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, ...)
mcp(...)

Arguments

Details

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

The null hypothesis is specified by a linear function $K β$, 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 β = m, whereas "less" corresponds to H_0: K β >= m and "greater" refers to H_0: K β <= 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(beta) 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.

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 β = m. The value of the linear function $K β$ 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.

Value

An object of class glht, more specifically a list with elements

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

Shayle R. Searle (1971), Linear Models. John Wiley & Sons, New York.

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)))