comparisons of approximate f test and a - repositories
TRANSCRIPT
COMPARISONS OF APPROXIMATE F TEST AND A PROCEDURE
FOR THE DEVELOPMENT OF AN ALTERNATIVE
by
BARBíjy JA ÎE KEENUM, B.S.
A THESIS
IN
MATHEMATICS
Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for
the Degree of
MASTER OF SCIENCES
Approved
Accepted
August, 1974
TTC
T3
V^tS-lM^4
ACKNOWLEDGEMENTS
I would like to thank Dr. James M. Davenport for hls
assistance in the preparation of this paper and computer
programs used for numerical calculations. I am grateful
to Dr. Truman Lewis for serving on my thesis committee^
I am deeply indebted to Dr. James M. Davenport, Dr^ Thomas
Boullion, Dr^ Benjamin Duran, Dr. Truman Lewis, and Mr.
Jack D. Towery for their guidance and encouragement in my
statistical studies.
ii
CONTENTS
ACKNOWLEDGEMENTS ii
TABLES iv
ILLUSTRATIONS V
I. INTRODUCTION 1
II. EARLIER COMPARITIVE STUDIES 4
III. ADDITIONAL COMPARITIVE STUDIES OF
PROCEDURES FOR TESTING 6 = ®i+®2 ®
IV. MOTIVATION FOR THE DEVELOPEMENT OF
AN IMPROVED PROCEDURE 38
V. CONCLUSIONS AND RECOMENDATIONS 43
REFERENCES 45
APPENDIX 47
111
TABLES
1. Probability of a Type-I Error for Satterthwaite's
Procedure at the Apparent .05 Level
and n^ = 1 13
2. Probability of a Type-I Error for Hoel's
Procedure at the Apparent .05 Level and
nj = 1 13
3. Probability of a Type-I Error for Dixon
aná Massey's Procedure at the Apparent •05 Level and n^ = 1 14
4. ProbcdDÍlity of a Type-I Error for Howe and
ítyers' Procedure at the Apparent .05 Level
and n^ = 1 14
5. Probability of Type-I Error for Cochran's
cuid Satterthwaite * s Procedures at the
Apparent .05 Level and n« = 1 35
IV
ILLUSTRATIONS
1. Probability of type-I error of Dixon and
Massey's procedure for various values of n.fTi^t and n^ = 1 as U -• » 17
2. Probêújility of type-I error of procedures for n, = 2, n, = 2, and n^ = 1 as U - oo 19
3. Probability of type-I error of procedures for n, = 6, n, = 6, and n^ = 1 as U -• <» 21
4. Probability of type-I error of procedures for n- = 18, n^ = 18, and n^ = 1 as U -> « 23
5. Probability of type-I error of procedures
for n, = 2, n^ = 6, and n^ = 1 as U -• » 25
6. Probability of type-I error of procedures for n- = 2, n^ = 18, and n^ = 1 as U -> » 27
7. Probability of type-I error of procedures for n, = 6, n^ = 2, and n^ = 1 as U ^ » 29
8 Probability of type-I error of procedures for n, = 18, n^ = 2, and n^ = 1 as U -• « 31
Chapter 1
Introduction
In many statistical investigations, it is desiraúsle
to test hypotheses involving linear combinations of var-
iauice and covariance components for which an F-test of the
usual type does not exist^ A simple example is the Behren-
Fisher problem of comparing the means of two normal pop-
ulations with unknown and not necessarily equal variance
parauneters^ The following stuây considers one specific
case, testing the relation
6 3 = 6 ^ + 02
versus e^ > 6, + Ø^ (1)
where 6., i = 1*2,3 denotes variances from populations
which are assumed to be normally distributed^
Several methods have been suggested for testing hypo-
thesis (1)• Due to the approximate nature of these pro-
cedures, the value of the probability of the type-I error
is not exact^ Also, the implementation of many tests may
be considered impractical in various situations^
This study attempts to determine which of several
tests is "best" with respect to the closeness of the size
to the desired a and the simplicity of the procedure^
Chapter 2 is a survey of established guidelines. In Chap-
ter 3, five procedures are considered which have been pro-
posed by
1. Satterthwaite
2. Hoel
3. Dixon and Massey
4. Howe and Myers
5. Cochran.
The true size of the tests has been calculated, assuming a
desired confidence interval of 95 percent. The probability
of a type-I error is found by the following equation which
is derived in Chapter 3:
a = Pr(type-I error]
= ^o PIO ' fa[n3,f]-'^l"Î9(u)du
where Q = Central F statistic with n^ and n,+no 3 1 2
degrees of freedom,
F . ^. = the a upper percentage point of the a vn^ f r /
F-distribution with n^, the numerator degrees
of freedom and f, the estimated denominator
degrees of freedom
(n,+n,) (1+u) k =- ^ ^ TT+ y n^+ryi
u Q
U = l/Q^ ~ ratio of variances (nuisêuice para-
meter)
v^ = meéui square estimate of the variance of the
ith population with n. degrees of freedom,
i = 1,2,3
u = Vj /v = ratio of the mean square estimates
g(u) = the density function of u.
