ConTestLRT: Likelihood-ratio-bar test for iht

In LeonardV/restriktor: Restricted Statistical Estimation and Inference for Linear Models

Likelihood-ratio-bar test for iht

Description

conTestLRT tests linear equality and/or inequality restricted hypotheses for linear models by LR-tests. It can be used directly and is called by the conTest function if test = "LRT".

Usage

## S3 method for class 'conLM'
conTestLRT(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, ...)
           
## S3 method for class 'conGLM'
conTestLRT(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, ...)           

## S3 method for class 'conMLM'
conTestLRT(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 conLM, conMLM or conGLM.
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.

Value

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.

Author(s)

Leonard Vanbrabant and Yves Rosseel

References

Silvapulle, M.J. and Sen, P.K. (2005). Constrained Statistical Inference. Wiley, New York

See Also

quadprog, conTest

Examples

## 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, test = "LRT")


# another way is to first fit the restricted model
fit_restr1 <- restriktor(fit1_lm, constraints = myConstraints1)

iht(fit_restr1, test = "LRT")


# 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, test = "LRT",
    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", test = "LRT",
    boot = "parametric", R = 9)

# or fit restricted linear model
fit2_con <- restriktor(fit2_lm, constraints = myConstraints2)

iht(fit2_con, test = "LRT")


# 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 = "LRT", boot = "parametric", R = 9)

## example 3:
# combination of equality and inequality constraints.
myConstraints3 <- ' group1  = group2
                    group3  < group4 '

iht(fit2_lm, constraints = myConstraints3, type = "B", 
    test = "LRT", 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.
fit3_con <- restriktor(fit2_lm, constraints = myConstraints3, 
                       se = "boot.model.based", B = 9)
iht(fit3_con, type = "B", test = "LRT", 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
fit4_lm <- lm(y ~ X*Z, data = DATA3)

# constraint syntax
myConstraints4 <- ' AVE := X + 16.86137*X.Z; 
                    AVE > 0 '

iht(fit4_lm, constraints = myConstraints4, test = "LRT")

# or
fit4_con <- restriktor(fit4_lm, constraints = ' AVE := X + 16.86137*X.Z; 
                                                AVE > 0 ')
iht(fit4_con, test = "LRT")

LeonardV/restriktor documentation built on April 12, 2024, 1:27 p.m.