R: Test Linear Hypothesis

linearHypothesis {car} | R Documentation

## Test Linear Hypothesis

### Description

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.

### Usage

    linearHypothesis(model, ...)

    lht(model, ...)

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

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

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

    ## S3 method for class '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 class 'polr'
    linearHypothesis(model, hypothesis.matrix, rhs=NULL, vcov.,
    	verbose=FALSE, ...)

    ## S3 method for class 'linearHypothesis.mlm'
    print(x, SSP=TRUE, SSPE=SSP,
        digits=getOption("digits"), ...)

    ## S3 method for class 'lme'
    linearHypothesis(model, hypothesis.matrix, rhs=NULL,
    		vcov.=NULL, singular.ok=FALSE, verbose=FALSE, ...)

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

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

    ## S3 method for class 'svyglm'
    linearHypothesis(model, ...)

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

    matchCoefs(model, pattern, ...)

    ## Default S3 method:
    matchCoefs(model, pattern, coef.=coef, ...)

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

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

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

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

### Arguments

`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.
`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.

### Details

`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:

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`.

### Value

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.

### Author(s)

Achim Zeileis and John Fox jfox@mcmaster.ca

### References

Fox, J. (2008) Applied Regression Analysis and Generalized Linear Models, Second Edition. Sage.

Fox, J. and Weisberg, S. (2011) An R Companion to Applied Regression, Second 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`

### Examples

    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")

    ## End(Not run)

    ## Not run:
    	library(lme4)
    	example(glmer)
    	linearHypothesis(gm1, matchCoefs(gm1, "period"))

    ## End(Not run)

[Package car version 2.1-4 Index]