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

Performs multi-parameter hypothesis tests for a vector of statistical parameters and compares nested statistical models obtained from multiply imputed data sets.

1 2 3 |

`model` |
A list of fitted statistical models (“full” model) as produced by |

`null.model` |
A list of fitted statistical models (“restricted” model) as produced by |

`method` |
A character string denoting the method by which the test is performed. Can be |

`use` |
A character string denoting Wald- or likelihood-based based tests. Can be either |

`ariv` |
A character string denoting how the ARIV is calculated. Can be |

`df.com` |
(optional) A number denoting the complete-data degrees of freedom for the hypothesis test. Only used if |

`data` |
(optional) A list of imputed data sets (see 'Details'). Only used if |

This function compares two nested statistical models fitted to multiply imputed data sets by pooling Wald-like or likelihood-ratio tests.

Pooling methods for Wald-like tests of multiple parameters were introduced by Rubin (1987) and further developed by Li, Raghunathan and Rubin (1991).
The pooled Wald test is referred to as *D_1* and can be used by setting `method = "D1"`

.
*D_1* is the multi-parameter equivalent of `testEstimates`

, that is, it tests multiple parameters simultaneously.
For *D_1*, the complete-data degrees of freedom are assumed to be infinite, but they can be adjusted for smaller samples by supplying `df.com`

(Reiter, 2007).

An alternative method for Wald-like hypothesis tests was suggested by Li, Meng, Raghunathan and Rubin (1991).
The procedure is called *D_2* and can be used by setting `method = "D2"`

.
*D_2* calculates the Wald-test directly for each data set and then pools the resulting *χ^2* values.
The source of these values is specified by the `use`

argument.
If `use = "wald"`

(the default), then a Wald test similar to *D_1* is performed.
If `use = "likelihood"`

, then the two models are compared with a likelihood-ratio test instead.

Pooling methods for likelihood-ration tests were suggested by Meng and Rubin (1992).
This procedure is referred to as *D_3* and can be used by setting `method = "D3"`

.
*D_3* compares the two models by pooling the likelihood-ratio test across multiply imputed data sets.

Finally, an improved method for pooling likelihood-ratio tests was recommended by Chan & Meng (2019).
This method is referred to as *D_4* and can be used by setting `method = "D4"`

.
*D_4* also compares models by pooling the likelihood-ratio test but does so in a more general and efficient manner.

The function supports different classes of statistical models depending on which `method`

is chosen.
*D_1* supports models that define `coef`

and `vcov`

methods (or similar) for extracting the parameter estimates and their estimated covariance matrix.
*D_2* can be used for the same models (if `use = "wald"`

and models that define a `logLik`

method (if `use = "likelihood"`

).
*D_3* supports linear models, linear mixed-effects models (see Laird, Lange, & Stram, 1987) with an arbitrary cluster structed if estimated with `lme4`

or a single cluster if estimated by `nlme`

, and structural equation models estimated with `lavaan`

(requires ML estimator, see 'Note').
Finally, *D_4* supports models that define a `logLik`

method but can fail if the data to which the model was fitted cannot be found.
In such a case, users can provide the list of imputed data sets directly by specifying the `data`

argument or refit with the `include.data`

argument in `with.mitml.list`

.
Support for other statistical models may be added in future releases.

The *D_4*, *D_3*, and *D_2* methods support different estimators of the relative increase in variance (ARIV), which can be specified with the `ariv`

argument.
If `ariv = "default"`

, the default estimators are used.
If `ariv = "positive"`

, the default estimators are used but constrained to take on strictly positive values.
This is useful if the estimated ARIV is negative.
If `ariv = "robust"`

, which is available only for *D_4*, the "robust" estimator proposed by Chan & Meng (2019) is used.
This method should be used with caution, because it requires much stronger assumptions and may result in liberal inferences if these assumptions are violated.

A list containing the results of the model comparison.
A `print`

method is used for more readable output.

The methods *D_4*, *D_3*, and the likelihood-based *D_2* assume that models were fit using maximum likelihood (ML).
Models fit using REML are automatically refit using ML.
Models fit in `'lavaan'`

using the MLR estimator or similar techniques that require scaled *chi^2* difference tests are currently not supported.

Simon Grund

Chan, K. W., & Meng, X.-L. (2019). Multiple improvements of multiple imputation likelihood ratio tests. ArXiv:1711.08822 [Math, Stat]. https://arxiv.org/abs/1711.08822

Laird, N., Lange, N., & Stram, D. (1987). Maximum likelihood computations with repeated measures: Application of the em algorithm. *Journal of the American Statistical Association, 82*, 97-105.

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.

Meng, X.-L., & Rubin, D. B. (1992). Performing likelihood ratio tests with multiply-imputed data sets. *Biometrika, 79*, 103-111.

Reiter, J. P. (2007). Small-sample degrees of freedom for multi-component significance tests with multiple imputation for missing data. *Biometrika, 94*, 502-508.

Rubin, D. B. (1987). *Multiple imputation for nonresponse in surveys*. Hoboken, NJ: Wiley.

`testEstimates`

, `testConstraints`

, `with.mitml.list`

, `anova.mitml.result`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | ```
data(studentratings)
fml <- ReadDis + SES ~ ReadAchiev + (1|ID)
imp <- panImpute(studentratings, formula = fml, n.burn = 1000, n.iter = 100, m = 5)
implist <- mitmlComplete(imp)
# * Example 1: multiparameter hypothesis test for 'ReadDis' and 'SES'
# This tests the hypothesis that both effects are zero.
require(lme4)
fit0 <- with(implist, lmer(ReadAchiev ~ (1|ID), REML = FALSE))
fit1 <- with(implist, lmer(ReadAchiev ~ ReadDis + (1|ID), REML = FALSE))
# apply Rubin's rules
testEstimates(fit1)
# multiparameter hypothesis test using D1 (default)
testModels(fit1, fit0)
# ... adjusting for finite samples
testModels(fit1, fit0, df.com = 47)
# ... using D2 ("wald", using estimates and covariance-matrix)
testModels(fit1, fit0, method = "D2")
# ... using D2 ("likelihood", using likelihood-ratio test)
testModels(fit1, fit0, method = "D2", use = "likelihood")
# ... using D3 (likelihood-ratio test, requires ML fit)
testModels(fit1, fit0, method = "D3")
# ... using D4 (likelihood-ratio test, requires ML fit)
testModels(fit1, fit0, method = "D4")
``` |

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