MULTIRESPONSE NORMAL ERROR DISTRIBUTIONS

A corresponding normal distribution is available for multiresponse data, that is, for interdependent observations of two or more measurable quantities. Such data are common in experiments with chemical mixtures, mechanical structures, and electric circuits as well as in population surveys and econometric studies. Modeling with multiresponse data is treated in Chapter 7 and in the software of Appendix C. 

The multiresponse counterpart of (1/ t ) is the inverse covariance matrix 27 , which exists only if 27 has full rank. This condition is achievable by selecting a linearly independent set of responses, as described in Chapter 7. Then the exponential function in Eq. (4.2-11) may be generalized to 

This expression goes to zero with increasing magnitude of any error component Eiu, since the argument is nonpositive whenever 27 exists see Problem 4.C. 

Setting the error density function proportional to the exponential function just given, and adjusting to unit total probability, one obtains 

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The statistical investigation of a model begins with the estimation of its parameters from observations. Chapters 4 and 5 give some background for this step. For single-response observations with independent normal error distributions and given relative precisions, Bayes theorem leads to the famous method of least squares. Multiresponse observations need more detailed treatment, to be discussed in Chapter 7. 

The methods of Chapter 6 are not appropriate for multiresponse investigations unless the responses have known relative precisions and independent, unbiased normal distributions of error. These restrictions come from the error model in Eq. (6.1-2). Single-response models were treated under these assumptions by Gauss (1809, 1823) and less completely by Legendre (1805), co-discoverer of the method of least squares. Aitken (1935) generalized weighted least squares to multiple responses with a specified error covariance matrix his method was extended to nonlinear parameter estimation by Bard and Lapidus (1968) and Bard (1974). However, least squares is not suitable for multiresponse problems unless information is given about the error covariance matrix we may consider such applications at another time. 

Box and Draper (1965) took another major step by deriving a posterior density function p 6 Y), averaged over S, for estimating a parameter vector 6 from a full matrix Y of multiresponse observations. The errors in the observations were assumed to be normally distributed with an unknown m X m covariance matrix S. Michael Box and Norman Draper (1972) gave a corresponding function for a data matrix Y of discrete blocks of responses and applied that function to design of multiresponse experiments. 

Box and Draper (1965) derived a density function for estimating the parameter vector 6 of a multiresponse model from a full data matrix Y, subject to errors normally distributed in the manner of Eq. (4.4-3) with a full unknown covariance matrix E. With this type of data, every event u has a full set of m responses, as illustrated in Table 7.1. The predictive density function for prospective data arrays Y from n independent events, consistent with Eqs. (7.1-1) and (7.1-3), is... 

When the v experimental errors are normally distributed with zero mean and those associated with the Mh and kh responses (e.g., in the differential method of kinetic analysis r and r j are statistically correlated, the parameters are estimated from the minimization of the following multiresponse objective criterion ... 

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