Chp4 - Simultaneous Inference and Other Topics in...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Simultaneous Inference and Other Topics in Regression Analysis • Simultaneous inference of ( β , β 1 ) and their linear functions • Regression through the origin • Effects of measurement error • Inverse prediction • Choice of X levels 1 Simultaneous Inference of β and β 1 We would like to get 1 − α confidence regions that the conclusions for both β and β 1 are correct. We call the set of estimates (or tests) of interest the family of estimates (or tests). A statement confidence coefficient is a probability statement about one parameter, which indicates the proportion of correct estimates that are obtained for repeated samples. A family confidence coefficient indicates the proportion of families of estimates that are entirely correct for re- peated samples. 2 S i n g l e c o n fi d e n c e i n t e r v a l s P r | b j − β j | s ( b j ) ≤ t ( 1 − α , 2 , n − 2 ) = 1 − α ; j = , 1 . ( 1 ) G e n e r a l l y , P r | b j − β j | s ( b j ) ≤ t ( 1 − α , 2 , n − 2 ) ; j = , 1 ( 2 ) 6 = 1 Y j = P r | b j − β j | s ( b j ) ≤ t ( 1 − α , 2 , n − 2 ) u n l e s s t h e y a r e i n d e p e n d e n t . 3 B o n f e r r o n i J o in t C o n fi d e n c e I n t e r v a l s I n t h e o r y w e c a n w o r k o u t t h e j o i n t p r o b a b i l i t y ( 2 ) , w h i c h i n v o l v e s a c o m p l i c a t e d t w o- d im e n s i o n i n t e g r a l . B u t w e c a n e a s i l y g e t a n l o w e r b o u n d u s i n g t h e f o l l o w i n g B o n f e r r o n i i n e q u a l i t y P r m [ k = 1 A k ! ≤ m X k = 1 P r ( A k ) ; P r m \ k = 1 A c k ! ≥ 1 − m X k = 1 P r ( A k ) T h e r e f o r e , w i t h A j = | b j − β j | s ( b j ) ≥ t ( 1 − α , 2 , n − 2 ) o , ( 2 ) ≥ 1 − 2 α . G e n e r a l l y f o r m ( c o r r e l a t e d ) t e s t s , w e c a n u s e α / m s i g n i fi c a n c e l e v e l f o r i n d i v i d u a l t e s t t o g u a r a n t e e a n o v e r a l l T y p e- I e r r o r α . 4 S im u l t a n e o u s E s t im a t i o n o fM e a n R e s p o n s e s G o a l : e s t im a t e t h e m e a n r e s p o n s e a t X l e v e l s { X h ; h = 1 , · · · ,m } . W o r k in g-H o t e l l in g P r o c e d u r e : T h e W o r k i n g-H o t e l l i n g c o n fi d e n c e b a n d c o v e r s t h e e n t i r e r e g r e s s i o n l i n e . S o f o r c o n fi d e n c e i n t e r v a l ˆ Y h ± W s ( ˆ Y h ) ; W = p 2 F ( 1 − α , 2 , n − 2 ) , a n d t h e f a m i l y c o n fi d e n c e c o e ffi c i e n t f o r t h e s e s im u l t a n e o u s e s t im a t e s w i l l b e a t l e a s t 1 − α . B o n f e r r o n i P r o c e d u r e : W i t h B o n f e r r o n i p r o c e d u r e , w e j u s t n e e d t o u s e α / m a s t h e i n d i v i d u a l T y p e- I e r r o r f o r e a c h X h . ˆ Y h ± B s ( ˆ Y h ) ; B = t ( 1 − α 2 m , n − 2 ) . 5 S im u l t a n e o u s P r e d i c t i o n I n t e r v a l s f o r N e w O b s e r v a t i o n s G o a l : p r e d i c t n e w o b s e r v a t i o...
View Full Document

This note was uploaded on 10/30/2011 for the course AMS 578 taught by Professor Finch,s during the Spring '08 term at SUNY Stony Brook.

Page1 / 14

Chp4 - Simultaneous Inference and Other Topics in...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online