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

Simultaneous inference well be concerned about Estimating 0 and 1 jointly.Estimating more than one mean response, E(Y), at a time.Predicting more than one new observation at a time.

Why simultaneous inference is importantA 95% confidence interval implies a 95% chance that the interval contains 0.A 95% confidence interval implies a 95% chance that the interval contains 1.If the intervals are independent, then have only a (0.950.95) 100 = 90.25% chance that both intervals are correct.(Intervals not independent, but point made.)

TerminologyFamily 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).

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

Bonferroni joint confidence intervals for 0 and 1GOAL: To formulate joint confidence intervals for 0 and 1 with a specified family confidence level.BASIC IDEA: Make statement confidence level for 0 higherMake statement confidence level for 1 higherSo that the family confidence level for (0 , 1) is at least (1-)100%.

Recall: Original confidence intervalsGoal is to adjust the t-multiples so that family confidence coefficient is 1-.That is, we need to find the * to put into the above formulas to achieve the desired family coefficient of 1- .

A little derivationLet A1 = the event that first confidence interval does not contain 0 (i.e., incorrect).So A1C = the event that first confidence interval contains 0 (i.e., correct).P(A1) = and P(A1C) = 1-

A little derivation (contd)Let A2 = the event that second confidence interval does not contain 1 (i.e., incorrect).So A2C = the event that second confidence interval contains 1 (i.e., correct).P(A2) = and P(A2C) = 1-

Becoming a not so little derivationWe want P(A1C and A2C) to be at least 1-.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 = 1 2 So, we need * to be set to /2.

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

Example: 90% family confidence intervalThe regression equation ispunt = 14.9 + 0.903 leg

Predictor Coef SE Coef T PConstant 14.91 31.37 0.48 0.644leg 0.9027 0.2101 4.30 0.001n=13 punterst(0.975, 11) = 2.201We are 90% confident that 0 is between -54.1 and 83.9 and 1 is between 0.44 and 1.36.

A couple of more points about Bonferroni intervalsBonferroni 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-(/g), so need to look up 1- (/2g).

Bonferroni intervals for more than one mean response at a timeTo estimate the mean response E(Yh) for g different Xh values with family confidence coefficient 1-:where:g is the number of confidence intervals in the family

Example: Mean punting distance for leg strengths of 140, 150, 160 lbs.Predicted Values for New ObservationsNew Fit SE Fit 95.0% CI 95.0% PI140 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 punterst(0.99, 11) = 2.718We are 94% confident that the mean responses for leg strengths of 140, 150, 160 pounds are

Two procedures for predicting g new observations simultaneouslyBonferroni procedureScheff procedureUse the procedure that gives the narrower prediction limits.

Bonferroni intervals for predicting more than one new obsn at a timeTo predict g new observations Yh for g different Xh values with family confidence coefficient 1-:where:g is the number of prediction intervals in the family

Scheff intervals for predicting more than one new obsn at a timeTo predict g new observations Yh for g different Xh values with family confidence coefficient 1-:where:g is the number of prediction intervals in the family

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

Example: Punting distance for leg strengths of 140 and 150 lbs.Predicted Values for New ObservationsNew Fit SE Fit 95.0% CI 95.0% PI140 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 punterss(pred(140)) = 17.28There is a 90% chance that the punting distances for leg strengths of 140 and 150 pounds will bes(pred(150)) = 17.21

Simultaneous prediction in MinitabStat >> 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.