Quadratic Inference Functions (QIF) are a quasi-likelihood based technique for building regression models and testing a variety of general linear hypotheses for correlated observations arising from longitudinal and clustered data. The QIF method is an alternative to Generalized Estimating Equations (GEE) for analyzing such data. The main advantage to QIF is that it is based on an optimization criterion, therby providing a simple approach to model building and testing.

QIF-LIB is an S-Plus library developed in conjunction with theory developed in Pilla and Loader (2005) while the core of the software provides an implementation of the Iteratively Reweighted Generalized Least Squares (IRGLS) algorithm developed by Loader and Pilla (2007). Currently supported features of the software include:

• Covariates: Time-varying and time-independent.
• Families: Gaussian, Binomial (logistic) and Poisson.
• Model Building: chi-square tests and AIC.
• Graphical output: trellis display of fitted values by subject.

The articles Pilla and Loader (2005) and Loader and Pilla (2007) are the definitive references for the theory and implementation underlying QIF-LIB, and should be cited in all reports, publications, presentations and any other work using the QIF-LIB.

Pilla and Loader (2005) developed new theory, derived large-sample properties, algorithms and model-building techniques for the Quadratic Inference Function, while correcting the asymptotic theory and numerical computations appearing in the original Biometrika article by Qu, Lindsay and Li (2000).

Loader and Pilla (2007) derived the IRGLS algorithm for estimation and testing with correlated data, including finding the QIF estimator, while establishing its converge properties.

Examples.

• Respiratory Health study.

See also the example functions in the references.

Download
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QIF-LIB is available for R and S/S-Plus on Unix/Linux operating systems. Windows compilers are not currently supported.

Changes
A few minor changes and additions since Loader and Pilla (2007) article was published.

• qiffit and irlsmod are now defined as S-4 style classes (J. Chambers, (1998). Programming with Data. Springer). From the user's perspective, the main change is that slots are extracted with @, not $.
• The new qifmod() function is synonymous with irlsmod().
• new anova.irlsmod() function (i.e., anova() method for "irlsmod" object) to present test results in an anova-like format. Note that models must be specified in a nested linear fashion --- see examples/laird.s for an example.
• The default optimization has been modified to use a hybrid irgls-conjugate gradient method, which improves convergence in some cases.
• new arguments to qiffit() to control the optimization:

      method="hybrid" or method="irls".
      maxit (limit on iterations).
      tol (convergnce tolerance)
  

Note, The convergence criterion is that N^{-1} Q(beta) changes by less than tol. The default tolerance is currently set to 1.0×10^-8, usually providing about four significant digits accuracy for parameter estimates. Reducing tol gives more precision in well-behaved cases; however, can lead to convergence problems in poorly behaved cases.

References

• Pilla, R.S. and Loader, C. (2005). On Large-Sample Estimation and Testing via Quadratic Inference Functions for Correlated Data. E-print Archive arXiv:math.ST/0505360.
• Loader, C. and Pilla, R.S. (2007). Iteratively Reweighted Generalized Least Squares for Estimation and Testing With Correlated Data: An Inference Function Framework, Journal of Computational & Graphical Statistics, 16: 925--945.

Any report, presentation, publication or other form of scholarly work using our methods and the QIF-LIB software should cite the above references.

History

• Version 1.0, November 20, 2007.
• Version 0.1 (R versions) March 27, 2005.
• Version 0.1, Released February 23, 2005.

Acknowledgments

This research has been supported by grants from

• National Science Foundation: DMS 03-06202 (C. Loader) and DMS 02-39053 (R.S. Pilla);
• Office of Naval Research: N00014-02-1-0316 (R. S. Pilla), N00014-04-1-0481 (R.S. Pilla and C. Loader) and N00014-06-1-0005 (R.S. Pilla).

Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Office of Naval Research or of the National Science Foundation.