Methods for General Linear Hypotheses
Simultaneous tests and confidence intervals for general linear hypotheses.
## S3 method for class 'glht'
summary(object, test = adjusted(), ...)
## S3 method for class 'glht'
confint(object, parm, level = 0.95, calpha = adjusted_calpha(),
...)
## S3 method for class 'glht'
coef(object, rhs = FALSE, ...)
## S3 method for class 'glht'
vcov(object, ...)
## S3 method for class 'confint.glht'
plot(x, xlim, xlab, ylim, ...)
## S3 method for class 'glht'
plot(x, ...)
univariate()
adjusted(type = c("single-step", "Shaffer", "Westfall", "free",
p.adjust.methods), ...)
Ftest()
Chisqtest()
adjusted_calpha(...)
univariate_calpha(...)object |
an object of class |
test |
a function for computing p values. |
parm |
additional parameters, currently ignored. |
level |
the confidence level required. |
calpha |
either a function computing the critical value or the critical value itself. |
rhs |
logical, indicating whether the linear function
K \hat{θ} or the right hand side
m ( |
type |
the multiplicity adjustment ( |
x |
an object of class |
xlim |
the |
ylim |
the y limits of the plot. |
xlab |
a label for the |
... |
additional arguments, such as |
The methods for general linear hypotheses as described by objects returned
by 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 K θ.
The test argument to summary takes a function specifying
the type of test to be applied. Classical Chisq (Wald test) or F statistics
for testing the global hypothesis H_0 are implemented in functions
Chisqtest and Ftest. Several approaches to multiplicity adjusted p
values for each of the linear hypotheses are implemented
in function adjusted. The type
argument to adjusted specifies the method to be applied:
"single-step" implements adjusted p values based on the joint
normal or t distribution of the linear function, and
"Shaffer" and "Westfall" implement logically constraint
multiplicity adjustments (Shaffer, 1986; Westfall, 1997).
"free" implements multiple testing procedures under free
combinations (Westfall et al, 1999).
In addition, all adjustment methods
implemented in p.adjust are available as well.
Simultaneous confidence intervals for linear functions can be computed
using method confint. Univariate confidence intervals
can be computed by specifying calpha = univariate_calpha()
to confint. The critical value can directly be specified as a scalar
to calpha as well. Note that plot(a) for some object a of class
glht is equivalent to 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 lm, are evaluated using the procedures
implemented in package 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).
Frank Bretz, Torsten Hothorn and Peter Westfall (2010), Multiple Comparisons Using R, CRC Press, Boca Raton.
Juliet P. Shaffer (1986), Modified sequentially rejective multiple test procedures. Journal of the American Statistical Association, 81, 826–831.
Peter H. Westfall (1997), Multiple testing of general contrasts using logical constraints and correlations. Journal of the American Statistical Association, 92, 299–306.
P. H. Westfall, R. D. Tobias, D. Rom, R. D. Wolfinger, Y. Hochberg (1999). 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.
Biometrical Journal, 50(3), 346–363;
See vignette("generalsiminf", package = "multcomp").
### 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 = 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))
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