,
Frank Bretz [aut],
Peter Westfall [aut],
Richard M. Heiberger [ctb],
Andre Schuetzenmeister [ctb],
Susan Scheibe [ctb]| Title: | Simultaneous Inference in General Parametric Models |
|---|---|
| Description: | Simultaneous tests and confidence intervals for general linear hypotheses in parametric models, including linear, generalized linear, linear mixed effects, and survival models. The package includes demos reproducing analyzes presented in the book "Multiple Comparisons Using R" (Bretz, Hothorn, Westfall, 2010, CRC Press). |
| Authors: | Torsten Hothorn [aut, cre] ,
Frank Bretz [aut],
Peter Westfall [aut],
Richard M. Heiberger [ctb],
Andre Schuetzenmeister [ctb],
Susan Scheibe [ctb] |
| Maintainer: | Torsten Hothorn <[email protected]> |
| License: | GPL-2 |
| Version: | 1.4-25 |
| Built: | 2024-03-16 05:09:39 UTC |
| Source: | https://github.com/r-forge/multcomp |
Indicators of 28 adverse events in a two-arm clinical trial.
data(adevent)A data frame with 160 observations on the following 29 variables.
E1a factor with levels no event event
E2a factor with levels no event event
E3a factor with levels no event event
E4a factor with levels no event event
E5a factor with levels no event event
E6a factor with levels no event event
E7a factor with levels no event event
E8a factor with levels no event event
E9a factor with levels no event event
E10a factor with levels no event event
E11a factor with levels no event event
E12a factor with levels no event event
E13a factor with levels no event event
E14a factor with levels no event event
E15a factor with levels no event event
E16a factor with levels no event event
E17a factor with levels no event event
E18a factor with levels no event event
E19a factor with levels no event event
E20a factor with levels no event event
E21a factor with levels no event event
E22a factor with levels no event event
E23a factor with levels no event event
E24a factor with levels no event event
E25a factor with levels no event event
E26a factor with levels no event event
E27a factor with levels no event event
E28a factor with levels no event event
groupgroup indicator.
The data is provided by Westfall et al. (1999, p. 242) and contains binary
indicators of 28 adverse events (E1,..., E28) for
two arms (group).
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.
A convenience function for univariate testing via z- and t-tests of estimated model coefficients
cftest(model, parm, test = univariate(), ...)model |
a fitted model. |
parm |
a vector of parameters to be tested, either a character vector of names or an integer. |
test |
a function for computing p values, see |
... |
additional arguments passed to |
The usual z- or t-tests are tested without adjusting for multiplicity.
An object of class summary.glht.
lmod <- lm(dist ~ speed, data = cars)
summary(lmod)
cftest(lmod)Cholesterol reduction for five treatments.
data("cholesterol")This data frame contains the following variables
treatment groups, a factor at levels 1time, 2times,
4times, drugD and drugE.
cholesterol reduction.
A clinical study was conducted to assess the effect of three formulations
of the same drug on reducing cholesterol. The formulations were
20mg at once (1time), 10mg twice a day (2times), and 5mg
four times a day (4times). In addition, two competing drugs
were used as control group (drugD and drugE). The purpose of
the study was to find which of the formulations, if any, is efficacious and how
these formulations compare with the existing drugs.
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., page 153.
### adjusted p-values for all-pairwise comparisons in a one-way layout
### set up ANOVA model
amod <- aov(response ~ trt, data = cholesterol)
### set up multiple comparisons object for all-pair comparisons
cht <- glht(amod, linfct = mcp(trt = "Tukey"))
### cf. Westfall et al. (1999, page 171)
summary(cht, test = univariate())
summary(cht, test = adjusted("Shaffer"))
summary(cht, test = adjusted("Westfall"))
### use only a subset of all pairwise hypotheses
K <- contrMat(table(cholesterol$trt), type="Tukey")
Ksub <- rbind(K[c(1,2,5),],
"D - test" = c(-1, -1, -1, 3, 0),
"E - test" = c(-1, -1, -1, 0, 3))
### reproduce results in Westfall et al. (1999, page 172)
### note: the ordering of our estimates here is different
amod <- aov(response ~ trt - 1, data = cholesterol)
summary(glht(amod, linfct = mcp(trt = Ksub[,5:1])),
test = adjusted("Westfall"))Extract information from glht, summary.glht or
confint.glht objects which is required to create
and plot compact letter displays of all pair-wise comparisons.
## S3 method for class 'summary.glht'
cld(object, level = 0.05, decreasing = FALSE, ...)
## S3 method for class 'glht'
cld(object, level = 0.05, decreasing = FALSE, ...)
