1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
|
\name{glht-methods}
\alias{summary.glht}
\alias{confint.glht}
\alias{coef.glht}
\alias{vcov.glht}
\alias{plot.glht}
\alias{plot.confint.glht}
\alias{univariate}
\alias{adjusted}
\alias{Ftest}
\alias{Chisqtest}
\alias{adjusted_calpha}
\alias{univariate_calpha}
\title{ Methods for General Linear Hypotheses }
\description{
Simultaneous tests and confidence intervals for general linear
hypotheses.
}
\usage{
\method{summary}{glht}(object, test = adjusted(), ...)
\method{confint}{glht}(object, parm, level = 0.95, calpha = adjusted_calpha(),
...)
\method{coef}{glht}(object, rhs = FALSE, ...)
\method{vcov}{glht}(object, ...)
\method{plot}{confint.glht}(x, xlim, xlab, ylim, ...)
\method{plot}{glht}(x, ...)
univariate()
adjusted(type = c("single-step", "Shaffer", "Westfall", "free",
p.adjust.methods), ...)
Ftest()
Chisqtest()
adjusted_calpha(...)
univariate_calpha(...)
}
\arguments{
\item{object}{ an object of class \code{\link{glht}}.}
\item{test}{ a function for computing p values.}
\item{parm}{ additional parameters, currently ignored.}
\item{level}{ the confidence level required.}
\item{calpha}{ either a function computing the critical value or the critical value itself.}
\item{rhs}{logical, indicating whether the linear function
\eqn{K \hat{\theta}} or the right hand side
\eqn{m} (\code{rhs = TRUE}) of the linear hypothesis
should be returned.}
\item{type}{ the multiplicity adjustment (\code{adjusted})
to be applied. See below and \code{\link{p.adjust}}.}
\item{x}{an object of class \code{\link{glht}} or \code{confint.glht}.}
\item{xlim}{the \code{x} limits \code{(x1, x2)} of the plot.}
\item{ylim}{the y limits of the plot.}
\item{xlab}{a label for the \code{x} axis.}
\item{...}{ additional arguments, such as \code{maxpts},
\code{abseps} or \code{releps} to
\code{\link[mvtnorm]{pmvnorm}} in \code{adjusted} or
\code{\link[mvtnorm]{qmvnorm}} in \code{confint}. Note
that additional arguments specified to \code{summary},
\code{confint}, \code{coef} and \code{vcov} methods
are currently ignored.}
}
\details{
The methods for general linear hypotheses as described by objects returned
by \code{\link{glht}} can be used to actually test the global
null hypothesis, each of the partial hypotheses and for
simultaneous confidence intervals for the linear function \eqn{K \theta}.
The \code{\link{coef}} and \code{\link{vcov}} methods compute the linear
function \eqn{K \hat{\theta}} and its covariance, respectively.
The \code{test} argument to \code{summary} takes a function specifying
the type of test to be applied. Classical Chisq (Wald test) or F statistics
for testing the global hypothesis \eqn{H_0} are implemented in functions
\code{Chisqtest} and \code{Ftest}. Several approaches to multiplicity adjusted p
values for each of the linear hypotheses are implemented
in function \code{adjusted}. The \code{type}
argument to \code{adjusted} specifies the method to be applied:
\code{"single-step"} implements adjusted p values based on the joint
normal or t distribution of the linear function, and
\code{"Shaffer"} and \code{"Westfall"} implement logically constraint
multiplicity adjustments (Shaffer, 1986; Westfall, 1997).
\code{"free"} implements multiple testing procedures under free
combinations (Westfall et al, 1999).
In addition, all adjustment methods
implemented in \code{\link{p.adjust}} are available as well.
Simultaneous confidence intervals for linear functions can be computed
using method \code{\link{confint}}. Univariate confidence intervals
can be computed by specifying \code{calpha = univariate_calpha()}
to \code{confint}. The critical value can directly be specified as a scalar
to \code{calpha} as well. Note that \code{plot(a)} for some object \code{a} of class
\code{glht} is equivalent to \code{plot(confint(a))}.
All simultaneous inference procedures implemented here control
the family-wise error rate (FWER). Multivariate
normal and t distributions, the latter one only for models of
class \code{\link{lm}}, are evaluated using the procedures
implemented in package \code{mvtnorm}. Note that the default
procedure is stochastic. Reproducible p-values and confidence
intervals require appropriate settings of seeds.
A more detailed description of the underlying methodology is
available from Hothorn et al. (2008) and Bretz et al. (2010).
}
\value{
\code{summary} computes (adjusted) p values for general linear hypotheses,
\code{\link{confint}} computes (adjusted) confidence intervals.
\code{\link{coef}} returns estimates of the linear function \eqn{K \theta}
and \code{\link{vcov}} its covariance.
}
\references{
Frank Bretz, Torsten Hothorn and Peter Westfall (2010),
\emph{Multiple Comparisons Using R}, CRC Press, Boca Raton.
Juliet P. Shaffer (1986),
Modified sequentially rejective multiple test procedures.
\emph{Journal of the American Statistical Association},
\bold{81}, 826--831.
Peter H. Westfall (1997),
Multiple testing of general contrasts using logical constraints
and correlations. \emph{Journal of the American Statistical Association},
\bold{92}, 299--306.
P. H. Westfall, R. D. Tobias, D. Rom, R. D. Wolfinger, Y. Hochberg (1999).
\emph{Multiple Comparisons and Multiple Tests Using the SAS System}.
Cary, NC: SAS Institute Inc.
Torsten Hothorn, Frank Bretz and Peter Westfall (2008),
Simultaneous Inference in General Parametric Models.
\emph{Biometrical Journal}, \bold{50}(3), 346--363;
See \code{vignette("generalsiminf", package = "multcomp")}.
}
\examples{
### set up a two-way ANOVA
amod <- aov(breaks ~ wool + tension, data = warpbreaks)
### set up all-pair comparisons for factor `tension'
wht <- glht(amod, linfct = mcp(tension = "Tukey"))
### 95\% simultaneous confidence intervals
plot(print(confint(wht)))
### the same (for balanced designs only)
TukeyHSD(amod, "tension")
### corresponding adjusted p values
summary(wht)
### all means for levels of `tension'
amod <- aov(breaks ~ tension, data = warpbreaks)
glht(amod, linfct = matrix(c(1, 0, 0,
1, 1, 0,
1, 0, 1), byrow = TRUE, ncol = 3))
### confidence bands for a simple linear model, `cars' data
plot(cars, xlab = "Speed (mph)", ylab = "Stopping distance (ft)",
las = 1)
### fit linear model and add regression line to plot
lmod <- lm(dist ~ speed, data = cars)
abline(lmod)
### a grid of speeds
speeds <- seq(from = min(cars$speed), to = max(cars$speed),
length.out = 10)
### linear hypotheses: 10 selected points on the regression line != 0
K <- cbind(1, speeds)
### set up linear hypotheses
cht <- glht(lmod, linfct = K)
### confidence intervals, i.e., confidence bands, and add them plot
cci <- confint(cht)
lines(speeds, cci$confint[,"lwr"], col = "blue")
lines(speeds, cci$confint[,"upr"], col = "blue")
### simultaneous p values for parameters in a Cox model
if (require("survival") && require("MASS")) {
data("leuk", package = "MASS")
leuk.cox <- coxph(Surv(time) ~ ag + log(wbc), data = leuk)
### set up linear hypotheses
lht <- glht(leuk.cox, linfct = diag(length(coef(leuk.cox))))
### adjusted p values
print(summary(lht))
}
}
\keyword{htest}
|
