car (version 3.1-2)

linearHypothesis: Test Linear Hypothesis


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), suppress.vcov.msg=FALSE, error.df, ...)

# 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 lmList linearHypothesis(model, ..., vcov.=vcov, coef.=coef)

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

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


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.



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.


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


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.


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.


For the default linearHypothesis method, if an F-test is requested and if error.df is missing, the error degrees of freedom will be computed by applying the df.residual function to the model; if df.residual returns NULL or NA, then a chi-square test will be substituted for the F-test (with a message to that effect.


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.


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.


a one-sided model formula using the ``data'' in idata and specifying the intra-subject design.


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


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.


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.


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.


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.


an optional character string to label the output.


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


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. For the "lmList" and "nlsList" methods, vcov. must be a function (defaulting to vcov) to be applied to each model in the list.


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. For the "lmList" and "nlsList" methods, coef. must be a function (defaulting to coef) to be applied to each model in the list.


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.


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.


an object produced by linearHypothesis.mlm.


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


minimum number of signficiant digits to print.


a regular expression to be matched against coefficient names.


for internal use by methods that call the default method.


arguments to pass down.


Achim Zeileis and John Fox


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.

Multinomial logit models fit by the multinom function in the nnet package invoke the default method, and the coefficient names are composed from the response-level names and conventional coefficient names, separated by a period ("."): see one of the examples below.

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:

  1. The transformation matrix can be specified directly via the P argument.

  2. 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.

See Also

anova, Anova, waldtest, hccm, vcovHC, vcovHAC, coef, vcov


Run this code
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, 

## 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)
## 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, 

## mixed-effects models examples:

if (FALSE)  # loads nlme package
	linearHypothesis(fm2, "age = 0")

if (FALSE)  # loads lme4 package
	linearHypothesis(gm1, matchCoefs(gm1, "period"))

if (require(nnet)){
  print(m <- multinom(partic ~ hincome + children, data=Womenlf))
  print(coefs <- as.vector(outer(c("", "parttime."), 
                            c("hincome", "childrenpresent"),
  linearHypothesis(m, coefs) # ominbus Wald test

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