## S3 method for class 'confint.glht'
cld(object, decreasing = FALSE, ...)object |
An object of class |
level |
Significance-level to be used to term a specific pair-wise comparison significant. |
decreasing |
logical. Should the order of the letters be increasing or decreasing? |
... |
additional arguments. |
This function extracts all the information from glht,
summary.glht or confint.glht objects that is required
to create a compact letter display of all pair-wise comparisons.
In case the contrast matrix is not of type "Tukey", an error
is issued. In case of confint.glht objects, a pair-wise comparison
is termed significant whenever a particular confidence interval contains 0.
Otherwise, p-values are compared to the value of "level".
Once, this information is extracted, plotting of all pair-wise
comparisons can be carried out.
An object of class cld, a list with items:
y |
Values of the response variable of the original model. |
yname |
Name of the response variable. |
x |
Values of the variable used to compute Tukey contrasts. |
weights |
Weights used in the fitting process. |
lp |
Predictions from the fitted model. |
covar |
A logical indicating whether the fitted model contained covariates. |
signif |
Vector of logicals indicating significant differences with hyphenated names that identify pair-wise comparisons. |
Hans-Peter Piepho (2004), An Algorithm for a Letter-Based Representation of All-Pairwise Comparisons, Journal of Computational and Graphical Statistics, 13(2), 456–466.
### multiple comparison procedures
### set up a one-way ANOVA
data(warpbreaks)
amod <- aov(breaks ~ tension, data = warpbreaks)
### specify all pair-wise comparisons among levels of variable "tension"
tuk <- glht(amod, linfct = mcp(tension = "Tukey"))
### extract information
tuk.cld <- cld(tuk)
### use sufficiently large upper margin
old.par <- par(mai=c(1,1,1.25,1), no.readonly = TRUE)
### plot
plot(tuk.cld)
par(old.par)
### now using covariates
data(warpbreaks)
amod2 <- aov(breaks ~ tension + wool, data = warpbreaks)
### specify all pair-wise comparisons among levels of variable "tension"
tuk2 <- glht(amod2, linfct = mcp(tension = "Tukey"))
### extract information
tuk.cld2 <- cld(tuk2)
### use sufficiently large upper margin
old.par <- par(mai=c(1,1,1.25,1), no.readonly = TRUE)
### plot using different colors
plot(tuk.cld2, col=c("black", "red", "blue"))
par(old.par)
### set up all pair-wise comparisons for count data
data(Titanic)
mod <- glm(Survived ~ Class, data = as.data.frame(Titanic), weights = Freq, family = binomial())
### specify all pair-wise comparisons among levels of variable "Class"
glht.mod <- glht(mod, mcp(Class = "Tukey"))
### extract information
mod.cld <- cld(glht.mod)
### use sufficiently large upper margin
old.par <- par(mai=c(1,1,1.5,1), no.readonly = TRUE)
### plot
plot(mod.cld)
par(old.par)Survival in a randomised trial comparing three treatments for Chronic Myelogeneous Leukemia (simulated data).
data("cml")A data frame with 507 observations on the following 7 variables.
centera factor with 54 levels indicating the study center.
treatmenta factor with levels trt1, trt2, trt3 indicating the treatment group.
sexsex (0 = female, 1 = male)
ageage in years
riskgrouprisk group (0 = low, 1 = medium, 2 = high)
statuscensoring status (FALSE = censored, TRUE = dead)
timesurvival or censoring time in days.
The data are simulated according to structure of the data by the German CML Study Group used in Hehlmann (1994).
R. Hehlmann, H. Heimpel, J. Hasford, H.J. Kolb, H. Pralle, D.K. Hossfeld, W. Queisser, H. Loeffler, A. Hochhaus, B. Heinze (1994), Randomized comparison of interferon-alpha with busulfan and hydroxyurea in chronic myelogenous leukemia. The German CML study group. Blood 84(12):4064-4077.
if (require("coxme")) {
data("cml")
### one-sided simultaneous confidence intervals for many-to-one
### comparisons of treatment effects concerning time of survival
### modeled by a frailty Cox model with adjustment for further
### covariates and center-specific random effect.
cml_coxme <- coxme(Surv(time, status) ~ treatment + sex + age + riskgroup + (1|center),
data = cml)
glht_coxme <- glht(model = cml_coxme, linfct = mcp(treatment = "Dunnett"),
alternative = "greater")
ci_coxme <- confint(glht_coxme)
exp(ci_coxme$confint)[1:2,]
}Computes contrast matrices for several multiple comparison procedures.
contrMat(n, type = c("Dunnett", "Tukey", "Sequen", "AVE",
"Changepoint", "Williams", "Marcus",
"McDermott", "UmbrellaWilliams", "GrandMean"),
base = 1)n |
a (possibly named) vector of sample sizes for each group. |
type |
type of contrast. |
base |
an integer specifying which group is considered the baseline group for Dunnett contrasts. |
Computes the requested matrix of contrasts for comparisons of mean levels.