The values for the probability of a type-I error are pre-
sented with a comparison of the procedures with respect
to the stability of a.
Because of the caution which must be taken when apply-
ing approximate tests and because of the tedium involved
in these tests, a simpler more precise method is desirable.
In Chapter 4, motivation for such a procedure is presented.
Chapter 2
Earlier Comparative Studies
Smith (1936) suggested using the test statistic
F' = — T ^ (2) v^+V2
for testing the hypothesis Ø^ = 6. + Ø^. It cam be shown,
based on the assumptions of normality of the populations,
that n.v./e., i = 1,2,3, is a chi-square statistic with n.
degrees of freedom. Therefore, it has been proposed that
F' is an approximate F statistic with n^ and f degrees of
freedom. The estimated denominator degrees of freedom f
was found by comparing the first two moments of the sta-
2 2 tistic s , where s ~ ^i "*" 2' ^^^ ^ ® moments of the
stemdard mean square variable. The resultant f is given
by
(Vl ^ V2)'
^s 3 T-
n^ "2
. (1 + u ) ^
u . 1 (3)
n^ "" n^
Satterthwaite (1941 & 1946) used Smith's F' test sta-
tistic and extended its use to generalized linear combi-
nations of variances. He cautioned against the use of this
method unless the difference of the degrees of freedom of
the chi-square estimates is small. Otherwise, the approx-
imate F test could be used. Welch (1947, 1949 & 1956),
Box (1954), Grunow (1951), and Cochran (1951) verified these
findings. Davenport and Webster (1972) investigated
Satterthwaite's test further. The following properties
of the size of the test were given which suggest guidelines
for its application:
1) AsU-^-OorU-*-*, a-^a.
2) As n^ increases, a departs from the nominal
a level.
3) As n, and n^ increase together, a approaches
a. A.
4) If U > 1 and n, > n^, then a is approximately
equal to a.
5) If U < 1 and n^ < n,, the function a is
approximately equal to the desired a.
6) If U > 1 and n, is much larger than n,, a
departs drastically from the given a-level.
Also a departs from the nominal a for U < 1
and n, much larger than n^*
The Behren-Fisher problem in which n^ = 1 is a special
case of the hypothesis being considered^ Many solutions to
this problem have been proposed. Mehta and Srinivasan
(1970) investigated those procedures derived by Banerjee
(1960), Fisher (1936), Pagurova (1968), and Wald (1955)
and an asymptotic series solution proposed by Welch (1947)
In their study Banerjee's and Fisher's tests were found
to be poor with respect to the deviation of the size of
the test from the nominal a; in addition the power test
performcuice was considered unacceptable. Wald's test is
limited to the case n, = n, and does not differ signif-
icantly from Welch's test. Pagurova's solution is a
generalization of Wald's test. Mehta and Srinivasan
found the difference of the true size and the nominal a
to be similar in Pagurova's and Welch's tests. Also, the
range of the values of the nuisance parêuneter U for which A.
a deviated from a was reported to be larger in Welch's
test. Neither test was found to be acceptable when n. or
n^ < 7. A modification of Pagurova's test was suggested
by Mehta and Srinivasan for small sample sizes; however,
this solution requires prior knowledge of the nuisance
parameter(U > 1 or U < 1). All of these tests are diff-
icult to apply.
Welch (1947) formulated a solution known as "Welch's
Approximate-t" test. It is of interest to note that
Satterthwaite's approximate F test statistic for n^ = 1
is the square of the statistic used in this procedure.
Wang (1971) made a comparison of the approximate-t test
and the asymptotic series solution proposed by Welch for
the case a = .05 and a = .01. Deviation of the true
probability of a type-I error from a was found to be
slightly larger for the t-test. However, Wang (1971)
suggested the use of the approximate t-test due to the
ease of using the usual t tables and the fact that tables
of critical values for the asymptotic series solution are
available for only selected values of a. Preference for
"Welch's Approximate t" test was also supported by
Davenport éuid Webster (1974). Comparison was made of this
test to those results reported by Mehta and Srinivasan
(1970). It was shown that in general, Welch's approximate
test was better than Banerjee's and Fisher's, and less
stable than the others with respect to a. However, it was
suggested that ease in carrying out the t-test might be
ample justification for the use of "Welch's Approximate
t" test.
