glmbb {glmbb}

All Hierarchical or Graphical Models for Generalized Linear Model

Description

Find all hierarchical submodels of specified GLM with information criterion (AIC, BIC, or AICc) within specified cutoff of minimum value. Alternatively, all such graphical models. Use branch and bound algorithm so we do not have to fit all models.

Usage

glmbb(big, little = ~ 1, family = poisson, data,
    criterion = c("AIC", "AICc", "BIC"), cutoff = 10,
    trace = FALSE, graphical = FALSE, BIC.option = c("length", "sum"), ...)

Arguments

big

an object of class "formula" specifying the largest model to be considered. Model specified must be hierarchical. (See also glm and formula and ‘Details’ section below.)

little

a formula specifying the smallest model to be considered. The response may be omitted and if not omitted is ignored (the response is taken from big). Default is ~ 1. Model specified must be nested within the model specified by big.

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.)

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the models. If not found in data, the variables are taken from environment(big), typically the environment from which glmbb is called.

criterion

a character string specifying the information criterion, must be one of "AIC" (Akaike Information Criterion, the default), "BIC" (Bayes Information Criterion) or "AICc" (AIC corrected for sample size). See section on AICc below.

cutoff

a nonnegative real number. This function finds all hierarchical models that are submodels of big and supermodels of little with information criterion less than or equal to the cutoff plus the minimum information criterion over all these models.

trace

logical. Emit debug info if TRUE.

graphical

logical. If TRUE search only over graphical models rather than hierarchical models.

BIC.option

a character string specifying the sample size n to be used in calculating BIC (ignored if criterion ! = "BIC"), must be either "length" or "sum" meaning either the length of the response vector (or number of rows if the response “vector” is actually a matrix) or the sum of the response (number of individuals classified if we are doing categorical data analysis). See section about BIC below. May be abbreviated.

...

additional named or unnamed arguments to be passed to glm.

Details

Typical value for big is something like foo ~ bar * baz * qux where foo is the response variable (or matrix when family is binomial or quasibinomial, see glm) and bar, baz, and qux are all the predictors that are considered for inclusion in models.

A model is hierarchical if it includes all lower-order interactions for each term. This is automatically what formulas with all variables connected by stars (*) do, like the example above. But other specifications are possible. For example, foo ~ (bar + baz + qux)^2 specifies the model with all main effects, and all two-way interactions, but no three-way interaction, and this is hierarchical.

A model m_1 is nested within a model m_1 if all terms in m_1 are also terms in m_2. The default little model ~ 1 is nested within every model except those specified to have no intercept by 0 + or some such (see formula).

The interaction graph of a model is the undirected graph whose node set is the predictor variables in the model and whose edge set has one edge for each pair of variables that are in an interaction term. A clique in a graph is a maximal complete subgraph. A model is graphical if it is hierarchical and has an interaction term for the variables in each clique. When graphical = TRUE only graphical models are considered.

Value

An object of class "glmbb" containing at least the following components:

data

the model frame, a data frame containing all the variables.

little

the argument little.

big

the argument big.

criterion

the argument criterion.

cutoff

the argument cutoff.

envir

an R environment object containing all of the fits done.

min.crit

the minimum value of the criterion.

graphical

the argument graphical.

BIC

It is unclear what the sample size, the n in the BIC penalty n \log(p) should be. Before version 0.4 of this package the BIC was taken to be the result of applying R generic function BIC to the fitted object produced by R function glm. This is generally wrong whenever we think we are doing categorical data analysis (Raftery, 1986; Kass and Raftery, 1995). Whether we consider the sampling scheme to be Poisson, multinomial, or product multinomial (and binomial is a special case of product multinomial) the sample size is the total number of individuals classified and is the only thing that is considered as going to infinity in the usual asymptotics for categorical data analysis. This the option BIC.option = "sum" should always be used for categorical data analysis.

AICc

AICc was derived by Hurvich and Tsai only for normal response models. Burnham and Anderson (2002, p. 378) recommend it for other models when no other small sample correction is known, but this is not backed up by any theoretical derivation.

References

Burnham, K. P. and Anderson, D. R. (2002) Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, second edition. Springer, New York.

Hand, D. J. (1981) Branch and bound in statistical data analysis. The Statistician, 30, 1–13.

Hurvich, C. M. and Tsai, C.-L. (1989) Regression and time series model selection in small samples. Biometrika, 76, 297–307.

Kass, R. E. and Raftery, A. E. (1995) Bayes factors. Journal of the American Statistical Association, 90, 773–795.

Raftery, A. E. (1986) A note on Bayes factors for log-linear contingency table models with vague prior information. Journal of the Royal Statistical Society, Series B, 48, 249–250.

See Also

family, formula, glm, isGraphical, isHierarchical

Examples

data(crabs)
gout <- glmbb(satell ~ (color + spine + width + weight)^3,
    criterion = "BIC", data = crabs)
summary(gout)