R: Input matrices for subselect search routines in generalized...

glmHmat {subselect} | R Documentation

## Input matrices for subselect search routines in generalized linear models

### Description

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.

### Usage

    ## S3 method for class 'glm'
    glmHmat(fitdglmmodel,...)

### Arguments

`fitdglmmodel` | A glm object containaing the estimates, and respective covariance matrix, of a generalized linear model.
`...` | further arguments for the method.

### Details

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.

glmHmat retrieves the values of the FI and H matrices from a glm object. These matrices may then be used as input to the search functions `anneal`, `genetic`, `improve` and `eleaps`.

### Value

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 `(FI %*% b %*% t(b) %*% FI)` where b is a vector of variable-coefficient estimates
`r` | The rank of the H matrix. Always set to one in glmHmat.
`call` | The function call which generated the output.

### References

[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.

### See Also

`anneal`, `genetic`, `improve`, `eleaps`, `glm`.

### Examples

    ##----------------------------------------------------------------

    ##----------------------------------------------------------------

    ##  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

[Package subselect version 0.14 Index]