The matrix of contrasts with appropriate row names is returned.
Frank Bretz, Torsten Hothorn and Peter Westfall (2010), Multiple Comparisons Using R, CRC Press, Boca Raton.
Frank Bretz, Alan Genz and Ludwig A. Hothorn (2001), On the numerical availability of multiple comparison procedures. Biometrical Journal, 43(5), 645–656.
n <- c(10,20,30,40)
names(n) <- paste("group", 1:4, sep="")
contrMat(n) # Dunnett is default
contrMat(n, base = 2) # use second level as baseline
contrMat(n, type = "Tukey")
contrMat(n, type = "Sequen")
contrMat(n, type = "AVE")
contrMat(n, type = "Changepoint")
contrMat(n, type = "Williams")
contrMat(n, type = "Marcus")
contrMat(n, type = "McDermott")
### Umbrella-protected Williams contrasts, i.e. a sequence of
### Williams-type contrasts with groups of higher order
### stepwise omitted
contrMat(n, type = "UmbrellaWilliams")
### comparison of each group with grand mean of all groups
contrMat(n, type = "GrandMean")Detergent durability in an incomplete two-way design.
data("detergent")This data frame contains the following variables
detergent, a factor at levels A, B,
C, D, and E.
block, a factor at levels B_1, ..., B_10.
response variable: number of plates washed before the foam disappears.
Plates were washed with five detergent varieties, in ten blocks. A complete design would have 50 combinations, here only three detergent varieties in each block were applied in a balanced incomplete block design. Note that there are six observations taken at each detergent level.
H. Scheffe (1959). The Analysis of Variance. New York: John Wiley & Sons, page 189.
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., page 189.
### set up two-way ANOVA without interactions
amod <- aov(plates ~ block + detergent, data = detergent)
### set up all-pair comparisons
dht <- glht(amod, linfct = mcp(detergent = "Tukey"))
### see Westfall et al. (1999, p. 190)
confint(dht)
### see Westfall et al. (1999, p. 192)
summary(dht, test = univariate())
## Not run:
summary(dht, test = adjusted("Shaffer"))
summary(dht, test = adjusted("Westfall"))
## End(Not run)Fatty acid content of different putative ecotypes of Bacillus simplex.
data("fattyacid")A data frame with 93 observations on the following 2 variables.
PEa factor with levels PE3, PE4, PE5, PE6, PE7, PE9 indicating the putative ecotype (PE).
FAa numeric vector indicating the content of fatty acid (FA).
The data give the fatty acid content for different putative ecotypes of Bacillus simplex. Variances of the values of fatty acid are heterogeneous among the putative ecotypes.
J. Sikorski, E. Brambilla, R. M. Kroppenstedt, B. J. Tindal (2008), The temperature adaptive fatty acid content in Bacillus simplex strains from ”Evolution Canyon“, Israel. Microbiology 154, 2416-2426.
if (require("sandwich")) {
data("fattyacid")
### all-pairwise comparisons of the means of fatty acid content
### FA between different putative ecotypes PE accounting for
### heteroscedasticity by using a heteroscedastic consistent
### covariance estimation
amod <- aov(FA ~ PE, data = fattyacid)
amod_glht <- glht(amod, mcp(PE = "Tukey"), vcov = vcovHC)
summary(amod_glht)
### simultaneous confidence intervals for the differences of
### means of fatty acid content between the putative ecotypes
confint(amod_glht)
}General linear hypotheses and multiple comparisons for parametric models, including generalized linear models, linear mixed effects models, and survival models.
## 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)model |
a fitted model,
for example an object returned by |
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 |
.
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 |
A general linear hypothesis refers to null hypotheses of the form
for some parametric model
model with parameter estimates coef(model).
The null hypothesis is specified by a linear function ,
the direction of the alternative and the right hand side .
Here, alternative equal to "two.sided" refers to
a null hypothesis , whereas
"less" corresponds to and
"greater" refers to
. The right hand side vector 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 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
. The value of the linear function
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 glht, more specifically a list with elements
model |
a fitted model, used in the call to |
linfct |
the matrix of linear functions |
rhs |
the vector of right hand side values |
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.
