Horseshoe Crab Mating Data

Description

Data on horseshoe crabs (Limulus polyphemus). Response is number of males surrounding a breeding female, color (factor), condition (factor), weight (quantitative), and width (quantitative) of the female.

Usage

data(crabs)

Format

A data frame with 173 observations on 6 variables. Individuals (rows of the data frame) are female horseshoe crabs. Variables other than satell refer to these females. The variables are

color

color. The colors given in Agresti are “light medium”, “medium”, “dark medium”, and “dark”. Here they are abbreviated to light, medium, dark, and darker, respectively.

spine

spine condition. The conditions given in Agresti are “both good”, “one worn or broken”, and “both worn or broken”. Here they are abbreviated to good, middle, bad, respectively.

width

carapace width in centimeters

satell

number of satellites, which males clustering around the female in addition to the male with which she is breeding.

weight

weight in grams.

y

shorthand for as.numeric(satell > 0).

Details

Quoting from the abstract of Brockmann (1996). “Horseshoe crabs arrive on the beach in pairs and spawn ... during ... high tides. Unattached males also come to the beach, crowd around the nesting couples and compete with attached males for fertilizations. Satellite males form large groups around some couples while ignoring others, resulting in a nonrandom distribution that cannot be explained by local environmental conditions or habitat selection.”

Source

Agresti, A. (2013) Categorical Data Analysis, Wiley, Hoboken, NJ., Section 4.3.2, http://users.stat.ufl.edu/~aa/cda/data.html

Brockmann, H. J. (1996) Satellite Male Groups in Horseshoe Crabs, Limulus polyphemus, Ethology, 102, 1–21.

Examples

  data(crabs)
  gout <- glm(satell ~ color + spine + width + weight, family = poisson,
      data = crabs)

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

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:

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)

Hierarchical and Graphical Models

Description

Say whether a formula corresponds to a hierarchical model or a graphical model. Or return a formula for a hierarchical or a graphical model.

Usage

asGraphical(formula)
isGraphical(formula)
asHierarchical(formula)
isHierarchical(formula)

Arguments

Details

A model is hierarchical if for every interaction it contains all the main effects and lower-order interactions for variables in that interaction.

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.

Value

For “is” functions, logical. TRUE if and only if the model is hierarchical or graphical, as the case may be.

For “as” functions, a formula for the smallest supermodel of the given model that is hierarchical or graphical, as the case may be.

Examples

isHierarchical(~ u * v)
isHierarchical(~ u : v)

isGraphical(~ u * v + u * w)
isGraphical(~ (u + v + w)^2)

asHierarchical(~ u:v + v:w)
asGraphical(~ (u + v + w)^2)

Summarize GLM Model Selection via Branch and Bound

Description

These functions are all methods for class glmbb or summary.glmbb objects.

Usage

## S3 method for class 'glmbb'
summary(object, cutoff, ...)

## S3 method for class 'summary.glmbb'
print(x, digits = max(3, getOption("digits") - 3),
    ...)

Arguments

Details

Let criterion denote the vector of criterion (AIC, BIC, or AICc) values for all of the models evaluated in the search. Those with criterion value greater than min(criterion) + cutoff are tossed.

We also define a vector weight by

w <- exp(- criterion / 2)
weight <- w / sum(w)

except that it is calculated differently to avoid overflow. These are so-called Akaike weights. They may or may not provide some guide as to how to deal with these models. For more see Burnham and Anderson (2002).

Value

summary.glmbb returns an object of class "summary.glmbb", a list with components

References

Burnham, K. P. and Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. Springer-Verlag, New York.

Examples

## For examples see those in help(glmbb)

Shorten a Hierarchical Formula

Description

Simplify a formula, assuming it is hierarchical, that is, an interaction implies all lower-order interactions and main effects involving the same variables are in the model.

Usage

tidy.formula.hierarchical(formula)

Arguments

Value

A character string coercible to a formula equivalent to the input.

Examples

tidy.formula.hierarchical(y ~ u + v + w + u:v + u:w + v:w + u:v:w)