PSYCHOMETRIKA--VOL. 39, NO. 4 DECEMBER, 1974

C O M M E N T ON ' T H E E S T I M A T I O N OF PROPORTIONS IN M GROUPS'

TOM LEONARD

UNIVERSITY OF WARWICK

Novick, Lewis and Jackson [1973] propose an interesting Bayesian method for the simultaneous estimation of the probabilities of several binomial distributions. They utilise arc-sine transformations and normal approximations and then refer to the normal theory method of Lindley [1972] which utilises the novel concept of 'exchangeability'. The posterior estimates for the probabilities shrink the usual proportions towards a central value based on collateral information, and should provide substantial improvements.

The purpose of our comment is to discuss the relative merits of other methods which are available whenever exchangeability of the probabilities is appropriate, The main methods are

(1) The Logistic Approach: Leonard [1972a] uses logistic transformations of the param- eters and proposes a simple exact method of estimation which is valid even if some of the cell frequencies are zero. This is because the logistic transformatior/s are to the parameters and not to the observed frequencies. The exact posterior modal estimates replace the usual observed logits in such a way that difficulties involving zeros are removed. The are-sine approximation is of course inaccurate in such cases.

Some illustrative approximations which use the observed logits are also proposed under the logistic approach, and these are not valid if any of the cell frequencies are zero. Mosteller has pointed out (personal communication with Novick) that the likelihood approximation used in this method is inferior to that under the arc-sine approach. This approximation is not however needed for the exact method.

A relative disadvantage of the arc-sine approach is that the transformed parameters are bounded between 0 and v/2, so that Lindley's normal theory should not strictly be applied. I t is not obvious that it suffices as an approximation. In the logistic case the trans- formed parameters may assume any real value, so that no approximations are needed under the prior formulation.

The main advantage of the logistic approach is that it easily generalisable to situations where more complex relationships are thought a priori to exist between the parameters, e.g. where they are related in an ordered fashion, or in regression mode ls - see Leonard [1973a and bl. The latter also extends the method to the situation involving several multi- nomial distributions by employing multivariate Iogit transformations. Holland [1973] shows that the arc-sine transformation is not generalisable to this case.

An important alternative approach to related multinomial problems is by Fienberg and Holland [1973l and Sutherland, Holland and Fienberg [1974]. Their estimates often approximate the posterior means and possess remarkably good asymptotic sampling theoretic properties. Like our exact method, this appears to be unaffected by the presence of zeros, and they indeed possess more sampling theoretic justification for their method in this case. We are not sure whether their method readily extends to many of the generalisa- tions discussed above. Another point is that under our approach we are able to introduce more prior information if we wish to.

Sutherland et al provide a most interesting analysis of a 14 X 14 occupational mobility table originally due to Karl Pearson, and demonstrate how to avoid difficulties involving

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zeros. Leonard [1975] provides an alternative to this analysis, and obtains some interesting results involving measures of association, as well as dealing with the zeros. He is able to assume a much more general prior covariance structure to that of Sutherland et al.

(2) The Arc-Sine Approach, 'Exact' Method: Leonard [1972b] removes one or two of the difficulties discussed in (1) by showing that normal approximations are unnecessary in the arc-sine situation. He proposes estimates based on the exact likelihood of the trans- formed parameters, and these are accurate even if some of the cell frequencies are zero. The estimates are easy to compute, and are 'exact' if we choose to ignore the boundedness of the parameters.

(3) Approximations to the Posterior Means: Leonard [1972b and 1973b] shows how to compute approximations to the posterior means under both the arc-sir~e and logistic ap- proaches, and provides some interesting numerical comparisons. The estimates are only valid if none of the cell frequencies are small, but are advantageous in the sense that they are based on the sensible quadratic loss function. The methods integrate the posterior means when the between-variance is known with respect to an approximation to the mar- ginal posterior distribution of the between-variance. The method in the logistic case is exact, given the (inferior) likelihood approximation, but in Lhe arc-sine case the boundedness of the parameters again plays its part, so that it is debatable which is the more accurate approach. When some of the cell frequencies are small, the exact method in (1) appears to be more reasonable.

(4) Approximations to the Marginal Posterior Distributions of the Parameters: I t is useful to make posterior probability statements about the parameters, as well as to estimate them. Leonard [1972b and 1973b] shows how to do this under either approach by employing integrations of a similar nature to those proposed in (3). Further convenient approximations, in the arc-sine situation, in the special case where the distributions are equally replicated, are as yet unpublished.

In the above discussion we have made little at tempt to say which approach actually provides 'best' estimates, and have not even defined 'best'. We have however described a few properties of the alternative methods, and hope that these will be of use to the practi- tioner.

The authors acknowledges the advice of Professor M. R. Novick and Dr. C. Lewis which helped him to obtain the solutions in (2)-(4) whilst working at the American College Testing Program. The research was financed by an A.C.T./I .T.P. Pre-doctoral Research Fellowship.

R E F E R E N C E S

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

Holland, P. Covariance stabilising transformations. Annals of Statistics 1, 1973, 84-92. Leonard, T. Bayesian methods for binomial data. Biometrika, 1972a, 59, 581-589. Leonard, T. Bayesian methods for binomial data: some computational methods and further

ideas. A. C. T. Technical Bulletin No. 7, Iowa City, Iowa: The American College Testing Program, 1972 (b).

Leonard, T. A. Bayesian method for histograms. Biometrika, 1973 (a), 60, 297-308. Leonard, T. Bayesian methods for the simultaneons estimation of several parameters.

Unpublished doctoral dissertation, Univelsity of London, 1973 (b). Leonard, T. Bayesian estimation methods for two-way contingency tables. Journal of the

Royal Statistical Society Series B, 1975, 37, (in press). Lindley, D. V. The estimation of many parameters. In V. P. Godambe and D. A. Sprott,

(Eds.), Foundations of Statistical Inference. Toronto: Holt, Rinehart and Winston, 1972, 435-455.

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Novick. M. R., Lewis C. and Jackson, P. H. The estimation of proportions in M groups. Psychometrika~ 1973, 38, 19--46.

Sutherland M., Holland, P. W. and Fienberg, S. E. Combining Bayes and frequency ap- proaches to estimate a multinomial parameter. In S. E. Fieaberg and A. Zellner, (Ed.), Studies in Bayesian Econometrics and Statistics, Amsterdam: North Holland Publishing Co., 1974, (in press).