glht
General Linear Hypotheses
General linear hypotheses and multiple comparisons for parametric models, including generalized linear models, linear mixed effects models, and survival models.
- Keywords
- htest
Usage
## S3 method for class 'matrix':
glht(model, linfct,
alternative = c("two.sided", "less", "greater"),
rhs = 0, ...)
## S3 method for class 'character':
glht(model, linfct, ...)
## S3 method for class 'expression':
glht(model, linfct, ...)
## S3 method for class 'mcp':
glht(model, linfct, ...)
mcp(..., interaction_average = FALSE, covariate_average = FALSE)
Arguments
- model
- a fitted model,
for example an object returned by
lm
,glm
, oraov
etc. It is - linfct
- a specification of the linear hypotheses to be tested. Linear functions can be specified by either the matrix of coefficients or by symbolic descriptions of one or more linear hypotheses. Multiple comparisons
- alternative
- a character string specifying the alternative hypothesis, must be one of '"two.sided"' (default), '"greater"' or '"less"'. You can specify just the initial letter.
- rhs
- an optional numeric vector specifying the right hand side of the hypothesis.
- interaction_average
- logical indicating if comparisons are averaging over interaction terms. Experimental!
- covariate_average
- logical indicating if comparisons are averaging over additional covariates. Experimental!
- ...
- additional arguments to function
modelparm
in allglht
methods. For functionmcp
, multiple comparisons are defined by matrices or
Details
A general linear hypothesis refers to null hypotheses of the form
$H_0: K \theta = m$ for some parametric model
model
with parameter estimates coef(model)
.
The null hypothesis is specified by a linear function $K \theta$,
the direction of the alternative and the right hand side $m$.
Here, alternative
equal to "two.sided"
refers to
a null hypothesis $H_0: K \theta = m$, whereas
"less"
corresponds to $H_0: K \theta \ge m$ and
"greater"
refers to
$H_0: K \theta \le m$. The right hand side vector $m$ can be defined
via the rhs
argument.
The generic method glht
dispatches on its second argument
(linfct
). There are three ways, and thus methods,
to specify linear functions to be tested:
1) The matrix
of coefficients $K$ can be specified directly
via the linfct
argument. In this case,
the number of columns of this matrix needs to correspond to the number of
parameters estimated by model
. It is assumed that
appropriate coef
and vcov
methods are available
for model
(modelparm
deals with some exceptions).
2) A symbolic description,
either a character
or expression
vector passed to glht
via its linfct
argument, can be used to define
the null hypothesis. A symbolic description must be interpretable as a valid
R expression consisting of both the left and right hand side
of a linear hypothesis.
Only the names of coef(model)
must be used as variable
names. The alternative is given by
the direction under the null hypothesis (=
or ==
refer to "two.sided"
, <=< code=""> means
"greater"
and >=
indicates
"less"
). Numeric vectors of length one
are valid values for the right hand side.
3) Multiple comparisons of means are defined by objects
of class mcp
as returned by the mcp
function.
For each factor, which is included in model
as independent variable,
a contrast matrix or a symbolic description of the contrasts
can be specified as arguments to mcp
. A symbolic
description may be a character
or expression
where the factor levels
are only used as variables names. In addition,
the type
argument to the contrast generating function
contrMat
may serve as a symbolic description of
contrasts as well.=<>
The mcp
function must be used with care when defining parameters
of interest in two-way ANOVA or ANCOVA models. Here, the definition
of treatment differences (such as Tukey's all-pair comparisons or Dunnett's
comparison with a control) might be problem specific.
Because it is impossible to determine the parameters of interest
automatically in this case, mcp
in multcomp
version 1.0-0 and higher generates comparisons for the main effects
only, ignoring covariates and interactions (older versions
automatically averaged over interaction terms). A warning is given. We refer to
Hsu (1996), Chapter 7, and Searle (1971), Chapter 7.3,
for further discussions and examples on this
issue.
glht
extracts the number of degrees of freedom
for models of class lm
(via modelparm
) and the
exact multivariate t distribution is evaluated. For all other
models, results rely on the normal approximation. Alternatively, the
degrees of freedom to be used for the evaluation of multivariate
t distributions can be given by the additional df
argument to
modelparm
specified via ...
