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Unformatted text preview: 7 The multivariate normal model Up until now all of our statistical models have been univariate models, that is, models for a single measurement on each member of a sample of individuals or each run of a repeated experiment. However, datasets are frequently multi variate , having multiple measurements for each individual or experiment. This chapter covers what is perhaps the most useful model for multivariate data, the multivariate normal model, which allows us to jointly estimate population means, variances and correlations of a collection of variables. After first cal culating posterior distributions under semiconjugate prior distributions, we show how the multivariate normal model can be used to impute data that are missing at random. 7.1 The multivariate normal density Example: Reading comprehension A sample of twentytwo children are given reading comprehension tests before and after receiving a particular instructional method. Each student i will then have two scores, Y i, 1 and Y i, 2 denoting the pre and postinstructional scores respectively. We denote each student’s pair of scores as a 2 × 1 vector Y i , so that Y i = Y i, 1 Y i, 2 = score on first test score on second test . Things we might be interested in include the population mean θ , E[ Y ] = E[ Y i, 1 ] E[ Y i, 2 ] = θ 1 θ 2 and the population covariance matrix Σ , Σ = Cov[ Y ] = E[ Y 2 1 ] E[ Y 1 ] 2 E[ Y 1 Y 2 ] E[ Y 1 ]E[ Y 2 ] E[ Y 1 Y 2 ] E[ Y 1 ]E[ Y 2 ] E[ Y 2 2 ] E[ Y 2 ] 2 = σ 2 1 σ 1 , 2 σ 1 , 2 σ 2 2 , P.D. Hoff, A First Course in Bayesian Statistical Methods , Springer Texts in Statistics, DOI 10.1007/9780387924076 7, c Springer Science+Business Media, LLC 2009 106 7 The multivariate normal model where the expectations above represent the unknown population averages. Having information about θ and Σ may help us in assessing the effectiveness of the teaching method, possibly evaluated with θ 2 θ 1 , or the consistency of the reading comprehension test, which could be evaluated with the correlation coefficient ρ 1 , 2 = σ 1 , 2 / p σ 2 1 σ 2 2 . The multivariate normal density Notice that θ and Σ are both functions of population moments , or population averages of powers of Y 1 and Y 2 . In particular, θ and Σ are functions of first and secondorder moments: firstorder moments: E[ Y 1 ] , E[ Y 2 ] secondorder moments: E[ Y 2 1 ] , E[ Y 1 Y 2 ] , E[ Y 2 2 ] Recall from Chapter 5 that a univariate normal model describes a population in terms of its mean and variance ( θ,σ 2 ), or equivalently its first two moments (E[ Y ] = θ, E[ Y 2 ] = σ 2 + θ 2 ). The analogous model for describing first and secondorder moments of multivariate data is the multivariate normal model....
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This note was uploaded on 11/24/2010 for the course STAT 201a taught by Professor Wu during the Spring '10 term at Pasadena City College.
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