Applied Linear Statistical Models. Simultaneous Inference Topics. Simultaneous Estimation of β 0 and β 1 Issues. Simultaneous Inference. Dr.

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1 Applied Linear Statistical Models Simultaneous Inference Dr. DH Jones Simultaneous Inference Topics Simultaneous estimation of β 0 and β 1 Bonferroni Metod Simultaneous estimation of several mean responses Working-Hotelling Metod Bonferroni Metod Simultaneous estimation for new observations Seffe s Metod Simultaneous Estimation of β 0 and β 1 Issues In data were X = 0 is meaningful, te intercept must be estimated as well as te slope Separate 95% confidence limits could be constructed, so 5% of samples would be incorrect for te slope and anoter, possibly te same, 5% would be incorrect for te intercept In fact, as muc as 10% of te samples could be incorrect for eiter or bot 1

2 parameters Family Confidence Coefficient Many problem require a series of confidence intervals (or tests) were it is desired to know te correctness of te entire set of intervals (or tests) Call te set of estimates te family of estimates A family confidence coefficient is te proportion of samples tat are correct for every member of te family of estimates, in repeated sampling and recalculation of eac estimate For example, in 95% family confidence intervals for joint estimation of te slope and te intercept, 5% of samples would be incorrect for eiter one or bot parameters Basic Idea: Bonferroni Joint Confidence Intervals Eac confidence interval is adjusted to be sligtly iger tan 1-α so tat te family as confidence coefficient 1-α Start wit ordinary 1-α confidence intervals for te slope and intercept Wat is te probability tat bot intervals are correct? Definition of te two events tat te intervals are incorrect A = β -CI wrong} β -CI wrong} 0 0 A = 1 1 2

3 Confidence of eac interval ( β0 ) ( A0) ( β1 ) ( A1) Pr -CI correct = Pr = 1 α Pr -CI correct = Pr = 1 α Confidence of te family of intervals ( β0 β 1 ) = ( 0 1) Pr -CI correct AND -CI correct Pr A A Lower bound on te family confidence Use te Bonferroni inequality (wic we prove sortly) ( A0 A1) ( A0) ( A1) Pr 1 Pr Pr = 1 α α = 1 2α Conclusion Even if eac parameter is estimated wit 1-α confidence, te family confidence could be as low as 1-2α To acieve 1-α family confidence, eac parameter sould be estimated wit 1-α/2 confidence CRITICAL VALUE, BONFERRONI FAMILY CONFIDENCE INTERVALS 3

4 General formulas for any level of confidence: b b } } ( 1 α 4; 2) ± Bs b 0 0 ± Bs b 1 1 B= t n Example: Westwood Data 90% Bonferroni Family Critical Value Critical value for family intervals: ( ) ( ) B= t ;8 = t.975;8 = Note critical value for individual intervals: ( ) t( ) t ;8 =.95;8 = Bonferroni Inequality ( A1 A2) ( A1) ( A2) Pr 1 Pr Pr Proof Use Venn's teorem to devide sample space into four disjoint events ( A1 A2) ( A1 A2) ( A1 A2) ( A1 A2) 1= Pr + Pr + Pr + Pr Terefore 4

5 ( A1 A2) = ( A1 A2) ( A1 A2) ( A1 A2) = 1 Pr( A1 A2) Pr( A1 A2) + Pr( A1 A2) + Pr( A1 A2) Pr 1 Pr Pr Pr + 2Pr ( A1 A2) Pr( A1 A2) ( 1) ( 2) ( 1 2) ( ) P( A ) = 1 P A P A + P A A 1 P A 1 2 SIMULTANEOUS ESTIMATION OF MEAN RESPONSES Working-Hotelling Metod Tese formulas work for an unlimited family of confidence limits. Consequently, tey yield a confidence band for te entire response line. Formulas Yˆ = b + b X Yˆ 0 1 ˆ } ± Ws Y ( α ) 2 W F n = 2 1 ;2, 2 Example: Westwood for batc sizes 30, 55, and 80 For confidence coefficient of.90 F(.90;2,8)=3.11 5

6 W 2 = 2(3.11) = 6.22; W = Derivation for X = 30 Y } ( ) 90% CI = 70.0 ± E 73.2 Summary of standard errors and resultant CI X Y ˆ s Yˆ } Lower Upper Bonferroni Metod Wen te number of confidence limits is small Bonferroni generally gives smaller limits tan Working-Hotelling (except for te following example) Formulas Yˆ ˆ } ± Bs Y ( α ) B= 2t 1 2 g; n 2 g is te number of confidence limits in te family 6

7 Example: Westwood for batc sizes 30, 55, and 80 For g = 3 estimates.90 confidence coefficient B = t(1-.10/2(3);8) = t(.983;8) = 2.56 Te last step from linear interpolation of tabled values Derivation for X = 30 } ( ) 90% CI = 70.0 ± E Y 73.3 Summary of standard errors and resultant CI X Y ˆ s Yˆ } Lower Upper SIMULTANEOUS ESTIMATION INTERVALS FOR NEW OBSERVATIONS Seffe Metod For g CI s wit family confidence of 1-α 7

8 Yˆ ± Ss Y ˆ new ( )} ( 1 α;, 2) 2 S gf g n = Bonferroni Metod For g CI s wit family confidence of 1-α Yˆ ± Bs Y ˆ new ( )} ( α ) B= 2t 1 2 g; n 2 Example: Westwood 95% for tree batces of size 30, 55, and 80. ( ) ( ) ( ) ( ) 2 S F S = 3.95;3,8 = = = 3.49 B= t ;8 = t.992;8 = 3.04 Te Bonferroni metod gives tigter bounds X Yˆ ˆ ( new )} s Y Seffe Lower Seffe Upper Bonf. Lower Bonf. Upper