Input matrices for subselect search routines in generalized linear models
glmHmat uses a glm object (fitdglmmodel) to build an estimate of
Fisher's Information (FI) matrix together with an auxiliarly rank-one positive-defenite
matrix (H), such that the positive eigenvalue of FI^{-1} H equals the value of Wald's statistic
for testing the global significance of fitdglmmodel. These matrices may be used as input to the
variable selection search routines anneal, genetic
improve or eleaps, usign the minimization of Wald's statistic
as criterion for discarding variables.
## S3 method for class 'glm' glmHmat(fitdglmmodel,...)
fitdglmmodel |
A glm object containaing the estimates, and respective covariance matrix, of a generalized linear model. |
... |
further arguments for the method. |
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.
A list with four items:
mat |
An estimate (FI) of Fisher's information matrix for the full model variable-coefficient estimates |
H |
A product of the form |
r |
The rank of the H matrix. Always set to one in glmHmat. |
call |
The function call which generated the output. |
[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. We consider the last 100 observations of ## the iris data set (versicolor an verginica species) and try ## to find the best variable subsets for models that take species ## as the response variable. data(iris) iris2sp <- iris[iris$Species != "setosa",] # Create the input matrices for the search routines in a logistic regression model modelfit <- glm(Species ~ Sepal.Length + Sepal.Width + Petal.Length + Petal.Width,iris2sp,family=binomial) Hmat <- glmHmat(modelfit) Hmat ## $mat ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## Sepal.Length 0.28340358 0.03263437 0.09552821 -0.01779067 ## Sepal.Width 0.03263437 0.13941541 0.01086596 0.04759284 ## Petal.Length 0.09552821 0.01086596 0.08847655 -0.01853044 ## Petal.Width -0.01779067 0.04759284 -0.01853044 0.03258730 ## $H ## Sepal.Length Sepal.Width Petal.Length Petal.Width ## Sepal.Length 0.11643732 0.013349227 -0.063924853 -0.050181400 ## Sepal.Width 0.01334923 0.001530453 -0.007328813 -0.005753163 ## Petal.Length -0.06392485 -0.007328813 0.035095164 0.027549918 ## Petal.Width -0.05018140 -0.005753163 0.027549918 0.021626854 ## $r ## [1] 1 ## $call ## glmHmat(fitdglmmodel = modelfit) # Search for the 3 best variable subsets of each dimensionality by an exausitve search eleaps(Hmat$mat,H=Hmat$H,r=1,criterion="Wald",nsol=3) ## $subsets ## , , Card.1 ## Var.1 Var.2 Var.3 ## Solution 1 4 0 0 ## Solution 2 1 0 0 ## Solution 3 3 0 0 ## , , Card.2 ## Var.1 Var.2 Var.3 ## Solution 1 1 3 0 ## Solution 2 3 4 0 ## Solution 3 2 4 0 ## , , Card.3 ## Var.1 Var.2 Var.3 ## Solution 1 2 3 4 ## Solution 2 1 3 4 ## Solution 3 1 2 3 ## $values ## card.1 card.2 card.3 ## Solution 1 4.894554 3.522885 1.060121 ## Solution 2 5.147360 3.952538 2.224335 ## Solution 3 5.161553 3.972410 3.522879 ## $bestvalues ## Card.1 Card.2 Card.3 ## 4.894554 3.522885 1.060121 ## $bestsets ## Var.1 Var.2 Var.3 ## Card.1 4 0 0 ## Card.2 1 3 0 ## Card.3 2 3 4 ## $call ## eleaps(mat = Hmat$mat, nsol = 3, criterion = "Wald", H = Hmat$H, ## r = 1) ## It should be stressed that, unlike other criteria in the ## subselect package, the Wald criterion is not bounded above by ## 1 and is a decreasing function of subset quality, so that the ## 3-variable subsets do, in fact, perform better than their smaller-sized ## counterparts. ## > ## > proc.time() ## [1] 0.680 0.064 0.736 0.000 0.000
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