| linear.hypothesis {car} | R Documentation |
Generic function for testing a linear hypothesis, and methods
for linear models, generalized linear models, multivariate linear
models, and other models that have methods for coef and vcov.
linear.hypothesis(model, ...)
lht(model, ...)
## Default S3 method:
linear.hypothesis(model, hypothesis.matrix, rhs=NULL,
test=c("Chisq", "F"), vcov.=NULL, verbose=FALSE, ...)
## S3 method for class 'lm':
linear.hypothesis(model, hypothesis.matrix, rhs=NULL,
test=c("F", "Chisq"), vcov.=NULL, white.adjust=FALSE, ...)
## S3 method for class 'glm':
linear.hypothesis(model, ...)
## S3 method for class 'mlm':
linear.hypothesis(model, hypothesis.matrix, rhs=NULL, SSPE, V,
test, idata, icontrasts=c("contr.sum", "contr.poly"), idesign, iterms,
P=NULL, title="", verbose=FALSE, ...)
## S3 method for class 'linear.hypothesis.mlm':
print(x, SSP=TRUE, SSPE=SSP,
digits=unlist(options("digits")), ...)
model |
fitted model object. The default method works for models
for which the estimated parameters can be retrieved by coef and
the corresponding estimated covariance matrix by vcov. See the
Details for more information. |
hypothesis.matrix |
matrix (or vector) giving linear combinations of coefficients by rows, or a character vector giving the hypothesis in symbolic form (see Details). |
rhs |
right-hand-side vector for hypothesis, with as many entries as
rows in the hypothesis matrix; can be omitted, in which case it defaults
to a vector of zeroes. For a multivariate linear model, rhs is a
matrix, defaulting to 0. |
idata |
an optional data frame giving a factor or factors defining the intra-subject model for multivariate repeated-measures data. See Details for an explanation of the intra-subject design and for further explanation of the other arguments relating to intra-subject factors. |
icontrasts |
names of contrast-generating functions to be applied by default to factors and ordered factors, respectively, in the within-subject ``data''; the contrasts must produce an intra-subject model matrix in which different terms are orthogonal. |
idesign |
a one-sided model formula using the ``data'' in idata and
specifying the intra-subject design. |
iterms |
the quoted name of a term, or a vector of quoted names of terms, in the intra-subject design to be tested. |
P |
transformation matrix to be applied to the repeated measures in
multivariate repeated-measures data; if NULL and no
intra-subject model is specified, no response-transformation is applied; if
an intra-subject model is specified via the idata, idesign,
and (optionally) icontrasts arguments, then P is generated
automatically from the iterms argument. |
SSPE |
in linear.hypothesis method for mlm objects:
optional error sum-of-squares-and-products matrix; if missing,
it is computed from the model. In print method for
linear.hypothesis.mlm objects: if TRUE,
print the sum-of-squares and cross-products matrix for error. |
test |
character string, "F" or "Chisq",
specifying whether to compute the finite-sample
F statistic (with approximate F distribution) or the large-sample
Chi-squared statistic (with asymptotic Chi-squared distribution). For a
multivariate linear model, the multivariate test statistic to report — one of
"Pillai", "Wilks", "Hotelling-Lawley", or "Roy",
with "Pillai" as the default. |
title |
an optional character string to label the output. |
V |
inverse of sum of squares and products of the model matrix; if missing it is computed from the model. |
vcov. |
a function for estimating the covariance matrix of the regression
coefficients, e.g., hccm, or an estimated covariance matrix
for model. See also white.adjust. |
white.adjust |
logical or character. Convenience interface to hccm
(instead of using the argument vcov). Can be set either to a character
specifying the type argument of hccm or TRUE,
in which case "hc3" is used implicitly. For backwards compatibility. |
verbose |
If TRUE, the hypothesis matrix and right-hand-side
vector (or matrix) are printed to standard output; if FALSE (the default),
the hypothesis is only printed in symbolic form. |
x |
an object produced by linear.hypothesis.mlm. |
SSP |
if TRUE (the default), print the sum-of-squares and
cross-products matrix for the hypothesis and the response-transformation matrix. |
digits |
minimum number of signficiant digits to print. |
... |
aruments to pass down. |
Computes either a finite sample F statistic or asymptotic Chi-squared
statistic for carrying out a Wald-test-based comparison between a model
and a linearly restricted model. The default method will work with any
model object for which the coefficient vector can be retrieved by
coef and the coefficient-covariance matrix by vcov (otherwise
the argument vcov. has to be set explicitely). For computing the
F statistic (but not the Chi-squared statistic) a df.residual
method needs to be available. If a formula method exists, it is
used for pretty printing.
The method for "lm" objects calls the default method, but it
changes the default test to "F", supports the convenience argument
white.adjust (for backwards compatibility), and enhances the output
by residual sums of squares. For "glm" objects just the default
method is called (bypassing the "lm" method).
The function lht also dispatches to linear.hypothesis.
The hypothesis matrix can be supplied as a numeric matrix (or vector), the rows of which specify linear combinations of the model coefficients, which are tested equal to the corresponding entries in the righ-hand-side vector, which defaults to a vector of zeroes.