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").
### 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", quietly = TRUE, warn.conflicts = FALSE)) {
confint(glht(Colr(breaks ~ wool + tension,
data = warpbreaks),
linfct = mcp("tension" = "Tukey")))
}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
|
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 .
The coef and vcov methods compute the linear
function and its covariance, respectively.
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 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).
summary computes (adjusted) p values for general linear hypotheses,
confint computes (adjusted) confidence intervals.
coef returns estimates of the linear function
and vcov its covariance.
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))
}Dose response of litter weights in rats.
data("litter")This data frame contains the following variables
dosages at four levels: 0, 5, 50,
500.
gestation time as covariate.
number of animals in litter as covariate.
response variable: average post-birth weights in the entire litter.
Pregnant mice were divided into four groups and the compound in four different doses was administered during pregnancy. Their litters were evaluated for birth weights.
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., page 109.
P. H. Westfall (1997). Multiple Testing of General Contrasts Using Logical Constraints and Correlations. Journal of the American Statistical Association, 92(437), 299–306.
### fit ANCOVA model to data
amod <- aov(weight ~ dose + gesttime + number, data = litter)
### define matrix of linear hypotheses for `dose'
doselev <- as.integer(levels(litter$dose))
K <- rbind(contrMat(table(litter$dose), "Tukey"),
otrend = c(-1.5, -0.5, 0.5, 1.5),
atrend = doselev - mean(doselev),
ltrend = log(1:4) - mean(log(1:4)))
### set up multiple comparison object
Kht <- glht(amod, linfct = mcp(dose = K), alternative = "less")
### cf. Westfall (1997, Table 2)
summary(Kht, test = univariate())
summary(Kht, test = adjusted("bonferroni"))
summary(Kht, test = adjusted("Shaffer"))
summary(Kht, test = adjusted("Westfall"))
summary(Kht, test = adjusted("single-step"))Calculation of correlation between test statistics from multiple marginal models using the score decomposition
mmm(...)
mlf(...)... |
A names argument list containing fitted models ( |
Estimated correlations of the estimated parameters of interest from the multiple marginal models are obtained using a stacked version of the i.i.d. decomposition of parameter estimates by means of score components (first derivatives of the log likelihood). The method is less conservative than the Bonferroni correction. The details are provided by Pipper, Ritz and Bisgaard (2012).
The implementation assumes that the model were fitted to the same data,
i.e., the rows of the matrices returned by estfun belong to the
same observations for each model.
The reference distribution is always multivariate normal, if you want
to use the multivariate t, please specify the corresponding degrees of
freedom as an additional df argument to glht.
Observations with missing values contribute zero to the score function.
Models have to be fitted using na.exclude as na.action
argument.
An object of class mmm or mlf, basically a named list of the
arguments with a special method for glht being available for
the latter. vcov, estfun, and
bread methods are available for objects of class
mmm.
Code for the computation of the joint covariance and sandwich matrices was contributed by Christian Ritz and Christian B. Pipper.
Christian Bressen Pipper, Christian Ritz and Hans Bisgaard (2012), A Versatile Method for Confirmatory Evaluation of the Effects of a Covariate in Multiple Models, Journal of the Royal Statistical Society, Series C (Applied Statistics), 61, 315–326.
