Generic function for testing a linear hypothesis, and methods
for linear models, generalized linear models, multivariate linear
models, linear and generalized linear mixed-effects models,
generalized linear models fit with `svyglm`

in the survey package,
robust linear models fit with `rlm`

in the MASS package,
and other models that have methods for `coef`

and `vcov`

.
For mixed-effects models, the tests are Wald chi-square tests for the fixed effects.

`linearHypothesis(model, ...)`lht(model, ...)

# S3 method for default
linearHypothesis(model, hypothesis.matrix, rhs=NULL,
test=c("Chisq", "F"), vcov.=NULL, singular.ok=FALSE, verbose=FALSE,
coef. = coef(model), ...)

# S3 method for lm
linearHypothesis(model, hypothesis.matrix, rhs=NULL,
test=c("F", "Chisq"), vcov.=NULL,
white.adjust=c(FALSE, TRUE, "hc3", "hc0", "hc1", "hc2", "hc4"),
singular.ok=FALSE, ...)

# S3 method for glm
linearHypothesis(model, ...)

# S3 method for nlsList
linearHypothesis(model, ..., vcov., coef.)

# S3 method for mlm
linearHypothesis(model, hypothesis.matrix, rhs=NULL, SSPE, V,
test, idata, icontrasts=c("contr.sum", "contr.poly"), idesign, iterms,
check.imatrix=TRUE, P=NULL, title="", singular.ok=FALSE, verbose=FALSE, ...)
# S3 method for polr
linearHypothesis(model, hypothesis.matrix, rhs=NULL, vcov.,
verbose=FALSE, ...)
# S3 method for linearHypothesis.mlm
print(x, SSP=TRUE, SSPE=SSP,
digits=getOption("digits"), ...)
# S3 method for lme
linearHypothesis(model, hypothesis.matrix, rhs=NULL,
vcov.=NULL, singular.ok=FALSE, verbose=FALSE, ...)
# S3 method for mer
linearHypothesis(model, hypothesis.matrix, rhs=NULL,
vcov.=NULL, test=c("Chisq", "F"), singular.ok=FALSE, verbose=FALSE, ...)
# S3 method for merMod
linearHypothesis(model, hypothesis.matrix, rhs=NULL,
vcov.=NULL, test=c("Chisq", "F"), singular.ok=FALSE, verbose=FALSE, ...)
# S3 method for svyglm
linearHypothesis(model, ...)

# S3 method for rlm
linearHypothesis(model, ...)

# S3 method for survreg
linearHypothesis(model, hypothesis.matrix, rhs=NULL,
test=c("Chisq", "F"), vcov., verbose=FALSE, ...)

matchCoefs(model, pattern, ...)

# S3 method for default
matchCoefs(model, pattern, coef.=coef, ...)

# S3 method for lme
matchCoefs(model, pattern, ...)

# S3 method for mer
matchCoefs(model, pattern, ...)

# S3 method for merMod
matchCoefs(model, pattern, ...)

# S3 method for mlm
matchCoefs(model, pattern, ...)

model

fitted model object. The default method of `linearHypothesis`

works for models
for which the estimated parameters can be retrieved by `coef`

and
the corresponding estimated covariance matrix by `vcov`

. See the
*Details* for more information.

hypothesis.matrix

matrix (or vector) giving linear combinations
of coefficients by rows, or a character vector giving the hypothesis
in symbolic form (see *Details*).

rhs

right-hand-side vector for hypothesis, with as many entries as
rows in the hypothesis matrix; can be omitted, in which case it defaults
to a vector of zeroes. For a multivariate linear model, `rhs`

is a
matrix, defaulting to 0. This argument isn't available for F-tests for linear mixed models.

singular.ok

if `FALSE`

(the default), a model with aliased
coefficients produces an error; if `TRUE`

, the aliased coefficients
are ignored, and the hypothesis matrix should not have columns for them.
For a multivariate linear model: will return the hypothesis and error SSP
matrices even if the latter is singular; useful for computing univariate
repeated-measures ANOVAs where there are fewer subjects than df for within-subject
effects.

idata

an optional data frame giving a factor or factors defining the
intra-subject model for multivariate repeated-measures data. See
*Details* for an explanation of the intra-subject design and for
further explanation of the other arguments relating to intra-subject factors.

icontrasts

names of contrast-generating functions to be applied by default to factors and ordered factors, respectively, in the within-subject ``data''; the contrasts must produce an intra-subject model matrix in which different terms are orthogonal.

idesign

a one-sided model formula using the ``data'' in `idata`

and
specifying the intra-subject design.

iterms

the quoted name of a term, or a vector of quoted names of terms, in the intra-subject design to be tested.

check.imatrix

check that columns of the intra-subject model matrix for
different terms are mutually orthogonal (default, `TRUE`

). Set to `FALSE`

only if you have *already* checked that the intra-subject model matrix is
block-orthogonal.

P

transformation matrix to be applied to the repeated measures in
multivariate repeated-measures data; if `NULL`

*and* no
intra-subject model is specified, no response-transformation is applied; if
an intra-subject model is specified via the `idata`

, `idesign`

,
and (optionally) `icontrasts`

arguments, then `P`

is generated
automatically from the `iterms`

argument.

