mitml (version 0.4-5)

testConstraints: Test functions and constraints of model parameters

Description

Performs hypothesis tests for arbitrary functions of the model parameters using the Delta method.

Usage

testConstraints(model, qhat, uhat, constraints, method = c("D1", "D2"),
  ariv = c("default", "positive"), df.com = NULL)

Value

A list containing the results of the model comparison. A print method is used for more readable output.

Arguments

model

A list of fitted statistical models as produced by with.mitml.list or similar.

qhat

A matrix or list containing the point estimates of the parameters for each imputed data set (see 'Details').

uhat

An array or list containing the variance-covariance matrix of the parameters for each imputed data set (see 'Details').

constraints

A character vector specifying constraints or functions of the vector of model parameters to be tested (see 'Details').

method

A character string denoting the method by which the test is performed. Can be "D1" or "D2" (see 'Details'). Default is "D1".

ariv

A character string denoting how the ARIV is calculated. Can be "default" or "positive" (see 'Details').

df.com

(optional) A single number or a numeric vector denoting the complete-data degrees of freedom for the hypothesis test (see 'Details'). Only used if method = "D1".

Author

Simon Grund

Details

This function performs tests of arbitrary functions (or constraints) of the model parameters using similar methods as testModels. The function relies on the Delta method (e.g., Casella & Berger, 2002) for testing functions of the parameters and assumes that their sampling distribution is approximately normal. The parameters can either be extracted automatically from the fitted statistical models (model) or provided manually as matrices, arrays, or lists (qhat and uhat, see 'Examples').

Constraints and other functions of the model parameters are specified in the constraints argument. The constraints must be supplied as a character vector, where each string denotes a function or a constraint to be tested (see 'Examples').

The Wald-like tests that are carried out by testConstraints are pooled across the imputed data sets with the \(D_1\) (Li, Raghunathan & Rubin, 1991) or \(D_2\) (Li, Meng, Raghunathan & Rubin, 1991) method, where \(D_1\) operates on the constrained point and variance estimates, and \(D_2\) operates on the Wald-statistics (for additional details, see testModels). The pooled estimates and standard errors reported in the output are always based on \(D_1\).

For \(D_1\), the complete-data degrees of freedom can be adjusted for smaller samples by specifying df.com (see testModels).

This function supports general statistical models that define coef and vcov methods (e.g., lm, glm, lavaan and others) as well as multilevel models estimated with lme4 or nlme and GEEs estimated with geepack. The arguments qhat and uhat provide a general method for pooling parameter estimates regardless of model type (see 'Examples'). Support for further models may be added in future releases.

The ariv argument determines how the average relative increase in variance (ARIV) is calculated (see testModels). If ariv = "default", the default estimators are used. If ariv = "positive", the default estimators are used but constrained to take on strictly positive values.

References

Casella, G., & Berger, R. L. (2002). Statistical inference (2nd. Ed.). Pacific Grove, CA: Duxbury.

Li, K.-H., Meng, X.-L., Raghunathan, T. E., & Rubin, D. B. (1991). Significance levels from repeated p-values with multiply-imputed data. Statistica Sinica, 1, 65-92.

Li, K. H., Raghunathan, T. E., & Rubin, D. B. (1991). Large-sample significance levels from multiply imputed data using moment-based statistics and an F reference distribution. Journal of the American Statistical Association, 86, 1065-1073.

See Also

testModels, with.mitml.list

Examples

data(studentratings)

fml <- MathDis + ReadDis + SchClimate ~ (1|ID)
imp <- panImpute(studentratings, formula = fml, n.burn = 1000, n.iter = 100, m = 5)

implist <- mitmlComplete(imp)

# fit simple regression model
fit.lm <- with(implist, lm(SchClimate ~ ReadDis + MathDis))

# apply Rubin's rules
testEstimates(fit.lm)

# * Example 1: test 'identity' function of two parameters (automatic)
# test equivalent to model comparison with a restricted model (without 'ReadDis'
# and 'MathDis')

cons <- c("ReadDis", "MathDis")
testConstraints(fit.lm, constraints = cons)

# ... adjusting for finite samples
testConstraints(fit.lm, constraints = cons, df.com = 749)

# ... using D2
testConstraints(fit.lm, constraints = cons, method = "D2")

# * Example 2: test for equality of two parameters
# tests the hypothesis that the coefficients pertaining to 'ReadDis' and 'MathDis'
# are equal (ReadDis = MathDis)

cons <- c("ReadDis-MathDis")
testConstraints(fit.lm, constraints = cons)

# * Example 3: test against a fixed value
# tests the hypothesis that the coefficient of "ReadDis" is equal to one
# (i.e., 'ReadDis' - 1 == 0)

cons <- c("ReadDis-1")
testConstraints(fit.lm, constraints = cons)

# * Example 4: test 'identity' function of two parameters (qhat, uhat)

fit.lm <- with(implist, lm(SchClimate ~ ReadDis + MathDis))

qhat <- sapply(fit.lm, coef)
uhat <- sapply(fit.lm, function(x) vcov(x), simplify = "array")

cons <- c("ReadDis", "MathDis")
testConstraints(qhat = qhat, uhat = uhat, constraints = cons)