### replicate analysis of Hasler & Hothorn (2011),
### A Dunnett-Type Procedure for Multiple Endpoints,
### The International Journal of Biostatistics: Vol. 7: Iss. 1, Article 3.
### DOI: 10.2202/1557-4679.1258
library("sandwich")
### see ?coagulation
if (require("SimComp")) {
data("coagulation", package = "SimComp")
### level "S" is the standard, "H" and "B" are novel procedures
coagulation$Group <- relevel(coagulation$Group, ref = "S")
### fit marginal models
(m1 <- lm(Thromb.count ~ Group, data = coagulation))
(m2 <- lm(ADP ~ Group, data = coagulation))
(m3 <- lm(TRAP ~ Group, data = coagulation))
### set-up Dunnett comparisons for H - S and B - S
### for all three models
g <- glht(mmm(Thromb = m1, ADP = m2, TRAP = m3),
mlf(mcp(Group = "Dunnett")), alternative = "greater")
### joint correlation
cov2cor(vcov(g))
### simultaneous p-values adjusted by taking the correlation
### between the score contributions into account
summary(g)
### simultaneous confidence intervals
confint(g)
### compare with
## Not run:
library("SimComp")
SimCiDiff(data = coagulation, grp = "Group",
resp = c("Thromb.count","ADP","TRAP"),
type = "Dunnett", alternative = "greater",
covar.equal = TRUE)
## End(Not run)
### use sandwich variance matrix
g <- glht(mmm(Thromb = m1, ADP = m2, TRAP = m3),
mlf(mcp(Group = "Dunnett")),
alternative = "greater", vcov = sandwich)
summary(g)
confint(g)
}
### attitude towards science data
data("mn6.9", package = "TH.data")
### one model for each item
mn6.9.y1 <- glm(y1 ~ group, family = binomial(),
na.action = na.omit, data = mn6.9)
mn6.9.y2 <- glm(y2 ~ group, family = binomial(),
na.action = na.omit, data = mn6.9)
mn6.9.y3 <- glm(y3 ~ group, family = binomial(),
na.action = na.omit, data = mn6.9)
mn6.9.y4 <- glm(y4 ~ group, family = binomial(),
na.action = na.omit, data = mn6.9)
### test all parameters simulaneously
summary(glht(mmm(mn6.9.y1, mn6.9.y2, mn6.9.y3, mn6.9.y4),
mlf(diag(2))))
### group differences
summary(glht(mmm(mn6.9.y1, mn6.9.y2, mn6.9.y3, mn6.9.y4),
mlf("group2 = 0")))
### alternative analysis of Klingenberg & Satopaa (2013),
### Simultaneous Confidence Intervals for Comparing Margins of
### Multivariate Binary Data, CSDA, 64, 87-98
### http://dx.doi.org/10.1016/j.csda.2013.02.016
### see supplementary material for data description
### NOTE: this is not the real data but only a subsample
influenza <- structure(list(
HEADACHE = c(1L, 0L, 0L, 1L, 0L, 0L, 1L, 1L, 1L,
0L, 0L, 1L, 0L, 1L, 0L, 1L, 1L, 1L, 1L, 1L, 1L, 0L, 0L, 0L, 0L,
1L, 1L), MALAISE = c(0L, 0L, 1L, 1L, 0L, 1L, 1L, 1L, 0L, 1L,
0L, 0L, 1L, 1L, 0L, 0L, 1L, 0L, 1L, 0L, 1L, 0L, 1L, 1L, 0L, 1L,
0L), PYREXIA = c(0L, 0L, 0L, 0L, 0L, 1L, 0L, 1L, 0L, 0L, 1L,
1L, 1L, 1L, 0L, 0L, 0L, 0L, 0L, 1L, 1L, 1L, 0L, 0L, 0L, 1L, 1L
), ARTHRALGIA = c(0L, 0L, 0L, 0L, 1L, 0L, 1L, 0L, 1L, 1L, 0L,
0L, 1L, 1L, 0L, 1L, 0L, 0L, 1L, 0L, 0L, 0L, 0L, 1L, 1L, 1L, 1L
), group = structure(c(2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L), .Label = c("pla", "trt"), class = "factor"), Freq = c(32L,
165L, 10L, 23L, 3L, 1L, 4L, 2L, 4L, 2L, 1L, 1L, 1L, 1L, 167L,
1L, 11L, 37L, 7L, 7L, 5L, 3L, 3L, 1L, 2L, 4L, 2L)), .Names = c("HEADACHE",
"MALAISE", "PYREXIA", "ARTHRALGIA", "group", "Freq"), row.names = c(1L,
2L, 3L, 5L, 9L, 36L, 43L, 50L, 74L, 83L, 139L, 175L, 183L, 205L,
251L, 254L, 255L, 259L, 279L, 281L, 282L, 286L, 302L, 322L, 323L,
366L, 382L), class = "data.frame")
influenza <- influenza[rep(1:nrow(influenza), influenza$Freq), 1:5]
### Fitting marginal logistic regression models
(head_logreg <- glm(HEADACHE ~ group, data = influenza,
family = binomial()))
(mala_logreg <- glm(MALAISE ~ group, data = influenza,
family = binomial()))
(pyre_logreg <- glm(PYREXIA ~ group, data = influenza,
family = binomial()))
(arth_logreg <- glm(ARTHRALGIA ~ group, data = influenza,
family = binomial()))
### Simultaneous inference for log-odds
xy.sim <- glht(mmm(head = head_logreg,
mala = mala_logreg,
pyre = pyre_logreg,
arth = arth_logreg),
mlf("grouptrt = 0"))
summary(xy.sim)
confint(xy.sim)
### Artificial examples
### Combining linear regression and logistic regression
set.seed(29)
y1 <- rnorm(100)
y2 <- factor(y1 + rnorm(100, sd = .1) > 0)
x1 <- gl(4, 25)
x2 <- runif(100, 0, 10)
m1 <- lm(y1 ~ x1 + x2)
m2 <- glm(y2 ~ x1 + x2, family = binomial())
