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Pseudo-Bayes smoothing of cell frequencies

psbayes datavar [priorvar] [if exp] [in range] [ , by(rowvar [colvar [layervar]]) generate(newvar) prob tabdisp_options ]

Description

psbayes takes datavar, which should be a set of frequencies, and shrinks or smooths it towards a set of frequencies implied by prior probabilities. This will have the effect of replacing sampling zeros by positive estimates whenever the priors are positive.

For a set of data frequencies n_i, summing to n, and a set of prior probabilities q_i, the smoothed estimates are n * p_i, where

n n_i k p_i = ----- --- + ----- q_i, n + k n n + k

and shrinkage is tuned by the constant

n - SUM ( n_i) k = --------------------. SUM (n_i - n * q_i)

These estimates minimise the total mean square error between estimated and estimand probabilities. For more details, see the References.

If priorvar is specified, it must sum to 1 for the data used. If priorvar is not specified, it is taken to be a set of equal probabilities.

Options

by(rowvar colvar layervar) indicates that datavar refers to a table with rows (and columns if specified (and layers if specified)) indexed by the variable(s) named, which will structure a display of cell estimates using tabdisp. If by() is not specified, cell estimates will be displayed according to observation numbers.

generate(newvar) generates a new variable containing results.

prob indicates that probabilities rather than estimated frequencies are to be shown (and if desired kept).

tabdisp_options are options of tabdisp. Default center format(%9.1f).

Examples

. psbayes f prior, by(row col) g(sf)

. contract foreign rep78, zero nomiss . psbayes _freq, by(foreign rep78) prob

Author

Nicholas J. Cox, University of Durham, U.K. n.j.cox@durham.ac.uk

References

Agresti, A. 2002. Categorical data analysis. Hoboken, NJ: John Wiley.

Bishop, Y.M.M., Fienberg, S.E. and Holland, P.W. 1975. Discrete multivariate analysis. Cambridge, MA: MIT Press.

Fienberg, S.E. and Holland, P.W. 1970. Methods for eliminating zero counts in contingency tables. In Patil, G.P. (ed.) Random counts in scientific work. Volume 1: Random counts in models and structures. Pennsylvania State University Press, University Park, 233-260.

Fienberg, S.E. and Holland, P.W. 1972. On the choice of flattening constants for estimating multinomial probabilities. Journal of Multivariate Analysis 2: 127-134.

Fienberg, S.E. and Holland, P.W. 1973. Simultaneous estimation of multinomial cell probabilities. Journal, American Statistical Association 68: 683-691.

Good, I.J. 1965. The estimation of probabilities: an essay on modern Bayesian methods. MIT Press, Cambridge, MA.

Sutherland, M., Holland, P.W. and Fienberg, S.E. 1975. Combining Bayes and frequency approaches to estimate a multinomial parameter. In Fienberg, S.E. and Zellner, A. (eds) Studies in Bayesian econometrics and statistics in honor of Leonard J. Savage. North-Holland, Amsterdam, 585-617.

Also see

On-line: help for tabdisp