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There is considerable posterior probability near 0 with most of the posterior mass less than , indicating the third axis provides little explanatory power. The analysis in ZWG nds only the rst axis to be signi cant, which qualitatively agrees with the Bayesian analysis when the magnitude of the second axis is calibrated by the KL information.

The KL information provides more information than signi cance, however, since it gives a way of determining if the axis is signi cant by observing the amount of posterior probability near 0 and quantifying the predictive power of the axis. The Gibbs Sampler also allows us to compute posterior variances for any of the parameters. For example, the posterior standard deviations of the and components for the rst axis are between and , suggesting that a 1i or 1j should be considered statistically nonzero if they are greater in absolute value than around This includes 14 of the 35 1i and 5 of the 7 1j.

This guides the investigator toward the signi cant components of the interaction. This example is based on the dataset from Oman The data consist of measures of tomato softness as the force required to compress a given tomato 0. The 21 rows, indexed by j, represent randomly selected mother plants and the 3 columns, indexed by i, represent genotypes. The observed data clearly exhibit heteroscedasticity across the genotypes. We reanalyzed these data with the AMMI model with one axis.


The goal here is to emphasize the exibility of the Bayesian method regarding unequal variances and random e ects. Our model is 4. In frequentist analyses, xed e ects a ect the distribution of the response through the mean vector while random e ects a ect the covariance matrix.

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In the Bayesian approach, xed e ects are estimated using the prior distribution described in equation 3. This prior distribution induces a structure similar to a random e ect. The variance of the random e ect may be estimated by looking at the posterior distribution of 2. The results are shown in Table 2. Our results are similar to Oman's even though the models are slightly di erent. The posterior distribution of 2 is shown in Figure 2. The posterior mean and variance of 2 are 1.

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  • The posterior distribution of 2 is clearly shifted away from 0, indicating that softness varies among the mother plants. Our estimates of i2 , though comparable, are slightly smaller than Oman's estimates. Although multiplicative e ects appear to be present from both the posterior distribution of and the KL information 7 shown in Figure 3, the overall multiplicative e ect is relatively small.

    The mean KL information is corresponding to normal distributions standard deviations apart. This is also consistent with Oman's results, who found a signi cant, but not large, interaction component in his model formulation. One interesting theoretical contrast between the Bayesian and frequentist analyses of the AMMI model occurs for small cell sizes K , which occur often in practice.

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    Many frequentist methods for determining the number of components are based on asymptotics in K see CCS and references therein , and thus require adjustments for small K. We hope to investigate this issue further in future work. From a practical perspective, the Bayesian analysis based on the Gibbs Sampler easily provides information about the parameters or any functional of the parameters, and is exible enough to estimate generalizations of the AMMI model such as unequal variances. To the best of our knowledge, no one has attempted a complete frequentist analysis of heteroscedastic data.

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    From a Bayesian computational perspective, we have developed a prior for sampling coordinate axes and provided an implementation using MCMC techniques. The proposed method is computationally intensive, although it is unclear if there are intuitively appealing alternatives for estimating the axes directly from the discussion in section 2. The proposed goodness of t criterion allows t of the interaction components to be examined in two ways.

    Using the posterior distribution of either the individual or the KL information it is possible to visually determine if any particular axes are statistically signi cant from 0.

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    This may be formalized to a Bayes factor using the Savage-Dickey density ratio to provide a hypothesis test. The posterior distribution of the KL information also allows interpretation of the magnitude of the distance. It is possible an interaction component may be present, but may be small enough to ignore.

    Ignoring the component allows a more parsimonious model while keeping the component allows better predictions. The second axis of the soybean data provides an example. A second principal component appears to be present in the data, but its actual e ect seems small when calibrated by the KL information.

    Whether the axis should be retained should be decided on substantive grounds. The authors would also like to thank Genzhou Liu and two anonymous referees for helpful comments.

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    Equation of state calculations by fast computing machines. Journal of Chemical Physics 21 Methun, B. In Multilevel Analysis of Educational Data. Bock, R. Academic Press. San Diego. Oman, S. Biometrika 78 Rao, C. Schervish, M. Theory of Statistics. Springer Verlag. New York. Tierney, L. Markov Chains for Exploring Posterior Distributions.

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    Annals of Statistics 22 Tukey, J. Answer to Query Biometrics 11 van Emden, M. An Analysis of Complexity. Mathematical Centre Tracts 35, Amsterdam. Verdinelli, I.