# function for informative hypothesis testing (iht) `iht` tests linear equality and/or inequality restricted hypotheses for linear models. ```r iht(...) conTest(object, constraints = NULL, type = "summary", test = "F", rhs = NULL, neq = 0, ...) conTestD(model = NULL, data = NULL, constraints = NULL, type = c("A","B"), R = 1000L, bootstrap.type = "bollen.stine", return.test = TRUE, neq.alt = 0, double.bootstrap = "standard", double.bootstrap.R = 249, double.bootstrap.alpha = 0.05, parallel = c("no", "multicore", "snow"), ncpus = 1L, cl = NULL, verbose = FALSE, ...) ``` ## Arguments - `object`: an object of class `lm` or `rlm`. In this case, the constraint syntax needs to be specified OR an object of class `restriktor`. The constraints are inherited from the fitted restriktor object and do not to be specified again. - `model`: lavaan model syntax specifying the model. See `model.syntax` for more information. - `constraints`: there are two ways to constrain parameters. First, the constraint syntax consists of one or more text-based descriptions, where the syntax can be specified as a literal string enclosed by single quotes. Only the names of `coef(model)` can be used as names. See details `restriktor` for more information. Second, the constraint syntax consists of a matrix $R$ (or a vector in case of one constraint) and defines the left-hand side of the constraint $R\theta \ge rhs$, where each row represents one constraint. The number of columns needs to correspond to the number of parameters estimated ($\theta$) by model. The rows should be linear independent, otherwise the function gives an error. For more information about constructing the matrix $R$ and $rhs$ see the details in the `restriktor` function. - `data`: the data frame containing the observed variables being used to fit the lavaan model. - `type`: hypothesis test type "A", "B", "C", "global", or "summary" (default). See details for more information. - `test`: test statistic; for information about the null-distribution see details. * for object of class lm; if "F" (default), the F-bar statistic (Silvapulle, 1996) is computed. If "LRT", a likelihood ratio test statistic (Silvapulle and Sen, 2005, chp 3.) is computed. If "score", a global score test statistic (Silvapulle and Silvapulle, 1995) is computed. Note that, in case of equality constraints only, the usual unconstrained F-, Wald-, LR- and score-test statistic is computed. * for object of class rlm; if "F" (default), a robust likelihood ratio type test statistic (Silvapulle, 1992a) is computed. If "Wald", a robust Wald test statistic (Silvapulle, 1992b) is computed. If "score", a global score test statistic (Silvapulle, and Silvapulle, 1995) is computed. Note that, in case of equality constraints only, unconstrained robust F-, Wald-, score-test statistics are computed. * for object of class glm; if "F" (default), the F-bar statistic (Silvapulle, 1996) is computed. If "LRT", a likelihood ratio test statistic (Silvapulle and Sen, 2005, chp 4.) is computed. If "score", a global score test statistic (Silvapulle and Silvapulle, 1995) is computed. Note that, in case of equality constraints only, the usual unconstrained F-, Wald-, LR- and score-test statistic is computed. - `rhs`: vector on the right-hand side of the constraints; $R\theta \ge rhs$. The length of this vector equals the number of rows of the constraints matrix $R$ and consists of zeros by default. Note: only used if constraints input is a matrix or vector. - `neq`: integer (default = 0) treating the number of constraints rows as equality constraints instead of inequality constraints. For example, if `neq = 2`, this means that the first two rows of the constraints matrix $R$ are treated as equality constraints. Note: only used if constraints input is a matrix or vector. - `neq.alt`: integer: number of equality constraints that are maintained under the alternative hypothesis (for hypothesis test type "B"). - `R`: Integer; number of bootstrap draws. The default value is set to 1000. - `bootstrap.type`: If `"parametric"`, the parametric bootstrap is used. If `"bollen.stine"`, the semi-nonparametric Bollen-Stine bootstrap is used. The default is set to `"bollen.stine"`. - `return.test`: Logical; if `TRUE`, the function returns bootstrapped test-values. - `double.bootstrap`: If `"standard"` (default) the genuine double bootstrap is used to compute an additional set of plug-in p-values for each bootstrap sample. If `"no"`, no double bootstrap is used. If `"FDB"`, the fast double bootstrap is used to compute second level LRT-values for each bootstrap sample. Note that the `"FDB"` is experimental and should not be used by inexperienced users. - `double.bootstrap.R`: Integer; number of double bootstrap draws. The default value is set to 249. - `double.bootstrap.alpha`: The significance level to compute the adjusted alpha based on the plugin p-values. Only used if `double.bootstrap = "standard"`. The default value is set to 0.05. - `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 `InformativeTesting` call. - `verbose`: Logical; if `TRUE`, information is shown at each bootstrap draw. - ``...``: futher options for the `iht` and/or `restriktor` function. See details for more information. ## 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). For the robust tests, we found that the results based on these mixtures of F-distributions approximate the tail probabilities better than their asymptotic distributions. Note that, in case of equality constraints only, the null-distribution of the (non-)robust F-test statistics are based on an F-distribution. The (non-)robust Wald- and (non-)robust score-test statistics are based on chi-square distributions. If object is of class `lm` or `rlm`, the `conTest` function internally calls the `restriktor` function. Arguments for the `restriktor` function can be passed on via the `...`. Additional arguments for the `conTest` function can also passed on via the `...