# Test functions and constraints of model parameters Performs hypothesis tests for arbitrary functions of the model parameters using the Delta method. ```r testConstraints(model, qhat, uhat, constraints, method = c("D1", "D2"), ariv = c("default", "positive"), df.com = NULL) ``` ## 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"`. ## 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. ## Returns A list containing the results of the model comparison. A `print` method is used for more readable output. ## 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. ## Author(s) Simon Grund ## See Also `testModels`, `with.mitml.list` ## Examples ```r 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) ```

testConstraints function