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STAT 440: Design and Analysis

创建日期：2024-04-03 16:32:39

所属分类：统计学statistics

标签： STAT 440

Design and Analysis

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STAT 440: Design and Analysis
of Experiment
4. Simultaneous Inferences
The inferences covered before are related to individual interest, e.g.,
100(1-)% . . for − , i ≠ j, is
�� − �� ± 1−2,− ( 1 + 1)
This inference has significance level exactly , or the error rate= , the correct rate = 1- .
However, in practice, most often, we are interested in simultaneous inferences, i.e., we might be interested in all the possible pair-wise
comparisons.
For cutting method data set, r=3
There are total 32 = 3 possible comparisons.
1 − 2,1 − 3,2 − 3
If the individual error rate (or significance level) is = 0.05, then, the overall significance level is 1 − 1 − 3 = 1 − 1 − 0.05 3 = 0.1426 ≫ 0.05
If we have r levels of means to compare, then the overall significance level is 1 − 1 − ≫
To adjust this, many multiple procedures have been proposed. The idea is to modify the C.I. width or the rejection region by adjusting
the critical values.
Why?
- Often, investigators consider a large number of unplanned comparisons. (e.g.,
all pairwise comparisons among treatments)
- There can be many unplanned comparisons.
We begin with linear combination of treatment means:
= ∑=1
Note that contrasts are special cases of L with ∑=1 = 0.
And ( − )’s are special cases of contrasts, hence they are linear combinations too.
Point Estimate of L: � = ∑=1 ̂ = ∑=1 ��
{�}2 = � = �
=1

2
2

= 2�
=1

2

, {�}2 = �
=1

2

Result:
(i) 100(1-)% C.I for = ∑=1 is
� ±
1−

2,− {�} = � ± 1−2,− �
=1

2

(ii) Test �0: = 0: ≠ 0
(iii) Test Statistic: = �−0
∑=1

2

= �
{�} ~ ( − )
(iv) Rejection region: < −
1−

2
,− > 1−
2
,−
I. Bonferroni Multiple Comparison Procedure:
- Need to know number of comparisons, say, g
- We use
2
instead of in each test / interval.
- Conservative, i.e., overall Type I error (significance level) < .
- Sample size need not to be equal.
Result:
(1) 100(1-)% C.I for Bonferroni simultaneous C.I. for g contrasts are: � ± ∗ {�}
Where =
1−

2
,− , � = ∑=1 �� , {�} = ∑=1 2
(2) Test: �0: = 0: ≠ 0
= � − 0
∑=1

2

= �
{�} ~ ( − )
Reject region: if > 1−
2
,− < - 1−
2
,−
Example: Cutting method data: 95% Bonferroni simultaneous C.I. for 1 − 2, 1 − 3, 2 − 3
Sol: g=3, = 0.05

1−

2,− = 1−0.052∗3 , 9 = 0.9917, 9 = .
95% Bonferroni simultaneous C.I for multiple comparisons are
For 1 − 2, �1� − �2� ± 1−
2
,− ( 11 + 12) = 7.4 − 26 ± . ∗ 3.85 ∗ 14 + 14 = −18.6 ± 4.07 →(−22.67,−14.53)
For 1 − 3, �1� − �3� ± 1−
2
,− ( 11 + 13) = 7.4 − 35.4 ± . ∗ 3.85 ∗ 14 + 14 = −28 ± 4.07 →(−32.07,−23.93)
For 2 − 3, �2� − �3� ± 1−
2
,− ( 12 + 13) = 26 − 35.4 ± . ∗ 3.85 ∗ 14 + 14 = −9.4 ± 4.07 →(−13.47,−5.33)
II. Scheffe Multiple Comparison Procedure
- Works for all possible contrasts (tests/intervals)
- Most conservative, relatively easy.
- Use ( − 1)(1−,−1,−) in place of 1−
2
,−
- For both equal or unequal sample sizes
- overall significance is exactly .
Result: for = ∑=1
(1) 100(1-)% C.I for Scheffe simultaneous C.I. for L is: � ± ∗ {�}
Where = ( − 1)(1−,−1,−) , � = ∑=1 �� , {�} = ∑=1 2
(2) Test: �0: = 0: ≠ 0
= �2−1 ∗{�}2 ~ (−1,−)
Reject 0 if > (1−,−1,−)
Fail to reject 0 if ≤ (1−,−1,−)
Example: Cutting method:
95% Scheffe Simultaneous C.I. for 1 − 2, 1 − 3, 2 − 3 are:
= ( − 1)(1−,−1,−) = (3 − 1)(0.95,2,9) = 3 − 1 ∗ 4.26 = 2.9189
(1−,−1,−) = (0.95,2,9) = 4.26
For 1 − 2,
�1� − �2� ± ∗ ( 11 + 12) = 7.4 − 26 ± . ∗ 3.85 ∗ 14 + 14 = −18.6 ± 4.049 → (−22.65,−14.55)
For 1 − 3,
�1� − �3� ± ∗ ( 11 + 13) = 7.4 − 35.4 ± . ∗ 3.85 ∗ 14 + 14 = −28 ± 4.049 → (−32.05,−23.95)
For 1 − 3,
�2� − �3� ± ∗ ( 12 + 13) = 26 − 35.4 ± . ∗ 3.85 ∗ 14 + 14 = −9.4 ± 4.049 → (−13.45,−5.35)
Simultaneous test on = − ( + )
Test: �0: = 0: ≠ 0
= �2−1 ∗{�}2 = ( �2�−12 �1�+�3� )2
3−1 ∗
−
1
2
2
4
+
1 2
4
+
−
1
2
2
4
= 14.656
2
= 7.328 ~ (−1,−)
(1−,−1,−) = (1−0.05,3−1,12−3) = (0.95, 2, 9) = 4.26
> 4.26 , Reject 0
But If = 0.01,(1−,−1,−) = (0.99, 2, 9) = 8.02
< 8.02, Fail to reject 0
III. Tukey Multiple Comparison Procedure
- Deal with pairwise comparisons only
- An exact solution to pairwise comparisons.
- Overall error rate (significance level) for all 2 possible comparisons is exactly .
- Assumes equal sample sizes, 1 = 2 = ⋯ = = , but it is conservative for unequal sample
size, i.e., the overall significance level is < if the sample size unequal, a good thing.
- Based on the idea of studentized range distribution
- Use 1
2
(1−, , −) in Table B.9.