Chapter 3
Additional Comparative Studies of
Procedures for Testing ^2~^í^^2
An investigation of the stability of the size of
five procedures for testing the hypothesis given in
equation (1) has been conducted in order to present com-
parative guidelines for their use. Those procedures due
to Satterthwaite, Hoel, and Dixon and Massey, and Howe
and Myers' infinite series solution make use of Smith's
F' statistic, but suggest different methods for estimating
f, the denominator degrees of freedom. These estimates
are as follows:
Satterthwaite
f . 11±J1)1 (4)
n^ n^
Dixon and Massey
^D-M - , 2 Í5> rTjTT n^+l
8
Hoel
n^+2 ^2+2
Howe and Myers
^H-M " ^ " 2/(m(u) - 1)
where
00
kl u k+1 . k! 1 k+1 m(u) = 1 + 1 -^ ^ ^ •' + . ^» . £, k 1+u k n -• n (^ + j) n (^ + j)
j=l ^ j=l ^
2 l+^
Due to the reciprocal relation of the F distribution.
F <" l' 2> ^rj.r^) '
the value of the type-I error for specific values of U,
n, ,^2 is the same as for /U when the roles of n, and
n^ are interchanged. Therefore, without loss of general-
ity, values of a are presented for U > 1 only. Also,
calculations of a were restricted to the Behren-Fisher
problem (n^ = D • It is believed due to the nature of
the tests, that the effect of an increased value of n^
10
on a for tests considered in this chapter is the saroe as
that reported by Davenport and Webster (1971) for
Satterthwaite's procedure. Specifically, as n^ increases,
a departs from the nominal a level. Also the results re-
ported may be extended to the alternate hypotheses
83 < e^ + e^
êUld
63 e^ + 63
by applying usual techniques.
The method employed for finding the true probêúsility
of the type-I error of the approximate F tests is that
used by Davenport and Webster (1971). The size of the
test is given by
" = ^<^' * ^tnj.f])
where F , f\ ^^ ^^® ^^^ upper percentage point of the
F-distribution with n^ and f degrees of freedom. Due to /v
the âependency of f on u, a is a function of u, so that
PÍF' > F^, ^i) = í P(A|u)g(u)du (7) ain^»IJ Q
11
where P(A|u) is the probability of the event F' > F , ^, ain.^ij
for a particular value of u êmd g(u) is the density
function of u. When hypothesis (1) is true
F' = .Q v^ + V2 Q
where
Q = V3(n^ + nj)
e n^v^ n^v^
is distributed as an F with n^ and n, + n^ degrees of
freedom. Therefore, it follows that
F« =
K^l , "2 2 QiQ-^ + e ) ["T[~ T^
in^ + n7) iv^ + vj)
Q(l + U) n^ +
(1 + u) (Hj + n^)
n,u
(8)
Note that equation (8) is a function of u and U^ It can
be shown that using this new statistic for F' in (7)
. ^ ^(l-u)ln,,fj (l+u)(n^+n2) a = I P Q > -*
0 (1+U) nj U + n.
u g(u)âu^
12
This method of evaluating a was employed in computer pro-
grams, integration being accomplished by numerical
techniques^ The probabilities of the type-I error are
given in tables 1 - 4 for specific values of U, n^ and
nj with n- = l
V V
13
TABLE 1
Probability of a Type-I Error for Satterthwaite's Procedure at the Apparent 05 Level and n- = 1
V = ®i/e.
n. n. 16 30
2 2
6
6
2
18
18
2
6
2
6
18
2
18
0^0352
0^0469
0^0468
0^0471
0^0544
0^0544
0^0496
0.0373
0.0537
0.0441
0.0480
0.0627
0.0494
0.0498
0.0432
0.0621
0.0451
0.0501
0.0709
0.0480
0.0501
0.0564
0.0695
0.0498
0.0518
0.0743
0^0495
0^0502
0^0589
0^0675
0^0507
0^0514
0^0705
0^0499
0^0501
TABLE 2
Probability of a Type-I Error for Hoel's Procedure at the Apparent •OS Level and n^ = 1
V = ®i/e.
