Download presentation

Presentation is loading. Please wait.

Published byMeagan Dee Modified over 2 years ago

1
Simultaneous inference Estimating (or testing) more than one thing at a time (such as β 0 and β 1 ) and feeling confident about it …

2
Simultaneous inference we’ll be concerned about … Estimating β 0 and β 1 jointly. Estimating more than one mean response, E(Y), at a time. Predicting more than one new observation at a time.

3
Why simultaneous inference is important A 95% confidence interval implies a 95% chance that the interval contains β 0. A 95% confidence interval implies a 95% chance that the interval contains β 1. If the intervals are independent, then have only a (0.95×0.95) ×100 = 90.25% chance that both intervals are correct. (Intervals not independent, but point made.)

4
Terminology Family of estimates (or tests): a set of estimates (or tests) which you want all to be simultaneously correct. Statement confidence level: the confidence level, as you know it, that is, for just one parameter. Family confidence level: the confidence level of the whole family of interval estimates (or tests).

5
Examples A 95% confidence interval for β 0 – the 95% is a statement confidence level. A 95% confidence interval for β 1 – the 95% is a statement confidence level. Consider family of interval estimates for β 0 and β 1. If a 90.25% chance that both intervals are simultaneously correct, then 90.25% is the family confidence level.

6
Bonferroni joint confidence intervals for β 0 and β 1 GOAL: To formulate joint confidence intervals for β 0 and β 1 with a specified family confidence level. BASIC IDEA: –Make statement confidence level for β 0 higher –Make statement confidence level for β 1 higher –So that the family confidence level for (β 0, β 1 ) is at least (1-α)×100%.

7
Recall: Original confidence intervals For β 0 : For β 1 : Goal is to adjust the t-multiples so that family confidence coefficient is 1-α. That is, we need to find the α* to put into the above formulas to achieve the desired family coefficient of 1- α.

8
A little derivation Let A 1 = the event that first confidence interval does not contain β 0 (i.e., incorrect). So A 1 C = the event that first confidence interval contains β 0 (i.e., correct). P(A 1 ) = α and P(A 1 C ) = 1- α

9
A little derivation (cont’d) Let A 2 = the event that second confidence interval does not contain β 1 (i.e., incorrect). So A 2 C = the event that second confidence interval contains β 1 (i.e., correct). P(A 2 ) = α and P(A 2 C ) = 1- α

10
Becoming a not so little derivation… A1A1 A2A2 A 1 or A 2 A 1 C and A 2 C We want P(A 1 C and A 2 C ) to be at least 1-α. P(A 1 C and A 2 C ) = 1 – P(A 1 or A 2 ) = 1 – [P(A 1 )+P(A 2 ) – P(A 1 and A 2 )] = 1 – P(A 1 ) – P(A 2 ) + P(A 1 and A 2 )] ≥ 1 – P(A 1 ) – P(A 2 ) = 1 – α – α = 1 – 2α So, we need α* to be set to α/2.

11
Bonferroni joint confidence intervals Typically, the t-multiple in this setting is called the Bonferroni multiple and is denoted by the letter B.

12
Example: 90% family confidence interval The regression equation is punt = 14.9 + 0.903 leg Predictor Coef SE Coef T P Constant 14.91 31.37 0.48 0.644 leg 0.9027 0.2101 4.30 0.001 n=13 punters t(0.975, 11) = 2.201 We are 90% confident that β 0 is between -54.1 and 83.9 and β 1 is between 0.44 and 1.36.

13
A couple of more points about Bonferroni intervals Bonferroni intervals are most useful when there are only a few interval estimates in the family (o.w., the intervals get too large). Can specify different statement confidence levels to get desired family confidence level. Bonferroni technique easily extends to g interval estimates. Set statement confidence levels at 1-(α/g), so need to look up 1- (α/2g).

14
Bonferroni intervals for more than one mean response at a time To estimate the mean response E(Y h ) for g different X h values with family confidence coefficient 1-α: where: g is the number of confidence intervals in the family

15
Example: Mean punting distance for leg strengths of 140, 150, 160 lbs. Predicted Values for New Observations New Fit SE Fit 95.0% CI 95.0% PI 140 141.28 4.88 (130.55,152.01) (103.23,179.33) 150 150.31 4.63 (140.13,160.49) (112.41,188.20) 160 159.33 5.28 (147.72,170.95) (121.03,197.64) n=13 punters t(0.99, 11) = 2.718 We are 94% confident that the mean responses for leg strengths of 140, 150, 160 pounds are …

16
Two procedures for predicting g new observations simultaneously Bonferroni procedure Scheffé procedure Use the procedure that gives the narrower prediction limits.

17
Bonferroni intervals for predicting more than one new obs’n at a time To predict g new observations Y h for g different X h values with family confidence coefficient 1-α: where: g is the number of prediction intervals in the family

18
Scheffé intervals for predicting more than one new obs’n at a time To predict g new observations Y h for g different X h values with family confidence coefficient 1-α: where: g is the number of prediction intervals in the family

19
Example: Punting distance for leg strengths of 140 and 150 lbs. n = 13 punters Bonferroni multiple: Suppose we want a 90% family confidence level. Scheffé multiple: Since B is smaller than S, the Bonferroni prediction intervals will be narrower … so use them here instead of the Scheffé intervals.

20
Example: Punting distance for leg strengths of 140 and 150 lbs. Predicted Values for New Observations New Fit SE Fit 95.0% CI 95.0% PI 140 141.28 4.88 (130.55,152.01) (103.23,179.33) 150 150.31 4.63 (140.13,160.49) (112.41,188.20) n=13 punters s(pred(140)) = 17.28 There is a 90% chance that the punting distances for leg strengths of 140 and 150 pounds will be… s(pred(150)) = 17.21

21
Simultaneous prediction in Minitab Stat >> Regression >> Regression … Specify predictor and response. Under Options …, In “Prediction intervals for new observations” box, specify a column name containing multiple X values. Specify confidence level. Click on OK. Results appear in session window.

Similar presentations

Presentation is loading. Please wait....

OK

Regression through the origin

Regression through the origin

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on accounting standard 10 fixed assets Ppt on blood stain pattern analysis history Ppt on types of abortion Free download ppt on exponents and powers Ppt on unity in diversity and organic farming Ppt on four wheel car steering system Ppt on market friendly statements Ppt on intellectual property Ppt on the portrait of a lady Ppt on layer 3 switching