.
glht
methods return a specification of the null hypothesis
$H_0: K \theta = m$. The value of the linear function
$K \theta$ can be extracted using the coef
method and
the corresponding covariance matrix is available from the
vcov
method. Various simultaneous and univariate tests and
confidence intervals are available from summary.glht
and confint.glht
methods, respectively.
A more detailed description of the underlying methodology is available from Hothorn et al. (2008) and Bretz et al. (2010).
Value
- An object of class
glht
, more specifically a list with elements model a fitted model, used in the call to glht
linfct the matrix of linear functions rhs the vector of right hand side values $m$ coef the values of the linear functions vcov the covariance matrix of the values of the linear functions df optionally, the degrees of freedom when the exact t distribution is used for inference alternative a character string specifying the alternative hypothesis type optionally, a character string giving the name of the specific procedure - with
print
,summary
,confint
,coef
andvcov
methods being available. When called withlinfct
being anmcp
object, an additional elementfocus
is available storing the names of the factors under test.
References
Frank Bretz, Torsten Hothorn and Peter Westfall (2010), Multiple Comparisons Using R, CRC Press, Boca Raton.
Shayle R. Searle (1971), Linear Models. John Wiley & Sons, New York.
Jason C. Hsu (1996), Multiple Comparisons. Chapman & Hall, London.
Torsten Hothorn, Frank Bretz and Peter Westfall (2008),
Simultaneous Inference in General Parametric Models.
Biometrical Journal, 50(3), 346--363;
See vignette("generalsiminf", package = "multcomp")
.
Examples
### multiple linear model, swiss data
lmod <- lm(Fertility ~ ., data = swiss)
### test of H_0: all regression coefficients are zero
### (ignore intercept)
### define coefficients of linear function directly
K <- diag(length(coef(lmod)))[-1,]
rownames(K) <- names(coef(lmod))[-1]
K
### set up general linear hypothesis
glht(lmod, linfct = K)
### alternatively, use a symbolic description
### instead of a matrix
glht(lmod, linfct = c("Agriculture = 0",
"Examination = 0",
"Education = 0",
"Catholic = 0",
"Infant.Mortality = 0"))
### multiple comparison procedures
### set up a one-way ANOVA
amod <- aov(breaks ~ tension, data = warpbreaks)
### set up all-pair comparisons for factor `tension'
### using a symbolic description (`type' argument
### to `contrMat()')
glht(amod, linfct = mcp(tension = "Tukey"))
### alternatively, describe differences symbolically
glht(amod, linfct = mcp(tension = c("M - L = 0",
"H - L = 0",
"H - M = 0")))
### alternatively, define contrast matrix directly
contr <- rbind("M - L" = c(-1, 1, 0),
"H - L" = c(-1, 0, 1),
"H - M" = c(0, -1, 1))
glht(amod, linfct = mcp(tension = contr))
### alternatively, define linear function for coef(amod)
### instead of contrasts for `tension'
### (take model contrasts and intercept into account)
glht(amod, linfct = cbind(0, contr %*% contr.treatment(3)))
### mix of one- and two-sided alternatives
warpbreaks.aov <- aov(breaks ~ wool + tension,
data = warpbreaks)
### contrasts for `tension'
K <- rbind("L - M" = c( 1, -1, 0),
"M - L" = c(-1, 1, 0),
"L - H" = c( 1, 0, -1),
"M - H" = c( 0, 1, -1))
warpbreaks.mc <- glht(warpbreaks.aov,
linfct = mcp(tension = K),
alternative = "less")
### correlation of first two tests is -1
cov2cor(vcov(warpbreaks.mc))
### use smallest of the two one-sided
### p-value as two-sided p-value -> 0.0232
summary(warpbreaks.mc)