The proportional fitting procedure (IPFP,) is an iterative algorithm for estimating expected cell values [M_ijk] of a contingency table such that the marginal conditions are met.
For iterative computation of the M_ij, using MLE for the model, other Algorithm exist , and most common are for 2D while am interested in that of 3D, Assuming model of independence is equivalent to the model that all odds ratios equal one. I need references of examples showing manual calculation, and how to do more advance ones on R.
Deming and Stephan (1940) first proposed the use of an iterative proportional fitting procedure to estimate cell probabilities in a contingency table subject to certain marginal constraints BUT FAIL TO PROVE THE CONVERGENCE! {note: The IPFP is also known as the RAS algorithm in economics and matrix raking or matrix scaling in computer science.}
A rigorous proof of convergence by means of differential geometry is due to Fienberg (1970) while a good mathematical foundations can be found in the book of Bishop et al. (1975). Also, convergence and error nature by Pukelsheim and Simeone (2009).
With improvement and innovations in Computing, and the use of Computers for Statistical Analysis and computations many approaches and partial success have been achieved, (algorithm and packages on R, SAS, Excel, etc). Example is the mipfp on the R platform (packages) ... https://rdrr.io/cran/mipfp/man/Ipfp.html.
This is where I ,ambitiously, intend to research and make contribution; more than geometrical proof of convergence, ...extending it not only to least squared approach but to situations where cells have zero counts, and the generalized approach.
The Algorithms which include Newton–Raphson method and the EM algorithm can be extended but in most cases, IPFP is preferred due to its computational speed, numerical stability and algebraic simplicity.
If unique MLEs exist, IPFP exhibits linear convergence in the worst case (Fienberg 1970), but exponential convergence has also been observed (Pukelsheim and Simeone 2009).
References:
Bishop, Y. M. M., Fienberg, S. E., Holland, P. W. (1975). Discrete Multivariate Analysis: Theory and Practice. MIT Press. ISBN 978-0-262-02113-5.
Deming, W. E., Stephan, F. F. (1940). On a Least Squares Adjustment of a Sampled Frequency Table When the Expected Marginal Totals are Known. Annals of Mathematical Statistics 11 (4): 427-444.
Fienberg, S. E. (1970). An Iterative Procedure for Estimation in Contingency Tables. Annals of Mathematical Statistics 41 (3): 907-917.
https://rdrr.io/cran/mipfp/man/Ipfp.html
Use table function; is you share the exact breaker you faced, probably I can help you out.
You were right Suraj Bhagat .. , thank you! Table function, table(), uses the cross-classifying factors to build a contingency table of the counts at each combination of factor levels. Am, however, interested in obtaining the maximum likelihood - expected counts for the cells.
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