Statistics 762 Nonlinear Statistical Models for Univariate and - - PowerPoint PPT Presentation

ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response

Statistics 762

Nonlinear Statistical Models for Univariate and Multivariate Response

Instructor: Peter Bloomfield Course home page:

http://www.stat.ncsu.edu/people/bloomfield/courses/st762/

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ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response

Introduction

Objective: provide a comprehensive treatment of modern approaches to modeling univariate and multivariate responses. Theme: univariate and multivariate models share common features; univariate models will be covered first, in detail; multivariate models will then be discussed.

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ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response

Point of departure: “Classical” linear regression

Y = response, or dependent variable (scalar) x = (p × 1) covariate, or independent variable.

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ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response

Covariates may be controlled (fixed) or merely observed (random), but in either case are known without error. The response Y is always viewed as random, because: it may have measurement error; it may vary from one subject to another; since subjects are sampled randomly from some population, this is also a source of randomness; it may vary randomly over time for a given subject.

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ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response

A regression model is a specification of the distribution of Y for each possible value of x. If x is fixed, this is just the distribution of Y as a function of x. If x is random, x and Y have a joint distribution, and the regression model is the conditional distribution of Y given x. In the random x case, a complete probability model requires also the marginal distribution of x, but that is irrelevant to the dependence of Y on x.

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ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response

Observed data are pairs (Yj, xj), j = 1, 2, . . . , n. Matrix notation: Y = (Y1, Y2, . . . , Yn)T , X =        xT

1

xT

2

. . . xT

n

       where Y is (n × 1) and X is (n × p).

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ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response

The linear regression model: Yj = xT

j β + ej,

j = 1, 2, . . . , n. In matrix notation, Y = Xβ + e where e = (e1, e2, . . . , en)T .

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ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response

Assumptions: (0) E(ej) = 0; no bias. (1) E(Yj| xj) = xT

j β; implies (0).

(2) e1, e2, . . . , en are identically distributed, with variance σ2. (3) e1, e2, . . . , en are independent. (4) e1, e2, . . . , en are normally distributed.

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ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response

In the random covariate case, (2) – (4) are interpreted conditionally

• n X.

Under (1) – (4), Y ∼ N

• Xβ, σ2In
• .

All the standard statistical theory of estimation and testing may be derived from this property.

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