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:
T
Y = (Y1 , Y2 , . . . , Yn ) ,
where Y is (n × 1) and X is (n × p).
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
xT1




X=


xT2
..
.






xTn
ST 762
Nonlinear Statistical Models for Univariate and Multivariate Response
The linear regression model:
Yj = xTj 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 ) = xTj β; implies (0).
(2) e1 , e2 , . . . , en are identically distributed, with variance σ 2 .
(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
on X.
Under (1) – (4),
Y ∼ N Xβ, σ 2 In .
All the standard statistical theory of estimation and testing may be
derived from this property.
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