"1
2
2
6
6
2
18
18
n^
2
6
2
6
18
2
18
0.0503
0^0549
0^0548
0^0500
0^0607
0.0607
0.0500
0.0529
0.0637
0.0505
0.0509
0.0718
0.0535
0.0502
0.0595
0.0744
0.0501
0.0528
0.0830
0.0505
0.0504
16
0.0714
0.0833
0.0521
0.0536
0.0888
0.0501
0.0504
30
0.0716 0.0801
0.0521
0.0527
0.0839
0.0502
0.0503
14
TABLE 3
Probability of a Type-I Error for Dixon and Massey's Procedure at the Apparent .05 Level and n^ = 1
n. n. 16 20
2
2
6
6
2
18
18
2
6
2
6
18
2
18
0.0572
0.0562
0.0561
0.0505
0^0599
0^0599
O^OSOO
0.0601
0.0652
0.0519
0.0516
0.0710
0.0530
0.0502
0.0679
0.0768
0.0520
0.0540
0.0830
0.0504
0.0506
0.0847
0.0919
0.0556
0.0567
0.0954
0.0505
0.0509
0.0884 0.0932
0.0564
0.0567
0.0953
0.0508
0.0509
TABLE 4
Probability of a Type-I Error for Howe and Myers' Procedure at the Apparent .05 Level and n^ = 1
V = ®i/e.
"1
2
2
6
6
2
18
18
n^
2
6
2
6
18
2
18
0.0582
0.0563
0.0562
0.0503
0.0606
0.0606
0.0500
0.0611
0.0657
0.0515
0.0511
0.0717
0.0535
0.0501
0.0683
0.0771
0.0508
0.0528
0.0834
0.0506
0.0503
16
0.0821
0.0897
0.0522
0.0532
0.0931
0.0501
0.0503
20
0.0835
0.0890
0.0521
0.0524
0.0912
0.0502
0.0502
15
Values for the size of the tests due to Satter-
thwaite, Hoel, Dixon and Massey, and Howe and Myers sup-
ports the following relationships between the test
statistic parêuneters and the type-I error a:
1. As n^ = n^ increases in values, a
approaches a. /s
2. For U > 1 and small n^,a is unstable;
the discrepancy increases as n^ in-
creases.
3. For U < 1 and small n~, as n in-
creases the difference of a and a
increases.
4. When U > 1 and n.. small, a is close
to a, also a approaches the nominal
a as n., increases. i /\
5. When U < 1 and n, small, a is close to
a, and a approaches the nominal a as
xiy increases.
The preceding trend was fcund to be most exaggerated
in the procedure proposed by Djxon and Massey. Type-I
errors for Dixon and Massey's test are presented in Figure 1.
Satterthwaite's procedure is the most conservative
of the tests considered, as secn in figures 2 - 8 .
It can be shown that
fg i H "< S ' D-M'-
M í 1
3 u>
II
!-•
P 0)
G
4-
8 .
P) t3 Di
S P) (0 01 (D
»< -
co •0 n 0 o (D o. C ^{ (D
»-h O H{
< P> H{ H-0 c cn < 0) »-• c: (D (0
0 M>
»«í H' ^ .
»-• • •
*\3 H 0 tx P> cr H-M H-rt
^<
0 hh
rt »<
(D 1
H
(D ^
0 H
0 H í
o H-X 0 3
û % û i • 1
P F F F F F
> > > > > >
Q) 0)
C
4-
8 •
»0 H{ 0 o (D Oi c n (D (0
M» 0 •1
3 H'
II
> ^
3 lO
II
to ^
p) 3 Oi
3 CJ
II
^ H-
vQ •
N) • •
0 tr p) cr H-
H-rt
•< 0
r t ^< »0 (D 1
H
(D ^l
0 H{
0 H»
I I I I I
F F F F F F ^
\) [) \) \) [) \)
0) (0
G
+ 8
.
^ >1 0 O (D Oi c h (D (0
Mí 0 H
3 M
II
o\ ^
P lO
II
o\ ^
P) 9 Oi
3 OJ
II
^ H-
^ •
U> • •
»X3
0 CT 0) 0* H-
H-r t
*<
0 M»
r t
»0 (D 1
H
(D l-{ b{ 0 H
0 Mí
íl )l I Él )l »
O 00
21
<if-L
o
\
\
\
/
/
/
/
\r\ OJ
o OJ
i ,
( >
H 1 V +> •H ^ ^
.:: •p u u
1 r J 1 1 îí 1 1 >» 0) æ n tt >4 0) 0)
= é c i
O
F T
P) 01
C!
+ 8
.
»0 h 0 0 (D Oi (S H (D 01
M> 0 n 3
!-•
II
M 00 ^
P lO
II
!-• 00
^
0) :3 Oi
3 u>
•fl H-
vQ .
i ^ • •
»d H{ 0 tr & H-
H-rt
*< 0 M»
rt
•T3 (D 1
H
(D
h 0 h
0 M)
P M M P
(D 01
G
4-
8 .
»0 M 0 O (D Oi C M (D 01
M» O »1
3 !-•
II
lO ^
3 lO
II
<r> «
(D 3 Oi
3 CJ
II
• ^
H-vQ •
U l • •
0 cr & H-»-• H-rt
O M»
rt *< »0 (D 1
H
(D H
0 ^
0 M»
\ ;i M i k ll ll .
p) 01
G
4-
8 .
•0 h 0 o (D Oi
c H (D 01
M» 0 H
3 M
II
lO ^
3 lo
II
M æ ^
P) 3 Oi
3 u>
^ H-
vQ .
<T> • •
^ ^ o cr & H-!-• H-rt
*< O Mi
r t ^< »0 (D 1
H
(D H{ H{ 0 t-{
0 H\
Ut 01
G
4-
8 •
»0 h 0 0 (D Oi
c tt (D 01
Mî 0 H
3 M
II
m *
3 lO
II
lO ^
01 3 Oi
3 u>
II
•fl H-
vQ .
^ * t
•d
0 tr g-H-»-• H-
4 O M)
r t *< »0 (D 1
H
(D t{ h 0 H
0 Mí
29
<y
I i a> •p •H
s X t 0> •p •p tf co
H « O X
>» 0) n n n >4 ti o>
Ê* a 1 o o> X >
•H O Q X
vo O
U\ U\ O
O l A O
u\ J^ O
O
o
7
(U 01
G
4"
8 .
tJ h 0 o (D Oi
c H (D 01
M) 0 h
3 !-•
II
M 00
^
3 ro
11
lO
(U 3 Oi
3 o>
II
^ H-vQ .
00 * .
t{ 0 cr & H^
H^ rt *<
O M>
r t »<
(D i
H
(D K
0 H
O M>
, '1 , 1
32
equality exists between f. and f when U = 0 (see o D—M
appendix for proof). It is also conjectured that
s i h-M- <9)
A well known fact is that given different functional
estimates of the denominator degrees of freedom f, say
fj and f / fj and f are functÍDn of u, such that
^l < ^2' ^^^^
then
F r í 1 < F r 2 i. (H)
l i e n c e , f o r a g i v e n v a l u e of u
P[Q > F r r , • K] < P[Q > F , 2 1 • ^^ ^ [ n ^ ^ f j ] ' ^ a í n ^ ^ f ^ ]
K]
which implies that a for the test with parameter f, is
less than a fdr the test with í^. This demonstrates
that the size of the test using Satterthwaite's procedure
is less than that proposed by Hoel and more conservative
than the alternative suggested by Dixon and Massey, ex-
cept when U = 0, where equalit - holds. Also given that
33
inequality (9 ) holds, the size of Satterthwaites test is
also less than or equal to the size of the procedure
suggested by Howe and Myers.
For Satterthwaite's procedure a is generally close
to the desired a level and the range of values of the
nuisance parameter for which there is large deviation
from the nominal a is less in most cases. However, the
case n. = 18, n^ = 2, U > 1 merits closer inspection.
Under these conditions the behavior of a due to Hoel's
test is better than Satterthwaite's (see figure 8). It
is therefore suspected that given U > 1, small n^ and
large n, or U < 1, small n.. and large n^t Hoel's pro-
cedure should be used.
The "Cochran t-test" is an asymptotically normal
solution of the Behren - Fisher problem. This pro-
cedure applies the test statistic
"'l" ""^ (12)
where X. is the arithmetic mean of the ith population,
i = 1,2. The critical point C is given by
V,t + V t^ c = ^ ^ ^ ^ (13) ^ V^ + V^ ^ '
where t. is the 100 (l-a/2) percentage point of the
34
t-distribution with n. degrees of freedom. /
Specific values of a resulting frora Satterthwaite's
test were evaluated for comparison with the size of the
"Cochran t-test" reported by Lauer and Han (1971). These
are presented in table 5 along with the maximum deviation
of a from a for each test over the given four values of U.
It is observed that Cochran's test is more conservative
than Satterthwaite's.
It is believed that further investigation is needed
in order to present specific guidelines for determining
when Cochran's t-test should be applied. However, it
appears that when n^ is much larger than n and U is
assumed to be greater than one, Cochran's test is "best"
with respect to the closeness of a to the desired a. For
all other cases, Satterthwaite's test appears better than
Cochran's. Therefore, if U < 1 or no prior knowledge of
its value is known, Satterthwaite's procedure should be
employed.
TABLE 5
Probability of Type-I Error for Cochran's and Satterthwaite's Procedures at the Apparent .05 Level and n^ = 1 (Satterthwaite's Values
Given in Parenthesis)
35
n, = 2 n^ = 2 Max a-a
v= 10 100
0.0126 (0.0352)
0.0171 (0.0432)
0.0247 (0.0527)
0.0438 (0.0576)
0.0374 0.0148
n, = 2 n^ = 4
v= 8 20 200
v=
v=
v=
0.0204 (0.0481)
3
0.0265 (0.0586)
6
0.0366 (0.0720)
1/2
0.0204 (0.0481)
0.0317 (0.0631)
n, = 2
12
0.0388 (0.0695)
n, = 2
24
0.0461 (0.0711)
n, = 4
2
0.0190 (0.0413)
0.0398 (0.0661)
n^ = 6
30
0.0447 (0.0675)
n^ = 12
60
0.0487 (0.0648)
n^ = 2
5
0.0242 (0.0450)
0.0486 (0.0566)
300
0.0495 (0.0554)
600
0.0500 (0.0536)
50
0.0432 (0.0520)
0.0296 0.0161
0.0235 0.0195
0.0134 0.0220
0.0310 0.0020
36
v=
v=
v=
v=
v=
v=
1
0.0241 (0.0442)
2
0.0312 (0.0502)
1/3
0.0265 (0.0588)
1
0.0308 (0.0470)
2
0.0364 (0.0503)
1/2
0.0312 (0.0502)
n, = 4
4
0.0301 (0.0493)
n3_ = 4
8
0.0401 (0.0553)
n, = 6
4/3
0.0210 (0.0452)
n^ = 6
4
0.0361 (0.0501)
n.. = 6
8
0.0432 (0.0526)
2
0.0306 (0.0470)
n^ = 4
10
0.0375 (0.0530)
n^ = 8
20
0.0449 (0.0547)
n^ = 2
10/3
0.0242 (0.0446)
n^ = 6
10
0.0419 (0.0517)
^2 = 2
20
0.0467 (0.0521)
n^ = 4
5
0.0360 (0.0489)
100
0.0481 *
200
0.0494 (0.0510)
100/3
0.0423 (0.0570)
100
0.0489 (0.0506)
200
0.0496 (0.0503)
5
0.0475 (0.0505)
0.0259 0.0058
0.0188 0.0053
0.0290 0.0088
0.0191 0.0030
0.0126 0.0074
0.0194 0.0030
*this value was not computed
37
v=
v=
v=
v=
1
0.0348 (0.0482)
1/6
0.0366
1/2
0.0364 (0.0503)
1
0.0394 (0.0491)
n^ = Q
4
0.0394 (0.0502)
n^ = 12
4/6
0.0261 (0.0567)
n^ = 12
2
0.0362 (0.0486)
n^ = 12
4
0.0428 (0.0501)
n^ = 8
10
0.0440 (0.0511)
n^ = 2
10/6
0.0248 (0.0489)
n^ = 6
5
0.0405 (0.0496)
n^ = 12
10
0.0461 (0.0505)
100
0.0492 (0.0503)
100/6
0.0397 (0.0495)
50
0.0485 (0.0502)
100
0.0495 (0.0501)
0.0152 0.0018
0.0252 0.0067
0.0138 0.0014
0.0106 0.0009
Chapter 4
Motivation for the Development of
an Improved Procedure
Available tests for evaluating hypothesis (1) often
involve approximate procedures. Therefore, a simple, con-
cise, approximate solution to the problem which is as good
as, if not better than existing ones is needed. A search
for such a procedure has been conducted. At present this
problem is unresolved; however, motivation for the develop-
ment of an improved test is presented.
As previously shown, the true probability of a type-
1 error a is given by
a = Jo P(Q > F„(n,,f)' k]g(u)du.
Note that Q is a central F statistic with n^ and n, + n^
degrees of freedom. By letting f = n^ + n^ the percentage
point is now F , „ 4.« \ ^^^ i^ from the same distribution
as Q and no longer a function of u. Let h = h(u;n,^n^)
be some function such that h is approximately equal to k,
hence k/h = 1. The probability of type-I error is
38
39
a = ÍÂ PlQ > F , ,• ^|u]q(u)du = a. ''O ^^ aín^^n^^+n^) h' '' ^ '
Therefore, the preceding results in a test for which the
size is significantly improved.
By fixing u, k may be treated as a function of U with
parameters n ^ and n^. It is then desired to find a
function of U, say q(U), which may serve as a weighting
function of k such that
KCu^nj^n^) = ÍQ k • q(U)dU.
Several possibilities for the weighting function have been
investigated, some of which are presented below.
1• A scaled F with n, and n^ degrees of freedom
X = u/U is distributed as an F with n^ and
n^ degrees of freedom. By fixing u and
treating U as the variable, usual techniques
of integral transformation are applied to the
F. The resultant weighting function of U =
u/x is
qj^(U;u,nj^,n2) = ' pp\ [ ] 3 -
Mflh^ (n +n )
U
40
0 < u < »
with n, > 0, n, > 0, u > 0.
2. A scaled F with n^ and n ^ degrees of freedom
for a given value of the nuisance parameter
U, transformation of the F density function
of u/U results in scaled F function. n.
q^(u?U,nj^,n2) =
H'Mh l (n^+n^)
U
0 < u < «>, with n, > 0 , n 2 > 0 , U > 0 .
An estimate of U is u, so U is used as the
varicúsle of integration and u the scaling
parameter resulting in the following function
q^^Uîu^n^^^n^) =
n^+n^ n U
n^u
^l 7"
[^l 1^2 r l"
(n^+n^)
2
u
0 < U < ~, with n ^ > 0 , n 2 > 0 , u > 0 .
Note that
q^^U^u^n^^n^) = q ^ (U^u^n^fnj^) .
41
3. A chi-square with u degrees of freedom
Since u is an estimate of U and Ely] = y
where y is distributed chi-square with y
degrees of freedom, it seems reasonable to
let u be the degrees of freedom of the chi-
square prior function.
4. Chi-square with n^u/^n^-^) degrees of freedom
u/U is distributed F, . , therefore (nj n )
E[u/U] = ^2/(^2-2) which implies E[u] =
^2^/(^2-2) and U ^n^Eíuj/^n^-^). Therefore
n^u/^n^-^) is chosen as the degrees of free-
dom of the chi-square function.
5. Normal with mean u aná arbitrary variance
All weighting functions considered, when applied as pre-
viously suggested, resulted in unstable values of a; hence,
did not improve the approximate test.
It is suggested that the technique of curve fitting
may be applied to obtain a function h^u^n^^n^). Also an
investigation of the denominator degrees of freedom of the
F' statistic might be of value. A new procedure has been
evaluated for which the estimated denominator degrees of
freedom f is given by
f = minlfjj^^^n^^+n^].
42
The probability of a type-I error is more conservative
than that which resxilts from Howe and Myers' infinite
series solution and greater than Satterthwaites. This
technique, however, does not result in an improved test.
Chapter 5
Conclusions and Recomendations
Procedures for testing the hypothesis 63 = Ôi + ^2
are approximate and therefore the value of the type-I
error is not exact. Instability of the size of the test
is related to the parameters n, n^^n^ and U = ^-t/^o ^"^
the chosen procedure. Consequently, the size of the sample
taken from the population should be carefully determined
with consideration given to the relation of 6, to B^,
when it is known. Also the test should be chosen with
respect to the available degrees of freedom and the value
of the nuisance pareuneter.
Given no prior knowledge of the value of the nuisance
parameter, large, equal sample sizes should be taken when-
ever possible. Satterthwaite's relatively simplistic pro-
cedure may then be applied with confidence due to the sta-
bility of the test when the degrees of freedom are large
and equal.
When the situation is such that only small equal
sample sizes are available, Mehta and Srinivasan's mod-
ification of Pagurova's procedure may be used. However
the procedure involves knowledge as to whether U is great-
er thcui or less than one. Howe and Myers (1970) have 43
44
suggested euiother procedure which is good when n. and n^
cure small and equal. For the case being considered the
test criterion C(v,, v^, a) déveloped by Howe and Myers
is such that, given the null hypothesis is true.
Prlv^ > C{V^,V2,OL)] = a.
C(v,,V2,cx) is a series expansion developed by successive
approximations and involves the upper lOOa percentage
point of the gamma distribution with parameters a = 3/2
and 3 = 1 . This procedure's complexity makes it imprac-
tical in m£uiy situations and the technique involved is
beyond the intended extent of this text; therefore, the
equation for C(v, v yOi) is not included here.
If the relationship of 6. cuid 6 is known, then a
larger scunple should be taken from the population with
greatest variance. Then Hoel's test may be applied with
confidence. If however, it is necessary for a much
larger sample size to be taken from the population with
least variability, application of Cochran's test is
suggested.
REFERENCES
Banerjee, S.K. (1960). "Approximate Confidence Interval
for Linear Functions of Means of k Populations When
the Population Variances are Not Equal," gankhva 22,
357-8.
Box, G.E.P. (1954). "Some Theorems on Quadradic Forms
Applied in the Study of Analysis of VaricUice Problems,
I. Effect of Inequality of Variance in the One-Way
Classification," Ann. Math. Sta^^. 25, 290-302.
Cochran, W.G. (1951). "Testing a Linear Relation Among
Variances," Biometrics 7, 17-32.
Davenport, James M., and Webster, J.T., (1972). "Type-I
Error and Power of the Test Involving a Satter-
thwaite's Approximate F Statistic," Techometrics 14,
555-69.
Fisher, R.A. (1936). "The Fidueial Argument in Statistical
Inference," Ann• Euaei 6, 62-71.
Grunow, D.G.C. "Test for the Significance of the
Difference between Means in Two Normal Populations
having Unequal Variances." Bíf metrik 38, 252-256.
Howe, R.B. and Myers, R.H. (1970) , "An Alternative to
Satterthwaite's Test Involving Positive Linear
Combinations of Variance Components." J. Am. Stat.
Assn. 65, 404-412.
Lauer, G.N. (1971), "Power of Cochran's Test in Behren-
Fisher Problems." Unpublished Ph.D. Thesis, Ames,
lowa, Library, lowa State University of Science
and Technology.
45
46
Mehta, J.S., and Sriniuasan, R. (1970). "On the Behrens-
Fisher Problem." Biometr: |c 57, 649-55.
Pagurova, V.I. (1968), "Tests of Comparison of Mean Values
Based on Two Normal Samples." (Russian.) No. 5 Com-
puting Center of the Academy of Sciences of the
U.S.S.R. Moscow.
Satterthwaite, F.E. (1941). "Synthesis of Variance."
Psvchometril^^ 6, 309-16.
Satterthwaite, F.E. (1946). "An Approximate Distribution
of Estimates of Variance Components." Biometrics 2,
110-4.
Smith, H.F. (1936), "The Problem of Comparing the Results
of Two Experiments with Unequal Errors." Journal of
the Cottngil fgr Sgienti^jc and n^ugtrifll Researgh 9, 211-2.
Wald, A. (1955), "Testing for Difference Between the Means
of Two Normal Populations with Unknown Standard
Deviations." Selected Papers in Stat^stic s and
Prf^bahilítY by A. Wald, New York: McGraw-Hill.
Wang, Y.Y. (1971). "Probabilities of the Type-I Errors
of the Welch Tests for the Behrens-Fisher Problem."
.T. Am- Stat. Assn. 66, 605-8.
Welch, B.L. (1947). "The Generalization of Students'
Problem when Several Different Population Variances
are Involved." Biometrika 34, 28-35.
Welch, B.L. (1949). "Further Notes on Mrs. Aspin's Tables
and on Certain Approximations to the Tabled Function."
Biometrika 36, 293-6.
Welch, B.L. (1956). "On Linear Combinations of Several
Variances." J, Amer. Stat. ^fg§^9f 51, 132-48.
48
Theorem 1. Let f be Satterthwaite' s aná f^ „ be S D—M
Dixon and Masseys functional estimates of the denominator
degrees of freedom of the F' statistic. Then f < f -M* Proof: Assiune the following:
u > 0, n- > 0, n^ > 0.
Suppose
^H^ ^S
then
(1+u)^ 2 1 -
n , + l ' n^+l
(1+u)^
n^ "2
Due to the previous assumptions, both denominators are
positive and (1+u)^ j^ 0. Therefore,
HÍ + i- < - H L + 1 n, n^ — n,+l n^+l
which implies
49
u2 1 n^(nj^+1) •*• njnipT) - ^'
But thi8 i8 i]npo88Íble.
Hence, f^ < f^_^.
Theorem 2. Let f be Satterthwaite's and f„ Hoel's
functional estimates of the denominator degrees of free-
dom of the F* statistic. Then f < f„. S — H
Proof: Assume the following
u > 0, n, > 0 and n^ > 0.
Suppo8e
^H < ^S
then
(1+u)^ . (1-m)^ . 2
!íl + i- ^ + ^ n, n^ nj +2 n^+^
Given the preceding assunptions the following steps are
obtained U8ing ordinary algebraic techniques.
50
< 0
<^*«>'^-n^ *k-4^
-2^ + í- —4 + —K < 0 " l "^2 "^T*^ ^2+2
2 2 j <^*''> nJn^+2) * n j (n^+^)
^4 2u (nj +nj n^+n ) j n ^TnJjTÎT "*" nj^n^^nj^+î) (n^+Z) **" n^ín^+^)
25i3 (n^^+2n^+2n2+n2^)u^ 2u îrjTK~+2T "*" h^n^in^+^) (^2+2) " ' n^ín^+Z)
2u (n^+n^n^+n^)
nj^n^ínj^+^) (n^+Z) ^ ^
2^3 ( n ^ - n 2 ) V ^u . ^ n^(n^+Í) * n^n2(n^+2)(n2+2) * n^ín^+^) "*
But this is impo88Íble
Hence fg — ^H*