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General Linear Model With correlated error terms = 2 V ≠ 2 I

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The General Linear Model ≠ 2 I

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Summary

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Example Simple Linear Model where variance is proportional to X 2.

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Testing and Confidence Intervals The Model: can be converted to the model

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Thus Simultaneous Confidence Intervals (using model (2))

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Example: Simple Linear Model with no intercept The model

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Thus

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Also Special cases

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General Linear Model Case 2: unknown

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The General Linear Model unknown

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Call this the Ordinary Least Squares (OLS) estimator of Note: Thus the Ordinary Least Squares (OLS) estimator of is always unbiased.

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This is the Optimal (UMVU) estimator of Note: This is also an unbiased estimator of The Optimal (UMVU) estimator of requires the knowledge of in order to calculate it.

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Theorem: Equivalence of OLS estimator with UMVU estimator

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Proof

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Application: Consider the general linear model with intercept In this case the error terms are equally correlated. Also in this case the OLS estimators are equivalent to the UMVU estimators

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Proof

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Design Matrix, X, not of full rank

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The General Linear Model

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If the rank of X is equal to p then the columns of X are linearly independent and there is a unique way of representing If the rank of X is strictly less than p then there is no unique way of representing

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Comment: Usually the situation where the rank of X, r < p, arises in the following instances. 1.The design of the study (the choice of the values of X 1, X 2, …, X p ) was not careful enough to ensure that X had full rank. 2.Observations were missing causing the model to be altered Elements of are deleted along with corresponding rows of X, reducing the number of linear independent rows from p to r. 3.The model was defined in such a way that: i = 1 x i1 + 2 x i2 + … + p x ip is not uniquely determined by 1, 2, …, p.

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Two Basic approaches: 1.Impose p – r linear restrictions on the parameters This allows us to reduce the number of parameters to r. will have a unique representation if the p – r restrictions are added. This technique is usually used with ANOVA, MANOVA, ANACOVA models. 2.Live with the singularity. Restrict our attention to linear combinations of the parameters that have unique estimators. The two approaches are essentially the same (lead to the same conclusions).

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Recall: Linear Equations theory Consider the system of linear equations M (A), the linear space spanned by the columns of A

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Then the general solution to the system of linear equations is

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Maximum Likelihood Estimation leads to the system of linear equations p equations in p unknowns called the Normal equations

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Theorem The Normal equations are consistent. Proof It can be shown that M (XX) M (X) M(X)M(X) M(XX)M(XX) Theorem The general solution to the Normal equations is

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Theorem is the same for all solutions of the Normal equations Proof: the general solution to the Normal equations is Since M (XX) M (X) there exists a p × n matrix L such that X = XXL or X = LXX

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Definition: (Estimability) The linear function of the parameter vector, is called estimable if there exists a vector such that Example The simple linear model

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Thus is the only estimable function of 0, 1.

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Theorem: The following conditions are equivalent 2. For some solution,, of the Normal equation,, is a linear (in ) unbiased estimate of M(XX)M(XX) M(X)M(X)

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Proof: Assume Then there exists a vector such that M(X)M(X) Thus Thus 1. implies 5. (as well as 4.) Now assume 4.

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Thus 4. implies 3. Thus 4. implies 2. and 1.

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Example: One-way ANOVA (Analysis of Variance) Suppose we have k normal populations Let y i1, y i2, …, y in denote a sample of n from Let ij = y ij - ( i ), then i1, i2, …, in denotes a sample of n from distribution. where 11, 12, …, kn are kn independent observations from N(0, 2 ) distribution.

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Matrix Notation Let

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Then the model is

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= then linear space spanned by the vectors M(X)M(X)

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Thus the estimable parameters are of the form: The common approach is to add the restriction This reduces the number of parameters to k, and converts the model to full rank.

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Properties of estimable functions: 1.All linear functions are estimable M (X) = E p = p-dimensional Euclidean space (which contains all p-dimansional vectors) Proof If rank(X) = p then 2. is estimable if Proof (unique for all solutions of the normal equations) Hence is estimable.

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3. If and are estimable then Proof since and are estimable then

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