### Note that the same explanatory variables are considered in both models
### but the resulting parameter estimates are on 2 different scales
### (original and log-odds scales)
### Simultaneous inference for the same parameter in the 2 model fits
summary(glht(mmm(m1 = m1, m2 = m2), mlf("x12 = 0")))
### Simultaneous inference for different parameters in the 2 model fits
summary(glht(mmm(m1 = m1, m2 = m2),
mlf(m1 = "x12 = 0", m2 = "x13 = 0")))
### Simultaneous inference for different and identical parameters in the 2
### model fits
summary(glht(mmm(m1 = m1, m2 = m2),
mlf(m1 = c("x12 = 0", "x13 = 0"), m2 = "x13 = 0")))
### Examples for binomial data
### Two independent outcomes
y1.1 <- rbinom(100, 1, 0.45)
y1.2 <- rbinom(100, 1, 0.55)
group <- factor(rep(c("A", "B"), 50))
m1 <- glm(y1.1 ~ group, family = binomial)
m2 <- glm(y1.2 ~ group, family = binomial)
summary(glht(mmm(m1 = m1, m2 = m2),
mlf("groupB = 0")))
### Two perfectly correlated outcomes
y2.1 <- rbinom(100, 1, 0.45)
y2.2 <- y2.1
group <- factor(rep(c("A", "B"), 50))
m1 <- glm(y2.1 ~ group, family = binomial)
m2 <- glm(y2.2 ~ group, family = binomial)
summary(glht(mmm(m1 = m1, m2 = m2),
mlf("groupB = 0")))
### use sandwich covariance matrix
summary(glht(mmm(m1 = m1, m2 = m2),
mlf("groupB = 0"), vcov = sandwich))Extract model parameters and their covariance matrix as well as degrees of freedom (if available) from a fitted model.
modelparm(model, coef., vcov., df, ...)model |
a fitted model,
for example an object returned by |
coef. |
an accessor function for the model parameters. Alternatively, the vector of coefficients. |
vcov. |
an accessor function for the covariance matrix of the model parameters. Alternatively, the covariance matrix directly. |
df |
an optional specification of the degrees of freedom to be used in subsequent computations. |
... |
additional arguments, currently ignored. |
One can't expect coef and vcov methods
for arbitrary models to
return a vector of fixed effects model parameters (coef)
and corresponding covariance matrix (vcov).
The coef. and vcov. arguments can be used to define
modified coef or vcov methods for a specific model.
Methods for lmer, fixest,
and survreg objects are available (internally).
For objects inheriting from class lm the degrees of
freedom are determined from model and the corresponding
multivariate t distribution is used by all methods to glht
objects. By default, the asymptotic multivariate normal distribution
is used in all other cases unless df is specified by the user.
An object of class modelparm with elements
coef |
model parameters |
vcov |
covariance matrix of model parameters |
df |
degrees of freedom |
Measurements on four endpoints in a two-arm clinical trial.
data(mtept)A data frame with 111 observations on the following 5 variables.
treatmenta factor with levels Drug Placebo
E1endpoint 1
E2endpoint 2
E3endpoint 3
E4endpoint 4
The data (from Westfall et al., 1999) contain measurements of patients
in treatment (Drug) and control (Placebo) groups, with four
outcome variables.
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.
Directly specify estimated model parameters and their covariance matrix.
parm(coef, vcov, df = 0)coef |
estimated coefficients. |
vcov |
estimated covariance matrix of the coefficients. |
df |
an optional specification of the degrees of freedom to be used in subsequent computations. |
When only estimated model parameters and the corresponding
covariance matrix is available for simultaneous inference
using glht (for example, when only the results
but not the original data are available or, even worse, when the model
has been fitted outside R), function parm sets up an
object glht is able to compute on (mainly
by offering coef and vcov methods).
Note that the linear function in glht can't
be specified via mcp since the model terms
are missing.
An object of class parm with elements
coef |
model parameters |
vcov |
covariance matrix of model parameters |
df |
degrees of freedom |
## example from
## Bretz, Hothorn, and Westfall (2002).
## On multiple comparisons in R. R News, 2(3):14-17.
beta <- c(V1 = 14.8, V2 = 12.6667, V3 = 7.3333, V4 = 13.1333)
Sigma <- 6.7099 * (diag(1 / c(20, 3, 3, 15)))
confint(glht(model = parm(beta, Sigma, 37),
linfct = c("V2 - V1 >= 0",
"V3 - V1 >= 0",
"V4 - V1 >= 0")),
level = 0.9)Plot information of glht, summary.glht or confint.glht
objects stored as cld objects together with a compact
letter display of all pair-wise comparisons.
