Simultaneous inference

1. Simultaneous inference Estimating (or testing) more than one thing at a time (such as β0 and β1) and feeling confident about it …

2. Simultaneous inference we’ll be concerned about … • Estimating β0 and β1 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 β0. • A 95% confidence interval implies a 95% chance that the interval contains β1. • 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 β0 – the 95% is a statement confidence level. • A 95% confidence interval for β1 – the 95% is a statement confidence level. • Consider family of interval estimates for β0 and β1. 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 β0 and β1 • GOAL: To formulate joint confidence intervals for β0 and β1 with a specified family confidence level. • BASIC IDEA: • Make statement confidence level for β0higher • Make statement confidence level for β1higher • So that the family confidence level for (β0 , β1) is at least (1-α)×100%.

7. For β0: For β1: Recall: Original confidence intervals 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 A1 = the event that first confidence interval does not contain β0 (i.e., incorrect). • So A1C= the event that first confidence interval contains β0 (i.e., correct). • P(A1) = α and P(A1C) = 1- α

9. A little derivation (cont’d) • Let A2 = the event that second confidence interval does not contain β1 (i.e., incorrect). • So A2C= the event that second confidence interval contains β1 (i.e., correct). • P(A2) = α and P(A2C) = 1- α

10. A1 A2 A1 or A2 A1C and A2C Becoming a not so little derivation… P(A1C and A2C) = 1 – P(A1 or A2) = 1 – [P(A1)+P(A2) – P(A1 and A2)] = 1 – P(A1) – P(A2) + P(A1 and A2)] ≥ 1 – P(A1) – P(A2) = 1 – α– α = 1 – 2α We want P(A1C and A2C) to be at least 1-α. 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 β0 is between -54.1 and 83.9 andβ1 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(Yh) for g different Xh 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) t(0.99, 11) = 2.718 n=13 punters 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 Yh for g different Xh 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 Yh for g different Xh 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. Suppose we want a 90% family confidence level. n = 13 punters Bonferroni multiple: 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) s(pred(150)) = 17.21 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…

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. Click on OK. • Results appear in session window.