Alternatively, the hypothesis can be specified symbolically as a character vector with one or more elements, each of which gives either a linear combination of coefficients, or a linear equation in the coefficients (i.e., with both a left and right side separated by an equals sign). Components of a linear expression or linear equation can consist of numeric constants, or numeric constants multiplying coefficient names (in which case the number precedes the coefficient, and may be separated from it by spaces or an asterisk); constants of 1 or -1 may be omitted. Spaces are always optional. Components are separated by plus or minus signs. See the examples below.
A linear hypothesis for a multivariate linear model (i.e., an object of
class "mlm") can optionally include an intra-subject transformation matrix
for a repeated-measures design.
If the intra-subject transformation is absent (the default), the multivariate
test concerns all of the corresponding coefficients for the response variables.
There are two ways to specify the transformation matrix for the
repeated meaures:
P
argument.
idata, with default contrasts given by the icontrasts
argument. An intra-subject model-matrix is generated from the one-sided formula
specified by the idesign argument; columns of the model matrix
corresponding to different terms in the intra-subject model must be orthogonal
(as is insured by the default contrasts). Note that the contrasts given in
icontrasts can be overridden by assigning specific contrasts to the
factors in idata.
The repeated-measures transformation matrix consists of the
columns of the intra-subject model matrix corresponding to the term or terms
in iterms. In most instances, this will be the simpler approach, and
indeed, most tests of interests can be generated automatically via the
Anova function.
For a univariate model, an object of class "anova"
which contains the residual degrees of freedom
in the model, the difference in degrees of freedom, Wald statistic
(either "F" or "Chisq") and corresponding p value.
For a multivariate linear model, an object of class
"linear.hypothesis.mlm", which contains sums-of-squares-and-product
matrices for the hypothesis and for error, degrees of freedom for the
hypothesis and error, and some other information.
The returned object normally would be printed.
Achim Zeileis and John Fox jfox@mcmaster.ca
Fox, J. (1997) Applied Regression, Linear Models, and Related Methods. Sage.
Hand, D. J., and Taylor, C. C. (1987) Multivariate Analysis of Variance and Repeated Measures: A Practical Approach for Behavioural Scientists. Chapman and Hall.
O'Brien, R. G., and Kaiser, M. K. (1985) MANOVA method for analyzing repeated measures designs: An extensive primer. Psychological Bulletin 97, 316–333.
anova, Anova, waldtest,
hccm, vcovHC, vcovHAC,
coef, vcov
mod.davis <- lm(weight ~ repwt, data=Davis)
## the following are equivalent:
linear.hypothesis(mod.davis, diag(2), c(0,1))
linear.hypothesis(mod.davis, c("(Intercept) = 0", "repwt = 1"))
linear.hypothesis(mod.davis, c("(Intercept)", "repwt"), c(0,1))
linear.hypothesis(mod.davis, c("(Intercept)", "repwt = 1"))
## use asymptotic Chi-squared statistic
linear.hypothesis(mod.davis, c("(Intercept) = 0", "repwt = 1"), test = "Chisq")
## the following are equivalent:
## use HC3 standard errors via white.adjust option
linear.hypothesis(mod.davis, c("(Intercept) = 0", "repwt = 1"),
white.adjust = TRUE)
## covariance matrix *function*
linear.hypothesis(mod.davis, c("(Intercept) = 0", "repwt = 1"), vcov = hccm)
## covariance matrix *estimate*
linear.hypothesis(mod.davis, c("(Intercept) = 0", "repwt = 1"),
vcov = hccm(mod.davis, type = "hc3"))
mod.duncan <- lm(prestige ~ income + education, data=Duncan)
## the following are all equivalent:
linear.hypothesis(mod.duncan, "1*income - 1*education = 0")
linear.hypothesis(mod.duncan, "income = education")
linear.hypothesis(mod.duncan, "income - education")
linear.hypothesis(mod.duncan, "1income - 1education = 0")
linear.hypothesis(mod.duncan, "0 = 1*income - 1*education")
linear.hypothesis(mod.duncan, "income-education=0")
linear.hypothesis(mod.duncan, "1*income - 1*education + 1 = 1")
linear.hypothesis(mod.duncan, "2income = 2*education")
mod.duncan.2 <- lm(prestige ~ type*(income + education), data=Duncan)
coefs <- names(coef(mod.duncan.2))
## test against the null model (i.e., only the intercept is not set to 0)
linear.hypothesis(mod.duncan.2, coefs[-1])
## test all interaction coefficients equal to 0
linear.hypothesis(mod.duncan.2, coefs[grep(":", coefs)], verbose=TRUE)
## a multivariate linear model for repeated-measures data
## see ?OBrienKaiser for a description of the data set used in this example.
mod.ok <- lm(cbind(pre.1, pre.2, pre.3, pre.4, pre.5,
post.1, post.2, post.3, post.4, post.5,
fup.1, fup.2, fup.3, fup.4, fup.5) ~ treatment*gender,
data=OBrienKaiser)
coef(mod.ok)
## specify the model for the repeated measures:
phase <- factor(rep(c("pretest", "posttest", "followup"), c(5, 5, 5)),
levels=c("pretest", "posttest", "followup"))
hour <- ordered(rep(1:5, 3))
idata <- data.frame(phase, hour)
idata
## test the four-way interaction among the between-subject factors
## treatment and gender, and the intra-subject factors
## phase and hour
linear.hypothesis(mod.ok, c("treatment1:gender1", "treatment2:gender1"),
title="treatment:gender:phase:hour", idata=idata, idesign=~phase*hour,
iterms="phase:hour")