ConTest: function for informative hypothesis testing (iht)

Description Usage Arguments Details Value Author(s) References See Also Examples

Description

conTest tests linear equality and/or inequality restricted hypotheses for linear models.

Usage

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conTest(object, constraints = NULL, type = "summary", test = "F", 
        rhs = NULL, neq = 0, ...)

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.

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 Rθ ≥ rhs, where each row represents one constraint. The number of columns needs to correspond to the number of parameters estimated (θ) 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.

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; Rθ ≥ 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.

...

futher options for the conTest and/or restriktor function. See details for more information.

Details

The following hypothesis tests are available:

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.

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.

Sigma

variance-covariance matrix of unrestricted model.

R2.org

unrestricted R-squared.

R2.reduced

restricted R-squared.

boot

same as input.

model.org

original model.

Author(s)

Leonard Vanbrabant and Yves Rosseel

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 & probability letters, 28, 137–141.

Silvapulle, M. (1996) Robust bounded influence tests against one-sided hypotheses in general parametric models. Statistics & probability letters, 31, 45–50.

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

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.

See Also

quadprog, conTest

Examples

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## 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; 
                    GroupPassive < GroupNo '

conTest(fit1.lm, myConstraints1)


# another way is to first fit the restricted model
fit.restr1 <- restriktor(fit1.lm, constraints = myConstraints1)

conTest(fit.restr1)

## Not run:  
  # Or in matrix notation.
  Amat1 <- rbind(c(-1, 0,  1),
                 c( 0, 1, -1))
  myRhs1 <- rep(0L, nrow(Amat1)) 
  myNeq1 <- 0
  
  conTest(fit1.lm, constraints = Amat1,
          rhs = myRhs1, neq = myNeq1)

## End(Not run)            

#########################
## 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
                    group2 < group3
                    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.
conTest(fit2.lm, constraints = myConstraints2, type = "A", 
        boot = "parametric", R = 9)


# or fit restricted linear model
fit2.con <- restriktor(fit2.lm, constraints = myConstraints2)

conTest(fit2.con)

## Not run:  
  # 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

  conTest(fit2.con, constraints = Amat2, rhs = myRhs2, neq = myNeq2, 
          type = "A", boot = "parametric", R = 9)

## End(Not run)            

## example 3: equality constraints only.
myConstraints3 <- ' group1 == group2
                    group2 == group3
                    group3 == group4 '

conTest(fit2.lm, constraints = myConstraints3)

# or
fit3.con <- restriktor(fit2.lm, constraints = myConstraints3)
conTest(fit3.con)


## example 4:
# combination of equality and inequality constraints.
myConstraints4 <- ' group1 == group2
                    group3  < group4 '

conTest(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)
conTest(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 '

conTest(fit5.lm, constraints = myConstraints5)

# or
fit5.con <- restriktor(fit5.lm, constraints = ' AVE := X + 16.86137*X.Z; 
                                                AVE > 0 ')
conTest(fit5.con)

restriktor documentation built on Feb. 25, 2020, 5:08 p.m.