# glht: General Linear Hypotheses In multcomp: Simultaneous Inference in General Parametric Models

 glht R Documentation

## General Linear Hypotheses

### Description

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

### 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, ...)
## S3 method for class 'mlf'
glht(model, linfct, ...)
mcp(..., interaction_average = FALSE, covariate_average = FALSE)
```

### Arguments

 `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.

### Details

A general linear hypothesis refers to null hypotheses of the form H_0: K θ = m for some parametric model `model` with parameter estimates `coef(model)`.

The null hypothesis is specified by a linear function K θ, 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 θ = m, whereas `"less"` corresponds to H_0: K θ ≥ m and `"greater"` refers to H_0: K θ ≤ 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"`, `<=` 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 θ = m. The value of the linear function K θ 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` 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.

### 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)

### more complex models: Continuous outcome logistic
### regression; parameters are log-odds ratios
if (require("tram")) {
confint(glht(Colr(breaks ~ wool + tension,
data = warpbreaks),
linfct = mcp("tension" = "Tukey")))
}
```

multcomp documentation built on Aug. 7, 2022, 5:14 p.m.