5.1 Polynomial Regression (extended atheoretical example)

5.2 std mtx theory

  - mtx forms leading to

        [start from mtx formulatioh]
	then multvar normal
	calculation of Var (Y) for a general vector Y and for AY

        \hat beta = (XTX)^-1 XTy, (mtx calc)
	\hat y = Hy
	resid = (I-H)y
	RSS = (y-yhat)T(y-yhat) = yT(I-H)y

	E[beta]
	Var(beta)
	E[yhat]
	Var(yhat)
	E[e]
	Var(e}

	specialize for 1 regressor
	and also for intercept-only model

        S^2
	T for a single beta; get se (beta) from diag Var(beta)

        SST = (y-bary)T(y-barY)  = yT(I- H1)y
	SSreg = (yhat - bary)T(hat-bary) = yT(H-H1)y
	RSS = yT(I-H)y

        so SSreg + RSS = yT(H-H1)y + yT(I-H)y = yT(I-H1)y = SST

        cochran's (?) thm: in a SS decomp of SST, if the mtcs of the
        qforms are orthogonal then the qforms are indep.

 	this sort of thing lets us compute that under H0=inctp only,

            F = (SSreg/p)/(RSS/(m-p-1)) ~ F

        and more generally for large/small mdoel the F stat has an F
        distrib under the null model

        (power is more work and depends on noncentral chi^2 and F
        distrib, but similar...)

        again R^2 = SSreg/SST = 1 - RSS/SST

        alt R^_asj compares two variance estimates, S^2_model vs
        S^2_incpt

	this is important for testing discrete predictors, eg.


5.3  Ancova (kidiq?)


WEDS

added var plots

additve covariates

maybe come bac to poly regr

interactions

QUIZ

  hat mtx facts?
  give a regression result and see if tey can interp?

EXERCISES