bayesglm {arm} | R Documentation |
Bayesian functions for generalized linear modeling with independent normal, t, or Cauchy prior distribution for the coefficients.
bayesglm (formula, family = gaussian, data, weights, subset, na.action, start = NULL, etastart, mustart, offset, control = glm.control(...), model = TRUE, method = "glm.fit", x = FALSE, y = TRUE, contrasts = NULL, prior.mean = 0, prior.scale = 2.5, prior.scale.for.intercept = 10, prior.df = 1, scaled = TRUE, n.iter = 100, ...) bayesglm.fit (x, y, weights = rep(1, nobs), start = NULL, etastart = NULL, mustart = NULL, offset = rep(0, nobs), family = gaussian(), control = glm.control(), intercept = TRUE, prior.mean=0, prior.scale=2.5, prior.scale.for.intercept.default=10, prior.df=1, scaled=TRUE)
formula |
a symbolic description of the model to be fit. The details of model specification are given below. |
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 model. If not found in data , the
variables are taken from environment(formula) ,
typically the environment from which glm is called. |
weights |
an optional vector of weights to be used in the fitting
process. Should be NULL or a numeric vector. |
subset |
an optional vector specifying a subset of observations to be used in the fitting process. |
na.action |
a function which indicates what should happen
when the data contain NA s. The default is set by
the na.action setting of options , and is
na.fail if that is unset. The “factory-fresh”
default is na.omit . Another possible value is
NULL , no action. Value na.exclude can be useful. |
start |
starting values for the parameters in the linear predictor. |
etastart |
starting values for the linear predictor. |
mustart |
starting values for the vector of means. |
offset |
this can be used to specify an a priori
known component to be included in the linear predictor
during fitting. This should be NULL or a numeric vector of
length either one or equal to the number of cases.
One or more offset terms can be included in the
formula instead or as well, and if both are specified their sum is
used. See model.offset . |
control |
a list of parameters for controlling the fitting
process. See the documentation for glm.control
for details. |
model |
a logical value indicating whether model frame should be included as a component of the returned value. |
method |
the method to be used in fitting the model.
The default method "glm.fit" uses iteratively reweighted
least squares (IWLS). The only current alternative is
"model.frame" which returns the model frame and does no fitting. |
x, y |
For glm :
logical values indicating whether the response vector and model
matrix used in the fitting process should be returned as components
of the returned value.
For glm.fit : x is a design matrix of dimension n
* p , and y is a vector of observations of length n .
|
contrasts |
an optional list. See the contrasts.arg
of model.matrix.default . |
intercept |
logical. Should an intercept be included in the null model? |
prior.mean |
prior mean for the coefficients:default is 0. Can be a vector of length equal to the number of predictors (including the intercept, if any). If it is a scalar, it is expanded to the length of this vector. |
prior.scale |
prior scale for the coefficients: default is 2.5. Can be a vector of length equal to the number of predictors (including the intercept, if any). If it is a scalar, it is expanded to the length of this vector. |
prior.scale.for.intercept |
prior scale for the intercept: default is 10. |
prior.df |
for t distribution: default is 1 (Cauchy). Set to Inf to get normal prior distributions. Can be a vector of length equal to the number of predictors (including the intercept, if any). If it is a scalar, it is expanded to the length of this vector. |
scaled |
if scaled = TRUE, then the prior distribution is rescaled: default is TRUE |
n.iter |
default is 100. |
prior.scale.for.intercept.default |
prior scale for the intercept: default is 10. |
... |
further arguments passed to or from other methods. |
The program is a simple alteration of glm()
that uses an approximate EM
algorithm to update the betas at each step using an augmented regression
to represent the prior information.
We use Student-t prior distributions for the coefficients. The prior distribution for the constant term is set so it applies to the value when all predictors are set to their mean values.
If scaled=TRUE, the scales for the prior distributions of the coefficients are determined as follows: For a predictor with only one value, we just use prior.scale. For a predictor with two values, we use prior.scale/range(x). For a predictor with more than two values, we use prior.scale/(2*sd(x)).
We include all the glm()
arguments but we haven't tested that all the
options (e.g., offests
, contrasts
,
deviance
for the null model) all work.
The new arguments here are: prior.mean
, prior.scale
,
prior.scale.for.intercept
, prior.df
, and
scaled
.
See glm
for details.
Andrew Gelman gelman@stat.columbia.edu; Yu-Sung Su ys463@columbia.edu; Maria Grazia Pittau grazia@stat.columbia.edu; Aleks Jakulin Jakulin@stat.columbia.edu
Andrew Gelman, Aleks Jakulin, Maria Grazia Pittau and Yu-Sung Su, A default prior distribution for logistic and other regression models, unpublished paper available at http://www.stat.columbia.edu/~gelman/standardize/
n <- 100 x1 <- rnorm (n) x2 <- rbinom (n, 1, .5) b0 <- 1 b1 <- 1.5 b2 <- 2 y <- rbinom (n, 1, invlogit(b0+b1*x1+b2*x2)) M1 <- glm (y ~ x1 + x2, family=binomial(link="logit")) display (M1) # (using the display() function from regression.R) M2 <- bayesglm (y ~ x1 + x2, family=binomial(link="logit"), prior.scale=Inf, prior.df=Inf) display (M2) # just a test: this should be identical to classical logit M3 <- bayesglm (y ~ x1 + x2, family=binomial(link="logit")) # default Cauchy prior with scale 2.5 display (M3) M4 <- bayesglm (y ~ x1 + x2, family=binomial(link="logit"), prior.scale=2.5, prior.df=1) # Same as M3, explicitly specifying Cauchy prior with scale 2.5 display (M4) M5 <- bayesglm (y ~ x1 + x2, family=binomial(link="logit"), prior.scale=2.5, prior.df=7) # t_7 prior with scale 2.5 display (M5) M6 <- bayesglm (y ~ x1 + x2, family=binomial(link="logit"), prior.scale=2.5, prior.df=Inf) # normal prior with scale 2.5 display (M6) # Create separation: set y=1 whenever x2=1 # Now it should blow up without the prior! y <- ifelse (x2==1, 1, y) M1 <- glm (y ~ x1 + x2, family=binomial(link="logit")) display (M1) M2 <- bayesglm (y ~ x1 + x2, family=binomial(link="logit"), prior.scale=Inf, prior.df=Inf) # Same as M1 display (M2) M3 <- bayesglm (y ~ x1 + x2, family=binomial(link="logit")) display (M3) M4 <- bayesglm (y ~ x1 + x2, family=binomial(link="logit"), prior.scale=2.5, prior.df=1) # Same as M3 display (M4) M5 <- bayesglm (y ~ x1 + x2, family=binomial(link="logit"), prior.scale=2.5, prior.df=7) display (M5) M6 <- bayesglm (y ~ x1 + x2, family=binomial(link="logit"), prior.scale=2.5, prior.df=Inf) display (M6) # bayesglm with gaussian family (bayes lm) sigma <- 5 y2 <- rnorm (n, b0+b1*x1+b2*x2, sigma) M7 <- bayesglm (y2 ~ x1 + x2, prior.scale=Inf, prior.df=Inf) display (M7)