R: Bayesian Model Averaging for generalized linear models.

bic.glm {BMA} | R Documentation

## Bayesian Model Averaging for generalized linear models.

### Description

Bayesian Model Averaging accounts for the model uncertainty inherent in the variable selection problem by averaging over the best models in the model class according to approximate posterior model probability.

### Usage

    bic.glm(x, ...)

    ## S3 method for class 'matrix'
    bic.glm(x, y, glm.family, wt = rep(1, nrow(x)),
        strict = FALSE, prior.param = c(rep(0.5, ncol(x))), OR = 20,
        maxCol = 30, OR.fix = 2, nbest = 150, dispersion = NULL,
        factor.type = TRUE, factor.prior.adjust = FALSE,
        occam.window = TRUE, call = NULL, ...)

    ## S3 method for class 'data.frame'
    bic.glm(x, y, glm.family, wt = rep(1, nrow(x)),
        strict = FALSE, prior.param = c(rep(0.5, ncol(x))), OR = 20,
        maxCol = 30, OR.fix = 2, nbest = 150, dispersion = NULL,
        factor.type = TRUE, factor.prior.adjust = FALSE,
        occam.window = TRUE, call = NULL, ...)

    ## S3 method for class 'formula'
    bic.glm(f, data, glm.family, wt = rep(1, nrow(data)),
        strict = FALSE, prior.param = c(rep(0.5, ncol(x))), OR = 20,
        maxCol = 30, OR.fix = 2, nbest = 150, dispersion = NULL,
        factor.type = TRUE, factor.prior.adjust = FALSE,
        occam.window = TRUE, ...)

### Arguments

`x` | a matrix or data.frame of independent variables.
`y` | a vector of values for the dependent variable.
`f` | a formula
`data` | a data frame containing the variables in the model.
`glm.family` | a description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. (See 'family' for details of family functions.)
`wt` | an optional vector of weights to be used.
`strict` | a logical indicating whether models with more likely submodels are eliminated. `FALSE` returns all models whose posterior model probability is within a factor of `1/OR` of that of the best model.
`prior.param` | a vector of values specifying the prior weights for each variable.
`OR` | a number specifying the maximum ratio for excluding models in Occam's window
`maxCol` | a number specifying the maximum number of columns in design matrix (excluding intercept) to be kept.
`OR.fix` | width of the window which keeps models after the leaps approximation is done. Because the leaps and bounds gives only an approximation to BIC, there is a need to increase the window at this first "cut" so as to assure that no good models are deleted. The level of this cut is at `1/(OR^OR.fix)`; the default value for `OR.fix` is 2.
`nbest` | a number specifying the number of models of each size returned to `bic.glm` by the modified leaps algorithm.
`dispersion` | a logical value specifying whether dispersion should be estimated or not. Default is `TRUE` unless glm.family is poisson or binomial
`factor.type` | a logical value specifying how variables of class "factor" are handled. A factor variable with d levels is turned into (d-1) dummy variables using a treatment contrast. If `factor.type = TRUE`, models will contain either all or none of these dummy variables. If `factor.type = FALSE`, models are free to select the dummy variables independently. In this case, factor.prior.adjust determines the prior on these variables.
`factor.prior.adjust` | a logical value specifying whether the prior distribution on dummy variables for factors should be adjusted when `factor.type=FALSE`. When `factor.prior.adjust=FALSE`, all dummy variables for variable `i` have prior equal to `prior.param[i]`. Note that this makes the prior probability of the union of these variables much higher than `prior.param[i]`. Setting `factor.prior.adjust=T` corrects for this so that the union of the dummies equals `prior.param[i]` (and hence the deletion of the factor has a prior of `1-prior.param[i]`). This adjustment changes the individual priors on each dummy variable to ' `1-(1-pp[i])^(1/(k+1))`.
`occam.window` | a logical value specifying if Occam's window should be used. If set to `FALSE` then all models selected by the modified leaps algorithm are returned.
`call` | used internally
`...` | unused

### Details

Bayesian Model Averaging accounts for the model uncertainty inherent in the variable selection problem by averaging over the best models in the model class according to approximate posterior model probability.

### Value

`bic.glm` returns an object of class `bic.glm`

The function `summary` is used to print a summary of the results. The function `plot` is used to plot posterior distributions for the coefficients. The function `imageplot` generates an image of the models which were averaged over.

