methods function

Methods for General Linear Hypotheses

Methods for General Linear Hypotheses

Simultaneous tests and confidence intervals for general linear hypotheses.

## S3 method for class 'glht' summary(object, test = adjusted(), ...) ## S3 method for class 'glht' confint(object, parm, level = 0.95, calpha = adjusted_calpha(), ...) ## S3 method for class 'glht' coef(object, rhs = FALSE, ...) ## S3 method for class 'glht' vcov(object, ...) ## S3 method for class 'confint.glht' plot(x, xlim, xlab, ylim, ...) ## S3 method for class 'glht' plot(x, ...) univariate() adjusted(type = c("single-step", "Shaffer", "Westfall", "free", p.adjust.methods), ...) Ftest() Chisqtest() adjusted_calpha(...) univariate_calpha(...)

Arguments

  • object: an object of class glht.
  • 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 Kθ^K \hat{\theta} or the right hand side mm (rhs = TRUE) of the linear hypothesis should be returned.
  • type: the multiplicity adjustment (adjusted) to be applied. See below and p.adjust.
  • x: an object of class glht or confint.glht.
  • xlim: the x limits (x1, x2) of the plot.
  • ylim: the y limits of the plot.
  • xlab: a label for the x axis.
  • ...: additional arguments, such as maxpts, abseps or releps to pmvnorm in adjusted or qmvnorm in confint. Note that additional arguments specified to summary, confint, coef and vcov methods are currently ignored.

Details

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 KθK \theta.

The coef and vcov methods compute the linear function Kθ^K \hat{\theta} 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 H0H_0 are implemented in functions Chisqtest and Ftest. Several approaches to multiplicity adjusted p values for each of the linear hypotheses are implemented in function adjusted. The type

argument to adjusted specifies the method to be applied: "single-step" implements adjusted p values based on the joint normal or t distribution of the linear function, and "Shaffer" and "Westfall" implement logically constraint multiplicity adjustments (Shaffer, 1986; Westfall, 1997). "free" implements multiple testing procedures under free combinations (Westfall et al, 1999). In addition, all adjustment methods implemented in p.adjust are available as well.

Simultaneous confidence intervals for linear functions can be computed using method confint. Univariate confidence intervals can be computed by specifying calpha = univariate_calpha()

to confint. The critical value can directly be specified as a scalar to calpha as well. Note that plot(a) for some object a of class glht is equivalent to plot(confint(a)).

All simultaneous inference procedures implemented here control the family-wise error rate (FWER). Multivariate normal and t distributions, the latter one only for models of class lm, are evaluated using the procedures implemented in package mvtnorm. Note that the default procedure is stochastic. Reproducible p-values and confidence intervals require appropriate settings of seeds.

A more detailed description of the underlying methodology is available from Hothorn et al. (2008) and Bretz et al. (2010).

Returns

summary computes (adjusted) p values for general linear hypotheses, confint computes (adjusted) confidence intervals. coef returns estimates of the linear function KθK \theta

and vcov its covariance.

References

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").

Examples

### 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)) }