# Wald-bar test for robust iht `conTestWald` tests linear equality and/or inequality restricted hypotheses for linear models by Wald-tests. It can be used directly and is called by the `conTest` function if `test = "Wald"`. ```r ## S3 method for class 'conRLM' conTestWald(object, type = "A", neq.alt = 0, boot = "no", R = 9999, p.distr = rnorm, parallel = "no", ncpus = 1L, cl = NULL, seed = 1234, verbose = FALSE, control = NULL, ...) ``` ## Arguments - `object`: an object of class `conRLM`. - `type`: hypothesis test type "A", "B", "C", "global", or "summary" (default). See details for more information. - `neq.alt`: integer: number of equality constraints that are maintained under the alternative hypothesis (for hypothesis test type "B"), see example 3. - `boot`: the null-distribution of these test-statistics (except under type "C", see details) takes the form of a mixture of F-distributions. The tail probabilities can be computed directly via bootstrapping; if `"parametric"`, the p-value is computed based on the parametric bootstrap. By default, samples are drawn from a normal distribution with mean zero and varance one. See `p.distr` for other distributional options. If `"model.based"`, a model-based bootstrap method is used. Instead of computing the p-value via simulation, the p-value can also be computed using the chi-bar-square weights. If `"no"`, the p-value is computed based on the weights obtained via simulation (`mix_weights = "boot"`) or using the multivariate normal distribution function (`mix_weights = "pmvnorm"`). Note that, these weights are already available in the restriktor objected and do not need to be estimated again. However, there are two exception for objects of class `conRLM`, namely for computing the p-value for the robust `test` = `"Wald"` and the robust `"score"`. In these cases the weights need to be recalculated. - `R`: integer; number of bootstrap draws for `boot`. The default value is set to 9999. - `p.distr`: random generation distribution for the parametric bootstrap. For all available distributions see `?distributions`. For example, if `rnorm`, samples are drawn from the normal distribution (default) with mean zero and variance one. If `rt`, samples are drawn from a t-distribution. If `rchisq`, samples are drawn from a chi-square distribution. The random generation distributional parameters will be passed in via . - `parallel`: the type of parallel operation to be used (if any). If missing, the default is set "no". - `ncpus`: integer: number of processes to be used in parallel operation: typically one would chose this to the number of available CPUs. - `cl`: an optional parallel or snow cluster for use if parallel = "snow". If not supplied, a cluster on the local machine is created for the duration of the conTest call. - `seed`: seed value. The default value is set to 1234. - `verbose`: logical; if TRUE, information is shown at each bootstrap draw. - `control`: a list of control arguments: * `absval` tolerance criterion for convergence (default = sqrt(.Machine$double.eps)). Only used for model of class lm. * `maxit` the maximum number of iterations for the optimizer (default = 10000). Only used for model of class mlm (not yet supported). * `tol` numerical tolerance value. Estimates smaller than `tol` are set to 0. * `chunk_size` the chi-bar-square weights are computed for samples of size `chunk_size = 5000L`. This process is repeated iteratively until the weights converges (see `convergenge_crit`) or the maximum is reached, i.e., mix_weights_bootstrap_limit. * `convergence_crit` the convergence criterion for the iterative process. The default is 1e-03. - ``...``: Additional arguments that can be passed to the p.distr function, or arguments for the restriktor or iht function. Consider, for example, the mix_weights_bootstrap_limit argument, which specifies the maximum number of bootstrap draws (default is 100.000) used to compute the chi-bar-square weights. If mix_weights_bootstrap_limit is set to 100.000, then in each iteration, a sample of size 5000 is added until the weights converge, or the maximum limit is reached. ## Details The following hypothesis tests are available: * Type A: Test H0: all constraints with equalities ("=") active against HA: at least one inequality restriction ("\>") strictly true. * Type B: Test H0: all constraints with inequalities ("\>") (including some equalities ("=")) active against HA: at least one restriction false (some equality constraints may be maintained). * Type C: Test H0: at least one restriction false ("\<") against HA: all constraints strikty true ("\>"). This test is based on the intersection-union principle (Silvapulle and Sen, 2005, chp 5.3). Note that, this test only makes sense in case of no equality constraints. * Type global: equal to Type A but H0 contains additional equality constraints. This test is analogue to the global F-test in lm, where all coefficients but the intercept equal 0. The null-distribution of hypothesis test Type C is based on a t-distribution (one-sided). Its power can be poor in case of many inequalty constraints. Its main role is to prevent wrong conclusions from significant results from hypothesis test Type A. The exact finite sample distributions of the non-robust F-, score- and LR-test statistics based on restricted OLS estimates and normally distributed errors, are a mixture of F-distributions under the null hypothesis (Wolak, 1987). In agreement with Silvapulle (1992), we found that the results based on these mixtures of F-distributions approximate the tail probabilities of the robust