## S3 method for class 'cld'
plot(x, type = c("response", "lp"), ...)x |
An object of class |
type |
Should the response or the linear predictor (lp) be plotted.
If there are any covariates, the lp is automatically used. To
use the response variable, set |
... |
Other optional print parameters which are passed to the plotting functions. |
This function plots the information stored in glht, summary.glht or
confint.glht objects. Prior to plotting, these objects have to be converted to
cld objects (see cld for details).
All types of plots include a compact letter display (cld) of all pair-wise comparisons.
Equal letters indicate no significant differences. Two levels are significantly
different, in case they do not have any letters in common.
If the fitted model contains any covariates, a boxplot of the linear predictor is
generated with the cld within the upper margin. Otherwise, three different types
of plots are used depending on the class of variable y of the cld object.
In case of class(y) == "numeric", a boxplot is generated using the response variable,
classified according to the levels of the variable used for the Tukey contrast
matrix. Is class(y) == "factor", a mosaic plot is generated, and the cld is printed
above. In case of class(y) == "Surv", a plot of fitted survival functions is generated
where the cld is plotted within the legend.
The compact letter display is computed using the algorithm of Piepho (2004).
Note: The user has to provide a sufficiently large upper margin which can be used to
depict the compact letter display (see examples).
Hans-Peter Piepho (2004), An Algorithm for a Letter-Based Representation of All-Pairwise Comparisons, Journal of Computational and Graphical Statistics, 13(2), 456–466.
glht
cld
cld.summary.glht
cld.confint.glht
cld.glht
boxplot
mosaicplot
plot.survfit
### multiple comparison procedures
### set up a one-way ANOVA
data(warpbreaks)
amod <- aov(breaks ~ tension, data = warpbreaks)
### specify all pair-wise comparisons among levels of variable "tension"
tuk <- glht(amod, linfct = mcp(tension = "Tukey"))
### extract information
tuk.cld <- cld(tuk)
### use sufficiently large upper margin
old.par <- par(mai=c(1,1,1.25,1), no.readonly=TRUE)
### plot
plot(tuk.cld)
par(old.par)
### now using covariates
amod2 <- aov(breaks ~ tension + wool, data = warpbreaks)
tuk2 <- glht(amod2, linfct = mcp(tension = "Tukey"))
tuk.cld2 <- cld(tuk2)
old.par <- par(mai=c(1,1,1.25,1), no.readonly=TRUE)
### use different colors for boxes
plot(tuk.cld2, col=c("green", "red", "blue"))
par(old.par)
### get confidence intervals
ci.glht <- confint(tuk)
### plot them
plot(ci.glht)
old.par <- par(mai=c(1,1,1.25,1), no.readonly=TRUE)
### use 'confint.glht' object to plot all pair-wise comparisons
plot(cld(ci.glht), col=c("white", "blue", "green"))
par(old.par)
### set up all pair-wise comparisons for count data
data(Titanic)
mod <- glm(Survived ~ Class, data = as.data.frame(Titanic),
weights = Freq, family = binomial())
### specify all pair-wise comparisons among levels of variable "Class"
glht.mod <- glht(mod, mcp(Class = "Tukey"))
### extract information
mod.cld <- cld(glht.mod)
### use sufficiently large upper margin
old.par <- par(mai=c(1,1,1.5,1), no.readonly=TRUE)
### plot
plot(mod.cld)
par(old.par)
### set up all pair-wise comparisons of a Cox-model
if (require("survival") && require("MASS")) {
### construct 4 classes of age
Melanoma$Cage <- factor(sapply(Melanoma$age, function(x){
if( x <= 25 ) return(1)
if( x > 25 & x <= 50 ) return(2)
if( x > 50 & x <= 75 ) return(3)
if( x > 75 & x <= 100) return(4) }
))
### fit Cox-model
cm <- coxph(Surv(time, status == 1) ~ Cage, data = Melanoma)
### specify all pair-wise comparisons among levels of "Cage"
cm.glht <- glht(cm, mcp(Cage = "Tukey"))
# extract information & plot
old.par <- par(no.readonly=TRUE)
### use mono font family
if (dev.interactive())
old.par <- par(family = "mono")
plot(cld(cm.glht), col=c("black", "red", "blue", "green"))
par(old.par)
}
if (require("nlme") && require("lme4")) {
data("ergoStool", package = "nlme")
stool.lmer <- lmer(effort ~ Type + (1 | Subject),
data = ergoStool)
glme41 <- glht(stool.lmer, mcp(Type = "Tukey"))
old.par <- par(mai=c(1,1,1.5,1), no.readonly=TRUE)
plot(cld(glme41))
par(old.par)
}Recovery time after surgery.
data("recovery")This data frame contains the following variables
blanket type, a factor at four levels: b0, b1,
b2, and b3.
response variable: recovery time after a surgical procedure.
