| ols {Design} | R Documentation |
Fits the usual weighted or unweighted linear regression model using the
same fitting routines used by lm, but also storing the variance-covariance
matrix var and using traditional dummy-variable coding for categorical
factors.
Also fits unweighted models using penalized least squares, with the same
penalization options as in the lrm function. For penalized estimation,
there is a fitter function call lm.pfit.
ols(formula, data, weights, subset, na.action=na.delete,
method="qr", model=FALSE,
x=FALSE, y=FALSE, se.fit=FALSE, linear.predictors=TRUE,
penalty=0, penalty.matrix, tol=1e-7, sigma,
var.penalty=c('simple','sandwich'), ...)
formula |
an S formula object, e.g.
Y ~ rcs(x1,5)*lsp(x2,c(10,20)) |
data |
name of an S data frame containing all needed variables. Omit this to use a data frame already in the S ``search list''. |
weights |
an optional vector of weights to be used in the fitting
process. If specified, weighted least squares is used with
weights weights (that is, minimizing sum(w*e^2));
otherwise ordinary least squares is used. |
subset |
an expression defining a subset of the observations to use in the fit. The default
is to use all observations. Specify for example age>50 & sex="male" or
c(1:100,200:300)
respectively to use the observations satisfying a logical expression or those having
row numbers in the given vector.
|
na.action |
specifies an S function to handle missing data. The default is the function na.delete,
which causes observations with any variable missing to be deleted. The main difference
between na.delete and the S-supplied function na.omit is that
na.delete makes a list
of the number of observations that are missing on each variable in the model.
The na.action is usally specified by e.g. options(na.action="na.delete").
|
method |
specifies a particular fitting method, or "model.frame" instead to return the model frame
of the predictor and response variables satisfying any subset or missing value
checks.
|
model |
default is FALSE. Set to TRUE to return the model frame
as element model of the fit object.
|
x |
default is FALSE. Set to TRUE to return the expanded design matrix as element x
(without intercept indicators) of the
returned fit object. Set both x=TRUE if you are going to use
the residuals function later to return anything other than ordinary residuals.
|
y |
default is FALSE. Set to TRUE to return the vector of response values
as element y of the fit.
|
se.fit |
default is FALSE. Set to TRUE to compute the estimated standard errors of
the estimate of X beta and store them in element se.fit
of the fit.
|
linear.predictors |
set to FALSE to cause predicted values not to be stored
|
penalty |
|
penalty.matrix |
see lrm
|
tol |
tolerance for information matrix singularity |
sigma |
If sigma is given, it is taken as the actual root mean squared error parameter for the model. Otherwise sigma is estimated from the data using the usual formulas (except for penalized models). It is often convenient to specify
sigma=1 for models with no error, when using fastbw to find an
approximate model that predicts predicted values from the full model with
a given accuracy.
|
var.penalty |
the type of variance-covariance matrix to be stored in the var
component of the fit when penalization is used. The default is the
inverse of the penalized information matrix. Specify
var.penalty="sandwich" to use the sandwich estimator (see below
under var), which limited simulation studies have shown yields
variances estimates that are too low.
|
... |
arguments to pass to lm.wfit or
lm.fit |
For penalized estimation, the penalty factor on the log likelihood is
-0.5 β' P β / σ^2, where P is defined above.
The penalized maximum likelihood estimate (penalized least squares
or ridge estimate) of beta is (X'X + P)^{-1} X'Y.
The maximum likelihood estimate of σ^2 is (sse + β'
P β) / n, where
sse is the sum of squared errors (residuals).
The effective.df.diagonal vector is the
diagonal of the matrix X'X/(sse/n) σ^{2} (X'X + P)^{-1}.
the same objects returned from lm (unless penalty or penalty.matrix
are given - then an
abbreviated list is returned since lm.pfit is used as a fitter)
plus the design attributes
(see Design).
Predicted values are always returned, in the element linear.predictors.
The vectors or matrix stored if y=TRUE or x=TRUE have rows deleted according to subset and
to missing data, and have names or row names that come from the
data frame used as input data. If penalty or penalty.matrix is given,
the var matrix
returned is an improved variance-covariance matrix
for the penalized regression coefficient estimates. If
var.penalty="sandwich" (not the default, as limited simulation
studies have found it provides variance estimates that are too low) it
is defined as
σ^{2} (X'X + P)^{-1} X'X (X'X + P)^{-1}, where P is
penalty factors * penalty.matrix, with a column and row of zeros
added for the
intercept. When var.penalty="simple" (the default), var is
σ^{2} (X'X + P)^{-1}.
The returned list has a vector stats with named elements
n, Model L.R., d.f., R2, Sigma. Model L.R. is the model
likelihood ratio chi-square statistic, and R2 is
R^2. For penalized estimation, d.f. is the
effective degrees of freedom, which is the sum of the elements of another
vector returned, effective.df.diagonal, minus one for the intercept.
Sigma is the penalized maximum likelihood estimate (see below).
Frank Harrell
Department of Biostatistics, Vanderbilt University
f.harrell@vanderbilt.edu
Design, Design.trans, anova.Design, summary.Design, predict.Design,
fastbw, validate, calibrate, plot.Design,
specs.Design, cph, lrm, which.influence, lm, summary.lm,
print.ols, residuals.ols, latex.ols, na.delete, na.detail.response,
naresid, datadist, pentrace, vif, abs.error.pred
set.seed(1) x1 <- runif(200) x2 <- sample(0:3, 200, TRUE) distance <- (x1 + x2/3 + rnorm(200))^2 d <- datadist(x1,x2) options(datadist="d") # No d -> no summary, plot without giving all details f <- ols(sqrt(distance) ~ rcs(x1,4) + scored(x2), x=TRUE) # could use d <- datadist(f); options(datadist="d") at this point, # but predictor summaries would not be stored in the fit object for # use with plot.Design, summary.Design. In that case, the original # dataset or d would need to be accessed later, or all variable values # would have to be specified to summary, plot anova(f) which.influence(f) summary(f) summary.lm(f) # will only work if penalty and penalty.matrix not used # Fit a complex model and approximate it with a simple one x1 <- runif(200) x2 <- runif(200) x3 <- runif(200) x4 <- runif(200) y <- x1 + x2 + rnorm(200) f <- ols(y ~ rcs(x1,4) + x2 + x3 + x4) pred <- fitted(f) # or predict(f) or f$linear.predictors f2 <- ols(pred ~ rcs(x1,4) + x2 + x3 + x4, sigma=1) # sigma=1 prevents numerical problems resulting from R2=1 fastbw(f2, aics=100000) # This will find the best 1-variable model, best 2-variable model, etc. # in predicting the predicted values from the original model options(datadist=NULL)