`. See for example `conTestF` for all available arguments. ## Returns An object of class conTest or conTestLavaan for which a print is available. ## References Robertson, T., Wright, F.T. and Dykstra, R.L. (1988). **Order Restricted Statistical Inference** New York: Wiley. Shapiro, A. (1988). Towards a unified theory of inequality-constrained testing in multivariate analysis. **International Statistical Review** 56 , 49--62. Silvapulle, M. (1992a). Robust tests of inequality constraints and one-sided hypotheses in the linear model. **Biometrika**, 79 , 621--630. 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. and Silvapulle, P. (1995). A score test against one-sided alternatives. **American statistical association**, 90 , 342--349. Silvapulle, M. (1996) On an F-type statistic for testing one-sided hypotheses and computation of chi-bar-squared weights. **Statistics and probability letters**, 28 , 137--141. 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 Vanbrabant, L., Van de Schoot, R., Van Loey, N.E.E. and Rosseel, Y. (2017). A General Procedure for Testing Inequality Constrained Hypotheses in SEM. **Methodology European Journal of Research Methods for the Behavioral and Social Sciences**, 13 , 61-70. Van de Schoot, R., Hoijtink, H., and Dekovic, M. (2010). Testing inequality constrained hypotheses in SEM models. **Structural Equation Modeling**, 17 , 443-463. Van de Schoot, R., Strohmeier, D. (2011). Testing informative hypotheses in SEM increases power: An illustration contrasting classical. **International Journal of Behavioral Development**, 35 , 180-190. Wolak, F. (1987). An exact test for multiple inequality and equality constraints in the linear regression model. **Journal of the American statistical association**, 82 , 782--793. ## Author(s) Leonard Vanbrabant and Yves Rosseel ## See Also quadprog , `conTest` ## Examples ```r ## 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 linear model fit1.lm <- lm(Age ~ -1 + Group, data = DATA1) # the variable names can be used to impose constraints on # the corresponding regression parameters. coef(fit1.lm) # 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 < GroupNo ' iht(fit1.lm, myConstraints1) # another way is to first fit the restricted model fit.restr1 <- restriktor(fit1.lm, constraints = myConstraints1) iht(fit.restr1) # Or in matrix notation. Amat1 <- rbind(c(-1, 0, 1), c( 0, 1, -1)) myRhs1 <- rep(0L, nrow(Amat1)) myNeq1 <- 0 iht(fit1.lm, constraints = Amat1, rhs = myRhs1, neq = myNeq1) ######################### ## Artificial examples ## ######################### # generate data n <- 10 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 linear model fit2.lm <- lm(y ~ -1 + group, data = DATA2) coef(fit2.lm) ## example 2: increasing means myConstraints2 <- ' group1 < group2 < group3 < group4 ' # compute F-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.lm, constraints = myConstraints2, type = "A", boot = "parametric", R = 9) # or fit restricted linear model fit2.con <- restriktor(fit2.lm, constraints = myConstraints2) iht(fit2.con) # 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", boot = "parametric", R = 9) ## example 3: equality constraints only. myConstraints3 <- ' group1 = group2 = group3 = group4 ' iht(fit2.lm, constraints = myConstraints3) # or fit3.con <- restriktor(fit2.lm, constraints = myConstraints3) iht(fit3.con) ## example 4: # combination of equality and inequality constraints. myConstraints4 <- ' group1 = group2 group3 < group4 ' iht(fit2.lm, constraints = myConstraints4, type = "B", neq.alt = 1) # fit 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. fit4.con <- restriktor(fit2.lm, constraints = myConstraints4, se = "boot.model.based", B = 9) iht(fit4.con, type = "B", neq.alt = 1) ## example 5: # 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 fit5.lm <- lm(y ~ X*Z, data = DATA3) # constraint syntax myConstraints5 <- ' AVE := X + 16.86137*X.Z; AVE > 0 ' iht(fit5.lm, constraints = myConstraints5) # or fit5.con <- restriktor(fit5.lm, constraints = ' AVE := X + 16.86137*X.Z; AVE > 0 ') iht(fit5.con) # testing equality and/or inequality restrictions in SEM: ######################### ### real data example ### ######################### # Multiple group path model for facial burns example. # model syntax with starting values. burns.model <- 'Selfesteem ~ Age + c(m1, f1)*TBSA + HADS + start(-.10, -.20)*TBSA HADS ~ Age + c(m2, f2)*TBSA + RUM + start(.10, .20)*TBSA ' # constraints syntax burns.constraints <- 'f2 > 0 ; m1 < 0 m2 > 0 ; f1 < 0 f2 > m2 ; f1 < m1' # we only generate 2 bootstrap samples in this example; in practice # you may wish to use a much higher number. # the double bootstrap was switched off; in practice you probably # want to set it to "standard". example6 <- iht(model = burns.model, data = FacialBurns, R = 2, constraints = burns.constraints, double.bootstrap = "no", group = "Sex") example6 ########################## ### artificial example ### ########################## # Simple ANOVA model with 3 groups (N = 20 per group) set.seed(1234) Y <- cbind(c(rnorm(20,0,1), rnorm(20,0.5,1), rnorm(20,1,1))) grp <- c(rep("1", 20), rep("2", 20), rep("3", 20)) Data <- data.frame(Y, grp) #create model matrix fit.lm <- lm(Y ~ grp, data = Data) mfit <- fit.lm$model mm <- model.matrix(mfit) Y <- model.response(mfit) X <- data.frame(mm[,2:3]) names(X) <- c("d1", "d2") Data.new <- data.frame(Y, X) # model model <- 'Y ~ 1 + a1*d1 + a2*d2' # fit without constraints fit <- lavaan::sem(model, data = Data.new) # constraints syntax: mu1 < mu2 < mu3 constraints <- ' a1 > 0 a1 < a2 ' # we only generate 10 bootstrap samples in this example; in practice # you may wish to use a much higher number, say > 1000. The double # bootstrap is not necessary in case of an univariate ANOVA model. example7 <- iht(model = model, data = Data.new, start = lavaan::parTable(fit), R = 10L, double.bootstrap = "no", constraints = constraints) example7 ```

iht function