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Stat 5100 Handout #10 (Ch. 4)

Notes: Simultaneous Inference, Inverse Prediction, and Regression Through Origin

Simultaneous Inference

Previously (HO#8, Ch. 2), we did single inference on β0 & β1 (separately).

What if we want simultaneous inference on g parameters?

Suppose β0 & β1 are of simultaneous interest

We need to adjust inference (hypothesis testing or confidence intervals)

Multiplicity

Let Aj = event that individual (1-α)100% CI for βj does not contain the true value of βj

P(A0) = P(A1) =

What about: P(not A0 and not A1)? Bonferroni inequality:

So we can “control” the overall α by:

What about simultaneous intervals for response Y at multiple (g) X-levels?

As before (Ch. 2), need to choose mean (conf. int.) or individual (pred. int.)

Can just use Bonferroni adjustment:

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But sometimes other methods give “tighter” intervals

1. Mean response: Working-Hotelling

2. Individual prediction: Scheffe

Note that Working-Hotelling intervals don’t depend on the number of Xh levels (g)

o This is because they give the 1-α intervals over all Xh levels

o This is referred to as the confidence band for the regression line

o These will be wider than CI at each X-level, because t < W

Summary on simultaneous inference

SAS Examples

Want g (1–α)×100% simultaneous

intervals on: Methods Notes

Single

predictor

Multiple

predictors

g coefficients (β’s) Bonferroni HO 10 p. 1 HO 9 p. 1 HO 11 p. 5

Population mean Y at g specific X-levels

(or X-profiles for multiple predictors)

Bonferroni or

Working-Hotelling

HO 10 p. 2 HO 9 p. 2 HO 11 p. 6

Individual predicted Y at g [new] specific

X-levels (or X-profiles for mult predictors)

Bonferroni or

Scheffe

HO 10 p. 2 HO 9 p. 3 HO 11 p. 6

Inverse Prediction

Regular prediction: If X = Xh, what is �̂� ?

Inverse prediction: What is Xh value necessary to achieve �̂� = Yh ?

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Why not just switch X & Y in inverse prediction?

Regression through Origin

For contextual reasons, sometimes regression line should be forced through the origin

Then model is:

Cautions for regression through origin:

∑𝑒𝑖

R2

Usually don’t care about β0 estimate (unless reg. line at X=0 is of primary interest)

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Possible problems in linear regression

Assumptions – maybe no remedial measure will fix

Interpretation – sometimes X vs. Y relationship looks counterintuitive

X-levels not always “fixed”

R2 can be abused

Extrapolation

Example from 9/3/04 Nature: