Statistics 305
CHAPTER 7 – SECTIONS 1, 2, AND 3
Formulae for inferences about population means, and relationships among population
means, in multiple sample studies.
Assume r > 2 populations and sample sizes n1, n2, …, nr from N ( µ i , σ 2 ) populations, (i = 1, 2,
…, r). The sample means are y1 , y 2 , ..., y r and the pooled estimate of σ 2 is s 2p .
I.
SINGLE SAMPLE INFERENCES − Confidence Intervals for µ i .
A. Ignoring the problem with multiple comparisons
y i ± t n(1−−rα / 2) s p / ni
(i = 1, 2, ..., r )
are 100 (1− α)% confidence intervals for µ i .
B. Simultaneous confidence intervals
y i ± k 2* s p / ni
(i = 1, 2, ..., r )
have overall simultaneous confidence 100(1−α)%.
( k 2* is read from Table B.8A. JMP doesn’t support this method. It is called the PillaiRamachandran Method.)
II.
TWO SAMPLE INFERENCES − Confidence Intervals for µ i − µ j .
A. Ignoring the problem of multiple comparisons
y i − y j ± t n(1−−rα / 2) s p
1
1
+
ni n j
(i, j = 1, 2, ..., r )
(JMP output − “comparison of each pair using student’s t”).
B. Simultaneous confidence intervals
yi − y j ±
q*
2
sp
1
1
+
ni
nj
(i, j = 1, 2, ..., r )
(q* in Table B.9)
(df = n − r)
(JMP output − “comparison for all pairs using Tukey-Kramer HSD.)
III.
CONFIDENCE INTERVAL AND SIGNIFICANCE TEST FOR A LINEAR
COMBINATION OF POPULATION MEANS.
For user selected numbers c1 , c2 , ..., cr the interest is in L = c1 µ1 + c2 µ 2 + ... + cr µ r .
The point estimate is Lˆ = c1 y1 + c 2 y 2 + ... + c r y r .
A. Confidence interval for L.
c12 c 22
c r2
(1−α / 2)
ˆ
+
+ ... +
Li ± t n−r
sp
.
n1 n2
nr
B. Significance test of H 0 : L = # . Test statistic is:
Lˆ − #
T=
sp
c12 c 22
c2
+
+ ... + r
n1 n2
nr
.
This is taken as a student’s t random variable having n − r degrees of freedom.
(JMP doesn’t compute L̂ ’s directly.)
NOTE: If all ci except cj are zero and cj = 1 then the formula reduces to the single
sample formula I. Similarly if two of the c’s are −1 and 1, and all others zero, then the
formula for confidence interval is II.
If several confidence intervals 100(1− α 1 )%, 100(1− α 2 )%, …, 100(1− α k )% for L1, L2,
…, Lk are made, the Bonferroni inequality gives a bound for overall (simultaneous)
k
confidence 100γ % where γ ≥ 1 − ∑ α i .
i =1
IV. SIGNIFICANCE TESTS OF THE FORM H 0 : µ i − µ j = 0 VS H a : µ i − µ j ≠ 0 USING
JMP OUTPUT.
A. Ignoring the problem of multiple comparisons the output labeled “comparison for each
pair using student’s t” gives relevant information.
The matrix of values “ABS(DIF)−LSD” contains
y i − y j − t n(1−−rα / 2) s p
1
1
.
+
ni n j
LSD = t n(1−−rα / 2) s p
1
1
.
+
ni n j
The LSD is defined as
2
If y i − y j − LSD > 0 then the p-value of the significance test is less than α. Otherwise
it is ≥ α (and the 100(1− α)% confidence interval contains zero).
Understanding JMP Student’s t Comparison of Pairs of Means Output Matrix.
To perform the significance test
H 0 : µi − µ j = 0
H a : µi − µ j ≠ 0
we compute the value
yi − y j
Tn− r =
sp
(n − r
1
1
+
ni n j
DF ) .
Then the p-value of the test is 2(1 − CDF ( Tn−r ) ) for CDF of the student’s t
distribution having n − r DF.
Let t n(1−−rα / 2) be such that CDF (t n(1−−rα / 2) ) = 1 − α / 2 .
If
Tn− r > t n(1−−rα / 2)
then
yi − y j
sp
1
1
+
ni n j
> t n(1−−rα / 2)
or
y i − y j > s p t n(1−−rα / 2)
Define LSD = s p t n(1−−rα / 2)
1
1
.
+
ni n j
1
1
, then if
+
ni n j
y i − y j > LSD
hence
y i − y j − LSD > 0 ,
the p-value of the test is ≤ α, otherwise it is not.
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B. Simultaneous significance tests H 0 : µ i − µ j = 0 vs. H a : µ i − µ j ≠ 0.
The output labeled “comparison of all pairs using Tukey-Kramer HSD” is relevant.
The matrix of values “ABS(DIF)−LSD” contains
yi − y j − q * s p
1
1
+
ni n j
(q* here is your textbooks’ tabled value ÷ 2 )
LSD = q * s p
1
1
+
ni n j
If y i − y j − LSD > 0, µ i − µ j ≠ 0 is indicated at level α, the probability of rejecting
one or more true H0’s is ≤ α.
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