SSPE

in `linearHypothesis`

method for `mlm`

objects:
optional error sum-of-squares-and-products matrix; if missing,
it is computed from the model. In `print`

method for
`linearHypothesis.mlm`

objects: if `TRUE`

,
print the sum-of-squares and cross-products matrix for error.

test

character string, `"F"`

or `"Chisq"`

,
specifying whether to compute the finite-sample
F statistic (with approximate F distribution) or the large-sample
Chi-squared statistic (with asymptotic Chi-squared distribution). For a
multivariate linear model, the multivariate test statistic to report --- one or more of
`"Pillai"`

, `"Wilks"`

, `"Hotelling-Lawley"`

, or `"Roy"`

,
with `"Pillai"`

as the default.

title

an optional character string to label the output.

V

inverse of sum of squares and products of the model matrix; if missing it is computed from the model.

vcov.

a function for estimating the covariance matrix of the regression
coefficients, e.g., `hccm`

, or an estimated covariance matrix
for `model`

. See also `white.adjust`

.

coef.

a vector of coefficient estimates. The default is to get the
coefficient estimates from the `model`

argument, but the user can input
any vector of the correct length.

white.adjust

logical or character. Convenience interface to `hccm`

(instead of using the argument `vcov.`

). Can be set either to a character value
specifying the `type`

argument of `hccm`

or `TRUE`

,
in which case `"hc3"`

is used implicitly. The default is `FALSE`

.

verbose

If `TRUE`

, the hypothesis matrix, right-hand-side
vector (or matrix), and estimated value of the hypothesis
are printed to standard output; if `FALSE`

(the default),
the hypothesis is only printed in symbolic form and the value of the hypothesis
is not printed.

x

an object produced by `linearHypothesis.mlm`

.

SSP

if `TRUE`

(the default), print the sum-of-squares and
cross-products matrix for the hypothesis and the response-transformation matrix.

digits

minimum number of signficiant digits to print.

pattern

a regular expression to be matched against coefficient names.

...

arguments to pass down.

For a univariate model, an object of class `"anova"`

which contains the residual degrees of freedom
in the model, the difference in degrees of freedom, Wald statistic
(either `"F"`

or `"Chisq"`

), and corresponding p value.
The value of the linear hypothesis and its covariance matrix are returned
respectively as `"value"`

and `"vcov"`

attributes of the object
(but not printed).

For a multivariate linear model, an object of class
`"linearHypothesis.mlm"`

, which contains sums-of-squares-and-product
matrices for the hypothesis and for error, degrees of freedom for the
hypothesis and error, and some other information.

The returned object normally would be printed.

`linearHypothesis`

computes either a finite-sample F statistic or asymptotic Chi-squared
statistic for carrying out a Wald-test-based comparison between a model
and a linearly restricted model. The default method will work with any
model object for which the coefficient vector can be retrieved by
`coef`

and the coefficient-covariance matrix by `vcov`

(otherwise
the argument `vcov.`

has to be set explicitly). For computing the
F statistic (but not the Chi-squared statistic) a `df.residual`

method needs to be available. If a `formula`

method exists, it is
used for pretty printing.

The method for `"lm"`

objects calls the default method, but it
changes the default test to `"F"`

, supports the convenience argument
`white.adjust`

(for backwards compatibility), and enhances the output
by the residual sums of squares. For `"glm"`

objects just the default
method is called (bypassing the `"lm"`

method). The `svyglm`

method
also calls the default method.

The function `lht`

also dispatches to `linearHypothesis`

.

The hypothesis matrix can be supplied as a numeric matrix (or vector), the rows of which specify linear combinations of the model coefficients, which are tested equal to the corresponding entries in the right-hand-side vector, which defaults to a vector of zeroes.

Alternatively, the hypothesis can be specified symbolically as a character vector with one or more elements, each of which gives either a linear combination of coefficients, or a linear equation in the coefficients (i.e., with both a left and right side separated by an equals sign). Components of a linear expression or linear equation can consist of numeric constants, or numeric constants multiplying coefficient names (in which case the number precedes the coefficient, and may be separated from it by spaces or an asterisk); constants of 1 or -1 may be omitted. Spaces are always optional. Components are separated by plus or minus signs. Newlines or tabs in hypotheses will be treated as spaces. See the examples below.

If the user sets the arguments `coef.`

and `vcov.`

, then the computations
are done without reference to the `model`

argument. This is like assuming
that `coef.`

is normally distibuted with estimated variance `vcov.`

and the `linearHypothesis`

will compute tests on the mean vector for
`coef.`

, without actually using the `model`

argument.