An object of class `bic.glm` is a list containing at least the following components:

`postprob` | the posterior probabilities of the models selected
`deviance` | the estimated model deviances
`label` | labels identifying the models selected
`bic` | values of BIC for the models
`size` | the number of independent variables in each of the models
`which` | a logical matrix with one row per model and one column per variable indicating whether that variable is in the model
`probne0` | the posterior probability that each variable is non-zero (in percent)
`postmean` | the posterior mean of each coefficient (from model averaging)
`postsd` | the posterior standard deviation of each coefficient (from model averaging)
`condpostmean` | the posterior mean of each coefficient conditional on the variable being included in the model
`condpostsd` | the posterior standard deviation of each coefficient conditional on the variable being included in the model
`mle` | matrix with one row per model and one column per variable giving the maximum likelihood estimate of each coefficient for each model
`se` | matrix with one row per model and one column per variable giving the standard error of each coefficient for each model
`reduced` | a logical indicating whether any variables were dropped before model averaging
`dropped` | a vector containing the names of those variables dropped before model averaging
`call` | the matched call that created the bma.lm object

### Note

If more than `maxcol` variables are supplied, then bic.glm does stepwise elimination of variables until `maxcol` variables are reached. `bic.glm` handles factor variables according to the `factor.type` parameter. If this is true then factor variables are kept in the model or dropped in entirety. If false, then each dummy variable can be kept or dropped independently. If `bic.glm` is used with a formula that includes interactions between factor variables, then `bic.glm` will create a new factor variable to represent that interaction, and this factor variable will be kept or dropped in entirety if `factor.type` is true. This can create interpretation problems if any of the corresponding main effects are dropped. Many thanks to Sanford Weisberg for making source code for leaps available.

### Author(s)

Chris Volinsky volinsky@research.att.com, Adrian Raftery raftery@stat.washington.edu, and Ian Painter ian.painter@gmail.com

### References

Raftery, Adrian E. (1995). Bayesian model selection in social research (with Discussion). Sociological Methodology 1995 (Peter V. Marsden, ed.), pp. 111-196, Cambridge, Mass.: Blackwells.

An earlier version, issued as Working Paper 94-12, Center for Studies in Demography and Ecology, University of Washington (1994) is available as a technical report from the Department of Statistics, University of Washington.

### See Also

`summary.bic.glm`, `print.bic.glm`, `plot.bic.glm`

### Examples

    ## Not run:
    ### logistic regression
    library("MASS")
    data(birthwt)
    y<- birthwt$lo
    x<- data.frame(birthwt[,-1])
    x$race<- as.factor(x$race)
    x$ht<- (x$ht>=1)+0
    x<- x[,-9]
    x$smoke <- as.factor(x$smoke)
    x$ptl<- as.factor(x$ptl)
    x$ht <- as.factor(x$ht)
    x$ui <- as.factor(x$ui)

    glm.out.FT <- bic.glm(x, y, strict = FALSE, OR = 20,
                          glm.family="binomial", factor.type=TRUE)
    summary(glm.out.FT)
    imageplot.bma(glm.out.FT)

    glm.out.FF <- bic.glm(x, y, strict = FALSE, OR = 20,
                          glm.family="binomial", factor.type=FALSE)
    summary(glm.out.FF)
    imageplot.bma(glm.out.FF)

    glm.out.TT <- bic.glm(x, y, strict = TRUE, OR = 20,
                          glm.family="binomial", factor.type=TRUE)
    summary(glm.out.TT)
    imageplot.bma(glm.out.TT)

    glm.out.TF <- bic.glm(x, y, strict = TRUE, OR = 20,
                          glm.family="binomial", factor.type=FALSE)
    summary(glm.out.TF)
    imageplot.bma(glm.out.TF)

    ## End(Not run)

    ## Not run:
    ### Gamma family
    library(survival)
    data(veteran)
    surv.t<- veteran$time
    x<- veteran[,-c(3,4)]
    x$celltype<- factor(as.character(x$celltype))
    sel<- veteran$status == 0
    x<- x[!sel,]
    surv.t<- surv.t[!sel]

    glm.out.va <- bic.glm(x, y=surv.t, glm.family=Gamma(link="inverse"),
        factor.type=FALSE)
    summary(glm.out.va)
    imageplot.bma(glm.out.va)
    plot(glm.out.va)

    ## End(Not run)

    ### Poisson family
    ### Yates (teeth) data.

    x<- rbind(
        c(0, 0, 0),
        c(0, 1, 0),
        c(1, 0, 0),
        c(1, 1, 1))

    y<-c(4, 16, 1, 21)
    n<-c(1,1,1,1)

    models<- rbind(
        c(1, 1, 0),
        c(1, 1, 1))

    glm.out.yates <- bic.glm( x, y, n, glm.family = poisson(),
                             factor.type=FALSE)
    summary(glm.out.yates)

    ## Not run:
    ### Gaussian
    library(MASS)
    data(UScrime)
    f <- formula(log(y) ~  log(M)+So+log(Ed)+log(Po1)+log(Po2)+log(LF)+
                           log(M.F)+ log(Pop)+log(NW)+log(U1)+log(U2)+
                           log(GDP)+log(Ineq)+log(Prob)+log(Time))
    glm.out.crime <- bic.glm(f, data = UScrime, glm.family = gaussian())
    summary(glm.out.crime)
    # note the problems with the estimation of the posterior standard
    # deviation (compare with bicreg example)

    ## End(Not run)

[Package BMA version 3.18.9 Index]