General linear hypotheses and multiple comparisons for parametric models, including generalized linear models, linear mixed effects models, and survival models.

- Keywords
- htest

- model
- a fitted model,
for example an object returned by
`lm`

,`glm`

, or`aov`

etc. It is assumed that`coef`

and`vcov`

methods are available for`model`

. For multiple comparisons of means, methods`model.matrix`

,`model.frame`

and`terms`

are expected to be available for`model`

as well. - 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
in AN(C)OVA models are specified by objects returned from
function
`mcp`

. - 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 all`glht`

methods. For function`mcp`

, multiple comparisons are defined by matrices or symbolic descriptions specifying contrasts of factor levels where the arguments correspond to factor names.

```
"glht"(model, linfct, alternative = c("two.sided", "less", "greater"), rhs = 0, ...)
"glht"(model, linfct, ...)
"glht"(model, linfct, ...)
"glht"(model, linfct, ...)
"glht"(model, linfct, ...)
mcp(..., interaction_average = FALSE, covariate_average = FALSE)
```

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.

4) The `lsm`

function in package `lsmeans`

offers a symbolic
interface for the definition of least-squares means for factor combinations
which is very helpful when more complex contrasts are of special interest.

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).

- An object of class
- 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

`glht`

, more specifically a list with elements
`print`

, `summary`

,
`confint`

, `coef`

and `vcov`

methods being available. When called with `linfct`

being an
`mcp`

object, an additional element `focus`

is available
storing the names of the factors under test.```
### 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)
```

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")`

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