A linear hypothesis for a multivariate linear model (i.e., an object of
class `"mlm"`

) can optionally include an intra-subject transformation matrix
for a repeated-measures design.
If the intra-subject transformation is absent (the default), the multivariate
test concerns all of the corresponding coefficients for the response variables.
There are two ways to specify the transformation matrix for the
repeated measures:

The transformation matrix can be specified directly via the

`P`

argument.A data frame can be provided defining the repeated-measures factor or factors via

`idata`

, with default contrasts given by the`icontrasts`

argument. An intra-subject model-matrix is generated from the one-sided formula specified by the`idesign`

argument; columns of the model matrix corresponding to different terms in the intra-subject model must be orthogonal (as is insured by the default contrasts). Note that the contrasts given in`icontrasts`

can be overridden by assigning specific contrasts to the factors in`idata`

. The repeated-measures transformation matrix consists of the columns of the intra-subject model matrix corresponding to the term or terms in`iterms`

. In most instances, this will be the simpler approach, and indeed, most tests of interests can be generated automatically via the`Anova`

function.

`matchCoefs`

is a convenience function that can sometimes help in formulating hypotheses; for example
`matchCoefs(mod, ":")`

will return the names of all interaction coefficients in the model `mod`

.

Fox, J. (2016)
*Applied Regression Analysis and Generalized Linear Models*,
Third Edition. Sage.

Fox, J. and Weisberg, S. (2019)
*An R Companion to Applied Regression*, Third Edition, Sage.

Hand, D. J., and Taylor, C. C. (1987)
*Multivariate Analysis of Variance and Repeated Measures: A Practical
Approach for Behavioural Scientists.* Chapman and Hall.

O'Brien, R. G., and Kaiser, M. K. (1985)
MANOVA method for analyzing repeated measures designs: An extensive primer.
*Psychological Bulletin* **97**, 316--333.

# NOT RUN { mod.davis <- lm(weight ~ repwt, data=Davis) ## the following are equivalent: linearHypothesis(mod.davis, diag(2), c(0,1)) linearHypothesis(mod.davis, c("(Intercept) = 0", "repwt = 1")) linearHypothesis(mod.davis, c("(Intercept)", "repwt"), c(0,1)) linearHypothesis(mod.davis, c("(Intercept)", "repwt = 1")) ## use asymptotic Chi-squared statistic linearHypothesis(mod.davis, c("(Intercept) = 0", "repwt = 1"), test = "Chisq") ## the following are equivalent: ## use HC3 standard errors via white.adjust option linearHypothesis(mod.davis, c("(Intercept) = 0", "repwt = 1"), white.adjust = TRUE) ## covariance matrix *function* linearHypothesis(mod.davis, c("(Intercept) = 0", "repwt = 1"), vcov = hccm) ## covariance matrix *estimate* linearHypothesis(mod.davis, c("(Intercept) = 0", "repwt = 1"), vcov = hccm(mod.davis, type = "hc3")) mod.duncan <- lm(prestige ~ income + education, data=Duncan) ## the following are all equivalent: linearHypothesis(mod.duncan, "1*income - 1*education = 0") linearHypothesis(mod.duncan, "income = education") linearHypothesis(mod.duncan, "income - education") linearHypothesis(mod.duncan, "1income - 1education = 0") linearHypothesis(mod.duncan, "0 = 1*income - 1*education") linearHypothesis(mod.duncan, "income-education=0") linearHypothesis(mod.duncan, "1*income - 1*education + 1 = 1") linearHypothesis(mod.duncan, "2income = 2*education") mod.duncan.2 <- lm(prestige ~ type*(income + education), data=Duncan) coefs <- names(coef(mod.duncan.2)) ## test against the null model (i.e., only the intercept is not set to 0) linearHypothesis(mod.duncan.2, coefs[-1]) ## test all interaction coefficients equal to 0 linearHypothesis(mod.duncan.2, coefs[grep(":", coefs)], verbose=TRUE) linearHypothesis(mod.duncan.2, matchCoefs(mod.duncan.2, ":"), verbose=TRUE) # equivalent lh <- linearHypothesis(mod.duncan.2, coefs[grep(":", coefs)]) attr(lh, "value") # value of linear function attr(lh, "vcov") # covariance matrix of linear function ## a multivariate linear model for repeated-measures data ## see ?OBrienKaiser for a description of the data set used in this example. mod.ok <- lm(cbind(pre.1, pre.2, pre.3, pre.4, pre.5, post.1, post.2, post.3, post.4, post.5, fup.1, fup.2, fup.3, fup.4, fup.5) ~ treatment*gender, data=OBrienKaiser) coef(mod.ok) ## specify the model for the repeated measures: phase <- factor(rep(c("pretest", "posttest", "followup"), c(5, 5, 5)), levels=c("pretest", "posttest", "followup")) hour <- ordered(rep(1:5, 3)) idata <- data.frame(phase, hour) idata ## test the four-way interaction among the between-subject factors ## treatment and gender, and the intra-subject factors ## phase and hour linearHypothesis(mod.ok, c("treatment1:gender1", "treatment2:gender1"), title="treatment:gender:phase:hour", idata=idata, idesign=~phase*hour, iterms="phase:hour") ## mixed-effects models examples: # } # NOT RUN { library(nlme) example(lme) linearHypothesis(fm2, "age = 0") # } # NOT RUN { # } # NOT RUN { library(lme4) example(glmer) linearHypothesis(gm1, matchCoefs(gm1, "period")) # } # NOT RUN { # }