model) maximum likelihood and Fisher
information estimates. In particular, if W_{allv} is the value of
the Wald statistic testing
the significance of the full covariate vector, b and FI are coefficient and Fisher
information estimates and H is an auxiliary rank-one matrix given by
H = FI %*% b %*% t(b) %*% FI, it follows that the value of Wald's statistic for the
excluded variables (W_{excv}) in a given subset is given by
$W_{excv} = W_{allv} - tr (FI_{indices}^{-1} H_{indices}) ,$ where
FI_{indices} and H_{indices} are the
portions of the FI and H matrices associated with the selected variables.
The value of the Wald statistic.
[1] Lawless, J. and Singhal, K. (1978). Efficient Screening of Nonnormal Regression Models, Biometrics, Vol. 34, 318-327.
[2] Lawless, J. and Singhal, K. (1987). ISMOD: An All-Subsets Regression Program for Generalized Models I. Statistical and Computational Background, Computer Methods and Programs in Biomedicine, Vol. 24, 117-124.
## --------------------------------------------------------------- ## An example of variable selection in the context of binary response ## regression models. The logarithms and original physical measurements ## of the "Leptograpsus variegatus crabs" considered in the MASS crabs ## data set are used to fit a logistic model that takes the sex of each crab ## as the response variable. library(MASS) data(crabs) lFL <- log(crabs$FL) lRW <- log(crabs$RW) lCL <- log(crabs$CL) lCW <- log(crabs$CW) logrfit <- glm(sex ~ FL + RW + CL + CW + lFL + lRW + lCL + lCW, crabs,family=binomial) ## Warning message: ## fitted probabilities numerically 0 or 1 occurred in: glm.fit(x = X, y = Y, ## weights = weights, start = start, etastart = etastart, lHmat <- glmHmat(logrfit) wald.coef(lHmat$mat,lHmat$H,c(1,6,7),tolsym=1E-06) ## [1] 2.286739 ## Warning message: ## The covariance/total matrix supplied was slightly asymmetric: ## symmetric entries differed by up to 6.57252030578093e-14. ## (less than the 'tolsym' parameter). ## It has been replaced by its symmetric part. ## in: validmat(mat, p, tolval, tolsym) ## --------------------------------------------------------------- ## 2) An example computing the value of the Wald statistic in a logistic ## model for five subsets produced when a probit model was originally ## considered library(MASS) data(crabs) lFL <- log(crabs$FL) lRW <- log(crabs$RW) lCL <- log(crabs$CL) lCW <- log(crabs$CW) probfit <- glm(sex ~ FL + RW + CL + CW + lFL + lRW + lCL + lCW, crabs,family=binomial(link=probit)) ## Warning message: ## fitted probabilities numerically 0 or 1 occurred in: glm.fit(x = X, y = Y, ## weights = weights, start = start, etastart = etastart) pHmat <- glmHmat(probfit) probresults <-eleaps(pHmat$mat,kmin=3,kmax=3,nsol=5,criterion="Wald",H=pHmat$H, r=1,tolsym=1E-10) ## Warning message: ## The covariance/total matrix supplied was slightly asymmetric: ## symmetric entries differed by up to 3.14059889205964e-12. ## (less than the 'tolsym' parameter). ## It has been replaced by its symmetric part. ## in: validmat(mat, p, tolval, tolsym) logrfit <- glm(sex ~ FL + RW + CL + CW + lFL + lRW + lCL + lCW, crabs,family=binomial) ## Warning message: ## fitted probabilities numerically 0 or 1 occurred in: glm.fit(x = X, y = Y, ## weights = weights, start = start, etastart = etastart) lHmat <- glmHmat(logrfit) wald.coef(lHmat$mat,H=lHmat$H,probresults$subsets,tolsym=1e-06) ## Card.3 ## Solution 1 2.286739 ## Solution 2 2.595165 ## Solution 3 2.585149 ## Solution 4 2.669059 ## Solution 5 2.690954 ## Warning message: ## The covariance/total matrix supplied was slightly asymmetric: ## symmetric entries differed by up to 6.57252030578093e-14. ## (less than the 'tolsym' parameter). ## It has been replaced by its symmetric part. ## in: validmat(mat, p, tolval, tolsym)
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