tests better than their asymptotic distributions. Therefore, all p-values for hypothesis test Type `"A"`, `"B"` and `"global"` are computed based on mixtures of F-distributions. ## Returns An object of class conTest, for which a print is available. More specifically, it is a list with the following items: - **CON**: a list with useful information about the constraints. - **Amat**: constraints matrix. - **bvec**: vector of right-hand side elements. - **meq**: number of equality constraints. - **meq.alt**: same as input neq.alt. - **iact**: number of active constraints. - **type**: same as input. - **test**: same as input. - **Ts**: test-statistic value. - **df.residual**: the residual degrees of freedom. - **pvalue**: tail probability for `Ts`. - **b.eqrestr**: equality restricted regression coefficients. Only available for `type = "A"` and `type = "global"`, else `b.eqrestr = NULL`. - **b.unrestr**: unrestricted regression coefficients. - **b.restr**: restricted regression coefficients. - **b.restr.alt**: restricted regression coefficients under HA if some equality constraints are maintained. Only available for `type = "B"` else `b.restr.alt = NULL`. - **Sigma**: variance-covariance matrix of unrestricted model. - **R2.org**: unrestricted R-squared, not available for objects of class `conGLM`. - **R2.reduced**: restricted R-squared, not available for objects of class `conGLM`. - **boot**: same as input. - **model.org**: original model. ## References Silvapulle, M. (1992b). Robust Wald-Type Tests of One-Sided Hypotheses in the Linear Model. **Journal of the American Statistical Association**, 87 , 156--161. Silvapulle, M. (1996) Robust bounded influence tests against one-sided hypotheses in general parametric models. **Statistics and probability letters**, 31 , 45--50. Silvapulle, M.J. and Sen, P.K. (2005). **Constrained Statistical Inference**. Wiley, New York ## Author(s) Leonard Vanbrabant and Yves Rosseel ## See Also quadprog , `conTest` ## Examples ```r library(MASS) ## example 1: # the data consist of ages (in months) at which an # infant starts to walk alone. # prepare data DATA1 <- subset(ZelazoKolb1972, Group != "Control") # fit unrestricted robust linear model fit1.rlm <- rlm(Age ~ -1 + Group, data = DATA1, method = "MM") # the variable names can be used to impose constraints on # the corresponding regression parameters. coef(fit1.rlm) # constraint syntax: assuming that the walking # exercises would not have a negative effect of increasing the # mean age at which a child starts to walk. myConstraints1 <- ' GroupActive < GroupPassive; GroupPassive < GroupNo ' iht(fit1.rlm, myConstraints1, test = "Wald") # another way is to first fit the restricted model fit.restr1 <- restriktor(fit1.rlm, constraints = myConstraints1) iht(fit.restr1, test = "Wald") # Or in matrix notation. Amat1 <- rbind(c(-1, 0, 1), c( 0, 1, -1)) myRhs1 <- rep(0L, nrow(Amat1)) myNeq1 <- 0 iht(fit1.rlm, constraints = Amat1, test = "Wald", rhs = myRhs1, neq = myNeq1) ######################### ## Artificial examples ## ######################### # generate data n <- 30 means <- c(1,2,1,3) nm <- length(means) group <- as.factor(rep(1:nm, each = n)) y <- rnorm(n * nm, rep(means, each = n)) DATA2 <- data.frame(y, group) # fit unrestricted robust linear model fit2.rlm <- rlm(y ~ -1 + group, data = DATA2, method = "MM") coef(fit2.rlm) ## example 2: increasing means myConstraints2 <- ' group1 < group2 < group3 < group4 ' # compute Wald-test for hypothesis test Type A and compute the tail # probability based on the parametric bootstrap. We only generate 9 # bootstrap samples in this example; in practice you may wish to # use a much higher number. iht(fit2.rlm, constraints = myConstraints2, type = "A", test = "Wald", boot = "parametric", R = 9) # or fit restricted robust linear model fit2.con <- restriktor(fit2.rlm, constraints = myConstraints2) iht(fit2.con, test = "Wald") # increasing means in matrix notation. Amat2 <- rbind(c(-1, 1, 0, 0), c( 0,-1, 1, 0), c( 0, 0,-1, 1)) myRhs2 <- rep(0L, nrow(Amat2)) myNeq2 <- 0 iht(fit2.con, constraints = Amat2, rhs = myRhs2, neq = myNeq2, type = "A", test = "Wald", boot = "parametric", R = 9) ## example 3: # combination of equality and inequality constraints. myConstraints3 <- ' group1 = group2 group3 < group4 ' iht(fit2.rlm, constraints = myConstraints3, type = "B", test = "Wald", neq.alt = 1) # fit robust resticted model and compute model-based bootstrapped # standard errors. We only generate 9 bootstrap samples in this # example; in practice you may wish to use a much higher number. # Note that, a warning message may be thrown because the number of # bootstrap samples is too low. fit3.con <- restriktor(fit2.rlm, constraints = myConstraints3, se = "boot.model.based", B = 9) iht(fit3.con, type = "B", test = "Wald", neq.alt = 1) ## example 4: # restriktor can also be used to define effects using the := operator # and impose constraints on them. For example, is the # average effect (AVE) larger than zero? # generate data n <- 30 b0 <- 10; b1 = 0.5; b2 = 1; b3 = 1.5 X <- c(rep(c(0), n/2), rep(c(1), n/2)) set.seed(90) Z <- rnorm(n, 16, 5) y <- b0 + b1*X + b2*Z + b3*X*Z + rnorm(n, 0, sd = 10) DATA3 = data.frame(cbind(y, X, Z)) # fit linear model with interaction fit3.rlm <- rlm(y ~ X*Z, data = DATA3, method = "MM") # constraint syntax myConstraints4 <- ' AVE := X + 16.86137*X.Z; AVE > 0 ' iht(fit3.rlm, constraints = myConstraints4, test = "Wald") # or fit3.con <- restriktor(fit3.rlm, constraints = ' AVE := X + 16.86137*X.Z; AVE > 0 ') iht(fit3.con, test = "Wald") ```

conTestWald function