11+Regression

2 ssx1 x1 ssx2 x2 ssx1 x2 example 5 suppose we want

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: x2 , 2 SSx1 x1 SSx2 x2 − SSx1 x2 c22 = SSx1 x1 , 2 SSx1 x1 SSx2 x2 − SSx1 x2 Alternative hypothesis is One-tailed test Ha = {βi < 0} (or Ha = {βi > 0}) and reject region is RR = {t < −tα } (or Ha = {t > tα }), where tα are based on (n − k − 1) degree of freedom and we can find it using Table VI (McClave and Benson). Two-tailed test Ha = {βi = 0} and reject region is RR = {|t| > tα/2 }. A 100(1 − α)% confidence interval. The confidence interval for a parameters βi is ˆ βi ± tα/2 sβi , ˆ where tα/2 is based on (n − 3) degree of freedom. Remark. If the test of H0 = {βi = 0} fails to reject for some i, this does not mean that we should omit the βi . Several conclusions are possible: 1. there is no relationship between y and xi ; 2. a type II error occurred; 3. a relationship between y and xi exists, but is more complex than straight-line relationship. Multiple coefficient of determination R2 for the straight-line model is SSE =1− R =1− SSyy 2 12 n ¯2 ν =1 (yν − yν ) . n (yν − y )2 ¯ ν =1 A. Zhensykbaev Regression Testing global usefulness of the model. Consider the global F -test to indicate that the model is useful for prediction. The null hypothesis is H0 = {β1 = β2 = · · · = βk = 0}. Alternative hypothesis is Ha = {at least one βi = 0}. Test statistics is R2 /k , (1 − R2 )/(n − k − 1) where n - sample size, k - number of variables in the model. F= RR = {F > Fα } where Fα is is defined from the Tables VIII-XI (McClave and Benson) with ν1 = k and ν2 = n − k − 1. Prediction. The prediction interval for the mean value of y for specific values of x1 ,..., xk of level (P = 1 − α) is k ˆ βi (xi − xi ) ± tα/2 σy , ¯ ˆ y+ ¯ i=1 where tα/2 is defined from Table VI with df = n − k − 1, σy = σ ˆ ˆ 1 + n k k cij (xi − xi )(xj − xj ), ¯ ¯ i=1 j =1 and for k = 2 the quantities c11 , c22 was defined early, c12 = c21 = SSx1 x2 . 2 SSx1 x1 SSx2 x2 − SSx1 x2 Example 5. Supp...
View Full Document

Ask a homework question - tutors are online