mitml (version 0.3-4)

testConstraints: Test functions and constraints of model parameters

Description

Performs hypothesis tests for arbitrary functions of a parameter vector using the Delta method.

Usage

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

Arguments

model

A list of fitted statistical models (see examples).

qhat, uhat

Two matrices/arrays or lists containing estimates of the parameter vector and its covariance matrix, respectively, for each imputed data set (see examples).

constraints

A character vector specifying constraints or functions of the vector of model parameters to be tested.

method

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

df.com

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

Value

Returns a list containing the results of the model comparison, the constrained estimates and standard errors, and the relative increase in variance due to nonresponse (Rubin, 1987). A print method is used for better readable console output.

Details

This function is similar in functionality to testModels but extended to arbitrary functions (or constraints) of the model parameters. The function is based on the Delta method (e.g., Casella & Berger, 2002) according to which any function of the parameters can be tested using Wald-like methods, assuming that their sampling distribution is approximately normal. It is assumed that the parameters can be extracted using coef and vcov methods from the fitted models (or similar; e.g., regression coefficients, fixed effects in multilevel models) In cases where this is not possible, hypothesis tests can be carried out using user-supplied matrices/arrays or lists (qhat and uhat, see examples).

Constraints and functions of the model parameters can be 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 can be aggregated across data sets by methods $D_1$ (Li, Raghunathan & Rubin, 1991) and $D_2$ (Li, Meng, Raghunathan & Rubin, 1991), where $D_1$ operates on the constrained estimates and standard errors, and $D_2$ operates on the Wald-statistics (for an explanation, see testModels).

For $D_1$, the complete-data degrees of freedom can be adjusted for smaller samples by specifying df.com.

Currently, the procedure supports statistical models that define coef and vcov methods (e.g., lm) and multilevel models estimated with lme4 or nlme. The arguments qhat and uhat allow for more general hypothesis tests regardless of model class. Support for further models may be added in future releases.

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, print=1:5)

# 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
# multi-parameter hypothesis test, equivalent to model comparison

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 effects
# tests the hypothesis that the effects of '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 effect of "ReadDis" is one (ReadDis=1)

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

# * Example 4: test 'identity' using arrays and list

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

cons <- c("ReadDis","MathDis")
qhat <- sapply(fit.lm, coef)
uhat <- sapply(fit.lm, function(x) vcov(x), simplify="array")
testConstraints(qhat=qhat, uhat=uhat, constraints=cons)