A company developed specialized heating blankets designed to help the body heat following a surgical procedure. Four types of blankets were tried on surgical patients with the aim of comparing the recovery time of patients. One of the blanket was a standard blanket that had been in use already in various hospitals.
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., page 66.
### set up one-way ANOVA
amod <- aov(minutes ~ blanket, data = recovery)
### set up multiple comparisons: one-sided Dunnett contrasts
rht <- glht(amod, linfct = mcp(blanket = "Dunnett"),
alternative = "less")
### cf. Westfall et al. (1999, p. 80)
confint(rht, level = 0.9)
### the same
rht <- glht(amod, linfct = mcp(blanket = c("b1 - b0 >= 0",
"b2 - b0 >= 0",
"b3 - b0 >= 0")))
confint(rht, level = 0.9)Systolic blood pressure, age and gender of 69 people.
data("sbp")A data frame with 69 observations on the following 3 variables.
gendera factor with levels male female
sbpsystolic blood pressure in mmHg
ageage in years
D. G. Kleinbaum, L. L. Kupper, K. E. Muller, A. Nizam, A. (1998), Applied Regression Analysis and Other Multivariable Methods, Duxbury Press, North Scituate, MA.
Damages on young trees caused by deer browsing.
data("trees513")A data frame with 2700 observations on the following 4 variables.
damagea factor with levels yes and no
indicating whether or not the trees has been damaged by game animals,
mostly roe deer.
speciesa factor with levels spruce, fir,
pine, softwood (other), beech, oak, ash/maple/elm/lime,
and hardwood (other).
latticea factor with levels 1, ..., 53,
essentially a number indicating the position of the sampled area.
plota factor with levels x_1, ..., x_5 where
x is the lattice. plot
is nested within lattice and is a replication for each lattice point.
In most parts of Germany, the natural or artificial regeneration of forests is difficult due to a high browsing intensity. Young trees suffer from browsing damage, mostly by roe and red deer. In order to estimate the browsing intensity for several tree species, the Bavarian State Ministry of Agriculture and Foresty conducts a survey every three years. Based on the estimated percentage of damaged trees, suggestions for the implementation or modification of deer management plans are made. The survey takes place in all 756 game management districts (‘Hegegemeinschaften’) in Bavaria. The data given here are from the game management district number 513 ‘Unterer Aischgrund’ (located in Frankonia between Erlangen and H\"ochstadt) in 2006. The data of 2700 trees include the species and a binary variable indicating whether or not the tree suffers from damage caused by deer browsing.
Bayerisches Staatsministerium fuer Landwirtschaft und Forsten (2006), Forstliche Gutachten zur Situation der Waldverjuengung 2006. https://www.stmelf.bayern.de/wald/
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").
summary(trees513)Industrial waste output in a manufactoring plant.
data("waste")This data frame contains the following variables
temperature, a factor at three levels: low, medium, high.
environment, a factor at five levels: env1 ... env5.
response variable: waste output in a manufacturing plant.
The data are from an experiment designed to study the effect of temperature
(temp) and environment (envir) on waste output in a manufactoring plant.
Two replicate measurements were taken at each temperature / environment combination.
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., page 177.
### set up two-way ANOVA with interactions
amod <- aov(waste ~ temp * envir, data=waste)
### comparisons of main effects only
K <- glht(amod, linfct = mcp(temp = "Tukey"))$linfct
K
glht(amod, K)
### comparisons of means (by averaging interaction effects)
low <- grep("low:envi", colnames(K))
med <- grep("medium:envi", colnames(K))
K[1, low] <- 1 / (length(low) + 1)
K[2, med] <- 1 / (length(low) + 1)
K[3, med] <- 1 / (length(low) + 1)
K[3, low] <- - 1 / (length(low) + 1)
K
confint(glht(amod, K))
### same as TukeyHSD
TukeyHSD(amod, "temp")
### set up linear hypotheses for all-pairs of both factors
wht <- glht(amod, linfct = mcp(temp = "Tukey", envir = "Tukey"))
### cf. Westfall et al. (1999, page 181)
summary(wht, test = adjusted("Shaffer"))