Simultaneous inference

1
Simultaneous inference

• Estimating (or testing) more than one thing at a
  time (such as ß0 and ß1) and feeling confident
  about it

2
Simultaneous inference well be concerned about

• Estimating ß0 and ß1 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 ß0.
• A 95 confidence interval implies a 95 chance
  that the interval contains ß1.
• 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.)

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).

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Examples

• A 95 confidence interval for ß0 the 95 is a
  statement confidence level.
• A 95 confidence interval for ß1 the 95 is a
  statement confidence level.
• Consider family of interval estimates for ß0 and
  ß1. If a 90.25 chance that both intervals are
  simultaneously correct, then 90.25 is the family
  confidence level.

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Bonferroni joint confidence intervals for ß0 and
ß1

• GOAL To formulate joint confidence intervals for
  ß0 and ß1 with a specified family confidence
  level.
• BASIC IDEA
• Make statement confidence level for ß0 higher
• Make statement confidence level for ß1 higher
• So that the family confidence level for (ß0 , ß1)
  is at least (1-a)100.

7
Recall Original confidence intervals
Goal is to adjust the t-multiples so that family
confidence coefficient is 1-a. That is, we need
to find the a to put into the above formulas to
achieve the desired family coefficient of 1- a.
8
A little derivation

• Let A1 the event that first confidence interval
  does not contain ß0 (i.e., incorrect).
• So A1C the event that first confidence interval
  contains ß0 (i.e., correct).
• P(A1) a and P(A1C) 1- a

9
A little derivation (contd)

• Let A2 the event that second confidence
  interval does not contain ß1 (i.e., incorrect).
• So A2C the event that second confidence
  interval contains ß1 (i.e., correct).
• P(A2) a and P(A2C) 1- a

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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 a a
1 2a
We want P(A1C and A2C) to be at least 1-a.
So, we need a to be set to a/2.
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Bonferroni joint confidence intervals
Typically, the t-multiple in this setting is
called the Bonferroni multiple and is denoted by
the letter B.
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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
n13 punters
t(0.975, 11) 2.201
We are 90 confident that ß0 is between -54.1 and
83.9 and ß1 is between 0.44 and 1.36.
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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-(a/g), so need to look up 1- (a/2g).

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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-a
where
g is the number of confidence intervals in the
family
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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
n13 punters
We are 94 confident that the mean responses for
leg strengths of 140, 150, 160 pounds are
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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 obsn at a time
To predict g new observations Yh for g different
Xh values with family confidence coefficient 1-a
where
g is the number of prediction intervals in the
family
18
Scheffé intervals for predicting more than one
new obsn at a time
To predict g new observations Yh for g different
Xh values with family confidence coefficient 1-a
where
g is the number of prediction intervals in the
family
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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.
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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
n13 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
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Simultaneous prediction in Minitab

• Stat gtgt Regression gtgt 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.