Result:
(1) 100(1-)% C.I for Tukey simultaneous C.I. for difference of means, = −
� ± ∗ {�}
Where = 1
2
(1−, , −), � = �� − ��
�
2 = ∑=1 2 2 = 2 1 + 1 , {�}2 = ( 1 + 1), {�} = ( 1 + 1)
(2) Test: �
0: − = 0
: − ≠ 0
Test statistics: = �{�} ~ (,−)
Reject 0 if > (1−,,−)
Fail to reject 0 if ≤ (1−,,−)
Example: Cutting Method:
95% Tukey Simultaneous C.I. for 1 − 2, 1 − 3, 2 − 3 are:
= 12 (1−, , −) = 12 (0.95, 3, 9) = 12 ∗ 3.95 = 2.79307
For 1 − 2,
�1� − �2� ± ∗ ( 11 + 12) = 7.4 − 26 ± 2.79307 ∗ 3.85 ∗ 14 + 14 = −18.6 ± 3.874 → (−22.47,−14.73)
For 1 − 3,
�1� − �3� ± ∗ ( 11 + 13) = 7.4 − 35.4 ± 2.79307 ∗ 3.85 ∗ 14 + 14 = −28 ± 3.874 → (−31.87,−24.13)
For 1 − 3,
�2� − �3� ± ∗ ( 12 + 13) = 26 − 35.4 ± 2.79307 ∗ 3.85 ∗ 14 + 14 = −9.4 ± 3.874 → (−13.27,−5.53)

Summary Table for multiple comparisons of cutting method data:
Comparison of three procedures:
In pairwise comparisons, Tukey is superior to the Bonferroni procedure.
5. Sample size Determination with Estimation Approach
Goal: To approximate the sample size needed to conduct the multiple comparisons of interest. This is a
trial and error approach. Iterative procedure is needed some times.
To determine the sample size here, things needed.
(1) The acceptable half width of C.I. for the specified contrasts.
(2) Critical values for the multiple inferences, Bonferroni, Scheffe, or Tukey?
Which one to choose depends upon the situation, just as what procedure you would use to make
multiple comparisons.
e.g., number of contrasts known → Bonferroni
number of contrasts unknown → Scheffe
only pairwise comparisons → Tukey, may be Bonferroni
(3) An estimate of . Be careful ! As in all other sample size determination problems, the sample size
could be very sensitive to . Trouble occurs if the estimation of is not good.
Example: A company owning a large fleet of trucks wishes to determine whether 4 different brands of snow tires have the same mean
tread life (in thousands of miles).
Suppose that the company is interested in the following = 0.05. =2 according to past experience.
(1) All pairwise comparisons, i.e., 42 = 6 , − , ≠
(2) A trend in the mean tread life:
= 1 + 42 − 2 + 32
= 1 + 2 + 43 − 3
{�}2 = 2�
=1

2

Assuming equal sample size, 1 = 2 = 3 = 4 = 4, we use Scheffe multiple comparison procedure is
(1) Half width of 95% C.I. for − :
∗ ��− �� = ∗ 2�
=1

2

= 2�
=1

2

= ∗

12 + −1 2 = 2

= 2 2

Where = ( − 1)(1 − , − 1, − ) = (4 − 1)(1 − 0.05, 4 − 1, 4 ∗ 4 − 4) = 3 ∗ 2.87 = 2.93
(2) Half width of 95% C.I. for = +

−
+

is
∗

12 2 + −12 2 + −12 2 + 12 2 = 2
(3) Half width for = ++

−
4
3

Try n=10, then
= ( − 1)(1 − , − 1, − ) = (4 − 1)(0.95, 4 − 1, 4 ∗ 10 − 4) = 3 ∗ (0.95,3,36) = 3 ∗ 2.87 = 2.93
− : 2 2 = 2 2 ∗ 2.9310 = 2.62
= 1 + 42 − 2 + 32 : 2 = 2 ∗ 2.9310 = 1.85
= 1 + 2 + 43 − 3: 43 = 43 ∗ 2.9310 = 2.14
The half width are all ≤ ∆= 3, so n=10.
There are situations that an unequal sample sizes design will work better.
Suppose that the company is only interested in comparing 1 − 4 , 2 − 4 , 3 − 4
Then if the sample size for 4 is bigger than 1,2,3. Then we can actually increase the precision.
Let 1 = 2 = 3 = , 4 = 2. The defined precision with confidence coefficient 0.9 is to be ± 1.
Then, half width for those 3 pairs are (Bonferroni)

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EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 MAS316/414 ACCT2522 MSIN0181 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